Finite Group Databasebeta

← #30612485

Subgroups — #30612485

Order 15616.

660 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462740C211yesmaximal, characteristic, complemented
2#21462736C211yesmaximal, characteristic, complemented
3#21461447C212yesmaximal, complemented
4#21461435C212yesmaximal, complemented
5#21461431C212yesmaximal, complemented
6#21461419C212yesmaximal, complemented
7C2×#16840626C211yesmaximal, characteristic, complemented
8#21461116C211yesmaximal, characteristic, complemented
9#21461116C211yesmaximal, characteristic, complemented
10#21461032C211yesmaximal, characteristic, complemented
11#21454590C211yesmaximal, characteristic
12#16840812C2212yes—
13#16840811C2212yes—
14#16840796C2212yes—
15#16840791—22no—
16#16840790C2212yes—
17#16840789—22no—
18#16840784—21no—
19#16840781—21no—
20#16840778C2211yescharacteristic
21#16840778C2211yescharacteristic
22#16840766—21no—
23D61⋊(C4⋊C8)—21no—
24C2×C4⋊#92420C2211yescharacteristic
25C2×C4⋊#92420C2211yescharacteristic
26#16840626C2212yes—
27#16840626C2212yes—
28#16840612C2212yes—
29#16840612C2212yes—
30#16840610C2212yes—
31#16840610C2212yes—
32#92420⋊C8C2212yes—
33#92420⋊C8C2212yes—
34C2×#14519799C411yescharacteristic
35#16840230C2211yescharacteristic
36#16840230C2211yescharacteristic
37#16839856C411yescharacteristic
38#16839856C2211yescharacteristic
39#16839849C411yescharacteristic
40#16839849C411yescharacteristic
41#16839849C2211yescharacteristic
42#16839849C2211yescharacteristic
43#16839844C411yescharacteristic
44#16839844C2211yescharacteristic
45#16839756C411yescharacteristic
46#16839756C411yescharacteristic
47C2×C8×Dic61C2211yescharacteristic
48C61×#427C411yescharacteristic
49C2×#11758213—22no—
50C2×D61⋊C8—22no—
51C22×#92420—21no—
52C22×#92420—21no—
53#14519844—22no—
54C4⋊#92420—22no—
55C4⋊#92420—21no—
56C4⋊#92420—21no—
57C4⋊#92420—21no—
58C4⋊#92420D411yescharacteristic
59C4⋊#92420D411yescharacteristic
60C4⋊#92420C2312yes—
61C4⋊#92420C2312yes—
62#14519834—22no—
63#14519834—22no—
64C22⋊#92420—21no—
65C22⋊#92420D412yes—
66#14519832—22no—
67C4×#92420—22no—
68C4×#92420C2312yes—
69C4×#92420C2312yes—
70C2×#11758222C2×C411yescharacteristic
71C2×#11758222C2×C411yescharacteristic
72#14519799—24no—
73C2×Dic244C2×C411yescharacteristic
74C2×#11758190C2×C411yescharacteristic
75C2×#11758190C2×C411yescharacteristic
76C2×D488C2×C411yescharacteristic
77C22×#92418C2×C411yescharacteristic
78#14519742C2×C411yescharacteristic
79#14519742C2×C411yescharacteristic
80#14519742C2311yescharacteristic
81#14519742C2311yescharacteristic
82#14519739C2×C411yescharacteristic
83#14519739C2×C411yescharacteristic
84#14519739C2311yescharacteristic
85#14519739C2311yescharacteristic
86#14519737—21no—
87#14519737—21no—
88#14519737—21no—
89#14519737—21no—
90#14519734—21no—
91#14519734—21no—
92C2×C4×Dic61C2311yescharacteristic
93C42×D61—21no—
94C42×D61—21no—
95#14519700C2×C411yescharacteristic
96#14519700C2×C411yescharacteristic
97#14519700C2×C411yescharacteristic
98#14519700C2×C411yescharacteristic
99#14519697C2×C411yescharacteristic
100#14519697C2×C411yescharacteristic
101#14519697C2×C411yescharacteristic
102#14519697C2×C411yescharacteristic
103C8×Dic61C2×C411yescharacteristic
104C8×Dic61C2×C411yescharacteristic
105C8×Dic61C2311yescharacteristic
106C8×Dic61C2311yescharacteristic
107C22×C8×C61C2×C411yescharacteristic
108#117×C61C2×C411yescharacteristic
109#117×C61C2×C411yescharacteristic
110#107×C61C2×C411yescharacteristic
111#107×C61C2×C411yescharacteristic
112#106×C61C2×C411yescharacteristic
113#106×C61C2×C411yescharacteristic
114C22×C4×C61C22×C411yescharacteristic
115C2×C22⋊D61—21no—
116C2×C22⋊D61—21no—
117C22×Dic61C2×D411yescharacteristic
118C22×Dic61#4111yescharacteristic
119#11758222—22no—
120#11758222—22no—
121#11758222—22no—
122#11758222M1614yescomplemented
123C2×D244C22×C411yescharacteristic
124C2×D244C22×C411yescharacteristic
125C2×C4×D61—21no—
126C2×C4×D61—21no—
127C2×C4×D61—21no—
128C2×C4×D61—21no—
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×Dic122C22×C411yescharacteristic
132C2×Dic122C22×C411yescharacteristic
133C2×#92420—22no—
134C2×#92420—22no—
135C2×#92420—22no—
136C2×#92420—81no—
137C2×#92420—81no—
138C2×#92420C2×D412yes—
139C2×#92420C2×D412yes—
140#11758213—82no—
141D61⋊C8—82no—
142C2×C8×C61—22no—
143C2×C8×C61C22×C411yescharacteristic
144C2×C8×C61C22×C411yescharacteristic
145C4⋊C4×C61C22×C411yescharacteristic
146C4⋊C4×C61C22×C411yescharacteristic
147C4⋊C4×C61C22×C411yescharacteristic
148C4⋊C4×C61C22×C411yescharacteristic
149C22⋊C4×C61—21no—
150C22⋊C4×C61—21no—
151C42×C61—21no—
152C42×C61—21no—
153#11758198—21no—
154#11758198—21no—
155#11758198—21no—
156#11758198—21no—
157#11758196—21no—
158#11758196—21no—
159#11758196—21no—
160#11758196—21no—
161C4×Dic61—21no—
162C4×Dic61—21no—
163C4×Dic61—21no—
164C4×Dic61—21no—
165C4×Dic61C22×C411yescharacteristic
166C4×Dic61C2×D411yescharacteristic
167C4×Dic61#4111yescharacteristic
168C4×Dic61C2411yescharacteristic
169C2×#92418—22no—
170C2×#92418C22×C411yescharacteristic
171C2×#92418C22×C411yescharacteristic
172Dic244—22no—
173D488—22no—
174#11758190—22no—
175#11758190—22no—
176C23×C61C4×D411yescharacteristic
177C22×D61—21no—
178C22×D61—21no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61C4×D411yescharacteristic
184C2×C4×C61C2×M1612yes—
185C2×C4×C61#13112yes—
186C2×C4×C61C23×C411yescharacteristic
187C22⋊D61—24no—
188C22⋊D61—42no—
189C2×Dic61—21no—
190C2×Dic61—21no—
191C2×Dic61—21no—
192C2×Dic61—21no—
193C2×Dic61—41no—
194C2×Dic61C4×D411yescharacteristic
195C2×Dic61C4×D411yescharacteristic
196C2×Dic61C22×D411yescharacteristic
197C2×Dic61C2×#4111yescharacteristic
198D244—21no—
199D244—21no—
200D244—21no—
201D244C2×M1612yes—
202C4×D61—24no—
203C4×D61—42no—
204C4×D61—41no—
205C4×D61—41no—
206C4×D61—41no—
207C4×D61—41no—
208C4×D61—81no—
209C4×D61—81no—
210Dic122—21no—
211Dic122—21no—
212Dic122—21no—
213Dic122C2×M1612yes—
214#92420—22no—
215#92420—22no—
216#92420—42no—
217#92420—42no—
218#92420—42no—
219C8×C61—21no—
220C8×C61—21no—
221#92418—21no—
222#92418—21no—
223#17881—611nomaximal, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61#43512yes—
227C22×C61C2×C4×D411yescharacteristic
228C2×D61—22no—
229C2×D61—41no—
230C2×D61—41no—
231C2×D61—41no—
232C2×D61—41no—
233C2×D61—41no—
234C4×C61—41no—
235C4×C61—41no—
236C4×C61—41no—
237C4×C61—41no—
238C4×C61#43512yes—
239C4×C61C22×M1611yescharacteristic
240C4×C61C2×#13111yescharacteristic
241Dic61—22no—
242Dic61—21no—
243Dic61—21no—
244Dic61—21no—
245Dic61—41no—
246Dic61#57611yescharacteristic
247Dic61#57811yescharacteristic
248#2989—611no—
249#2988—611no—
250C4×#135—611no—
251#1588—612no—
252#1585—612no—
253#1584—612no—
254#1581—612no—
255C2×#335—611no—
256#1477—611no—
257#1477—611no—
258#1452—611no—
259C2×C61—22no—
260C2×C61C2×#43511yescharacteristic
261C2×C61#295111yescharacteristic
262C2×C61#295711yescharacteristic
263D61—42no—
264D61—81no—
265C2×#135—611no—
266C4×#41—611no—
267C4×#41—611no—
268#446—612no—
269C8×Q8—612no—
270C4×Q16—611no—
271C4×Q16—611no—
272C4×SD16—611no—
273C4×SD16—611no—
274C4×SD16—611no—
275C4×SD16—611no—
276C4×D8—611no—
277C4×D8—611no—
278#436—612no—
279#435—1222no—
280C8×D4—612no—
281C8×D4—1222no—
282#432—1221no—
283#431—611no—
284#431—611no—
285#431—1221no—
286#427#692311yescharacteristic
287#423—1221no—
288C2×C4⋊C8—611no—
289C2×C4⋊C8—611no—
290C2×C4⋊C8—1221no—
291#416—611no—
292#416—611no—
293C2×C4×C8—611no—
294#335—612no—
295#335—612no—
296Q8⋊C8—612no—
297Q8⋊C8—612no—
298D4⋊C8—612no—
299D4⋊C8—612no—
300#322—612no—
301#322—612no—
302C61#1788111yescharacteristic, complemented, Sylow
303C2×#41—611no—
304C2×#41—611no—
305#135—1224no—
306C2×Q16—611no—
307C2×SD16—611no—
308C2×SD16—611no—
309C2×D8—611no—
310C2×M16—1222no—
311C22×C8—611no—
312C22×C8—1222no—
313C22×C8—1221no—
314C22×C8—1221no—
315C22×C8C2×#692311yescharacteristic
316C4×Q8—611no—
317C4×Q8—611no—
318C4×Q8—611no—
319C4×Q8—611no—
320C4×D4—611no—
321C4×D4—611no—
322C4×D4—611no—
323C4×D4—611no—
324C4×D4—1221no—
325C4×D4—1221no—
326C4×D4—1221no—
327C4×D4—1221no—
328#117—1221no—
329#117—1221no—
330#117C2×#692311yescharacteristic
331#117C2×#692311yescharacteristic
332C2×C42—611no—
333C2×C42—1221no—
334C2×C42—1221no—
335#107C2×#692311yescharacteristic
336#107C2×#692311yescharacteristic
337#106C2×#692311yescharacteristic
338#106C2×#692311yescharacteristic
339C4⋊C8—611no—
340C4⋊C8—611no—
341C4⋊C8—612no—
342C4⋊C8—612no—
343C4⋊C8—1221no—
344C4⋊C8—1221no—
345C4⋊C8—1221no—
346C4⋊C8—1222no—
347C4⋊C8—1222no—
348#103—611no—
349#103—611no—
350#103—611no—
351#103—611no—
352#102—611no—
353#102—611no—
354#102—611no—
355#102—611no—
356C22⋊C8—612no—
357C22⋊C8—1222no—
358C22⋊C8—1222no—
359C22⋊C8—1221no—
360#97—1222no—
361C4×C8—612no—
362C4×C8—611no—
363C4×C8—611no—
364C4×C8—611no—
365C4×C8—611no—
366C4×C8—612no—
367C4×C8—1222no—
368#41—614no—
369#41—1222no—
370#41—1222no—
371#41—1222no—
372C2×Q8—611no—
373C2×Q8—611no—
374C2×D4—611no—
375C2×D4—611no—
376C2×D4—1221no—
377C2×D4—1221no—
378C22×C4—611no—
379C22×C4—611no—
380C22×C4—1221no—
381C22×C4—1221no—
382C22×C4—1221no—
383C22×C4—1221no—
384C22×C4—1221no—
385C22×C4—1221no—
386C22×C4C22×#692311yescharacteristic
387Q16—1222no—
388SD16—1222no—
389SD16—1222no—
390D8—1222no—
391M16—4882no—
392C2×C8—22no—
393C2×C8—612no—
394C2×C8—612no—
395C2×C8—611no—
396C2×C8—611no—
397C2×C8—1222no—
398C2×C8—1222no—
399C2×C8—1222no—
400C2×C8—1222no—
401C2×C8—4882no—
402C2×C8—4881no—
403C2×C8—4881no—
404C2×C8C22×#692311yescharacteristic
405C2×C8C22×#692311yescharacteristic
406C4⋊C4—1221no—
407C4⋊C4—1221no—
408C4⋊C4—1221no—
409C4⋊C4—1221no—
410C4⋊C4C22×#692311yescharacteristic
411C4⋊C4C22×#692311yescharacteristic
412C4⋊C4C22×#692311yescharacteristic
413C4⋊C4C22×#692311yescharacteristic
414C22⋊C4—21no—
415C22⋊C4—21no—
416C22⋊C4—1221no—
417C22⋊C4—1221no—
418C22⋊C4—1221no—
419C22⋊C4—1221no—
420C42—21no—
421C42—21no—
422C42—611no—
423C42—611no—
424C42—611no—
425C42—611no—
426C42—1221no—
427C42—1221no—
428C42—1221no—
429C42—1221no—
430C23—1221no—
431C23—1221no—
432C23D4×#692311yescharacteristic
433Q8—612no—
434Q8—1221no—
435Q8—1221no—
436Q8—1221no—
437D4—612no—
438D4—1224no—
439D4—1221no—
440D4—1221no—
441D4—1221no—
442D4—2442no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—611no—
448C2×C4—611no—
449C2×C4—611no—
450C2×C4—611no—
451C2×C4—1221no—
452C2×C4—1221no—
453C2×C4—1221no—
454C2×C4—1224no—
455C2×C4—1221no—
456C2×C4—4881no—
457C2×C4—4881no—
458C2×C4—2441no—
459C2×C4—2441no—
460C2×C4—2441no—
461C2×C4—2441no—
462C2×C4—2441no—
463C2×C4—2442no—
464C2×C4D4×#692311yescharacteristic
465C2×C4D61⋊M1612yes—
466C2×C4#1451986512yes—
467C2×C4C23×#692311yescharacteristic
468C8—21no—
469C8—21no—
470C8—1222no—
471C8—1222no—
472C8—1221no—
473C8—1221no—
474C8—2442no—
475C8—2442no—
476C8—2442no—
477C22—21no—
478C22—21no—
479C22—1222no—
480C22—2441no—
481C22—2441no—
482C22—2441no—
483C22—2441no—
484C22—2441no—
485C22#1684079212yes—
486C22C2×D4×#692311yescharacteristic
487C4—41no—
488C4—41no—
489C4—41no—
490C4—41no—
491C4—611no—
492C4—611no—
493C4—1221no—
494C4—1221no—
495C4—1221no—
496C4—1222no—
497C4—2441no—
498C4#1684079212yes—
499C4C2×D61⋊M1611yescharacteristic
500C4C2×#1451986511yescharacteristic
501C2—22no—
502C2—4881no—
503C2—2442no—
504C2C2×#1684079211yescharacteristic
505C2#2146258111yescharacteristic
506C2#2146259811yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#427C4
socleC22×C61C2×C4×D4
prime radicalC2×#14519799C4
cosocleC4×Dic61C24
hypercenter#427#6923
hypocenterC61#17881
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
1. #21462740C2semidirect
2. #21462736C2semidirect
3. #21461447C2semidirect
4. #21461435C2semidirect
5. #21461431C2semidirect
6. #21461419C2semidirect
7. C2×#16840626C2semidirect
8. #21461116C2semidirect
9. #21461116C2semidirect
10. #21461032C2semidirect
122. #11758222M16semidirect
302. C61#17881semidirect

Maximal subgroups

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

subgroupindex
#214627402
#214627362
#214614472
#214614352
#214614312
#214614192
C2×#168406262
#214611162
#214610322
#214545902
#1788161