Finite Group Databasebeta

← #30612197

Subgroups — #30612197

Order 15616.

634 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462514C211yesmaximal, characteristic
2#21462464C211yesmaximal, characteristic
3#21461396C211yesmaximal, characteristic, complemented
4#21461395C211yesmaximal, characteristic, complemented
5#21461393C211yesmaximal, characteristic, complemented
6#21461392C211yesmaximal, characteristic, complemented
7#21461388C211yesmaximal, characteristic, complemented
8#21461386C211yesmaximal, characteristic, complemented
9#21461384C211yesmaximal, characteristic, complemented
10#21461383C211yesmaximal, characteristic, complemented
11#21461376C211yesmaximal, characteristic, complemented
12#21461371C211yesmaximal, characteristic, complemented
13#21461370C211yesmaximal, characteristic, complemented
14C2×#16840623C211yesmaximal, characteristic, complemented
15#21454779C211yesmaximal, characteristic
16C2×#14519864—21no—
17#16840787—21no—
18#16840786—21no—
19#16840784C2211yescharacteristic
20#16840783C2211yescharacteristic
21#16840781C2211yescharacteristic
22#16840778C2211yescharacteristic
23#16840769—21no—
24#16840768—21no—
25#16840768—21no—
26#16840766C2211yescharacteristic
27#16840765C2211yescharacteristic
28#16840765C2211yescharacteristic
29#16840765C2211yescharacteristic
30D61⋊(C4⋊C8)C2211yescharacteristic
31C2×C4⋊#92420C2211yescharacteristic
32C2×C4⋊#92420C2211yescharacteristic
33C2×C4⋊#92420C2211yescharacteristic
34#16840735—21no—
35#16840726—21no—
36#16840626C2211yescharacteristic
37#16840626C2211yescharacteristic
38#16840626C2211yescharacteristic
39#16840626C2211yescharacteristic
40#16840625C2211yescharacteristic
41#16840625C2211yescharacteristic
42#16840625C2211yescharacteristic
43#16840625C2211yescharacteristic
44#16840624C2211yescharacteristic
45#16840624C2211yescharacteristic
46#16840623C2211yescharacteristic
47#16840623C2211yescharacteristic
48#16840623C2211yescharacteristic
49#16840623C2211yescharacteristic
50#16840623C2211yescharacteristic
51#16840623C2211yescharacteristic
52C4×#11758224C2211yescharacteristic
53#16839891C411yescharacteristic
54#16839890C411yescharacteristic
55#16839838C411yescharacteristic
56#16839837C411yescharacteristic
57#16839834C411yescharacteristic
58#16839833C411yescharacteristic
59#16839830C411yescharacteristic
60#16839814C2211yescharacteristic
61#16839812C2211yescharacteristic
62#16839812C2211yescharacteristic
63#16839714C2211yescharacteristic
64M16×Dic61C2211yescharacteristic
65C2×C8×Dic61C2211yescharacteristic
66C61×#435C411yescharacteristic
67#14519864—44no—
68C2×#11758213—21no—
69C2×#11758217—21no—
70C2×#11758217—21no—
71C2×#11758217—21no—
72C2×D61⋊C8—21no—
73C22×#92420—21no—
74C22×#92420—21no—
75C22×#92420—21no—
76#14519844—21no—
77#14519844—21no—
78#14519844—21no—
79#14519844—21no—
80C4⋊#92420—21no—
81C4⋊#92420—21no—
82C4⋊#92420—21no—
83C4⋊#92420—21no—
84C4⋊#92420C2311yescharacteristic
85C4⋊#92420C2311yescharacteristic
86C4⋊#92420C2311yescharacteristic
87C4⋊#92420C2311yescharacteristic
88C4⋊#92420C2311yescharacteristic
89C4⋊#92420C2311yescharacteristic
90C4⋊#92420C2311yescharacteristic
91C4⋊#92420C2311yescharacteristic
92#14519834—21no—
93#14519834—21no—
94C22⋊#92420—21no—
95C22⋊#92420—21no—
96#14519832—21no—
97#14519832—21no—
98C4×#92420—21no—
99C4×#92420—21no—
100C2×#11758224D411yescharacteristic
101C2×#11758194C2×C411yescharacteristic
102C2×#11758189C2×C411yescharacteristic
103C2×#11758189C2×C411yescharacteristic
104C22×#92418C2×C411yescharacteristic
105C4×Dic122C2×C411yescharacteristic
106#14519742Q811yescharacteristic
107D4×Dic61Q811yescharacteristic
108C4×C22⋊D61C2×C411yescharacteristic
109C4×C22⋊D61C2×C411yescharacteristic
110#14519737Q811yescharacteristic
111#14519737Q811yescharacteristic
112#14519735D411yescharacteristic
113#14519733D411yescharacteristic
114#14519733D411yescharacteristic
115C2×C4×Dic61C2311yescharacteristic
116C2×C4×Dic61C2311yescharacteristic
117C42×D61C2311yescharacteristic
118C4⋊#92418C2×C411yescharacteristic
119#14519706C2×C411yescharacteristic
120#14519706C2×C411yescharacteristic
121#14519706C2×C411yescharacteristic
122#14519706C2×C411yescharacteristic
123C22⋊#92418C2×C411yescharacteristic
124C22⋊#92418C2×C411yescharacteristic
125#14519689C2×C411yescharacteristic
126#14519689C2×C411yescharacteristic
127#14519688C2×C411yescharacteristic
128#14519687C2×C411yescharacteristic
129#14519687C2×C411yescharacteristic
130C8×Dic61C2×C411yescharacteristic
131C8×Dic61C2×C411yescharacteristic
132C8×Dic61C2311yescharacteristic
133C8×Dic61C2311yescharacteristic
134C8×Dic61C2311yescharacteristic
135C8×Dic61C2311yescharacteristic
136C2×M16×C61C2×C411yescharacteristic
137C22×C8×C61C2×C411yescharacteristic
138C4×D4×C61C2×C411yescharacteristic
139C4⋊C8×C61C2×C411yescharacteristic
140C22⋊C8×C61C2×C411yescharacteristic
141C22⋊C8×C61C2×C411yescharacteristic
142#97×C61C2×C411yescharacteristic
143C2×D4×C61C4⋊C411yescharacteristic
144C22×C4×C61C22×C411yescharacteristic
145C22×C4×C61C22×C411yescharacteristic
146C2×C22⋊D61C4⋊C411yescharacteristic
147C2×C22⋊D61C4⋊C411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×Q811yescharacteristic
151C22×Dic61C2×Q811yescharacteristic
152#11758224—22no—
153#11758224—22no—
154C2×C4×D61C22×C411yescharacteristic
155C2×C4×D61C2×D411yescharacteristic
156C2×C4×D61C2×Q811yescharacteristic
157C2×Dic122C4⋊C411yescharacteristic
158#11758217—42no—
159#11758217—42no—
160#11758217—42no—
161C2×#92420—21no—
162C2×#92420—21no—
163C2×#92420—21no—
164C2×#92420—21no—
165C2×#92420—21no—
166C2×#92420—21no—
167C2×#92420—21no—
168C2×#92420—21no—
169C2×#92420—42no—
170C2×#92420—42no—
171C2×#92420—42no—
172#11758213—42no—
173D61⋊C8—42no—
174M16×C61—22no—
175C2×C8×C61—22no—
176C2×C8×C61C22×C411yescharacteristic
177C2×C8×C61C22×C411yescharacteristic
178C2×C8×C61C22×C411yescharacteristic
179C2×C8×C61C22×C411yescharacteristic
180C4⋊C4×C61C4⋊C411yescharacteristic
181C22⋊C4×C61C4⋊C411yescharacteristic
182C22⋊C4×C61C4⋊C411yescharacteristic
183C42×C61C22×C411yescharacteristic
184C22⋊Dic61C4⋊C411yescharacteristic
185C22⋊Dic61C4⋊C411yescharacteristic
186#11758198C4⋊C411yescharacteristic
187#11758198C4⋊C411yescharacteristic
188C4⋊Dic61C4⋊C411yescharacteristic
189#11758196C4⋊C411yescharacteristic
190#11758196C4⋊C411yescharacteristic
191#11758196C4⋊C411yescharacteristic
192#11758196C4⋊C411yescharacteristic
193C4×Dic61C22×C411yescharacteristic
194C4×Dic61C22×C411yescharacteristic
195C4×Dic61C2×D411yescharacteristic
196C4×Dic61C2×D411yescharacteristic
197C4×Dic61C2×D411yescharacteristic
198C4×Dic61C2×Q811yescharacteristic
199C4×Dic61C2×Q811yescharacteristic
200C4×Dic61C2×Q811yescharacteristic
201C4×Dic61C2411yescharacteristic
202#11758194—41no—
203C2×#92418—41no—
204C2×#92418C22×C411yescharacteristic
205C2×#92418C22×C411yescharacteristic
206C2×#92418C22×C411yescharacteristic
207C2×#92418C22×C411yescharacteristic
208#11758189—22no—
209#11758189—41no—
210C23×C61C2×C4⋊C411yescharacteristic
211C23×C61C2×C4⋊C411yescharacteristic
212C22×D61C2×C4⋊C411yescharacteristic
213D4×C61—22no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61C2×C4⋊C411yescharacteristic
219C2×C4×C61C2×C4⋊C411yescharacteristic
220C2×C4×C61C2×C4⋊C411yescharacteristic
221C2×C4×C61C2×C4⋊C411yescharacteristic
222C2×C4×C61C23×C411yescharacteristic
223C22⋊D61—22no—
224C22⋊D61—22no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61C2×C4⋊C411yescharacteristic
230C2×Dic61C2×C4⋊C411yescharacteristic
231C2×Dic61C2×C4⋊C411yescharacteristic
232C2×Dic61C2×C4⋊C411yescharacteristic
233C2×Dic61C2×C4⋊C411yescharacteristic
234C2×Dic61AGL(1,ℤ/8)12yes—
235C2×Dic61#13712yes—
236C2×Dic61C22×D411yescharacteristic
237C2×Dic61C22×Q811yescharacteristic
238C4×D61—21no—
239C4×D61—21no—
240C4×D61—21no—
241C4×D61—21no—
242C4×D61—41no—
243Dic122—22no—
244#92420—41no—
245#92420—41no—
246#92420—41no—
247#92420—41no—
248#92420—41no—
249#92420—41no—
250#92420—41no—
251#92420—41no—
252C8×C61—21no—
253C8×C61—21no—
254C8×C61—21no—
255C8×C61#13112yescomplemented
256#92418—21no—
257#92418—21no—
258#92418—21no—
259#92418—21no—
260#17830—611nomaximal, Sylow
261C22×C61—21no—
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—21no—
265C22×C61—21no—
266C22×C61#42812yes—
267C22×C61C22×C4⋊C411yescharacteristic
268C2×D61—21no—
269C2×D61—21no—
270C2×D61—21no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—21no—
274C4×C61—21no—
275C4×C61C2×#13111yescharacteristic
276C4×C61#56811yescharacteristic
277Dic61—21no—
278Dic61—21no—
279Dic61—21no—
280Dic61—21no—
281Dic61—21no—
282Dic61#42812yes—
283Dic61C2×AGL(1,ℤ/8)11yescharacteristic
284Dic61C2×#13711yescharacteristic
285#2965—611no—
286#2925—611no—
287#2906—611no—
288#1574—611no—
289#1574—611no—
290#1573—611no—
291#1573—611no—
292#1571—611no—
293#1570—611no—
294#1569—611no—
295#1569—611no—
296#1567—611no—
297#1565—611no—
298#1565—611no—
299C2×#334—611no—
300C2×C61—22no—
301C2×C61—41no—
302C2×C61#290611yescharacteristic
303C2×C61C2×#42811yescharacteristic
304C2×C61#291611yescharacteristic
305D61—41no—
306C2×#131—1221no—
307C4×#41—611no—
308#446—611no—
309#446—611no—
310#436—611no—
311#436—611no—
312#435—611no—
313#435—611no—
314#435—611no—
315#435#692311yescharacteristic
316#433—1221no—
317#433—1221no—
318#432—611no—
319#432—611no—
320#431—611no—
321#431—611no—
322#431—611no—
323#431—611no—
324#431—611no—
325#424—1221no—
326#424—1221no—
327#424—1221no—
328#423—611no—
329#423—611no—
330#423—611no—
331#423—611no—
332C2×C4⋊C8—611no—
333C2×C4⋊C8—611no—
334C2×C4⋊C8—611no—
335C2×C4⋊C8—611no—
336#408—1221no—
337#405—1221no—
338C4×M16—611no—
339C4×M16—611no—
340C2×C4×C8—611no—
341#335—611no—
342#335—611no—
343#335—611no—
344#335—611no—
345#335—611no—
346#335—611no—
347#335—611no—
348#335—611no—
349#334—611no—
350#334—611no—
351#334—611no—
352#334—611no—
353#334—611no—
354#334—611no—
355#334—611no—
356#334—611no—
357C61#1783011yescharacteristic, complemented, Sylow
358C2×#41—611no—
359#131—2444no—
360C2×M16—611no—
361C2×M16—611no—
362C2×M16—611no—
363C2×M16—1221no—
364C2×M16—1221no—
365C2×M16—1221no—
366C2×M16—1221no—
367C2×M16C2×#692311yescharacteristic
368C22×C8—611no—
369C22×C8—1221no—
370C22×C8—1221no—
371C22×C8—1221no—
372C22×C8—1221no—
373C22×C8C2×#692311yescharacteristic
374C4×Q8—611no—
375C4×Q8—611no—
376C4×D4—611no—
377C4×D4—611no—
378C4×D4—611no—
379C4×D4—611no—
380C4×D4—611no—
381C4×D4C2×#692311yescharacteristic
382#117—611no—
383#117—611no—
384#117—611no—
385C2×C42—611no—
386C2×C42—611no—
387C2×C42—611no—
388C4⋊C8—611no—
389C4⋊C8—611no—
390C4⋊C8—611no—
391C4⋊C8—611no—
392C4⋊C8—611no—
393C4⋊C8—611no—
394C4⋊C8—611no—
395C4⋊C8—611no—
396C4⋊C8—611no—
397C4⋊C8—611no—
398C4⋊C8—611no—
399C4⋊C8—611no—
400C4⋊C8—611no—
401C4⋊C8—1221no—
402C4⋊C8—1221no—
403C4⋊C8—1221no—
404C4⋊C8—1221no—
405C4⋊C8—1221no—
406C4⋊C8—1221no—
407C4⋊C8—1221no—
408C4⋊C8—1221no—
409C4⋊C8C2×#692311yescharacteristic
410C22⋊C8—611no—
411C22⋊C8—611no—
412C22⋊C8—611no—
413C22⋊C8—611no—
414C22⋊C8—1221no—
415C22⋊C8—1221no—
416C22⋊C8—1221no—
417C22⋊C8—1221no—
418C22⋊C8C2×#692311yescharacteristic
419C22⋊C8C2×#692311yescharacteristic
420#97—611no—
421#97—611no—
422#97—611no—
423#97—1221no—
424#97—1221no—
425#97C2×#692311yescharacteristic
426C4×C8—611no—
427C4×C8—611no—
428C4×C8—611no—
429C4×C8—611no—
430C4×C8—611no—
431C4×C8—611no—
432C4×C8—1221no—
433C4×C8—1221no—
434#41—1222no—
435#41—1222no—
436C2×Q8—611no—
437C2×D4—611no—
438C2×D4—611no—
439C2×D4C4⋊#692311yescharacteristic
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4C22×#692311yescharacteristic
448C22×C4C22×#692311yescharacteristic
449M16—22no—
450M16—1222no—
451M16—2441no—
452M16—2441no—
453M16—2442no—
454M16—2442no—
455M16—2442no—
456M16—2442no—
457C2×C8—22no—
458C2×C8—611no—
459C2×C8—611no—
460C2×C8—611no—
461C2×C8—611no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—2441no—
471C2×C8—2442no—
472C2×C8—2442no—
473C2×C8—2442no—
474C2×C8—2442no—
475C2×C8C22×#692311yescharacteristic
476C2×C8C22×#692311yescharacteristic
477C2×C8C22×#692311yescharacteristic
478C2×C8C22×#692311yescharacteristic
479C4⋊C4—611no—
480C4⋊C4—611no—
481C4⋊C4—611no—
482C4⋊C4—611no—
483C4⋊C4—611no—
484C4⋊C4C4⋊#692311yescharacteristic
485C22⋊C4—611no—
486C22⋊C4—611no—
487C22⋊C4—611no—
488C22⋊C4—611no—
489C22⋊C4C4⋊#692311yescharacteristic
490C22⋊C4C4⋊#692311yescharacteristic
491C42—611no—
492C42—611no—
493C42—611no—
494C42—611no—
495C42—611no—
496C42—611no—
497C42—611no—
498C42—611no—
499C42—611no—
500C42C22×#692311yescharacteristic
501C23—611no—
502C23C2×C4⋊#692311yescharacteristic
503C23C2×C4⋊#692311yescharacteristic
504Q8—1222no—
505D4—22no—
506D4—1222no—
507D4—1222no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—612no—
513C2×C4—612no—
514C2×C4—611no—
515C2×C4—611no—
516C2×C4—611no—
517C2×C4—611no—
518C2×C4—611no—
519C2×C4—611no—
520C2×C4—611no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1221no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4C2×C4⋊#692311yescharacteristic
531C2×C4C2×C4⋊#692311yescharacteristic
532C2×C4C2×C4⋊#692311yescharacteristic
533C2×C4C2×C4⋊#692311yescharacteristic
534C2×C4C23×#692311yescharacteristic
535C8—21no—
536C8—21no—
537C8—21no—
538C8—1221no—
539C8—1221no—
540C8—1221no—
541C8—1221no—
542C8—2441no—
543C8—2441no—
544C8—2441no—
545C8—2441no—
546C8—2441no—
547C8—2441no—
548C8—2441no—
549C8—2441no—
550C8#1451986412yescomplemented
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—1221no—
557C22—1221no—
558C22—1221no—
559C22#1684077312yes—
560C22C22×C4⋊#692311yescharacteristic
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—611no—
566C4—611no—
567C4—612no—
568C4—1221no—
569C4—1221no—
570C4—1221no—
571C4—1221no—
572C4—1221no—
573C4C2×#1451986411yescharacteristic
574C4#1684084111yescharacteristic
575C2—22no—
576C2—41no—
577C2—2441no—
578C2#2146246411yescharacteristic
579C2C2×#1684077311yescharacteristic
580C2#2146248011yescharacteristic
subgroupG/H
centerC22C22×C4⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#435C4
socleC22×C61C22×C4⋊C4
prime radicalC2×#11758224D4
cosocleC4×Dic61C24
hypercenter#435#6923
hypocenterC61#17830
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
3. #21461396C2semidirect
4. #21461395C2semidirect
5. #21461393C2semidirect
6. #21461392C2semidirect
7. #21461388C2semidirect
8. #21461386C2semidirect
9. #21461384C2semidirect
10. #21461383C2semidirect
11. #21461376C2semidirect
12. #21461371C2semidirect
13. #21461370C2semidirect
14. C2×#16840623C2semidirect
255. C8×C61#131semidirect
357. C61#17830semidirect
550. C8#14519864semidirect

Maximal subgroups

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

subgroupindex
#214625142
#214624642
#214613962
#214613952
#214613932
#214613922
#214613882
#214613862
#214613842
#214613832
#214613762
#214613712
#214613702
C2×#168406232
#214547792
#1783061