Finite Group Databasebeta

← #27155225

Subgroups — #27155225

Order 12416.

1648 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13095512C211yesmaximal, characteristic, complemented
2C2×#41×D97C211yesmaximal, characteristic, complemented
3#20261404C211yesmaximal, characteristic, complemented
4Q8×C22⋊D97C211yesmaximal, characteristic, complemented
5#20261397C211yesmaximal, characteristic, complemented
6#20261387C213yesmaximal, complemented
7#20261381C213yesmaximal, complemented
8#20261378C213yesmaximal, complemented
9#20261369C213yesmaximal, complemented
10#20261284C213yesmaximal, complemented
11#20261248C213yesmaximal, complemented
12#20261245C213yesmaximal, complemented
13C4×#13095512C211yesmaximal, characteristic, complemented
14C4×D4×D97C213yesmaximal, complemented
15#548×C97C211yesmaximal, characteristic, complemented
16C2×#13095512—28no—
17C2×#13095512C2211yescharacteristic
18C2×#13095512C2213yescomplemented
19C2×#13095512C2211yescharacteristic, complemented
20C2×#13095512C2213yescomplemented
21C2×#13095511C2213yescomplemented
22C2×#13095510C2213yescomplemented
23#41×D97—28no—
24C2×Q8×D97C2211yescharacteristic, complemented
25C22×D388C2213yescomplemented
26C2×D4×D97C2213yes—
27C2×D4×D97C2213yescomplemented
28C22×C4×D97C2213yescomplemented
29C22×C4×D97C2213yescomplemented
30#15690808C2213yescomplemented
31#15690806C2213yescomplemented
32#15690798C2213yescomplemented
33#15690795C2216yescomplemented
34#15690792C2216yescomplemented
35#15690790C2213yescomplemented
36#15690789C2213yescomplemented
37#15690786C2213yescomplemented
38#15690785C2213yescomplemented
39#15690784C2213yescomplemented
40#15690783C2216yescomplemented
41#15690783C2216yescomplemented
42#15690781C2213yescomplemented
43#15690780C2216yescomplemented
44#15690778C2216yescomplemented
45Q8×Dic97C2211yescharacteristic, complemented
46C4×D388C2213yescomplemented
47C4×D388C2213yescomplemented
48D4×Dic97C2213yescomplemented
49#15690772C2213yescomplemented
50#15690772C2213yescomplemented
51C4×C22⋊D97C2213yes—
52C4×C22⋊D97C2213yescomplemented
53#15690770C2211yescharacteristic
54#15690770C2213yescomplemented
55#15690770C2216yescomplemented
56#15690768C2213yescomplemented
57C4⋊C4×D97C2213yescomplemented
58C22⋊C4×D97C2213yescomplemented
59C22⋊C4×D97C2213yescomplemented
60C42×D97C2213yescomplemented
61C2×#41×C97C2211yescharacteristic, complemented
62C22×Q8×C97C2211yescharacteristic, complemented
63#124×C97C2213yescomplemented
64C22⋊Q8×C97C2213yescomplemented
65#121×C97C2213yescomplemented
66C4×Q8×C97C2211yescharacteristic, complemented
67C4×D4×C97C2213yescomplemented
68C23×D97C2313yes—
69C23×D97C2313yes—
70#13095512—24no—
71#13095512—212no—
72#13095512—212no—
73#13095512D418yescomplemented
74#13095511—212no—
75#13095510—212no—
76C2×Dic194C2313yes—
77Q8×D97—24no—
78C2×D388—212no—
79C2×D388C2313yes—
80C2×D388C2316yes—
81C2×D388C2313yes—
82C2×D388C2313yes—
83C2×C22⋊D97C2313yes—
84C2×C22⋊D97C2316yes—
85C2×C22⋊D97C2316yes—
86C2×C22⋊D97C2311yescharacteristic
87D4×D97—212no—
88D4×D97—212no—
89C22×Dic97C2311yescharacteristic
90C22×Dic97C2313yes—
91C2×C4×D97—212no—
92C2×C4×D97—212no—
93C2×C4×D97C2313yes—
94C2×C4×D97C2313yes—
95C2×C4×D97C2313yes—
96C2×C4×D97C2311yescharacteristic
97C2×C4×D97C2313yes—
98C2×C4×D97C2313yes—
99C2×C4×D97C2313yes—
100C2×C4×D97C2313yes—
101C2×C4×D97C2313yes—
102C4⋊Dic97C2313yes—
103C4⋊Dic97C2313yes—
104#13095490C2311yescharacteristic
105#13095490C2313yes—
106#13095490C2316yes—
107C22⋊Dic97C2313yes—
108C22⋊Dic97C2313yes—
109#13095488C2311yescharacteristic
110#13095488C2313yes—
111#13095488C2313yes—
112#13095488C2313yes—
113#13095488C2316yes—
114#13095488C2316yes—
115C4×Dic97C2311yescharacteristic
116C4×Dic97C2313yes—
117C4×Dic97C2313yes—
118#41×C97—24no—
119C2×Q8×C97—24no—
120C2×Q8×C97C2313yes—
121C2×Q8×C97C2311yescharacteristic
122C2×D4×C97C2313yes—
123C2×D4×C97C2313yes—
124C22×C4×C97C2313yes—
125C22×C4×C97C2313yes—
126C4⋊C4×C97C2313yes—
127C4⋊C4×C97C2313yes—
128C22⋊C4×C97C2311yescharacteristic
129C22⋊C4×C97C2313yes—
130C22⋊C4×C97C2316yes—
131C42×C97C2313yes—
132C23×C97C2413yes—
133C23×C97C2411yescharacteristic
134C22×D97—26no—
135C22×D97—26no—
136C22×D97—26no—
137C22×D97—26no—
138C22×D97—43no—
139C22×D97—43no—
140C22×D97C2413yes—
141C22×D97C2413yes—
142C22×D97C2411yescharacteristic
143Q8×C97—26no—
144Q8×C97C2×D414yescomplemented
145D4×C97—26no—
146D4×C97—26no—
147C2×C4×C97—26no—
148C2×C4×C97—26no—
149C2×C4×C97C2411yescharacteristic
150C2×C4×C97C2413yes—
151C2×C4×C97C2413yes—
152C2×C4×C97C2411yescharacteristic
153C2×C4×C97C2413yes—
154C22⋊D97—212no—
155C22⋊D97—26no—
156C22⋊D97—46no—
157C22⋊D97#4114yescomplemented
158C2×Dic97—22no—
159C2×Dic97—26no—
160C2×Dic97C2413yes—
161C2×Dic97C2413yes—
162C2×Dic97C2411yescharacteristic
163C2×Dic97C2411yescharacteristic
164C2×Dic97C2411yescharacteristic
165D388—212no—
166D388—26no—
167D388—26no—
168D388C2×D4112yescomplemented
169C4×D97—26no—
170C4×D97—26no—
171C4×D97—26no—
172C4×D97—26no—
173C4×D97—22no—
174C4×D97—26no—
175C4×D97—26no—
176C4×D97—43no—
177C4×D97C2×D4112yescomplemented
178Dic194—26no—
179C22×C97—21no—
180C22×C97—21no—
181C22×C97—23no—
182C22×C97—23no—
183C22×C97—23no—
184C22×C97C2×#4112yescomplemented
185C22×C97C2511yescharacteristic
186C2×D97—21no—
187C2×D97—23no—
188C2×D97—23no—
189C2×D97—21no—
190C2×D97—23no—
191C2×D97—23no—
192C2×D97—23no—
193C2×D97—412no—
194C2×D97—412no—
195C2×D97C22×D416yes—
196C2×D97C2×#4112yescomplemented
197C4×C97—21no—
198C4×C97—23no—
199C4×C97—23no—
200C4×C97—21no—
201C4×C97C22×D416yes—
202Dic97—23no—
203Dic97—21no—
204Dic97—23no—
205Dic97C22×D412yes—
206Dic97C2×#4112yescomplemented
207C2×C97—22no—
208C2×C97—43no—
209C2×C97C23×D411yescharacteristic
210C2×C97C22×#4111yescharacteristic
211C2×C97#58511yescharacteristic
212D97—22no—
213D97—26no—
214D97—43no—
215D4×#41—971nomaximal, Sylow
216C97D4×#4111yescharacteristic, complemented, Sylow
217C22×#41—971no—
218C22×#41—971no—
219#550—971no—
220D4×Q8—971no—
221#548—971no—
222#548D9711yescharacteristic, complemented
223#547—973no—
224#546—973no—
225#546—973no—
226D42—973no—
227#532—973no—
228#525—973no—
229#525—973no—
230C4×#41—971no—
231C2×C4×D4—973no—
232C2×#41—971no—
233C2×#41—973no—
234C2×#41—973no—
235C2×#41—971no—
236C2×#41—973no—
237C2×#41—973no—
238C2×#41—1948no—
239C2×#41—1948no—
240C2×#41C2×D9711yescharacteristic, complemented
241C22×Q8—971no—
242C22×Q8C2×D9711yescharacteristic, complemented
243C22×D4—973no—
244C22×D4—973no—
245C22×D4—973no—
246C23×C4—973no—
247C23×C4—973no—
248C4⋊Q8—973no—
249#127—973no—
250#124—973no—
251#124C2×D9713yescomplemented
252#123—976no—
253#123—976no—
254C22⋊Q8—973no—
255C22⋊Q8—973no—
256C22⋊Q8—973no—
257C22⋊Q8C2×D9713yescomplemented
258#121—973no—
259#121—973no—
260#121—973no—
261#121—976no—
262#121—976no—
263#121C2×D9713yescomplemented
264#120—976no—
265#120—976no—
266C4×Q8—971no—
267C4×Q8C2×D9711yescharacteristic, complemented
268C4×D4—973no—
269C4×D4—973no—
270C4×D4—973no—
271C4×D4—973no—
272C4×D4—973no—
273C4×D4—971no—
274C4×D4—973no—
275C4×D4—976no—
276C4×D4—973no—
277C4×D4—973no—
278C4×D4C2×D9713yescomplemented
279#117—973no—
280C2×C4⋊C4—973no—
281C2×C22⋊C4—973no—
282C2×C22⋊C4—973no—
283C2×C42—973no—
284C24—973no—
285C24—973no—
286#41—24no—
287#41—978no—
288#41—19412no—
289#41—19412no—
290#41—19412no—
291#41—19412no—
292#41—1944no—
293C2×Q8—24no—
294C2×Q8—973no—
295C2×Q8—1944no—
296C2×Q8C22×D9713yes—
297C2×Q8C22×D9711yescharacteristic
298C2×D4—971no—
299C2×D4—976no—
300C2×D4—973no—
301C2×D4—976no—
302C2×D4—973no—
303C2×D4—973no—
304C2×D4—976no—
305C2×D4—973no—
306C2×D4—19412no—
307C2×D4—19412no—
308C2×D4—19412no—
309C2×D4C22×D9713yes—
310C2×D4C22×D9713yes—
311C22×C4—973no—
312C22×C4—971no—
313C22×C4—973no—
314C22×C4—973no—
315C22×C4—973no—
316C22×C4—973no—
317C22×C4—973no—
318C22×C4—971no—
319C22×C4—973no—
320C22×C4—973no—
321C22×C4—973no—
322C22×C4—19412no—
323C22×C4—19412no—
324C22×C4C22×D9713yes—
325C22×C4C22×D9713yes—
326C4⋊C4—971no—
327C4⋊C4—973no—
328C4⋊C4—976no—
329C4⋊C4—973no—
330C4⋊C4—973no—
331C4⋊C4C22×D9713yes—
332C4⋊C4C22×D9713yes—
333C22⋊C4—973no—
334C22⋊C4—973no—
335C22⋊C4—971no—
336C22⋊C4—973no—
337C22⋊C4—973no—
338C22⋊C4—973no—
339C22⋊C4—976no—
340C22⋊C4—976no—
341C22⋊C4C22×D9711yescharacteristic
342C22⋊C4C22×D9713yes—
343C22⋊C4C22×D9716yes—
344C42—971no—
345C42—973no—
346C42—973no—
347C42C22×D9713yes—
348C23—971no—
349C23—973no—
350C23—973no—
351C23—3883no—
352C23—3883no—
353C23—1946no—
354C23—1946no—
355C23—1946no—
356C23—1946no—
357C23C23×D9713yes—
358C23C23×D9711yescharacteristic
359Q8—26no—
360Q8—1946no—
361Q8D4×D9714yescomplemented
362D4—26no—
363D4—26no—
364D4—974no—
365D4—9712no—
366D4—3886no—
367D4—19412no—
368D4—1946no—
369D4—1946no—
370D4—19412no—
371D4—1946no—
372C2×C4—26no—
373C2×C4—26no—
374C2×C4—971no—
375C2×C4—971no—
376C2×C4—9712no—
377C2×C4—973no—
378C2×C4—971no—
379C2×C4—973no—
380C2×C4—3883no—
381C2×C4—1942no—
382C2×C4—1946no—
383C2×C4—1946no—
384C2×C4—1946no—
385C2×C4—1946no—
386C2×C4—1942no—
387C2×C4—1946no—
388C2×C4—1946no—
389C2×C4—1946no—
390C2×C4C23×D9711yescharacteristic
391C2×C4C23×D9713yes—
392C2×C4C23×D9711yescharacteristic
393C2×C4C23×D9713yes—
394C2×C4C23×D9713yes—
395C22—21no—
396C22—21no—
397C22—23no—
398C22—23no—
399C22—23no—
400C22—976no—
401C22—972no—
402C22—38812no—
403C22—38812no—
404C22—1943no—
405C22—1943no—
406C22—1943no—
407C22—1941no—
408C22—1943no—
409C22—1943no—
410C22—1941no—
411C22#41×D9712yescomplemented
412C22C24×D9711yescharacteristic
413C4—21no—
414C4—23no—
415C4—23no—
416C4—21no—
417C4—972no—
418C4—972no—
419C4—1941no—
420C4—1943no—
421C4—1943no—
422C4C2×D4×D9716yes—
423C2—22no—
424C2—43no—
425C2—3883no—
426C2—1942no—
427C2—1946no—
428C2C22×D4×D9711yescharacteristic
429C2C2×#41×D9711yescharacteristic
430C2#2026157211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D97
commutator, socle, cosocleC22×C97C25
Fitting#548×C97C2
hypercenter#548D97
hypocenterC97D4×#41
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C22×#13095512C2semidirect
2. C2×#41×D97C2semidirect
3. #20261404C2semidirect
4. Q8×C22⋊D97C2semidirect
5. #20261397C2semidirect
6. #20261387C2semidirect
7. #20261381C2semidirect
8. #20261378C2semidirect
9. #20261369C2semidirect
10. #20261284C2semidirect
11. #20261248C2semidirect
12. #20261245C2semidirect
13. C4×#13095512C2semidirect
14. C4×D4×D97C2semidirect
15. #548×C97C2semidirect
18. C2×#13095512C22semidirect
19. C2×#13095512C22semidirect
20. C2×#13095512C22semidirect
21. C2×#13095511C22semidirect
22. C2×#13095510C22semidirect
24. C2×Q8×D97C22semidirect
25. C22×D388C22semidirect
27. C2×D4×D97C22semidirect
28. C22×C4×D97C22semidirect
29. C22×C4×D97C22semidirect
30. #15690808C22semidirect
31. #15690806C22semidirect
32. #15690798C22semidirect
33. #15690795C22semidirect
34. #15690792C22semidirect
35. #15690790C22semidirect
36. #15690789C22semidirect
37. #15690786C22semidirect
38. #15690785C22semidirect
39. #15690784C22semidirect
40. #15690783C22semidirect
41. #15690783C22semidirect
42. #15690781C22semidirect
43. #15690780C22semidirect
44. #15690778C22semidirect
45. Q8×Dic97C22semidirect
46. C4×D388C22semidirect
47. C4×D388C22semidirect
48. D4×Dic97C22semidirect
49. #15690772C22semidirect
50. #15690772C22semidirect
52. C4×C22⋊D97C22semidirect
54. #15690770C22semidirect
55. #15690770C22semidirect
56. #15690768C22semidirect
57. C4⋊C4×D97C22semidirect
58. C22⋊C4×D97C22semidirect
59. C22⋊C4×D97C22semidirect
60. C42×D97C22semidirect
61. C2×#41×C97C22semidirect
62. C22×Q8×C97C22semidirect
63. #124×C97C22semidirect
64. C22⋊Q8×C97C22semidirect
65. #121×C97C22semidirect
66. C4×Q8×C97C22semidirect
67. C4×D4×C97C22semidirect
73. #13095512D4semidirect
144. Q8×C97C2×D4semidirect
157. C22⋊D97#41semidirect
168. D388C2×D4semidirect
177. C4×D97C2×D4semidirect
184. C22×C97C2×#41semidirect
196. C2×D97C2×#41semidirect
206. Dic97C2×#41semidirect
216. C97D4×#41semidirect
222. #548D97semidirect
240. C2×#41C2×D97semidirect
242. C22×Q8C2×D97semidirect
251. #124C2×D97semidirect
257. C22⋊Q8C2×D97semidirect
263. #121C2×D97semidirect
267. C4×Q8C2×D97semidirect
278. C4×D4C2×D97semidirect
361. Q8D4×D97semidirect
411. C22#41×D97semidirect

Maximal subgroups

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

subgroupindex
C22×#130955122
C2×#41×D972
#202614042
Q8×C22⋊D972
#202613972
#202613872
#202613812
#202613782
#202613692
#202612842
#202612482
#202612452
C4×#130955122
C4×D4×D972
#548×C972
D4×#4197