Finite Group Databasebeta

← #29683265

Subgroups — #29683265

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#21454590C211yesmaximal, characteristic
2#21454469C211yesmaximal, characteristic, complemented
3#21454246C211yesmaximal, characteristic, complemented
4#21446467C211yesmaximal, characteristic, complemented
5#21446465C211yesmaximal, characteristic, complemented
6#21446420C211yesmaximal, characteristic, complemented
7#21446147C211yesmaximal, characteristic
8#21446145C211yesmaximal, characteristic
9#21446095C211yesmaximal, characteristic, complemented
10#21446093C211yesmaximal, characteristic, complemented
11#21445720C211yesmaximal, characteristic, complemented
12#21445628C211yesmaximal, characteristic, complemented
13#21445628C211yesmaximal, characteristic, complemented
14#21445569C211yesmaximal, characteristic, complemented
15C61×#1922C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic
17C2×#14519785C2211yescharacteristic
18C2×#14519783C2211yescharacteristic
19#16840230C2211yescharacteristic
20#16840230C2211yescharacteristic
21#16839856C411yescharacteristic, complemented
22#16839856C411yescharacteristic, complemented
23#16839849C411yescharacteristic, complemented
24#16839849C411yescharacteristic, complemented
25#16839849C411yescharacteristic, complemented
26#16839849C411yescharacteristic, complemented
27#16839844C411yescharacteristic, complemented
28#16839844C411yescharacteristic, complemented
29#16839822—21no—
30#16839822—21no—
31#16839818—21no—
32#16839818—21no—
33C2×#14519712C2211yescharacteristic
34C2×#14519712C2211yescharacteristic
35#16839789C2211yescharacteristic
36C2×#14519705C2211yescharacteristic
37C2×#14519705C2211yescharacteristic
38C2×#14519703C2211yescharacteristic
39C2×#14519703C2211yescharacteristic
40#16839756C2211yescharacteristic
41#16839756C2211yescharacteristic
42#16839730C2211yescharacteristic
43#16839728C2211yescharacteristic
44#16839719—21no—
45#16839718C2211yescharacteristic
46#16839716—21no—
47M16×Dic61C2211yescharacteristic
48M16×Dic61C2211yescharacteristic
49C4×#11758189—21no—
50C2×#14519687C2211yescharacteristic
51#97×D61—21no—
52C2×C8×Dic61C2211yescharacteristic
53#16839608C2211yescharacteristic
54#16839527C2211yescharacteristic
55#16839521C2211yescharacteristic
56#16839521C2211yescharacteristic
57#16839517C2211yescharacteristic
58#16839516C2211yescharacteristic
59#16839516C2211yescharacteristic
60C61×#427C2211yescharacteristic
61C61×#424C2211yescharacteristic
62C61×#405C2211yescharacteristic
63C61×#341C2211yescharacteristic
64C61×#339C2211yescharacteristic
65C61×#339C2211yescharacteristic
66C61×#336C2211yescharacteristic
67C2×#11758222C2311yescharacteristic
68C2×#11758222C2311yescharacteristic
69#14519799—22no—
70#14519799—22no—
71C2×Dic244C2×C411yescharacteristic
72C2×#11758190C2×C411yescharacteristic
73C2×#11758190C2×C411yescharacteristic
74C2×D488C2×C411yescharacteristic
75#14519785—22no—
76#14519785—22no—
77#14519783—22no—
78#14519783—22no—
79C2×#11758194C2311yescharacteristic
80C2×#11758194C2311yescharacteristic
81C2×#11758189—21no—
82C2×#11758189—21no—
83C22×#92418C2311yescharacteristic
84C22×#92418C2311yescharacteristic
85C2×C8×D61—21no—
86C2×C8×D61—21no—
87#14519742C2×C411yescharacteristic
88#14519742C2×C411yescharacteristic
89#14519742C2×C411yescharacteristic
90#14519742C2×C411yescharacteristic
91#14519739C2×C411yescharacteristic
92#14519739C2×C411yescharacteristic
93#14519739C2×C411yescharacteristic
94#14519739C2×C411yescharacteristic
95#14519737—21no—
96#14519737—21no—
97#14519737—21no—
98#14519737—21no—
99#14519734—21no—
100#14519734—21no—
101C2×C4×Dic61C2311yescharacteristic
102C2×C4×Dic61C2311yescharacteristic
103C42×D61—21no—
104C42×D61—21no—
105#14519712—22no—
106#14519712—22no—
107C4⋊#92418—21no—
108C4⋊#92418—21no—
109#14519705—22no—
110#14519705—22no—
111#14519703—22no—
112#14519703—22no—
113#14519700C2×C411yescharacteristic
114#14519700C2×C411yescharacteristic
115#14519700C2×C411yescharacteristic
116#14519700C2×C411yescharacteristic
117#14519697C2×C411yescharacteristic
118#14519697C2×C411yescharacteristic
119#14519697C2×C411yescharacteristic
120#14519697C2×C411yescharacteristic
121#14519689—21no—
122#14519689—21no—
123#14519689—21no—
124#14519689—21no—
125#14519688—21no—
126#14519687—21no—
127#14519687—21no—
128#14519687D411yescharacteristic
129#14519687D411yescharacteristic
130#14519687D411yescharacteristic
131#14519687D411yescharacteristic
132C4×#92418—21no—
133C8×Dic61—21no—
134C8×Dic61—21no—
135C8×Dic61C2×C411yescharacteristic
136C8×Dic61C2×C411yescharacteristic
137C8×Dic61C2×C411yescharacteristic
138C8×Dic61C2×C411yescharacteristic
139#14519684—21no—
140#14519684—21no—
141C2×M16×C61C2311yescharacteristic
142C2×M16×C61C2311yescharacteristic
143C22×C8×C61C2311yescharacteristic
144C22×C8×C61C2311yescharacteristic
145#117×C61C2311yescharacteristic
146#117×C61C2311yescharacteristic
147C2×C42×C61C2311yescharacteristic
148#107×C61C2×C411yescharacteristic
149#107×C61C2×C411yescharacteristic
150#106×C61C2×C411yescharacteristic
151#106×C61C2×C411yescharacteristic
152C4⋊C8×C61—21no—
153C4⋊C8×C61—21no—
154#97×C61—21no—
155C4×C8×C61—21no—
156C22×C4×C61—21no—
157C22×C4×C61C2411yescharacteristic
158C2×C22⋊D61—21no—
159C2×C22⋊D61—21no—
160C22×Dic61—21no—
161C22×Dic61#4111yescharacteristic
162C22×Dic61#4111yescharacteristic
163#11758222—21no—
164#11758222—21no—
165#11758222—21no—
166#11758222—21no—
167#11758222—21no—
168#11758222—21no—
169#11758222#4112yes—
170#11758222#4112yes—
171C2×D244C22×C411yescharacteristic
172C2×D244C22×C411yescharacteristic
173C2×C4×D61—21no—
174C2×C4×D61—21no—
175C2×C4×D61—21no—
176C2×C4×D61—21no—
177C2×C4×D61—21no—
178C2×C4×D61—21no—
179C2×Dic122C22×C411yescharacteristic
180C2×Dic122C22×C411yescharacteristic
181M16×C61—21no—
182M16×C61—21no—
183M16×C61—21no—
184M16×C61#4112yes—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61C22×C411yescharacteristic
192C2×C8×C61C22×C411yescharacteristic
193C2×C8×C61C2×D411yescharacteristic
194C2×C8×C61C2×D411yescharacteristic
195C4⋊C4×C61C22×C411yescharacteristic
196C4⋊C4×C61C22×C411yescharacteristic
197C4⋊C4×C61C22×C411yescharacteristic
198C4⋊C4×C61C22×C411yescharacteristic
199C22⋊C4×C61—21no—
200C22⋊C4×C61—21no—
201C42×C61—21no—
202C42×C61—21no—
203C42×C61—21no—
204C42×C61—21no—
205#11758198—21no—
206#11758198—21no—
207#11758198—21no—
208#11758198—21no—
209#11758196—21no—
210#11758196—21no—
211#11758196—21no—
212#11758196—21no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C4×Dic61—21no—
217C4×Dic61—21no—
218C4×Dic61—21no—
219C4×Dic61C22×C411yescharacteristic
220C4×Dic61C22×C411yescharacteristic
221C4×Dic61C2×D411yescharacteristic
222C4×Dic61C2×D411yescharacteristic
223#11758194—21no—
224#11758194—21no—
225#11758194—21no—
226#11758194#4112yes—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418—21no—
231C2×#92418—21no—
232C2×#92418—21no—
233C2×#92418C22×C411yescharacteristic
234C2×#92418C22×C411yescharacteristic
235C2×#92418C2×D411yescharacteristic
236C2×#92418C2×D411yescharacteristic
237Dic244—22no—
238D488—22no—
239#11758190—22no—
240#11758190—22no—
241#11758189—24no—
242#11758189—24no—
243C8×D61—24no—
244C8×D61—24no—
245C23×C61C2×#4111yescharacteristic
246C22×D61—21no—
247C22×D61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61C23×C411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C22⋊D61—22no—
262C22⋊D61—22no—
263C22⋊D61—41no—
264C22⋊D61—41no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61C4×D411yescharacteristic
273C2×Dic61C4×D411yescharacteristic
274C2×Dic61C4×D411yescharacteristic
275C2×Dic61C4×D411yescharacteristic
276D244—21no—
277D244—21no—
278D244—21no—
279D244#11712yes—
280C4×D61—22no—
281C4×D61—22no—
282C4×D61—41no—
283C4×D61—41no—
284C4×D61—81no—
285C4×D61—81no—
286C4×D61—81no—
287C4×D61—81no—
288Dic122—21no—
289Dic122—21no—
290Dic122—21no—
291Dic122#11712yes—
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×C61C2×C4×D411yescharacteristic
308C22×C61#52911yescharacteristic
309C22×C61#52911yescharacteristic
310C2×D61—22no—
311C2×D61—41no—
312C2×D61—41no—
313C2×D61—41no—
314C2×D61—41no—
315C2×D61—41no—
316C4×C61—41no—
317C4×C61—41no—
318C4×C61—41no—
319C4×C61—41no—
320C4×C61—81no—
321C4×C61C2×#11711yescharacteristic
322C4×C61C4×#4111yescharacteristic
323C4×C61#52511yescharacteristic
324C4×C61#52511yescharacteristic
325Dic61—21no—
326Dic61—21no—
327Dic61—21no—
328Dic61—21no—
329Dic61—22no—
330Dic61—41no—
331Dic61—81no—
332C4×#135—611no—
333#2923—611no—
334#2875—611no—
335#1929—611no—
336#1929—611no—
337#1922—611no—
338#1922D6111yescharacteristic, complemented
339#1876—611no—
340#1876—611no—
341#1866—611no—
342#1866—611no—
343C4×#108—611no—
344#1759—611no—
345#1759—611no—
346#1746—611no—
347C2×C61—21no—
348C2×C61—21no—
349C2×C61#229011yescharacteristic
350C2×C61#295711yescharacteristic
351C2×C61#295711yescharacteristic
352D61—42no—
353D61—81no—
354C2×#135—611no—
355C2×#131—611no—
356C2×#131—611no—
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×#108—611no—
373#427C2×D6111yescharacteristic
374#424—611no—
375#424C2×D6111yescharacteristic
376C2×#104—611no—
377C2×#104—611no—
378C2×#104—611no—
379C2×#104—611no—
380#416—611no—
381#416—611no—
382#408—611no—
383#408—611no—
384#405—611no—
385#405—1221no—
386#405—1221no—
387#405C2×D6111yescharacteristic
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#341C2×D6111yescharacteristic
397#339—611no—
398#339—611no—
399#339C2×D6111yescharacteristic
400#339C2×D6111yescharacteristic
401#336—611no—
402#336—611no—
403#336—611no—
404#336C2×D6111yescharacteristic
405C61#2114011yescharacteristic, complemented, Sylow
406C2×#41—611no—
407C2×#41—611no—
408#135—1222no—
409#135—1222no—
410C2×Q16—611no—
411C2×SD16—611no—
412C2×SD16—611no—
413C2×D8—611no—
414#131—1222no—
415#131—1222no—
416#131—1222no—
417#131—1222no—
418C2×M16—611no—
419C2×M16—611no—
420C2×M16—1221no—
421C2×M16—1221no—
422C2×M16C22×D6111yescharacteristic
423C2×M16C22×D6111yescharacteristic
424C22×C8—611no—
425C22×C8—611no—
426C22×C8—1221no—
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—1221no—
443#117—1221no—
444#117C22×D6111yescharacteristic
445#117C22×D6111yescharacteristic
446C2×C42—611no—
447C2×C42—611no—
448C2×C42—1221no—
449C2×C42—1221no—
450C2×C42C22×D6111yescharacteristic
451#108—1222no—
452#108—1222no—
453#107C4×D6111yescharacteristic
454#107C4×D6111yescharacteristic
455#106C4×D6111yescharacteristic
456#106C4×D6111yescharacteristic
457C4⋊C8—21no—
458C4⋊C8—21no—
459C4⋊C8—1221no—
460C4⋊C8—1221no—
461#104—1222no—
462#104—1222no—
463#104—1222no—
464#104—1222no—
465#103—611no—
466#103—611no—
467#103—611no—
468#103—611no—
469#102—611no—
470#102—611no—
471#102—611no—
472#102—611no—
473C22⋊C8—1221no—
474C22⋊C8—1221no—
475C22⋊C8—1221no—
476C22⋊C8—1221no—
477#97—21no—
478#97—611no—
479#97—611no—
480#97—611no—
481#97—611no—
482#97—1221no—
483#97—1221no—
484#97—1221no—
485C4×C8—21no—
486C4×C8—611no—
487C4×C8—611no—
488C4×C8—611no—
489C4×C8—611no—
490C4×C8—1221no—
491C4×C8—1221no—
492C4×C8—1221no—
493#95—1221no—
494#95—1221no—
495#41—612no—
496#41—612no—
497#41—1221no—
498#41—1221no—
499#41—1221no—
500#41—1221no—
501#41—1221no—
502#41—1221no—
503C2×Q8—611no—
504C2×Q8—611no—
505C2×D4—611no—
506C2×D4—611no—
507C2×D4—1221no—
508C2×D4—1221no—
509C22×C4—21no—
510C22×C4—611no—
511C22×C4—611no—
512C22×C4—1221no—
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—21no—
525M16—21no—
526M16—21no—
527M16—612no—
528M16—1224no—
529M16—1224no—
530M16—1221no—
531M16—1221no—
532M16—1221no—
533M16#1175822612yes—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—611no—
541C2×C8—611no—
542C2×C8—611no—
543C2×C8—611no—
544C2×C8—1224no—
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—1221no—
557C4⋊C4—1221no—
558C4⋊C4—1221no—
559C4⋊C4—1221no—
560C4⋊C4C2×C4×D6111yescharacteristic
561C4⋊C4C2×C4×D6111yescharacteristic
562C4⋊C4C2×C4×D6111yescharacteristic
563C4⋊C4C2×C4×D6111yescharacteristic
564C22⋊C4—21no—
565C22⋊C4—21no—
566C22⋊C4—1221no—
567C22⋊C4—1221no—
568C22⋊C4—1221no—
569C22⋊C4—1221no—
570C42—21no—
571C42—21no—
572C42—21no—
573C42—21no—
574C42—611no—
575C42—611no—
576C42—611no—
577C42—611no—
578C42—1221no—
579C42—1221no—
580C42—1221no—
581C42—1221no—
582C42—1221no—
583C42—1221no—
584C23—1221no—
585C23—1221no—
586C23#41×D6111yescharacteristic
587Q8—612no—
588Q8—1221no—
589Q8—1221no—
590Q8—1221no—
591D4—612no—
592D4—1222no—
593D4—1222no—
594D4—1221no—
595D4—1221no—
596D4—1221no—
597D4—2441no—
598D4—2441no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—611no—
607C2×C4—611no—
608C2×C4—611no—
609C2×C4—611no—
610C2×C4—1221no—
611C2×C4—1222no—
612C2×C4—1222no—
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×C4#41×D6111yescharacteristic
629C2×C4#41×D6111yescharacteristic
630C2×C4C2×#1175822611yescharacteristic
631C8—21no—
632C8—21no—
633C8—21no—
634C8—21no—
635C8—22no—
636C8—41no—
637C8—1221no—
638C8—1221no—
639C8—1222no—
640C8—1221no—
641C8—1221no—
642C8—2441no—
643C22—21no—
644C22—21no—
645C22—1222no—
646C22—2441no—
647C22—2441no—
648C22—2441no—
649C22—2441no—
650C22—2441no—
651C22C4×D4×D6111yescharacteristic
652C22#1684029511yescharacteristic
653C22#1684029711yescharacteristic
654C4—41no—
655C4—41no—
656C4—41no—
657C4—41no—
658C4—81no—
659C4—1222no—
660C4—1221no—
661C4—1221no—
662C4—1221no—
663C4—1221no—
664C4—4881no—
665C4—2441no—
666C4#117×D6111yescharacteristic
667C4C4×#1175822611yescharacteristic
668C4#1684027011yescharacteristic
669C4#1684027511yescharacteristic
670C2—21no—
671C2—21no—
672C2—4881no—
673C2—2442no—
674C2#2144825711yescharacteristic
675C2#2145473111yescharacteristic
676C2#2145473311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1922C2
socleC22×C61C2×C4×D4
prime radicalC2×#14519799C22
cosocleC22×C4×C61C24
hypercenter#1922D61
hypocenterC61#21140
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1922

Complements

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

NH ≅ G/Ntype
2. #21454469C2semidirect
3. #21454246C2semidirect
4. #21446467C2semidirect
5. #21446465C2semidirect
6. #21446420C2semidirect
9. #21446095C2semidirect
10. #21446093C2semidirect
11. #21445720C2semidirect
12. #21445628C2semidirect
13. #21445628C2semidirect
14. #21445569C2semidirect
15. C61×#1922C2semidirect
21. #16839856C4semidirect
22. #16839856C4semidirect
23. #16839849C4semidirect
24. #16839849C4semidirect
25. #16839849C4semidirect
26. #16839849C4semidirect
27. #16839844C4semidirect
28. #16839844C4semidirect
338. #1922D61semidirect
405. C61#21140semidirect

Maximal subgroups

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

subgroupindex
#214545902
#214544692
#214542462
#214464672
#214464652
#214464202
#214461472
#214461452
#214460952
#214460932
#214457202
#214456282
#214455692
C61×#19222
#2114061