Finite Group Databasebeta

← #29683252

Subgroups — #29683252

Order 15616.

780 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454615C211yesmaximal, characteristic
2#21454469C211yesmaximal, characteristic, complemented
3#21454245C211yesmaximal, characteristic, complemented
4#21446468C211yesmaximal, characteristic, complemented
5#21446466C211yesmaximal, characteristic, complemented
6#21446421C211yesmaximal, characteristic, complemented
7#21446415C211yesmaximal, characteristic, complemented
8#21446146C211yesmaximal, characteristic
9#21446144C211yesmaximal, characteristic
10#21446096C211yesmaximal, characteristic, complemented
11#21445727C211yesmaximal, characteristic, complemented
12#104×Dic61C211yesmaximal, characteristic, complemented
13#21445623C211yesmaximal, characteristic, complemented
14#21445566C211yesmaximal, characteristic, complemented
15C61×#1866C211yesmaximal, characteristic, complemented
16C2×#14519797C2211yescharacteristic
17C2×#14519784C2211yescharacteristic
18#41×Dic61C2211yescharacteristic
19#16840230C2211yescharacteristic
20#16839871C411yescharacteristic, complemented
21#16839871C411yescharacteristic, complemented
22#16839866C411yescharacteristic, complemented
23#16839866C411yescharacteristic, complemented
24#16839865C411yescharacteristic, complemented
25#16839865C411yescharacteristic, complemented
26#16839859C411yescharacteristic, complemented
27#16839859C411yescharacteristic, complemented
28#16839822—21no—
29#16839822—21no—
30#16839818—21no—
31#16839818—21no—
32C2×#14519713C2211yescharacteristic
33#16839798C2211yescharacteristic
34#16839789C2211yescharacteristic
35C2×#14519705C2211yescharacteristic
36C2×#14519704C2211yescharacteristic
37C2×#14519703C2211yescharacteristic
38#16839759C2211yescharacteristic
39#16839730C2211yescharacteristic
40#16839728C2211yescharacteristic
41#16839720—21no—
42#16839720C2211yescharacteristic
43#16839717—21no—
44#16839717C2211yescharacteristic
45M16×Dic61—21no—
46M16×Dic61C2211yescharacteristic
47M16×Dic61C2211yescharacteristic
48C2×#14519687—21no—
49C2×#14519687C2211yescharacteristic
50C2×#14519687C2211yescharacteristic
51C42×Dic61C2211yescharacteristic
52#16839526C2211yescharacteristic
53#16839526C2211yescharacteristic
54#16839523C2211yescharacteristic
55#16839522C2211yescharacteristic
56#16839521C2211yescharacteristic
57#16839517C2211yescharacteristic
58#16839516C2211yescharacteristic
59#16839516C2211yescharacteristic
60C2×#131×C61C2211yescharacteristic
61C2×#108×C61C2211yescharacteristic
62C61×#424C2211yescharacteristic
63C2×#104×C61C2211yescharacteristic
64C61×#416C2211yescharacteristic
65C61×#339C2211yescharacteristic
66C61×#336C2211yescharacteristic
67C2×#11758222C2311yescharacteristic
68#14519797—22no—
69#14519797—22no—
70C2×#11758202C2×C411yescharacteristic
71C2×#11758200C2×C411yescharacteristic
72C2×#11758201C2×C411yescharacteristic
73C2×#11758199C2×C411yescharacteristic
74#14519784—22no—
75#14519784—22no—
76C2×#11758194—21no—
77C2×#11758194C2311yescharacteristic
78C2×#11758194C2311yescharacteristic
79C22×#92418—21no—
80C22×#92418C2311yescharacteristic
81C22×#92418C2311yescharacteristic
82Q8×Dic61C2×C411yescharacteristic
83Q8×Dic61C2×C411yescharacteristic
84#14519742C2×C411yescharacteristic
85#14519742C2×C411yescharacteristic
86D4×Dic61—21no—
87D4×Dic61—21no—
88D4×Dic61C2×C411yescharacteristic
89D4×Dic61C2×C411yescharacteristic
90#14519739C2×C411yescharacteristic
91#14519739C2×C411yescharacteristic
92#14519737—21no—
93#14519737—21no—
94#14519736—21no—
95#14519736C2311yescharacteristic
96#14519734—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61—21no—
100C2×C4×Dic61C2311yescharacteristic
101C2×C4×Dic61C2311yescharacteristic
102C42×D61—21no—
103#14519713—22no—
104#14519710C2×C411yescharacteristic
105#14519710C2×C411yescharacteristic
106#14519708C2×C411yescharacteristic
107#14519708C2×C411yescharacteristic
108C4⋊#92418—21no—
109C4⋊#92418—21no—
110#14519705—22no—
111#14519704—22no—
112#14519703—22no—
113#14519702C2×C411yescharacteristic
114#14519702C2×C411yescharacteristic
115#14519699C2×C411yescharacteristic
116#14519699C2×C411yescharacteristic
117#14519689—21no—
118#14519689—21no—
119#14519689—21no—
120#14519689—21no—
121#14519687—21no—
122#14519687—21no—
123#14519687—21no—
124#14519687—21no—
125#14519687C2×C411yescharacteristic
126#14519687C2×C411yescharacteristic
127#14519687C2×C411yescharacteristic
128#14519687C2×C411yescharacteristic
129#14519687D411yescharacteristic
130#14519687D411yescharacteristic
131#14519687D411yescharacteristic
132#14519687D411yescharacteristic
133C8×Dic61—21no—
134C8×Dic61—21no—
135C8×Dic61—21no—
136C8×Dic61—21no—
137C2×#41×C61C2311yescharacteristic
138#131×C61—22no—
139#131×C61—22no—
140C2×M16×C61—21no—
141C2×M16×C61C2311yescharacteristic
142C2×M16×C61C2311yescharacteristic
143C22×C8×C61—21no—
144C22×C8×C61C2311yescharacteristic
145C22×C8×C61C2311yescharacteristic
146#117×C61C2311yescharacteristic
147C2×C42×C61C2311yescharacteristic
148#108×C61—22no—
149C4⋊C8×C61—21no—
150C4⋊C8×C61—21no—
151#104×C61—22no—
152#103×C61C2×C411yescharacteristic
153#103×C61C2×C411yescharacteristic
154#102×C61C2×C411yescharacteristic
155#102×C61C2×C411yescharacteristic
156#41×C61—21no—
157#41×C61—21no—
158#41×C61#4112yes—
159#41×C61#4112yes—
160C2×Q8×C61C22×C411yescharacteristic
161C2×D4×C61—21no—
162C2×D4×C61C22×C411yescharacteristic
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61C2411yescharacteristic
166C2×C22⋊D61—21no—
167C22×Dic61—21no—
168C22×Dic61—21no—
169C22×Dic61—21no—
170C22×Dic61#4111yescharacteristic
171C22×Dic61#4111yescharacteristic
172#11758222—21no—
173#11758222—21no—
174#11758222—21no—
175#11758222—21no—
176C2×D244C22×C411yescharacteristic
177C2×C4×D61—21no—
178C2×C4×D61—21no—
179C2×C4×D61—21no—
180C2×Dic122C22×C411yescharacteristic
181M16×C61—24no—
182M16×C61—21no—
183M16×C61—21no—
184M16×C61—21no—
185M16×C61#4112yes—
186C2×C8×C61—24no—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61—21no—
192C2×C8×C61—21no—
193C2×C8×C61C22×C411yescharacteristic
194C2×C8×C61C22×C411yescharacteristic
195C2×C8×C61C2×D411yescharacteristic
196C2×C8×C61C2×D411yescharacteristic
197C4⋊C4×C61C22×C411yescharacteristic
198C4⋊C4×C61C22×C411yescharacteristic
199C22⋊C4×C61—21no—
200C42×C61—21no—
201C42×C61—21no—
202C42×C61—21no—
203C22⋊Dic61—21no—
204C22⋊Dic61—21no—
205C22⋊Dic61—21no—
206#11758202—22no—
207#11758201—22no—
208#11758200—22no—
209#11758199—22no—
210#11758198—21no—
211#11758198—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61C22×C411yescharacteristic
215C4⋊Dic61C22×C411yescharacteristic
216#11758196—21no—
217#11758196—21no—
218C4×Dic61—21no—
219C4×Dic61—21no—
220C4×Dic61—21no—
221C4×Dic61—21no—
222C4×Dic61—21no—
223C4×Dic61—21no—
224C4×Dic61—21no—
225C4×Dic61—42no—
226C4×Dic61—42no—
227C4×Dic61C22×C411yescharacteristic
228C4×Dic61C22×C411yescharacteristic
229C4×Dic61C2×D411yescharacteristic
230C4×Dic61C2×D411yescharacteristic
231#11758194—24no—
232#11758194—21no—
233#11758194—21no—
234#11758194—21no—
235#11758194#4112yes—
236C2×#92418—24no—
237C2×#92418—21no—
238C2×#92418—21no—
239C2×#92418—21no—
240C2×#92418—21no—
241C2×#92418—21no—
242C2×#92418—21no—
243C2×#92418C22×C411yescharacteristic
244C2×#92418C22×C411yescharacteristic
245C2×#92418C2×D411yescharacteristic
246C2×#92418C2×D411yescharacteristic
247C23×C61—21no—
248C23×C61C2×#4111yescharacteristic
249C22×D61—21no—
250Q8×C61—21no—
251Q8×C61#11712yes—
252D4×C61—22no—
253D4×C61—22no—
254D4×C61—21no—
255D4×C61#11712yes—
256C2×C4×C61—21no—
257C2×C4×C61—22no—
258C2×C4×C61—22no—
259C2×C4×C61—22no—
260C2×C4×C61—21no—
261C2×C4×C61—21no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61C23×C411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C22⋊D61—41no—
271C22⋊D61—41no—
272C2×Dic61—22no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—81no—
281C2×Dic61—81no—
282C2×Dic61C4×D411yescharacteristic
283C2×Dic61C4×D411yescharacteristic
284C2×Dic61C4×D411yescharacteristic
285C2×Dic61C4×D411yescharacteristic
286D244—21no—
287D244—21no—
288C4×D61—41no—
289C4×D61—41no—
290C4×D61—81no—
291C4×D61—81no—
292Dic122—21no—
293Dic122—21no—
294C8×C61—21no—
295C8×C61—21no—
296C8×C61—21no—
297C8×C61—22no—
298C8×C61—21no—
299C8×C61—41no—
300#92418—21no—
301#92418—21no—
302#92418—21no—
303#92418—22no—
304#92418—21no—
305#92418—41no—
306#21139—611nomaximal, Sylow
307C22×C61—21no—
308C22×C61—21no—
309C22×C61—22no—
310C22×C61—41no—
311C22×C61—41no—
312C22×C61C2×C4×D411yescharacteristic
313C22×C61#52911yescharacteristic
314C22×C61#52911yescharacteristic
315C2×D61—41no—
316C2×D61—41no—
317C2×D61—41no—
318C4×C61—22no—
319C4×C61—42no—
320C4×C61—41no—
321C4×C61—41no—
322C4×C61C2×#11711yescharacteristic
323C4×C61C4×#4111yescharacteristic
324C4×C61#52511yescharacteristic
325C4×C61#52511yescharacteristic
326Dic61—21no—
327Dic61—21no—
328Dic61—21no—
329Dic61—21no—
330Dic61—42no—
331Dic61—41no—
332Dic61—41no—
333Dic61—41no—
334#2943—611no—
335#2923—611no—
336#2875—611no—
337#1929—611no—
338#1929—611no—
339#1922—611no—
340#1922—611no—
341#1876—611no—
342#1876—611no—
343#1866—611no—
344#1866D6111yescharacteristic, complemented
345#1778—611no—
346C4×#104—611no—
347C4×#104—611no—
348#1745—611no—
349C2×C61—21no—
350C2×C61—21no—
351C2×C61—42no—
352C2×C61#229011yescharacteristic
353C2×C61#295611yescharacteristic
354C2×C61#295611yescharacteristic
355D61—81no—
356C2×#135—611no—
357C2×#131—611no—
358C2×#131C2×D6111yescharacteristic
359C4×#41—611no—
360C4×#41—611no—
361#442—611no—
362#442—611no—
363#441—611no—
364#441—611no—
365#440—611no—
366#440—611no—
367#440—611no—
368#440—611no—
369#433—1221no—
370#433—1221no—
371#432—1221no—
372#432—1221no—
373C2×#108—611no—
374C2×#108C2×D6111yescharacteristic
375#427—611no—
376#424—611no—
377#424C2×D6111yescharacteristic
378C2×#104—611no—
379C2×#104—611no—
380C2×#104—611no—
381C2×#104C2×D6111yescharacteristic
382#416—611no—
383#416C2×D6111yescharacteristic
384#408—611no—
385#408—611no—
386#405—611no—
387#405—611no—
388#405—1221no—
389#405—1221no—
390C4×M16—611no—
391C4×M16—611no—
392C4×M16—1221no—
393C2×#97—611no—
394C2×#97—611no—
395C2×#97—1221no—
396C43—611no—
397#341—611no—
398#341—611no—
399#339—611no—
400#339—611no—
401#339—611no—
402#339C2×D6111yescharacteristic
403#336—611no—
404#336—611no—
405#336—611no—
406#336C2×D6111yescharacteristic
407C61#2113911yescharacteristic, complemented, Sylow
408C2×#41—611no—
409C2×#41C22×D6111yescharacteristic
410#135—1222no—
411#135—1222no—
412C2×Q16—611no—
413C2×SD16—611no—
414C2×SD16—611no—
415C2×D8—611no—
416#131—22no—
417#131—22no—
418#131—1222no—
419#131—1222no—
420C2×M16—21no—
421C2×M16—611no—
422C2×M16—611no—
423C2×M16—1221no—
424C2×M16C22×D6111yescharacteristic
425C2×M16C22×D6111yescharacteristic
426C22×C8—21no—
427C22×C8—611no—
428C22×C8—611no—
429C22×C8—1221no—
430C22×C8C22×D6111yescharacteristic
431C22×C8C22×D6111yescharacteristic
432C4×Q8—611no—
433C4×Q8—611no—
434C4×Q8—611no—
435C4×Q8—611no—
436C4×D4—611no—
437C4×D4—611no—
438C4×D4—611no—
439C4×D4—611no—
440C4×D4—1221no—
441C4×D4—1221no—
442C4×D4—1221no—
443C4×D4—1221no—
444#117—611no—
445#117—1221no—
446#117—1221no—
447#117C22×D6111yescharacteristic
448C2×C42—611no—
449C2×C42—611no—
450C2×C42—1221no—
451C2×C42—1221no—
452C2×C42—1221no—
453C2×C42—1221no—
454C2×C42C22×D6111yescharacteristic
455#108—22no—
456#108—1222no—
457#107—611no—
458#107—611no—
459#106—611no—
460#106—611no—
461C4⋊C8—21no—
462C4⋊C8—21no—
463C4⋊C8—1221no—
464C4⋊C8—1221no—
465#104—22no—
466#104—1222no—
467#104—1222no—
468#104—1222no—
469#103—611no—
470#103—611no—
471#103C4×D6111yescharacteristic
472#103C4×D6111yescharacteristic
473#102—611no—
474#102—611no—
475#102C4×D6111yescharacteristic
476#102C4×D6111yescharacteristic
477C22⋊C8—1221no—
478C22⋊C8—1221no—
479C22⋊C8—1221no—
480C22⋊C8—1221no—
481#97—611no—
482#97—611no—
483#97—611no—
484#97—611no—
485#97—611no—
486#97—611no—
487#97—611no—
488#97—611no—
489#97—1221no—
490#97—1221no—
491#97—1221no—
492#97—1221no—
493C4×C8—1221no—
494C4×C8—1221no—
495C4×C8—1221no—
496C4×C8—1221no—
497#41—21no—
498#41—21no—
499#41—1221no—
500#41—1221no—
501#41—1221no—
502#41—1221no—
503#41#1175822412yes—
504#41#1175822612yes—
505C2×Q8—611no—
506C2×Q8C2×C4×D6111yescharacteristic
507C2×D4—21no—
508C2×D4—611no—
509C2×D4—1221no—
510C2×D4C2×C4×D6111yescharacteristic
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4—1221no—
521C22×C4C23×D6111yescharacteristic
522Q16—1222no—
523SD16—1222no—
524SD16—1222no—
525D8—1222no—
526M16—24no—
527M16—21no—
528M16—21no—
529M16—21no—
530M16—612no—
531M16—1224no—
532M16—1221no—
533M16—1221no—
534M16—1221no—
535M16#1175822412yes—
536C2×C8—24no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—611no—
544C2×C8—611no—
545C2×C8—611no—
546C2×C8—611no—
547C2×C8—1224no—
548C2×C8—1221no—
549C2×C8—1221no—
550C2×C8—1221no—
551C2×C8—1221no—
552C2×C8—1221no—
553C2×C8—1221no—
554C2×C8C2×C4×D6111yescharacteristic
555C2×C8C2×C4×D6111yescharacteristic
556C2×C8D4×D6111yescharacteristic
557C2×C8D4×D6111yescharacteristic
558C4⋊C4—611no—
559C4⋊C4—611no—
560C4⋊C4—1221no—
561C4⋊C4—1221no—
562C4⋊C4—1221no—
563C4⋊C4—1221no—
564C4⋊C4C2×C4×D6111yescharacteristic
565C4⋊C4C2×C4×D6111yescharacteristic
566C22⋊C4—21no—
567C22⋊C4—1221no—
568C22⋊C4—1221no—
569C22⋊C4—1221no—
570C22⋊C4—1221no—
571C22⋊C4—1221no—
572C42—21no—
573C42—21no—
574C42—21no—
575C42—611no—
576C42—611no—
577C42—611no—
578C42—611no—
579C42—1221no—
580C42—1221no—
581C42—1221no—
582C42—1221no—
583C42—1221no—
584C42—1221no—
585C42—1221no—
586C42—2442no—
587C42—2442no—
588C23—21no—
589C23—1221no—
590C23#41×D6111yescharacteristic
591Q8—21no—
592Q8—1221no—
593Q8—1221no—
594Q8#1451973512yes—
595D4—22no—
596D4—22no—
597D4—21no—
598D4—1221no—
599D4—1221no—
600D4—2441no—
601D4—2441no—
602D4#1451973512yes—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—611no—
612C2×C4—611no—
613C2×C4—611no—
614C2×C4—611no—
615C2×C4—1222no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—4881no—
620C2×C4—4881no—
621C2×C4—4881no—
622C2×C4—4881no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4C22×C4×D6111yescharacteristic
630C2×C4C2×D4×D6111yescharacteristic
631C2×C4#41×D6111yescharacteristic
632C2×C4C2×#1175822411yescharacteristic
633C2×C4C2×#1175822411yescharacteristic
634C2×C4C2×#1175822611yescharacteristic
635C8—21no—
636C8—21no—
637C8—21no—
638C8—22no—
639C8—21no—
640C8—41no—
641C8—1221no—
642C8—1221no—
643C8—1221no—
644C8—1221no—
645C8—1222no—
646C8—2441no—
647C22—21no—
648C22—21no—
649C22—22no—
650C22—41no—
651C22—41no—
652C22—2441no—
653C22—2441no—
654C22—2441no—
655C22C4×D4×D6111yescharacteristic
656C22#1684028911yescharacteristic
657C22#1684029811yescharacteristic
658C4—22no—
659C4—42no—
660C4—41no—
661C4—41no—
662C4—1221no—
663C4—1221no—
664C4—1221no—
665C4—1221no—
666C4—2442no—
667C4—2441no—
668C4—2441no—
669C4—2441no—
670C4C2×#1451973511yescharacteristic
671C4C4×#1175822411yescharacteristic
672C4#1684026811yescharacteristic
673C4#1684027511yescharacteristic
674C2—21no—
675C2—21no—
676C2—42no—
677C2—4881no—
678C2#2144825511yescharacteristic
679C2#2145472411yescharacteristic
680C2#2145472711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1866C2
socleC22×C61C2×C4×D4
prime radicalC2×#14519797C22
cosocleC22×C4×C61C24
hypercenter#1866D61
hypocenterC61#21139
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1866

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21454469C2semidirect
3. #21454245C2semidirect
4. #21446468C2semidirect
5. #21446466C2semidirect
6. #21446421C2semidirect
7. #21446415C2semidirect
10. #21446096C2semidirect
11. #21445727C2semidirect
12. #104×Dic61C2semidirect
13. #21445623C2semidirect
14. #21445566C2semidirect
15. C61×#1866C2semidirect
20. #16839871C4semidirect
21. #16839871C4semidirect
22. #16839866C4semidirect
23. #16839866C4semidirect
24. #16839865C4semidirect
25. #16839865C4semidirect
26. #16839859C4semidirect
27. #16839859C4semidirect
344. #1866D61semidirect
407. C61#21139semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214546152
#214544692
#214542452
#214464682
#214464662
#214464212
#214464152
#214461462
#214461442
#214460962
#214457272
#104×Dic612
#214456232
#214455662
C61×#18662
#2113961