Finite Group Databasebeta

← #30612352

Subgroups — #30612352

Order 15616.

714 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#21462736C211yesmaximal, characteristic, complemented
2#21462734C211yesmaximal, characteristic, complemented
3#21461416C212yesmaximal, complemented
4#21461413C212yesmaximal, complemented
5#21461403C212yesmaximal, complemented
6#21461402C212yesmaximal, complemented
7C2×#16840624C211yesmaximal, characteristic, complemented
8#21461116C211yesmaximal, characteristic, complemented
9#21461114C211yesmaximal, characteristic, complemented
10#21461022C211yesmaximal, characteristic, complemented
11#21454587C211yesmaximal, characteristic
12#16840811C2212yes—
13Q8×#92420C2212yes—
14#16840790C2212yes—
15#16840789—22no—
16D4×#92420—22no—
17D4×#92420C2212yes—
18#16840781—21no—
19#16840778—21no—
20#16840778C2211yescharacteristic
21#16840778C2211yescharacteristic
22D61⋊(C4⋊C8)—21no—
23C2×C4⋊#92420—21no—
24C2×C4⋊#92420C2211yescharacteristic
25C2×C4⋊#92420C2211yescharacteristic
26#16840624C2212yes—
27#16840624C2212yes—
28#16840612C2212yes—
29Q8⋊#92420C2212yes—
30#16840610C2212yes—
31D4⋊#92420C2212yes—
32#16840603C2212yes—
33C8×#92420C2212yes—
34C2×#14519797C411yescharacteristic
35#41×Dic61C2211yescharacteristic
36#16840230C2211yescharacteristic
37#16839854C411yescharacteristic
38#16839854C2211yescharacteristic
39#16839853C411yescharacteristic
40#16839853C2211yescharacteristic
41#16839847C411yescharacteristic
42#16839847C2211yescharacteristic
43#16839842C411yescharacteristic
44#16839842C2211yescharacteristic
45#16839798C411yescharacteristic
46#16839759C411yescharacteristic
47C2×C8×Dic61C2211yescharacteristic
48C61×#416C411yescharacteristic
49C2×D61⋊C8—22no—
50C22×#92420—21no—
51C22×#92420—21no—
52C22×#92420—22no—
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—
63C22⋊#92420—21no—
64C22⋊#92420—22no—
65C22⋊#92420D412yes—
66C4×#92420—22no—
67C4×#92420—22no—
68C4×#92420C2312yes—
69C4×#92420C2312yes—
70C2×#11758222C2×C411yescharacteristic
71#14519797C818yescomplemented
72C2×#11758202C2×C411yescharacteristic
73C2×#11758200C2×C411yescharacteristic
74C2×#11758201C2×C411yescharacteristic
75C2×#11758199C2×C411yescharacteristic
76C22×#92418C2×C411yescharacteristic
77Q8×Dic61C2×C411yescharacteristic
78Q8×Dic61C2311yescharacteristic
79#14519742C2×C411yescharacteristic
80#14519742C2311yescharacteristic
81D4×Dic61—21no—
82D4×Dic61—21no—
83D4×Dic61C2×C411yescharacteristic
84D4×Dic61C2311yescharacteristic
85#14519739C2×C411yescharacteristic
86#14519739C2311yescharacteristic
87#14519737—21no—
88#14519737—21no—
89#14519736—21no—
90#14519736C2×C411yescharacteristic
91#14519734—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61C2311yescharacteristic
94C42×D61—21no—
95#14519710C2×C411yescharacteristic
96#14519710C2×C411yescharacteristic
97#14519708C2×C411yescharacteristic
98#14519708C2×C411yescharacteristic
99#14519702C2×C411yescharacteristic
100#14519702C2×C411yescharacteristic
101#14519699C2×C411yescharacteristic
102#14519699C2×C411yescharacteristic
103C8×Dic61C2×C411yescharacteristic
104C8×Dic61C2×C411yescharacteristic
105C8×Dic61C2311yescharacteristic
106C8×Dic61C2311yescharacteristic
107C2×#41×C61C2×C411yescharacteristic
108C22×C8×C61C2×C411yescharacteristic
109#117×C61C2×C411yescharacteristic
110#103×C61C2×C411yescharacteristic
111#103×C61C2×C411yescharacteristic
112#102×C61C2×C411yescharacteristic
113#102×C61C2×C411yescharacteristic
114#41×C61—22no—
115#41×C61C2×C814yescomplemented
116C2×Q8×C61C22×C411yescharacteristic
117C2×D4×C61—21no—
118C2×D4×C61C22×C411yescharacteristic
119C22×C4×C61—21no—
120C22×C4×C61C22×C411yescharacteristic
121C2×C22⋊D61—21no—
122C22×Dic61—21no—
123C22×Dic61—21no—
124C22×Dic61C2×D411yescharacteristic
125C22×Dic61#4111yescharacteristic
126#11758222—22no—
127#11758222C2×C814yescomplemented
128C2×D244C22×C411yescharacteristic
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×C4×D61—21no—
132C2×Dic122C22×C411yescharacteristic
133C2×#92420—22no—
134C2×#92420—22no—
135C2×#92420—22no—
136C2×#92420—82no—
137C2×#92420—81no—
138C2×#92420—81no—
139C2×#92420C2×D412yes—
140C2×#92420C2×D412yes—
141D61⋊C8—82no—
142C2×C8×C61—22no—
143C2×C8×C61C22×C411yescharacteristic
144C2×C8×C61C22×C411yescharacteristic
145C4⋊C4×C61C22×C411yescharacteristic
146C4⋊C4×C61C22×C411yescharacteristic
147C22⋊C4×C61—21no—
148C42×C61—21no—
149C22⋊Dic61—21no—
150C22⋊Dic61—21no—
151C22⋊Dic61—21no—
152#11758202C2×C814yescomplemented
153#11758201C2×C814yescomplemented
154#11758200C2×C814yescomplemented
155#11758199C2×C814yescomplemented
156#11758198—21no—
157#11758198—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160C4⋊Dic61C22×C411yescharacteristic
161C4⋊Dic61C22×C411yescharacteristic
162#11758196—21no—
163#11758196—21no—
164C4×Dic61—21no—
165C4×Dic61—21no—
166C4×Dic61—21no—
167C4×Dic61—21no—
168C4×Dic61—21no—
169C4×Dic61C22×C411yescharacteristic
170C4×Dic61C2×D411yescharacteristic
171C4×Dic61#4111yescharacteristic
172C4×Dic61C2411yescharacteristic
173C2×#92418C2×C814yescomplemented
174C2×#92418C22×C411yescharacteristic
175C2×#92418C22×C411yescharacteristic
176C23×C61—21no—
177C23×C61C4×D411yescharacteristic
178C22×D61—21no—
179Q8×C61—21no—
180Q8×C61C22×C812yes—
181D4×C61—24no—
182D4×C61—21no—
183D4×C61C22×C812yes—
184C2×C4×C61—21no—
185C2×C4×C61—24no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61C4×D411yescharacteristic
189C2×C4×C61C22×C812yes—
190C2×C4×C61#13112yes—
191C2×C4×C61C23×C411yescharacteristic
192C22⋊D61—24no—
193C2×Dic61—21no—
194C2×Dic61—21no—
195C2×Dic61—21no—
196C2×Dic61—21no—
197C2×Dic61—21no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61—81no—
202C2×Dic61C4×D411yescharacteristic
203C2×Dic61C4×D411yescharacteristic
204C2×Dic61C22×D411yescharacteristic
205C2×Dic61C2×#4111yescharacteristic
206D244—21no—
207D244C22×C812yes—
208C4×D61—24no—
209C4×D61—41no—
210C4×D61—41no—
211C4×D61—81no—
212Dic122—21no—
213Dic122C22×C812yes—
214#92420—22no—
215#92420—22no—
216#92420—42no—
217#92420—42no—
218#92420—42no—
219C8×C61—21no—
220C8×C61—21no—
221#92418C22×C812yes—
222#92418C22×C812yes—
223#17858—611nomaximal, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—22no—
227C22×C61—41no—
228C22×C61—41no—
229C22×C61C8×D412yes—
230C22×C61C2×C4×D411yescharacteristic
231C2×D61—22no—
232C2×D61—41no—
233C2×D61—41no—
234C4×C61—22no—
235C4×C61—41no—
236C4×C61—41no—
237C4×C61C8×D412yes—
238C4×C61C23×C811yescharacteristic
239C4×C61C2×#13111yescharacteristic
240Dic61—22no—
241Dic61—21no—
242Dic61—21no—
243Dic61—21no—
244Dic61—41no—
245Dic61—41no—
246Dic61#57711yescharacteristic
247Dic61#57711yescharacteristic
248#2988—611no—
249#2988—611no—
250C4×#135—611no—
251#1580—612no—
252#1579—612no—
253C8×SD16—612no—
254C8×SD16—612no—
255C2×#334—611no—
256#1477—611no—
257#1477—611no—
258#1450—611no—
259C2×C61—22no—
260C2×C61—42no—
261C2×C61C2×C8×D411yescharacteristic
262C2×C61#295211yescharacteristic
263C2×C61#295711yescharacteristic
264D61—42no—
265C2×#135—611no—
266C4×#41—611no—
267C4×#41—611no—
268C8×Q8—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—
278C8×D4—612no—
279C8×D4—612no—
280C8×D4—1222no—
281C8×D4—1222no—
282#431—611no—
283#431—611no—
284#431—1221no—
285#431—1221no—
286#427—611no—
287C2×C4⋊C8—611no—
288C2×C4⋊C8—611no—
289C2×C4⋊C8—1221no—
290C2×C4⋊C8—1221no—
291#416—611no—
292#416#692311yescharacteristic
293C2×C4×C8—611no—
294#334—612no—
295#334—612no—
296Q8⋊C8—612no—
297Q8⋊C8—612no—
298D4⋊C8—612no—
299D4⋊C8—612no—
300#322—612no—
301C82—612no—
302C61#1785811yescharacteristic, complemented, Sylow
303C2×#41—611no—
304C2×#41C2×#692311yescharacteristic
305#135—618no—
306C2×Q16—611no—
307C2×SD16—611no—
308C2×SD16—611no—
309C2×D8—611no—
310C22×C8—611no—
311C22×C8—1222no—
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—611no—
329#117—1221no—
330#117—1221no—
331#117C2×#692311yescharacteristic
332C2×C42—611no—
333C2×C42—1221no—
334C2×C42—1221no—
335#107—611no—
336#107—611no—
337#106—611no—
338#106—611no—
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#103C2×#692311yescharacteristic
351#103C2×#692311yescharacteristic
352#102—611no—
353#102—611no—
354#102C2×#692311yescharacteristic
355#102C2×#692311yescharacteristic
356C22⋊C8—612no—
357C22⋊C8—1222no—
358C22⋊C8—1222no—
359C22⋊C8—1221no—
360C4×C8—612no—
361C4×C8—611no—
362C4×C8—611no—
363C4×C8—611no—
364C4×C8—611no—
365C4×C8—612no—
366C4×C8—1222no—
367C4×C8—1222no—
368#41—22no—
369#41—614no—
370#41—1222no—
371#41D61⋊C814yescomplemented
372C2×Q8—611no—
373C2×Q8C22×#692311yescharacteristic
374C2×D4—21no—
375C2×D4—611no—
376C2×D4—1221no—
377C2×D4C22×#692311yescharacteristic
378C22×C4—21no—
379C22×C4—611no—
380C22×C4—611no—
381C22×C4—1221no—
382C22×C4—1221no—
383C22×C4—1221no—
384C22×C4—1221no—
385C22×C4—1221no—
386C22×C4C22×#692311yescharacteristic
387Q16—614no—
388SD16—614no—
389SD16—614no—
390D8—614no—
391C2×C8—22no—
392C2×C8—614no—
393C2×C8—611no—
394C2×C8—611no—
395C2×C8—612no—
396C2×C8—612no—
397C2×C8—1222no—
398C2×C8—1222no—
399C2×C8—1222no—
400C2×C8—4882no—
401C2×C8—4882no—
402C2×C8—4881no—
403C2×C8—4881no—
404C2×C8C22×#692311yescharacteristic
405C2×C8C22×#692311yescharacteristic
406C4⋊C4—611no—
407C4⋊C4—611no—
408C4⋊C4—1221no—
409C4⋊C4—1221no—
410C4⋊C4—1221no—
411C4⋊C4—1221no—
412C4⋊C4C22×#692311yescharacteristic
413C4⋊C4C22×#692311yescharacteristic
414C22⋊C4—21no—
415C22⋊C4—1221no—
416C22⋊C4—1221no—
417C22⋊C4—1221no—
418C22⋊C4—1221no—
419C22⋊C4—1221no—
420C42—21no—
421C42—611no—
422C42—611no—
423C42—611no—
424C42—611no—
425C42—1221no—
426C42—1221no—
427C42—1221no—
428C42—1221no—
429C42—1221no—
430C23—21no—
431C23—1221no—
432C23D4×#692311yescharacteristic
433Q8—21no—
434Q8—612no—
435Q8—1221no—
436Q8C2×D61⋊C812yes—
437D4—24no—
438D4—21no—
439D4—612no—
440D4—1224no—
441D4—1221no—
442D4C2×D61⋊C812yes—
443C2×C4—21no—
444C2×C4—24no—
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—1221no—
457C2×C4—4881no—
458C2×C4—4881no—
459C2×C4—2441no—
460C2×C4—2441no—
461C2×C4—2441no—
462C2×C4—2441no—
463C2×C4—2441no—
464C2×C4D4×#692311yescharacteristic
465C2×C4C2×D61⋊C812yes—
466C2×C4#1451986412yes—
467C2×C4C23×#692311yescharacteristic
468C8—21no—
469C8—21no—
470C8—612no—
471C8—612no—
472C8—1222no—
473C8—1222no—
474C8—2442no—
475C8—2442no—
476C8—2442no—
477C22—21no—
478C22—21no—
479C22—22no—
480C22—41no—
481C22—41no—
482C22—1222no—
483C22—2441no—
484C22—2441no—
485C22#1684078912yes—
486C22C2×D4×#692311yescharacteristic
487C4—22no—
488C4—41no—
489C4—41no—
490C4—611no—
491C4—611no—
492C4—1221no—
493C4—1221no—
494C4—1221no—
495C4—1222no—
496C4—2441no—
497C4—2441no—
498C4#1684078912yes—
499C4C22×D61⋊C811yescharacteristic
500C4C2×#1451986411yescharacteristic
501C2—22no—
502C2—42no—
503C2—2442no—
504C2C2×#1684078911yescharacteristic
505C2#2146258211yescharacteristic
506C2#2146259611yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC4×C61C23×C8
FrattiniC2×C4C23×#6923
FittingC61×#416C4
socleC22×C61C2×C4×D4
prime radicalC2×#14519797C4
cosocleC4×Dic61C24
hypercenter#416#6923
hypocenterC61#17858
LCSG ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #21462736C2semidirect
2. #21462734C2semidirect
3. #21461416C2semidirect
4. #21461413C2semidirect
5. #21461403C2semidirect
6. #21461402C2semidirect
7. C2×#16840624C2semidirect
8. #21461116C2semidirect
9. #21461114C2semidirect
10. #21461022C2semidirect
71. #14519797C8semidirect
115. #41×C61C2×C8semidirect
127. #11758222C2×C8semidirect
152. #11758202C2×C8semidirect
153. #11758201C2×C8semidirect
154. #11758200C2×C8semidirect
155. #11758199C2×C8semidirect
173. C2×#92418C2×C8semidirect
302. C61#17858semidirect
371. #41D61⋊C8semidirect

Maximal subgroups

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

subgroupindex
#214627362
#214627342
#214614162
#214614132
#214614032
#214614022
C2×#168406242
#214611162
#214611142
#214610222
#214545872
#1785861