Finite Group Databasebeta

← #29683263

Subgroups — #29683263

Order 15616.

764 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454587C211yesmaximal, 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#108×Dic61C211yesmaximal, characteristic, complemented
12#21445631C211yesmaximal, characteristic, complemented
13#21445628C211yesmaximal, characteristic, complemented
14#21445569C211yesmaximal, characteristic, complemented
15C61×#1866C211yesmaximal, characteristic, complemented
16C2×#14519797C2211yescharacteristic
17C2×#14519784C2211yescharacteristic
18#41×Dic61C2211yescharacteristic
19#16840230C2211yescharacteristic
20#16839854C411yescharacteristic, complemented
21#16839854C411yescharacteristic, complemented
22#16839853C411yescharacteristic, complemented
23#16839853C411yescharacteristic, complemented
24#16839847C411yescharacteristic, complemented
25#16839847C411yescharacteristic, complemented
26#16839842C411yescharacteristic, complemented
27#16839842C411yescharacteristic, 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×C8×Dic61C2211yescharacteristic
51#16839608C2211yescharacteristic
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×Dic61C2311yescharacteristic
99C2×C4×Dic61C2311yescharacteristic
100C42×D61—21no—
101#14519713—22no—
102#14519710C2×C411yescharacteristic
103#14519710C2×C411yescharacteristic
104#14519708C2×C411yescharacteristic
105#14519708C2×C411yescharacteristic
106C4⋊#92418—21no—
107C4⋊#92418—21no—
108#14519705—22no—
109#14519704—22no—
110#14519703—22no—
111#14519702C2×C411yescharacteristic
112#14519702C2×C411yescharacteristic
113#14519699C2×C411yescharacteristic
114#14519699C2×C411yescharacteristic
115#14519689—21no—
116#14519689—21no—
117#14519689—21no—
118#14519689—21no—
119#14519687—21no—
120#14519687—21no—
121#14519687—21no—
122#14519687—21no—
123#14519687D411yescharacteristic
124#14519687D411yescharacteristic
125#14519687D411yescharacteristic
126#14519687D411yescharacteristic
127C8×Dic61—21no—
128C8×Dic61—21no—
129C8×Dic61—21no—
130C8×Dic61—21no—
131C8×Dic61C2×C411yescharacteristic
132C8×Dic61C2×C411yescharacteristic
133C8×Dic61C2×C411yescharacteristic
134C8×Dic61C2×C411yescharacteristic
135#14519684—21no—
136#14519684—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×Dic61C22×C411yescharacteristic
226C4×Dic61C22×C411yescharacteristic
227C4×Dic61C2×D411yescharacteristic
228C4×Dic61C2×D411yescharacteristic
229#11758194—24no—
230#11758194—21no—
231#11758194—21no—
232#11758194—21no—
233#11758194#4112yes—
234C2×#92418—24no—
235C2×#92418—21no—
236C2×#92418—21no—
237C2×#92418—21no—
238C2×#92418—21no—
239C2×#92418—21no—
240C2×#92418—21no—
241C2×#92418C22×C411yescharacteristic
242C2×#92418C22×C411yescharacteristic
243C2×#92418C2×D411yescharacteristic
244C2×#92418C2×D411yescharacteristic
245C23×C61—21no—
246C23×C61C2×#4111yescharacteristic
247C22×D61—21no—
248Q8×C61—21no—
249Q8×C61#11712yes—
250D4×C61—22no—
251D4×C61—22no—
252D4×C61—21no—
253D4×C61#11712yes—
254C2×C4×C61—21no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61C23×C411yescharacteristic
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C22⋊D61—41no—
269C22⋊D61—41no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—81no—
279C2×Dic61—81no—
280C2×Dic61C4×D411yescharacteristic
281C2×Dic61C4×D411yescharacteristic
282C2×Dic61C4×D411yescharacteristic
283C2×Dic61C4×D411yescharacteristic
284D244—21no—
285D244—21no—
286C4×D61—41no—
287C4×D61—41no—
288C4×D61—81no—
289C4×D61—81no—
290Dic122—21no—
291Dic122—21no—
292C8×C61—21no—
293C8×C61—21no—
294C8×C61—21no—
295C8×C61—22no—
296C8×C61—21no—
297C8×C61—41no—
298#92418—21no—
299#92418—21no—
300#92418—21no—
301#92418—22no—
302#92418—21no—
303#92418—41no—
304#21140—611nomaximal, Sylow
305C22×C61—21no—
306C22×C61—21no—
307C22×C61—22no—
308C22×C61—41no—
309C22×C61—41no—
310C22×C61C2×C4×D411yescharacteristic
311C22×C61#52911yescharacteristic
312C22×C61#52911yescharacteristic
313C2×D61—41no—
314C2×D61—41no—
315C2×D61—41no—
316C4×C61—22no—
317C4×C61—41no—
318C4×C61—41no—
319C4×C61—81no—
320C4×C61C2×#11711yescharacteristic
321C4×C61C4×#4111yescharacteristic
322C4×C61#52511yescharacteristic
323C4×C61#52511yescharacteristic
324Dic61—21no—
325Dic61—21no—
326Dic61—21no—
327Dic61—21no—
328Dic61—41no—
329Dic61—41no—
330Dic61—41no—
331Dic61—81no—
332C4×#135—611no—
333#2923—611no—
334#2875—611no—
335#1929—611no—
336#1929—611no—
337#1922—611no—
338#1922—611no—
339#1876—611no—
340#1876—611no—
341#1866—611no—
342#1866D6111yescharacteristic, complemented
343C4×#108—611no—
344#1759—611no—
345#1759—611no—
346#1746—611no—
347C2×C61—21no—
348C2×C61—21no—
349C2×C61—42no—
350C2×C61#229011yescharacteristic
351C2×C61#295711yescharacteristic
352C2×C61#295711yescharacteristic
353D61—81no—
354C2×#135—611no—
355C2×#131—611no—
356C2×#131C2×D6111yescharacteristic
357C4×#41—611no—
358C4×#41—611no—
359C4×Q16—611no—
360C4×Q16—611no—
361C4×SD16—611no—
362C4×SD16—611no—
363C4×SD16—611no—
364C4×SD16—611no—
365C4×D8—611no—
366C4×D8—611no—
367#433—1221no—
368#433—1221no—
369#432—1221no—
370#432—1221no—
371C2×#108—611no—
372C2×#108C2×D6111yescharacteristic
373#427—611no—
374#424—611no—
375#424C2×D6111yescharacteristic
376C2×#104—611no—
377C2×#104—611no—
378C2×#104—611no—
379C2×#104C2×D6111yescharacteristic
380#416—611no—
381#416C2×D6111yescharacteristic
382#408—611no—
383#408—611no—
384#405—611no—
385#405—611no—
386#405—1221no—
387#405—1221no—
388C4×M16—611no—
389C4×M16—611no—
390C4×M16—1221no—
391C2×#97—611no—
392C2×#97—1221no—
393C2×C4×C8—611no—
394#376—611no—
395#341—611no—
396#341—611no—
397#339—611no—
398#339—611no—
399#339—611no—
400#339C2×D6111yescharacteristic
401#336—611no—
402#336—611no—
403#336—611no—
404#336C2×D6111yescharacteristic
405C61#2114011yescharacteristic, complemented, Sylow
406C2×#41—611no—
407C2×#41C22×D6111yescharacteristic
408#135—1222no—
409#135—1222no—
410C2×Q16—611no—
411C2×SD16—611no—
412C2×SD16—611no—
413C2×D8—611no—
414#131—22no—
415#131—22no—
416#131—1222no—
417#131—1222no—
418C2×M16—21no—
419C2×M16—611no—
420C2×M16—611no—
421C2×M16—1221no—
422C2×M16C22×D6111yescharacteristic
423C2×M16C22×D6111yescharacteristic
424C22×C8—21no—
425C22×C8—611no—
426C22×C8—611no—
427C22×C8—1221no—
428C22×C8C22×D6111yescharacteristic
429C22×C8C22×D6111yescharacteristic
430C4×Q8—611no—
431C4×Q8—611no—
432C4×Q8—611no—
433C4×Q8—611no—
434C4×D4—611no—
435C4×D4—611no—
436C4×D4—611no—
437C4×D4—611no—
438C4×D4—1221no—
439C4×D4—1221no—
440C4×D4—1221no—
441C4×D4—1221no—
442#117—611no—
443#117—1221no—
444#117—1221no—
445#117C22×D6111yescharacteristic
446C2×C42—611no—
447C2×C42—611no—
448C2×C42—1221no—
449C2×C42—1221no—
450C2×C42C22×D6111yescharacteristic
451#108—22no—
452#108—1222no—
453#107—611no—
454#107—611no—
455#106—611no—
456#106—611no—
457C4⋊C8—21no—
458C4⋊C8—21no—
459C4⋊C8—1221no—
460C4⋊C8—1221no—
461#104—22no—
462#104—1222no—
463#104—1222no—
464#104—1222no—
465#103—611no—
466#103—611no—
467#103C4×D6111yescharacteristic
468#103C4×D6111yescharacteristic
469#102—611no—
470#102—611no—
471#102C4×D6111yescharacteristic
472#102C4×D6111yescharacteristic
473C22⋊C8—1221no—
474C22⋊C8—1221no—
475C22⋊C8—1221no—
476C22⋊C8—1221no—
477#97—611no—
478#97—611no—
479#97—611no—
480#97—611no—
481#97—1221no—
482#97—1221no—
483#97—1221no—
484#97—1221no—
485C4×C8—611no—
486C4×C8—611no—
487C4×C8—611no—
488C4×C8—611no—
489C4×C8—1221no—
490C4×C8—1221no—
491C4×C8—1221no—
492C4×C8—1221no—
493#95—1221no—
494#95—1221no—
495#41—21no—
496#41—21no—
497#41—1221no—
498#41—1221no—
499#41—1221no—
500#41—1221no—
501#41#1175822412yes—
502#41#1175822612yes—
503C2×Q8—611no—
504C2×Q8C2×C4×D6111yescharacteristic
505C2×D4—21no—
506C2×D4—611no—
507C2×D4—1221no—
508C2×D4C2×C4×D6111yescharacteristic
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—611no—
512C22×C4—611no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4C23×D6111yescharacteristic
520Q16—1222no—
521SD16—1222no—
522SD16—1222no—
523D8—1222no—
524M16—24no—
525M16—21no—
526M16—21no—
527M16—21no—
528M16—612no—
529M16—1224no—
530M16—1221no—
531M16—1221no—
532M16—1221no—
533M16#1175822412yes—
534C2×C8—24no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—611no—
542C2×C8—611no—
543C2×C8—611no—
544C2×C8—611no—
545C2×C8—1224no—
546C2×C8—1221no—
547C2×C8—1221no—
548C2×C8—1221no—
549C2×C8—1221no—
550C2×C8—1221no—
551C2×C8—1221no—
552C2×C8C2×C4×D6111yescharacteristic
553C2×C8C2×C4×D6111yescharacteristic
554C2×C8D4×D6111yescharacteristic
555C2×C8D4×D6111yescharacteristic
556C4⋊C4—611no—
557C4⋊C4—611no—
558C4⋊C4—1221no—
559C4⋊C4—1221no—
560C4⋊C4—1221no—
561C4⋊C4—1221no—
562C4⋊C4C2×C4×D6111yescharacteristic
563C4⋊C4C2×C4×D6111yescharacteristic
564C22⋊C4—21no—
565C22⋊C4—1221no—
566C22⋊C4—1221no—
567C22⋊C4—1221no—
568C22⋊C4—1221no—
569C22⋊C4—1221no—
570C42—21no—
571C42—21no—
572C42—21no—
573C42—611no—
574C42—611no—
575C42—611no—
576C42—611no—
577C42—1221no—
578C42—1221no—
579C42—1221no—
580C42—1221no—
581C42—1221no—
582C42—1221no—
583C42—1221no—
584C23—21no—
585C23—1221no—
586C23#41×D6111yescharacteristic
587Q8—21no—
588Q8—1221no—
589Q8—1221no—
590Q8#1451973512yes—
591D4—22no—
592D4—22no—
593D4—21no—
594D4—1221no—
595D4—1221no—
596D4—2441no—
597D4—2441no—
598D4#1451973512yes—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—611no—
608C2×C4—611no—
609C2×C4—611no—
610C2×C4—611no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—4881no—
615C2×C4—4881no—
616C2×C4—4881no—
617C2×C4—4881no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4C22×C4×D6111yescharacteristic
626C2×C4C2×D4×D6111yescharacteristic
627C2×C4#41×D6111yescharacteristic
628C2×C4C2×#1175822411yescharacteristic
629C2×C4C2×#1175822411yescharacteristic
630C2×C4C2×#1175822611yescharacteristic
631C8—21no—
632C8—21no—
633C8—21no—
634C8—22no—
635C8—21no—
636C8—41no—
637C8—1221no—
638C8—1221no—
639C8—1221no—
640C8—1221no—
641C8—1222no—
642C8—2441no—
643C22—21no—
644C22—21no—
645C22—22no—
646C22—41no—
647C22—41no—
648C22—2441no—
649C22—2441no—
650C22—2441no—
651C22C4×D4×D6111yescharacteristic
652C22#1684028911yescharacteristic
653C22#1684029811yescharacteristic
654C4—22no—
655C4—41no—
656C4—41no—
657C4—81no—
658C4—1221no—
659C4—1221no—
660C4—1221no—
661C4—1221no—
662C4—4881no—
663C4—2441no—
664C4—2441no—
665C4—2441no—
666C4C2×#1451973511yescharacteristic
667C4C4×#1175822411yescharacteristic
668C4#1684026811yescharacteristic
669C4#1684027511yescharacteristic
670C2—21no—
671C2—21no—
672C2—42no—
673C2—4881no—
674C2#2144825511yescharacteristic
675C2#2145473111yescharacteristic
676C2#2145473311yescharacteristic
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#21140
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. #108×Dic61C2semidirect
12. #21445631C2semidirect
13. #21445628C2semidirect
14. #21445569C2semidirect
15. C61×#1866C2semidirect
20. #16839854C4semidirect
21. #16839854C4semidirect
22. #16839853C4semidirect
23. #16839853C4semidirect
24. #16839847C4semidirect
25. #16839847C4semidirect
26. #16839842C4semidirect
27. #16839842C4semidirect
342. #1866D61semidirect
405. C61#21140semidirect

Maximal subgroups

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

subgroupindex
#214545872
#214544692
#214542452
#214464682
#214464662
#214464212
#214464152
#214461462
#214461442
#214460962
#108×Dic612
#214456312
#214456282
#214455692
C61×#18662
#2114061