Finite Group Databasebeta

← #30612705

Subgroups — #30612705

Order 15616.

638 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462734C211yesmaximal, characteristic, complemented
2#21462709C211yesmaximal, characteristic, complemented
3#21461448C211yesmaximal, characteristic, complemented
4#21461438C211yesmaximal, characteristic, complemented
5#21461426C211yesmaximal, characteristic, complemented
6#21461420C211yesmaximal, characteristic, complemented
7#21461415C211yesmaximal, characteristic, complemented
8#21461409C211yesmaximal, characteristic, complemented
9SD16×#92420C211yesmaximal, characteristic, complemented
10#21461399C211yesmaximal, characteristic, complemented
11#21461392C211yesmaximal, characteristic, complemented
12#21461150C211yesmaximal, characteristic, complemented
13#21461124C211yesmaximal, characteristic, complemented
14#21461040C211yesmaximal, characteristic, complemented
15#21455872C211yesmaximal, characteristic
16Q8×#92420C2211yescharacteristic
17Q8×#92420C2211yescharacteristic
18#16840797C2211yescharacteristic
19#16840796C2211yescharacteristic
20#16840794—21no—
21#16840791—21no—
22#16840790C2211yescharacteristic
23D4×#92420—21no—
24D4×#92420—21no—
25D4×#92420C2211yescharacteristic
26D4×#92420C2211yescharacteristic
27D4×#92420C2211yescharacteristic
28#16840783C2211yescharacteristic
29#16840779C2211yescharacteristic
30#16840778—21no—
31#16840778C2211yescharacteristic
32C2×C4⋊#92420—21no—
33C2×C4⋊#92420C2211yescharacteristic
34#16840736—21no—
35#16840735—21no—
36#16840626C2211yescharacteristic
37#16840626C2211yescharacteristic
38#16840623C2211yescharacteristic
39#16840623C2211yescharacteristic
40Q8⋊#92420C2211yescharacteristic
41Q8⋊#92420C2211yescharacteristic
42#16840610C2211yescharacteristic
43#16840610C2211yescharacteristic
44D4⋊#92420C2211yescharacteristic
45D4⋊#92420C2211yescharacteristic
46D4⋊#92420C2211yescharacteristic
47D4⋊#92420C2211yescharacteristic
48#92420⋊C8C2211yescharacteristic
49#92420⋊C8C2211yescharacteristic
50#16840603C2211yescharacteristic
51C8×#92420C2211yescharacteristic
52#16840307C2211yescharacteristic
53#41×Dic61C2211yescharacteristic
54#16840087C411yescharacteristic
55#16840083C411yescharacteristic
56#16840055C411yescharacteristic
57#16840050C411yescharacteristic
58#16839960C411yescharacteristic
59#16839955C411yescharacteristic
60#16839907C411yescharacteristic
61SD16×Dic61C2211yescharacteristic
62#16839847C2211yescharacteristic
63D8×Dic61C2211yescharacteristic
64#16839842C2211yescharacteristic
65#16839812C2211yescharacteristic
66C61×#449C411yescharacteristic
67C2×#11758213—21no—
68C2×#11758217—21no—
69C2×D61⋊C8—21no—
70C22×#92420—21no—
71C22×#92420—21no—
72C22×#92420—21no—
73C22×#92420—21no—
74#14519844—21no—
75C4⋊#92420—21no—
76C4⋊#92420—21no—
77C4⋊#92420—21no—
78C4⋊#92420D411yescharacteristic
79C4⋊#92420D411yescharacteristic
80C4⋊#92420C2311yescharacteristic
81C4⋊#92420C2311yescharacteristic
82C4⋊#92420C2311yescharacteristic
83C4⋊#92420C2311yescharacteristic
84#14519834—21no—
85#14519834—21no—
86C22⋊#92420—21no—
87C22⋊#92420—21no—
88C22⋊#92420—21no—
89C22⋊#92420—21no—
90C22⋊#92420—21no—
91C22⋊#92420—21no—
92C22⋊#92420D411yescharacteristic
93C22⋊#92420D411yescharacteristic
94#14519832—21no—
95C4×#92420—21no—
96C4×#92420—21no—
97C4×#92420C2311yescharacteristic
98C4×#92420C2311yescharacteristic
99C4×#92420C2311yescharacteristic
100C4×#92420C2311yescharacteristic
101C2×#11758224—21no—
102C2×#11758201C2×C411yescharacteristic
103C2×#11758199C2×C411yescharacteristic
104#14519776C2×C411yescharacteristic
105#14519773C2×C411yescharacteristic
106#14519763—21no—
107#14519751C2×C411yescharacteristic
108#14519749—21no—
109Q8×Dic61C2×C411yescharacteristic
110Q8×Dic61C2311yescharacteristic
111D4×Dic61—21no—
112D4×Dic61—21no—
113D4×Dic61C2×C411yescharacteristic
114D4×Dic61C2311yescharacteristic
115D4×Dic61C2311yescharacteristic
116#14519739C2311yescharacteristic
117#14519737—21no—
118#14519736—21no—
119#14519736C2×C411yescharacteristic
120C2×C4×Dic61—21no—
121C2×C4×Dic61C2311yescharacteristic
122#14519711C2×C411yescharacteristic
123#14519710C2×C411yescharacteristic
124#14519709C2×C411yescharacteristic
125#14519708C2×C411yescharacteristic
126#14519706C2×C411yescharacteristic
127#14519706C2×C411yescharacteristic
128#14519701C2×C411yescharacteristic
129#14519699C2×C411yescharacteristic
130#14519699C2×C411yescharacteristic
131#14519698C2×C411yescharacteristic
132#14519698C2×C411yescharacteristic
133#14519698C2×C411yescharacteristic
134C22⋊#92418C2×C411yescharacteristic
135C8×Dic61C2311yescharacteristic
136C8×Dic61C2311yescharacteristic
137C2×#41×C61C2×C411yescharacteristic
138C2×SD16×C61C2×C411yescharacteristic
139C2×D8×C61C2×C411yescharacteristic
140#121×C61C2×C411yescharacteristic
141#103×C61C2×C411yescharacteristic
142#102×C61C2×C411yescharacteristic
143C22⋊C8×C61C2×C411yescharacteristic
144#41×C61—22no—
145#41×C61—22no—
146C2×Q8×C61C22×C411yescharacteristic
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61C22×C411yescharacteristic
150C2×D4×C61C22×C411yescharacteristic
151C22×C4×C61—21no—
152C22×C4×C61C22×C411yescharacteristic
153C2×C22⋊D61—21no—
154C2×C22⋊D61—21no—
155C2×C22⋊D61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61#4111yescharacteristic
161#11758224—81no—
162#11758224—81no—
163C2×D244C22×C411yescharacteristic
164C2×C4×D61—21no—
165C2×Dic122—21no—
166#11758217—81no—
167C2×#92420—21no—
168C2×#92420—21no—
169C2×#92420—21no—
170C2×#92420—21no—
171C2×#92420—21no—
172C2×#92420—21no—
173C2×#92420—81no—
174C2×#92420—81no—
175C2×#92420—81no—
176C2×#92420—81no—
177C2×#92420C2×D411yescharacteristic
178C2×#92420C2×D411yescharacteristic
179C2×#92420C2×D411yescharacteristic
180C2×#92420C2×D411yescharacteristic
181#11758213—81no—
182D61⋊C8—81no—
183SD16×C61—22no—
184D8×C61—22no—
185C2×C8×C61C22×C411yescharacteristic
186C2×C8×C61C22×C411yescharacteristic
187C4⋊C4×C61C22×C411yescharacteristic
188C22⋊C4×C61—21no—
189C22⋊Dic61—21no—
190C22⋊Dic61—21no—
191C22⋊Dic61—21no—
192C22⋊Dic61—21no—
193#11758201—22no—
194#11758199—22no—
195#11758198—21no—
196#11758198—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61C22×C411yescharacteristic
200C4⋊Dic61C22×C411yescharacteristic
201C4⋊Dic61C22×C411yescharacteristic
202#11758196—21no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61C22×C411yescharacteristic
207C4×Dic61C2×D411yescharacteristic
208C4×Dic61#4111yescharacteristic
209C4×Dic61C2411yescharacteristic
210C2×#92418C22×C411yescharacteristic
211C2×#92418C22×C411yescharacteristic
212C23×C61—21no—
213C23×C61—21no—
214C23×C61C4×D411yescharacteristic
215C22×D61—21no—
216Q8×C61—21no—
217Q8×C61#13112yes—
218D4×C61—24no—
219D4×C61—21no—
220D4×C61—21no—
221D4×C61—21no—
222D4×C61—81no—
223D4×C61#13112yes—
224C2×C4×C61—24no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61C4×D411yescharacteristic
230C2×C4×C61C23×C411yescharacteristic
231C22⋊D61—81no—
232C22⋊D61—81no—
233C22⋊D61—81no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—81no—
245C2×Dic61C4×D411yescharacteristic
246C2×Dic61C4×D411yescharacteristic
247C2×Dic61C22×D411yescharacteristic
248C2×Dic61C2×#4111yescharacteristic
249D244—21no—
250D244—21no—
251C4×D61—41no—
252C4×D61—41no—
253Dic122—81no—
254#92420—21no—
255#92420—21no—
256#92420—21no—
257#92420—21no—
258#92420—41no—
259#92420—41no—
260#92420—41no—
261#92420—41no—
262#92420—41no—
263#92420—41no—
264C8×C61—21no—
265C8×C61—21no—
266#92418—21no—
267#92418—21no—
268#17920—611nomaximal, Sylow
269C22×C61—21no—
270C22×C61—22no—
271C22×C61—21no—
272C22×C61—21no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61C2×C4×D411yescharacteristic
279C2×D61—41no—
280C2×D61—41no—
281C2×D61—41no—
282C4×C61—21no—
283C4×C61—22no—
284C4×C61—41no—
285C4×C61C2×#13111yescharacteristic
286C4×C61C2×#13111yescharacteristic
287Dic61—21no—
288Dic61—21no—
289Dic61—21no—
290Dic61—41no—
291Dic61—41no—
292Dic61—41no—
293Dic61#57611yescharacteristic
294Dic61#57711yescharacteristic
295#3106—611no—
296#2988—611no—
297#2981—611no—
298#1588—611no—
299#1586—611no—
300#1583—611no—
301#1581—611no—
302#1580—611no—
303SD16⋊C8—611no—
304C8×SD16—611no—
305C8×D8—611no—
306#1573—611no—
307#1488—611no—
308#1480—611no—
309#1454—611no—
310C2×C61—41no—
311C2×C61—42no—
312C2×C61—81no—
313C2×C61#292911yescharacteristic
314C2×C61#294611yescharacteristic
315C2×C61#295711yescharacteristic
316D61—81no—
317#532—611no—
318C4×#41—611no—
319#482—611no—
320#482—611no—
321#477—611no—
322#476—611no—
323#460—611no—
324#459—611no—
325#449—611no—
326#449#692311yescharacteristic
327C8×Q8—611no—
328C8×Q8—611no—
329C4×SD16—611no—
330C4×SD16—611no—
331C4×D8—611no—
332C4×D8—611no—
333#436—611no—
334#436—611no—
335#435—1221no—
336#435—1221no—
337C8×D4—611no—
338C8×D4—611no—
339C8×D4—611no—
340C8×D4—611no—
341C8×D4—1221no—
342C8×D4—1221no—
343#432—611no—
344#431—611no—
345#431—611no—
346#431—611no—
347#431—1221no—
348C2×C4⋊C8—611no—
349C2×C4⋊C8—1221no—
350#408—1221no—
351#408—1221no—
352#335—611no—
353#335—611no—
354#334—611no—
355#334—611no—
356Q8⋊C8—611no—
357Q8⋊C8—611no—
358D4⋊C8—611no—
359D4⋊C8—611no—
360D4⋊C8—611no—
361D4⋊C8—611no—
362D4⋊C8—611no—
363D4⋊C8—611no—
364#322—611no—
365#322—611no—
366#322—611no—
367C82—611no—
368C61#1792011yescharacteristic, complemented, Sylow
369C2×#41—1221no—
370C2×#41C2×#692311yescharacteristic
371C2×SD16—611no—
372C2×SD16C2×#692311yescharacteristic
373C2×D8—611no—
374C2×D8C2×#692311yescharacteristic
375C2×M16—1221no—
376C2×M16—1221no—
377C22×C8—1221no—
378C22×C8—1221no—
379C22×C8—1221no—
380C22×C8—1221no—
381C22×C8—1221no—
382C4⋊Q8—611no—
383#127—611no—
384#124—1221no—
385#121—611no—
386#121—1221no—
387#121C2×#692311yescharacteristic
388C4×Q8—611no—
389C4×Q8—611no—
390C4×D4—611no—
391C4×D4—611no—
392C4×D4—611no—
393C4×D4—611no—
394C4×D4—1221no—
395C4×D4—1221no—
396C4×D4—1221no—
397#117—611no—
398#117—1221no—
399C2×C42—611no—
400C2×C42—1221no—
401#107—611no—
402#107—611no—
403#106—611no—
404#106—611no—
405C4⋊C8—611no—
406C4⋊C8—611no—
407C4⋊C8—611no—
408C4⋊C8—611no—
409C4⋊C8—611no—
410C4⋊C8—611no—
411C4⋊C8—611no—
412C4⋊C8—611no—
413C4⋊C8—1221no—
414C4⋊C8—1221no—
415C4⋊C8—1221no—
416C4⋊C8—1221no—
417#103—611no—
418#103C2×#692311yescharacteristic
419#102—611no—
420#102—611no—
421#102—611no—
422#102—611no—
423#102—611no—
424#102C2×#692311yescharacteristic
425C22⋊C8—611no—
426C22⋊C8—611no—
427C22⋊C8—611no—
428C22⋊C8—1221no—
429C22⋊C8—1221no—
430C22⋊C8—1221no—
431C22⋊C8—1221no—
432C22⋊C8—1221no—
433C22⋊C8—1221no—
434C22⋊C8—1221no—
435C22⋊C8—1221no—
436C22⋊C8C2×#692311yescharacteristic
437#97—1221no—
438C4×C8—611no—
439C4×C8—611no—
440C4×C8—611no—
441C4×C8—611no—
442C4×C8—611no—
443C4×C8—611no—
444C4×C8—1221no—
445C4×C8—1221no—
446#41—22no—
447#41—22no—
448#41—4881no—
449#41—4881no—
450C2×Q8—1221no—
451C2×Q8C22×#692311yescharacteristic
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—611no—
455C2×D4—1221no—
456C2×D4—1221no—
457C2×D4—1221no—
458C2×D4C22×#692311yescharacteristic
459C2×D4C22×#692311yescharacteristic
460C22×C4—21no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4C22×#692311yescharacteristic
468SD16—22no—
469SD16—1222no—
470D8—22no—
471D8—1222no—
472M16—4881no—
473M16—4881no—
474C2×C8—611no—
475C2×C8—611no—
476C2×C8—611no—
477C2×C8—611no—
478C2×C8—611no—
479C2×C8—611no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—4881no—
487C2×C8—4881no—
488C2×C8—4881no—
489C2×C8—4881no—
490C2×C8—4881no—
491C2×C8C22×#692311yescharacteristic
492C2×C8C22×#692311yescharacteristic
493C4⋊C4—611no—
494C4⋊C4—611no—
495C4⋊C4—611no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4C22×#692311yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—1221no—
502C22⋊C4—1221no—
503C22⋊C4—1221no—
504C22⋊C4—1221no—
505C22⋊C4—1221no—
506C22⋊C4—1221no—
507C42—611no—
508C42—611no—
509C42—611no—
510C42—611no—
511C42—1221no—
512C42—1221no—
513C42—1221no—
514C23—21no—
515C23—21no—
516C23—1221no—
517C23D4×#692311yescharacteristic
518Q8—21no—
519Q8—4881no—
520Q8#1451986412yes—
521D4—24no—
522D4—21no—
523D4—21no—
524D4—21no—
525D4—81no—
526D4—1221no—
527D4—1221no—
528D4—4881no—
529D4—4881no—
530D4—4881no—
531D4#1451986412yes—
532C2×C4—24no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—611no—
538C2×C4—611no—
539C2×C4—611no—
540C2×C4—611no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—4881no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4D4×#692311yescharacteristic
555C2×C4C23×#692311yescharacteristic
556C8—21no—
557C8—21no—
558C8—1221no—
559C8—1221no—
560C8—1221no—
561C8—1221no—
562C8—1221no—
563C8—1221no—
564C8—2441no—
565C8—2441no—
566C8—2441no—
567C8—2441no—
568C8—2441no—
569C8—2441no—
570C22—21no—
571C22—22no—
572C22—21no—
573C22—21no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—2441no—
580C22—2441no—
581C22—2441no—
582C22C2×D4×#692311yescharacteristic
583C4—21no—
584C4—22no—
585C4—41no—
586C4—611no—
587C4—611no—
588C4—1221no—
589C4—1221no—
590C4—1221no—
591C4—2441no—
592C4—2441no—
593C4—2441no—
594C4C2×#1451986411yescharacteristic
595C4C2×#1451986411yescharacteristic
596C2—41no—
597C2—42no—
598C2—81no—
599C2—4881no—
600C2#2146253411yescharacteristic
601C2#2146257611yescharacteristic
602C2#2146259811yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#449C4
socleC22×C61C2×C4×D4
prime radical#21455872C2
cosocleC4×Dic61C24
hypercenter#449#6923
hypocenterC61#17920
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #21462734C2semidirect
2. #21462709C2semidirect
3. #21461448C2semidirect
4. #21461438C2semidirect
5. #21461426C2semidirect
6. #21461420C2semidirect
7. #21461415C2semidirect
8. #21461409C2semidirect
9. SD16×#92420C2semidirect
10. #21461399C2semidirect
11. #21461392C2semidirect
12. #21461150C2semidirect
13. #21461124C2semidirect
14. #21461040C2semidirect
368. C61#17920semidirect

Maximal subgroups

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

subgroupindex
#214627342
#214627092
#214614482
#214614382
#214614262
#214614202
#214614152
#214614092
SD16×#924202
#214613992
#214613922
#214611502
#214611242
#214610402
#214558722
#1792061