Finite Group Databasebeta

← #27150592

Subgroups — #27150592

Order 12416.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D97C211yesmaximal, characteristic
2#20261211C211yesmaximal, characteristic, complemented
3#20261207C211yesmaximal, characteristic
4#20260837C211yesmaximal, characteristic, complemented
5#20260836C211yesmaximal, characteristic, complemented
6#20260834C211yesmaximal, characteristic, complemented
7#20260826C211yesmaximal, characteristic, complemented
8C4×#13095498C211yesmaximal, characteristic, complemented
9C4×#13095499C211yesmaximal, characteristic, complemented
10#20260820C211yesmaximal, characteristic, complemented
11#20260772C211yesmaximal, characteristic, complemented
12#20260742C211yesmaximal, characteristic, complemented
13#20260738C211yesmaximal, characteristic
14#20260692C211yesmaximal, characteristic, complemented
15#440×C97C211yesmaximal, characteristic, complemented
16C2×Q8×D97C2211yescharacteristic
17C2×D4×D97C2211yescharacteristic
18C2×#13095500C2211yescharacteristic
19C2×#13095499C2211yescharacteristic
20C2×#13095498C2211yescharacteristic
21SD16×D97C418yescomplemented
22C2×#13095494C2211yescharacteristic
23C2×C8×D97C2211yescharacteristic
24Q8×Dic97C2211yescharacteristic, complemented
25C4×Dic194—21no—
26C4×Dic194C2211yescharacteristic, complemented
27#15690775—21no—
28#15690775C2211yescharacteristic, complemented
29C4×D388C2211yescharacteristic
30D4×Dic97C2211yescharacteristic
31#15690772C2211yescharacteristic
32C4×C22⋊D97—21no—
33#15690770—21no—
34#15690768—21no—
35#15690768C2211yescharacteristic
36#15690768C2211yescharacteristic
37#15690767—21no—
38#15690767C2211yescharacteristic
39C2×#13095488—21no—
40C22⋊C4×D97—21no—
41#15690744C2211yescharacteristic, complemented
42#15690741C2211yescharacteristic, complemented
43#15690741C2211yescharacteristic, complemented
44#15690735C2211yescharacteristic, complemented
45#15690734C2211yescharacteristic, complemented
46#15690733C2211yescharacteristic, complemented
47#15690732C2211yescharacteristic
48#15690731C2211yescharacteristic
49#15690730C2211yescharacteristic
50#15690720C2211yescharacteristic, complemented
51C4×#11715895C2211yescharacteristic, complemented
52C8×Dic97C2211yescharacteristic, complemented
53C2×SD16×C97C2211yescharacteristic
54C4×Q8×C97C2211yescharacteristic, complemented
55C4×D4×C97C2211yescharacteristic
56#107×C97C2211yescharacteristic, complemented
57#103×C97C2211yescharacteristic, complemented
58#102×C97C2211yescharacteristic
59#97×C97C2211yescharacteristic, complemented
60C23×D97—21no—
61C2×Dic194—21no—
62C2×Dic194C2311yescharacteristic
63Q8×D97—22no—
64Q8×D97C2×C414yescomplemented
65C2×D388C2311yescharacteristic
66C2×C22⋊D97—21no—
67D4×D97—22no—
68D4×D97C2×C414yes—
69C22×Dic97—21no—
70C2×C4×D97—21no—
71C2×C4×D97—21no—
72C2×C4×D97—21no—
73C2×C4×D97C2311yescharacteristic
74#13095500C2×C414yescomplemented
75#13095499C2×C414yescomplemented
76#13095498C2×C414yescomplemented
77#13095494—22no—
78C2×#11715895C2311yescharacteristic
79C2×#11715895C2311yescharacteristic
80C8×D97C2×C414yescomplemented
81C4⋊Dic97—21no—
82C4⋊Dic97C2311yescharacteristic
83C4⋊Dic97C2311yescharacteristic
84#13095490—21no—
85#13095490—21no—
86#13095490D411yescharacteristic
87#13095490D411yescharacteristic
88C22⋊Dic97—21no—
89#13095488—21no—
90#13095488—21no—
91#13095488—21no—
92#13095488—21no—
93#13095488D411yescharacteristic
94#13095488D411yescharacteristic
95C4×Dic97—21no—
96C4×Dic97—21no—
97C4×Dic97—21no—
98C4×Dic97—21no—
99C4×Dic97—21no—
100C4×Dic97C2311yescharacteristic
101C2×Q8×C97C2311yescharacteristic
102C2×D4×C97C2311yescharacteristic
103C22×C4×C97—21no—
104SD16×C97C2×C414yescomplemented
105C2×C8×C97C2311yescharacteristic
106C2×C8×C97C2311yescharacteristic
107C4⋊C4×C97—21no—
108C4⋊C4×C97C2311yescharacteristic
109C4⋊C4×C97C2311yescharacteristic
110C22⋊C4×C97—21no—
111C42×C97—21no—
112C42×C97C2311yescharacteristic
113C23×C97—21no—
114C22×D97—21no—
115C22×D97—24no—
116C22×D97—81no—
117C22×D97—81no—
118C22×D97C2×D411yescharacteristic
119Q8×C97—21no—
120Q8×C97C22×C412yes—
121D4×C97—21no—
122D4×C97C22×C412yes—
123C2×C4×C97—21no—
124C2×C4×C97—21no—
125C2×C4×C97—21no—
126C2×C4×C97—81no—
127C2×C4×C97C2×D411yescharacteristic
128C2×C4×C97C2×D411yescharacteristic
129C2×C4×C97C2411yescharacteristic
130C22⋊D97—24no—
131C2×Dic97—21no—
132C2×Dic97—21no—
133C2×Dic97—21no—
134C2×Dic97—81no—
135C2×Dic97C2×D411yescharacteristic
136C2×Dic97C2×D411yescharacteristic
137C2×Dic97C2×D411yescharacteristic
138D388—21no—
139D388C22×C412yes—
140C4×D97—24no—
141C4×D97—81no—
142C4×D97—81no—
143C4×D97C22×C412yes—
144C4×D97#4112yes—
145Dic194—21no—
146Dic194—24no—
147Dic194C22×C412yes—
148C8×C97—21no—
149C8×C97C22×C412yes—
150#11715895—21no—
151#11715895C22×C412yes—
152C22×C97—22no—
153C22×C97—41no—
154C22×C97—41no—
155C22×C97C22×D411yescharacteristic
156C2×D97—21no—
157C2×D97—21no—
158C2×D97—22no—
159C2×D97—41no—
160C2×D97—41no—
161C2×D97—84no—
162C2×D97C4×D412yes—
163C4×C97—22no—
164C4×C97—21no—
165C4×C97—21no—
166C4×C97—41no—
167C4×C97—41no—
168C4×C97C23×C411yescharacteristic
169C4×C97C2×#4111yescharacteristic
170Dic97—22no—
171Dic97—21no—
172Dic97—21no—
173Dic97—41no—
174Dic97—41no—
175Dic97C4×D412yes—
176C2×C97—42no—
177C2×C97C2×C4×D411yescharacteristic
178C2×C97#57711yescharacteristic
179C2×C97#57711yescharacteristic
180D97—22no—
181D97—42no—
182#2949—971nomaximal, Sylow
183C97#294911yescharacteristic, complemented, Sylow
184C22×SD16—971no—
185#519—971no—
186#518—971no—
187#440—971no—
188#440—971no—
189#440—971no—
190#440D9711yescharacteristic, complemented
191C4×SD16—971no—
192C4×SD16—971no—
193C4×SD16—971no—
194C4×SD16—971no—
195#427—971no—
196#419—971no—
197#418—971no—
198#405—971no—
199C22×Q8—971no—
200C22×D4—971no—
201C2×SD16—971no—
202C2×SD16—971no—
203C2×SD16—978no—
204C2×SD16—971no—
205C2×SD16C2×D9711yescharacteristic
206C2×M16—971no—
207C22×C8—971no—
208C4×Q8—971no—
209C4×Q8—971no—
210C4×Q8—971no—
211C4×Q8—1941no—
212C4×Q8—1941no—
213C4×Q8C2×D9711yescharacteristic, complemented
214C4×D4—971no—
215C4×D4—971no—
216C4×D4—971no—
217C4×D4—1941no—
218C4×D4—1941no—
219C4×D4C2×D9711yescharacteristic
220#117—971no—
221#117—971no—
222#117—971no—
223#117—1941no—
224#117—1941no—
225C2×C22⋊C4—1941no—
226C2×C22⋊C4—1941no—
227#107—971no—
228#107C2×D9711yescharacteristic, complemented
229#106—971no—
230#106—971no—
231#103—971no—
232#103—971no—
233#103—971no—
234#103C2×D9711yescharacteristic, complemented
235#102—971no—
236#102—971no—
237#102—971no—
238#102C2×D9711yescharacteristic
239#97—971no—
240#97C2×D9711yescharacteristic, complemented
241C4×C8—971no—
242C4×C8—971no—
243C24—1941no—
244C2×Q8—974no—
245C2×Q8—971no—
246C2×Q8—1941no—
247C2×Q8—1942no—
248C2×Q8C22×D9711yescharacteristic
249C2×D4—974no—
250C2×D4—971no—
251C2×D4—1942no—
252C2×D4—1941no—
253C2×D4C22×D9711yescharacteristic
254C22×C4—21no—
255C22×C4—971no—
256C22×C4—1941no—
257C22×C4—1941no—
258C22×C4—1941no—
259C22×C4—1941no—
260SD16—974no—
261SD16—974no—
262SD16—974no—
263SD16C4×D9714yescomplemented
264M16—1942no—
265C2×C8—971no—
266C2×C8—974no—
267C2×C8—971no—
268C2×C8C22×D9711yescharacteristic
269C2×C8C22×D9711yescharacteristic
270C4⋊C4—21no—
271C4⋊C4—971no—
272C4⋊C4—971no—
273C4⋊C4—971no—
274C4⋊C4—971no—
275C4⋊C4—1941no—
276C4⋊C4—1941no—
277C4⋊C4—1941no—
278C4⋊C4C22×D9711yescharacteristic
279C4⋊C4C22×D9711yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—971no—
282C22⋊C4—971no—
283C22⋊C4—1941no—
284C22⋊C4—1941no—
285C22⋊C4—1941no—
286C22⋊C4—1941no—
287C22⋊C4—1941no—
288C42—21no—
289C42—971no—
290C42—1941no—
291C42—1941no—
292C42—1941no—
293C42—1941no—
294C42—1941no—
295C42C22×D9711yescharacteristic
296C23—21no—
297C23—7761no—
298C23—7761no—
299C23—971no—
300C23—1944no—
301C23—1941no—
302Q8—21no—
303Q8—972no—
304Q8—1944no—
305Q8—1941no—
306Q8C2×C4×D9712yes—
307D4—21no—
308D4—972no—
309D4—1944no—
310D4—1941no—
311D4C2×C4×D9712yes—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—81no—
316C2×C4—7761no—
317C2×C4—7761no—
318C2×C4—7761no—
319C2×C4—971no—
320C2×C4—972no—
321C2×C4—972no—
322C2×C4—971no—
323C2×C4—971no—
324C2×C4—1944no—
325C2×C4—1941no—
326C2×C4—1941no—
327C2×C4—1941no—
328C2×C4D4×D9711yescharacteristic
329C2×C4D4×D9711yescharacteristic
330C2×C4C23×D9711yescharacteristic
331C8—21no—
332C8—972no—
333C8—1941no—
334C8C2×C4×D9712yes—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—7764no—
339C22—972no—
340C22—3881no—
341C22—3881no—
342C22—1942no—
343C22—1941no—
344C22—1941no—
345C22C2×D4×D9711yescharacteristic
346C4—22no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—972no—
352C4—3881no—
353C4—3881no—
354C4—1942no—
355C4—1941no—
356C4—1941no—
357C4C22×C4×D9711yescharacteristic
358C4#41×D9711yescharacteristic
359C2—42no—
360C2—3882no—
361C2—1942no—
362C2C4×D4×D9711yescharacteristic
363C2#2026154411yescharacteristic
364C2#2026154511yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutatorC4×C97C23×C4
FrattiniC2×C4C23×D97
Fitting#440×C97C2
socleC22×C97C22×D4
prime radicalC2×D4×D97C22
cosocleC2×C4×C97C24
hypercenter#440D97
hypocenterC97#2949
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. #20261211C2semidirect
4. #20260837C2semidirect
5. #20260836C2semidirect
6. #20260834C2semidirect
7. #20260826C2semidirect
8. C4×#13095498C2semidirect
9. C4×#13095499C2semidirect
10. #20260820C2semidirect
11. #20260772C2semidirect
12. #20260742C2semidirect
14. #20260692C2semidirect
15. #440×C97C2semidirect
21. SD16×D97C4semidirect
24. Q8×Dic97C22semidirect
26. C4×Dic194C22semidirect
28. #15690775C22semidirect
41. #15690744C22semidirect
42. #15690741C22semidirect
43. #15690741C22semidirect
44. #15690735C22semidirect
45. #15690734C22semidirect
46. #15690733C22semidirect
50. #15690720C22semidirect
51. C4×#11715895C22semidirect
52. C8×Dic97C22semidirect
54. C4×Q8×C97C22semidirect
56. #107×C97C22semidirect
57. #103×C97C22semidirect
59. #97×C97C22semidirect
64. Q8×D97C2×C4semidirect
74. #13095500C2×C4semidirect
75. #13095499C2×C4semidirect
76. #13095498C2×C4semidirect
80. C8×D97C2×C4semidirect
104. SD16×C97C2×C4semidirect
183. C97#2949semidirect
190. #440D97semidirect
213. C4×Q8C2×D97semidirect
228. #107C2×D97semidirect
234. #103C2×D97semidirect
240. #97C2×D97semidirect
263. SD16C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D972
#202612112
#202612072
#202608372
#202608362
#202608342
#202608262
C4×#130954982
C4×#130954992
#202608202
#202607722
#202607422
#202607382
#202606922
#440×C972
#294997