Finite Group Databasebeta

← #27155389

Subgroups — #27155389

Order 12416.

1536 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690857C211yesmaximal, characteristic, complemented
2C2×#41×D97C211yesmaximal, characteristic, complemented
3#20261405C213yesmaximal, complemented
4#20261404C211yesmaximal, characteristic, complemented
5#20261403C213yesmaximal, complemented
6#20261397C211yesmaximal, characteristic, complemented
7#20261394C213yesmaximal, complemented
8#20261392C211yesmaximal, characteristic, complemented
9#20261391C213yesmaximal, complemented
10#20261301C213yesmaximal, complemented
11#20261298C213yesmaximal, complemented
12#20261245C213yesmaximal, complemented
13#20261244C213yesmaximal, complemented
14#20261211C211yesmaximal, characteristic, complemented
15#550×C97C211yesmaximal, characteristic, complemented
16#15690857—28no—
17C2×#13095512C2211yescharacteristic
18C2×#13095511C2213yes—
19C2×#13095511C2213yes—
20C2×#13095510C2213yes—
21C2×#13095510C2213yes—
22C2×#13095510C2213yescomplemented
23#41×D97—28no—
24C22×Dic194C2213yescomplemented
25C2×Q8×D97C2211yescharacteristic
26C2×Q8×D97C2211yescharacteristic, complemented
27C2×D4×D97C2213yes—
28C22×C4×D97C2213yes—
29#15690806C2213yescomplemented
30#15690806C2216yescomplemented
31#15690798C2213yescomplemented
32#15690797C2213yescomplemented
33#15690796C2216yescomplemented
34#15690795C2211yescharacteristic
35#15690792C2213yes—
36#15690790C2213yescomplemented
37#15690789C2213yescomplemented
38#15690788C2213yescomplemented
39#15690787C2213yescomplemented
40#15690785C2213yescomplemented
41#15690784C2213yescomplemented
42#15690783C2213yescomplemented
43#15690782C2216yescomplemented
44#15690782C2216yescomplemented
45#15690782C2213yescomplemented
46#15690781C2213yescomplemented
47#15690781C2213yescomplemented
48#15690780C2213yes—
49#15690778C2213yes—
50#15690778C2216yes—
51Q8×Dic97C2211yescharacteristic, complemented
52C4×Dic194C2213yescomplemented
53#15690775C2213yescomplemented
54C4×D388C2213yescomplemented
55D4×Dic97C2216yescomplemented
56C4×C22⋊D97C2213yescomplemented
57C4×C22⋊D97C2213yescomplemented
58#15690770C2216yescomplemented
59#15690768C2213yes—
60#15690767C2213yes—
61C2×#41×C97C2211yescharacteristic
62C2×#41×C97C2211yescharacteristic, complemented
63#127×C97C2213yescomplemented
64#121×C97C2213yescomplemented
65#121×C97C2213yescomplemented
66C4×Q8×C97C2211yescharacteristic, complemented
67C4×D4×C97C2213yescomplemented
68C23×D97C2313yes—
69#13095512—24no—
70#13095511—212no—
71#13095511—212no—
72#13095510—212no—
73#13095510—212no—
74#13095510—212no—
75C2×Dic194—212no—
76C2×Dic194C2313yes—
77C2×Dic194C2313yes—
78C2×Dic194C2313yes—
79Q8×D97—24no—
80Q8×D97D418yescomplemented
81C2×D388C2311yescharacteristic
82C2×D388C2313yes—
83C2×D388C2313yes—
84C2×C22⋊D97C2313yes—
85C2×C22⋊D97C2316yes—
86C2×C22⋊D97C2313yes—
87C2×C22⋊D97C2316yes—
88C2×C22⋊D97C2313yes—
89C2×C22⋊D97C2313yes—
90D4×D97—212no—
91C22×Dic97C2316yes—
92C22×Dic97C2313yes—
93C2×C4×D97—212no—
94C2×C4×D97C2313yes—
95C2×C4×D97C2313yes—
96C2×C4×D97C2313yes—
97C2×C4×D97C2311yescharacteristic
98C2×C4×D97C2313yes—
99C4⋊Dic97C2313yes—
100C4⋊Dic97C2313yes—
101#13095490C2313yes—
102#13095490C2313yes—
103#13095490C2313yes—
104#13095490C2311yescharacteristic
105C22⋊Dic97C2313yes—
106C22⋊Dic97C2313yes—
107C22⋊Dic97C2316yes—
108#13095488C2311yescharacteristic
109#13095488C2313yes—
110#13095488C2316yes—
111#13095488C2311yescharacteristic
112#13095488C2313yes—
113#13095488C2311yescharacteristic
114#13095488C2313yes—
115C4×Dic97C2313yes—
116C4×Dic97C2316yes—
117#41×C97—24no—
118#41×C97—24no—
119C2×Q8×C97C2311yescharacteristic
120C2×D4×C97C2313yes—
121C2×D4×C97C2313yes—
122C2×D4×C97C2316yes—
123C2×D4×C97C2313yes—
124C22×C4×C97C2313yes—
125C22×C4×C97C2313yes—
126C4⋊C4×C97C2313yes—
127C4⋊C4×C97C2311yescharacteristic
128C22⋊C4×C97C2313yes—
129C22⋊C4×C97C2313yes—
130C42×C97C2313yes—
131C23×C97C2413yes—
132C23×C97C2413yes—
133C22×D97—26no—
134C22×D97—43no—
135C22×D97—43no—
136C22×D97C2413yes—
137C22×D97C2411yescharacteristic
138C22×D97C2411yescharacteristic
139Q8×C97C2×D414yes—
140D4×C97—26no—
141D4×C97—26no—
142D4×C97—46no—
143D4×C97—43no—
144C2×C4×C97—26no—
145C2×C4×C97—26no—
146C2×C4×C97C2411yescharacteristic
147C2×C4×C97C2413yes—
148C2×C4×C97C2411yescharacteristic
149C2×C4×C97C2411yescharacteristic
150C2×C4×C97C2413yes—
151C22⋊D97—26no—
152C22⋊D97—212no—
153C22⋊D97—26no—
154C22⋊D97—43no—
155C22⋊D97—43no—
156C22⋊D97—46no—
157C2×Dic97—26no—
158C2×Dic97—212no—
159C2×Dic97C2411yescharacteristic
160C2×Dic97C2413yes—
161C2×Dic97C2413yes—
162C2×Dic97C2411yescharacteristic
163C2×Dic97C2413yes—
164D388—22no—
165D388—26no—
166D388—43no—
167C4×D97—22no—
168C4×D97—26no—
169C4×D97—26no—
170C4×D97—26no—
171C4×D97C2×D4112yes—
172Dic194—26no—
173Dic194—26no—
174Dic194C2×D4112yes—
175C22×C97—23no—
176C22×C97—23no—
177C22×C97—23no—
178C22×C97—23no—
179C22×C97—23no—
180C22×C97—23no—
181C22×C97C2511yescharacteristic
182C2×D97—21no—
183C2×D97—21no—
184C2×D97—21no—
185C2×D97—23no—
186C2×D97—21no—
187C2×D97—23no—
188C2×D97—21no—
189C2×D97—23no—
190C2×D97—412no—
191C2×D97C22×D412yes—
192C4×C97—21no—
193C4×C97—21no—
194C4×C97—23no—
195C4×C97—21no—
196C4×C97C22×D416yes—
197Dic97—23no—
198Dic97—21no—
199Dic97—21no—
200Dic97—23no—
201Dic97C22×D416yes—
202C2×C97—43no—
203C2×C97—43no—
204C2×C97C23×D411yescharacteristic
205C2×C97#58511yescharacteristic
206C2×C97#58511yescharacteristic
207D97—22no—
208D97—41no—
209D97—43no—
210#3487—971nomaximal, Sylow
211C97#348711yescharacteristic, complemented, Sylow
212C2×2−1+4—971no—
213C22×#41—971no—
214#550—971no—
215#550—973no—
216#550—973no—
217#550D9711yescharacteristic, complemented
218#548—971no—
219#548—973no—
220#548—971no—
221#548—973no—
222#535—973no—
223#535—973no—
224#525—973no—
225#525—973no—
226#519—971no—
2272−1+4—1948no—
228C2×#41—971no—
229C2×#41—973no—
230C2×#41—973no—
231C2×#41—973no—
232C2×#41—973no—
233C2×#41—973no—
234C2×#41—1948no—
235C2×#41C2×D9711yescharacteristic
236C2×#41C2×D9711yescharacteristic, complemented
237C22×Q8—971no—
238C22×Q8—971no—
239C22×Q8—973no—
240C22×D4—973no—
241C23×C4—973no—
242#127—973no—
243#127—976no—
244#127C2×D9713yescomplemented
245#124—973no—
246#124—976no—
247#124—973no—
248#123—971no—
249#123—973no—
250C22⋊Q8—973no—
251C22⋊Q8—973no—
252C22⋊Q8—973no—
253C22⋊Q8—973no—
254#121—973no—
255#121—973no—
256#121—973no—
257#121—976no—
258#121—976no—
259#121—973no—
260#121—973no—
261#121—973no—
262#121C2×D9713yescomplemented
263#121C2×D9713yescomplemented
264#120—976no—
265#120—973no—
266#120—973no—
267C4×Q8—971no—
268C4×Q8—973no—
269C4×Q8—973no—
270C4×Q8C2×D9711yescharacteristic, complemented
271C4×D4—976no—
272C4×D4—973no—
273C4×D4—973no—
274C4×D4—976no—
275C4×D4—973no—
276C4×D4C2×D9713yescomplemented
277#117—973no—
278#117—973no—
279C24—973no—
280#41—24no—
281#41—24no—
282#41—19412no—
283#41—19412no—
284#41—19412no—
285#41—19412no—
286#41—19412no—
287#41—1944no—
288C2×Q8—978no—
289C2×Q8—973no—
290C2×Q8—973no—
291C2×Q8—973no—
292C2×Q8—19412no—
293C2×Q8—1944no—
294C2×Q8C22×D9711yescharacteristic
295C2×D4—976no—
296C2×D4—973no—
297C2×D4—973no—
298C2×D4—973no—
299C2×D4—976no—
300C2×D4—973no—
301C2×D4—973no—
302C2×D4—971no—
303C2×D4—973no—
304C2×D4—19412no—
305C2×D4C22×D9713yes—
306C2×D4C22×D9713yes—
307C2×D4C22×D9716yes—
308C2×D4C22×D9713yes—
309C22×C4—973no—
310C22×C4—976no—
311C22×C4—973no—
312C22×C4—973no—
313C22×C4—973no—
314C22×C4—973no—
315C22×C4—971no—
316C22×C4—19412no—
317C22×C4C22×D9713yes—
318C22×C4C22×D9713yes—
319C4⋊C4—973no—
320C4⋊C4—973no—
321C4⋊C4—973no—
322C4⋊C4—971no—
323C4⋊C4—973no—
324C4⋊C4—973no—
325C4⋊C4C22×D9713yes—
326C4⋊C4C22×D9711yescharacteristic
327C22⋊C4—973no—
328C22⋊C4—973no—
329C22⋊C4—976no—
330C22⋊C4—973no—
331C22⋊C4—971no—
332C22⋊C4—971no—
333C22⋊C4—971no—
334C22⋊C4—973no—
335C22⋊C4—976no—
336C22⋊C4—973no—
337C22⋊C4C22×D9713yes—
338C22⋊C4C22×D9713yes—
339C42—976no—
340C42—973no—
341C42C22×D9713yes—
342C23—971no—
343C23—973no—
344C23—971no—
345C23—3883no—
346C23—3883no—
347C23—1946no—
348C23C23×D9713yes—
349C23C23×D9713yes—
350Q8—9712no—
351Q8—1946no—
352Q8—1946no—
353Q8D4×D9714yes—
354D4—26no—
355D4—26no—
356D4—46no—
357D4—43no—
358D4—3883no—
359D4—3883no—
360D4—3886no—
361D4—3883no—
362D4—1942no—
363D4—19412no—
364D4—1946no—
365D4—1946no—
366D4—1946no—
367C2×C4—26no—
368C2×C4—26no—
369C2×C4—973no—
370C2×C4—9712no—
371C2×C4—971no—
372C2×C4—973no—
373C2×C4—973no—
374C2×C4—971no—
375C2×C4—1946no—
376C2×C4—1942no—
377C2×C4—1946no—
378C2×C4—19412no—
379C2×C4—1946no—
380C2×C4—1946no—
381C2×C4C23×D9713yes—
382C2×C4C23×D9711yescharacteristic
383C2×C4C23×D9711yescharacteristic
384C2×C4C23×D9711yescharacteristic
385C2×C4C23×D9713yes—
386C22—23no—
387C22—23no—
388C22—23no—
389C22—23no—
390C22—23no—
391C22—23no—
392C22—972no—
393C22—38812no—
394C22—1941no—
395C22—1943no—
396C22—1941no—
397C22—1943no—
398C22—1941no—
399C22—1943no—
400C22—1941no—
401C22—1941no—
402C22C24×D9711yescharacteristic
403C4—21no—
404C4—21no—
405C4—23no—
406C4—21no—
407C4—976no—
408C4—1941no—
409C4—1943no—
410C4—1941no—
411C4—1943no—
412C4C2×D4×D9716yes—
413C2—43no—
414C2—43no—
415C2—3883no—
416C2—3881no—
417C2—1942no—
418C2C22×D4×D9711yescharacteristic
419C2#2026157011yescharacteristic
420C2#2026157111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D97
commutator, socle, cosocleC22×C97C25
Fitting#550×C97C2
hypercenter#550D97
hypocenterC97#3487
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#15690857C2semidirect
2. C2×#41×D97C2semidirect
3. #20261405C2semidirect
4. #20261404C2semidirect
5. #20261403C2semidirect
6. #20261397C2semidirect
7. #20261394C2semidirect
8. #20261392C2semidirect
9. #20261391C2semidirect
10. #20261301C2semidirect
11. #20261298C2semidirect
12. #20261245C2semidirect
13. #20261244C2semidirect
14. #20261211C2semidirect
15. #550×C97C2semidirect
22. C2×#13095510C22semidirect
24. C22×Dic194C22semidirect
26. C2×Q8×D97C22semidirect
29. #15690806C22semidirect
30. #15690806C22semidirect
31. #15690798C22semidirect
32. #15690797C22semidirect
33. #15690796C22semidirect
36. #15690790C22semidirect
37. #15690789C22semidirect
38. #15690788C22semidirect
39. #15690787C22semidirect
40. #15690785C22semidirect
41. #15690784C22semidirect
42. #15690783C22semidirect
43. #15690782C22semidirect
44. #15690782C22semidirect
45. #15690782C22semidirect
46. #15690781C22semidirect
47. #15690781C22semidirect
51. Q8×Dic97C22semidirect
52. C4×Dic194C22semidirect
53. #15690775C22semidirect
54. C4×D388C22semidirect
55. D4×Dic97C22semidirect
56. C4×C22⋊D97C22semidirect
57. C4×C22⋊D97C22semidirect
58. #15690770C22semidirect
62. C2×#41×C97C22semidirect
63. #127×C97C22semidirect
64. #121×C97C22semidirect
65. #121×C97C22semidirect
66. C4×Q8×C97C22semidirect
67. C4×D4×C97C22semidirect
80. Q8×D97D4semidirect
211. C97#3487semidirect
217. #550D97semidirect
236. C2×#41C2×D97semidirect
244. #127C2×D97semidirect
262. #121C2×D97semidirect
263. #121C2×D97semidirect
270. C4×Q8C2×D97semidirect
276. C4×D4C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908572
C2×#41×D972
#202614052
#202614042
#202614032
#202613972
#202613942
#202613922
#202613912
#202613012
#202612982
#202612452
#202612442
#202612112
#550×C972
#348797