Finite Group Databasebeta

← #27150468

Subgroups — #27150468

Order 12416.

544 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13095495C211yesmaximal, characteristic, complemented
2C2×M16×D97C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D97C211yesmaximal, characteristic
4#20260810C212yesmaximal, complemented
5#20260809C212yesmaximal, complemented
6#20260803C212yesmaximal, complemented
7#20260798C212yesmaximal, complemented
8#20260758C211yesmaximal, characteristic, complemented
9#20260698C211yesmaximal, characteristic, complemented
10M16×Dic97C211yesmaximal, characteristic, complemented
11#407×C97C211yesmaximal, characteristic, complemented
12C22×C22⋊D97C411yescharacteristic
13C22×C4×D97C2211yescharacteristic
14C2×#13095495—24no—
15C2×#13095495C2211yescharacteristic
16C2×#13095495C2211yescharacteristic
17C2×#13095495C2212yescomplemented
18C2×#13095494C2212yescomplemented
19M16×D97—24no—
20C22×#11715895C2212yescomplemented
21C2×C8×D97C2212yescomplemented
22C4×C22⋊D97C411yescharacteristic
23C4×C22⋊D97C411yescharacteristic
24C4×C22⋊D97C411yescharacteristic
25C4×C22⋊D97C411yescharacteristic
26C4×C22⋊D97C2211yescharacteristic
27C4×C22⋊D97C2211yescharacteristic
28C4×C22⋊D97C2211yescharacteristic
29C4×C22⋊D97C2211yescharacteristic
30C2×#13095490C411yescharacteristic
31C2×C22⋊Dic97C411yescharacteristic
32C2×#13095488C411yescharacteristic
33C2×C4×Dic97C2211yescharacteristic
34#15690739C2212yescomplemented
35#15690739C2212yescomplemented
36#15690722C2212yescomplemented
37#15690722C2212yescomplemented
38#15690720C2212yescomplemented
39C8×Dic97C2212yescomplemented
40C23×C4×C97C2211yescharacteristic
41C2×M16×C97C2211yescharacteristic
42C2×M16×C97C2211yescharacteristic
43C22⋊C8×C97C2212yescomplemented
44C22⋊C8×C97C2212yescomplemented
45C23×D97C2×C411yescharacteristic
46C2×C22⋊D97—24no—
47C2×C22⋊D97C2×C411yescharacteristic
48C2×C22⋊D97C2×C411yescharacteristic
49C2×C22⋊D97C2×C411yescharacteristic
50C2×C22⋊D97C2×C411yescharacteristic
51C22×Dic97C2×C411yescharacteristic
52C22×Dic97C2×C411yescharacteristic
53C22×Dic97C2×C411yescharacteristic
54C2×C4×D97—22no—
55C2×C4×D97—22no—
56C2×C4×D97C2×C411yescharacteristic
57C2×C4×D97C2×C411yescharacteristic
58C2×C4×D97C2311yescharacteristic
59C2×C4×D97C2311yescharacteristic
60#13095495—22no—
61#13095495—24no—
62#13095495D414yescomplemented
63#13095494—24no—
64C2×#11715895—24no—
65C2×#11715895C2312yes—
66C2×#11715895C2312yes—
67C8×D97—24no—
68#13095490C2×C411yescharacteristic
69#13095490C2×C411yescharacteristic
70#13095490C2×C411yescharacteristic
71#13095490C2×C411yescharacteristic
72C22⋊Dic97C2×C411yescharacteristic
73C22⋊Dic97C2×C411yescharacteristic
74C22⋊Dic97C2×C411yescharacteristic
75C22⋊Dic97C2×C411yescharacteristic
76#13095488C2×C411yescharacteristic
77#13095488C2×C411yescharacteristic
78#13095488C2×C411yescharacteristic
79#13095488C2×C411yescharacteristic
80C4×Dic97C2×C411yescharacteristic
81C4×Dic97C2×C411yescharacteristic
82C4×Dic97C2311yescharacteristic
83C4×Dic97C2311yescharacteristic
84C24×C97C2×C411yescharacteristic
85C22×C4×C97—22no—
86C22×C4×C97—22no—
87C22×C4×C97C2×C411yescharacteristic
88C22×C4×C97C2×C411yescharacteristic
89C22×C4×C97C2×C411yescharacteristic
90C22×C4×C97C2311yescharacteristic
91C22×C4×C97C2311yescharacteristic
92C22×C4×C97C2311yescharacteristic
93M16×C97—22no—
94M16×C97—22no—
95C2×C8×C97C2312yes—
96C2×C8×C97C2312yes—
97C23×C97—22no—
98C23×C97—22no—
99C23×C97—41no—
100C23×C97C22×C411yescharacteristic
101C23×C97C22×C411yescharacteristic
102C23×C97C22×C411yescharacteristic
103C22×D97—22no—
104C22×D97—22no—
105C22×D97—41no—
106C22×D97C22×C411yescharacteristic
107C22×D97C22×C411yescharacteristic
108C2×C4×C97—22no—
109C2×C4×C97—22no—
110C2×C4×C97—22no—
111C2×C4×C97—21no—
112C2×C4×C97—21no—
113C2×C4×C97—21no—
114C2×C4×C97—21no—
115C2×C4×C97C22×C411yescharacteristic
116C2×C4×C97C22×C411yescharacteristic
117C2×C4×C97C22×C411yescharacteristic
118C2×C4×C97C2×D412yes—
119C2×C4×C97#4112yes—
120C2×C4×C97C2411yescharacteristic
121C22⋊D97—22no—
122C22⋊D97—22no—
123C22⋊D97—22no—
124C22⋊D97M1614yescomplemented
125C2×Dic97—22no—
126C2×Dic97—22no—
127C2×Dic97—41no—
128C2×Dic97C22×C411yescharacteristic
129C2×Dic97C22×C411yescharacteristic
130C2×Dic97C22×C411yescharacteristic
131C2×Dic97C22×C411yescharacteristic
132C2×Dic97C22×C411yescharacteristic
133C2×Dic97C22×C411yescharacteristic
134C4×D97—22no—
135C4×D97—22no—
136C4×D97—21no—
137C4×D97—21no—
138C4×D97—21no—
139C4×D97—21no—
140C8×C97—22no—
141C8×C97—22no—
142#11715895—22no—
143#11715895C2×D414yescomplemented
144C22×C97—21no—
145C22×C97—21no—
146C22×C97—21no—
147C22×C97—21no—
148C22×C97—21no—
149C22×C97—21no—
150C22×C97—21no—
151C22×C97—44no—
152C22×C97C4×D412yes—
153C22×C97C2×M1612yescomplemented
154C22×C97C23×C411yescharacteristic
155C2×D97—21no—
156C2×D97—21no—
157C2×D97—21no—
158C2×D97—21no—
159C2×D97—21no—
160C2×D97—44no—
161C2×D97C2×M1612yescomplemented
162C4×C97—21no—
163C4×C97—21no—
164C4×C97C4×D412yes—
165C4×C97C22×D411yescharacteristic
166C4×C97C2×#4111yescharacteristic
167Dic97—21no—
168Dic97—21no—
169Dic97—21no—
170Dic97—21no—
171Dic97—21no—
172Dic97C2×M1612yescomplemented
173C2×C97—22no—
174C2×C97—22no—
175C2×C97—41no—
176C2×C97C2×C4×D411yescharacteristic
177C2×C97C22×M1611yescharacteristic
178C2×C97#56811yescharacteristic
179D97—22no—
180D97—41no—
181D4×M16—971nomaximal, Sylow
182C97D4×M1611yescharacteristic, complemented, Sylow
183C22×M16—971no—
184C22×M16—971no—
185C2×C4×D4—971no—
186#436—972no—
187#435—972no—
188#435—972no—
189C8×D4—972no—
190#423—971no—
191#407—971no—
192#407D9711yescharacteristic, complemented
193C4×M16—971no—
194C22×D4—971no—
195C23×C4—971no—
196C23×C4C2×D9711yescharacteristic
197C2×M16—971no—
198C2×M16—971no—
199C2×M16—972no—
200C2×M16—972no—
201C2×M16—1944no—
202C2×M16—1944no—
203C2×M16C2×D9711yescharacteristic
204C2×M16C2×D9711yescharacteristic
205C22×C8—972no—
206C22×C8—972no—
207C4×D4—971no—
208C4×D4—971no—
209C4×D4—971no—
210C4×D4—971no—
211C4×D4—971no—
212C4×D4—971no—
213C4×D4—971no—
214C4×D4—971no—
215C2×C4⋊C4—971no—
216C2×C22⋊C4—971no—
217C2×C22⋊C4—971no—
218C2×C42—971no—
219C4⋊C8—972no—
220C4⋊C8—972no—
221C22⋊C8—972no—
222C22⋊C8—972no—
223C22⋊C8C2×D9712yescomplemented
224C22⋊C8C2×D9712yescomplemented
225#97—972no—
226C4×C8—972no—
227C24—971no—
228C24C4×D9711yescharacteristic
229C2×D4—971no—
230C2×D4—971no—
231C2×D4—971no—
232C2×D4—971no—
233C2×D4—1944no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—971no—
237C22×C4—971no—
238C22×C4—971no—
239C22×C4—971no—
240C22×C4—971no—
241C22×C4—971no—
242C22×C4—971no—
243C22×C4—1942no—
244C22×C4—1942no—
245C22×C4C4×D9711yescharacteristic
246C22×C4C4×D9711yescharacteristic
247C22×C4C4×D9711yescharacteristic
248C22×C4C22×D9711yescharacteristic
249C22×C4C22×D9711yescharacteristic
250C22×C4C22×D9711yescharacteristic
251M16—22no—
252M16—22no—
253M16—974no—
254M16—1944no—
255M16—1944no—
256M16—1942no—
257C2×C8—972no—
258C2×C8—972no—
259C2×C8—1944no—
260C2×C8—1944no—
261C2×C8C22×D9712yes—
262C2×C8C22×D9712yes—
263C4⋊C4—971no—
264C4⋊C4—971no—
265C4⋊C4—971no—
266C4⋊C4—971no—
267C22⋊C4—971no—
268C22⋊C4—971no—
269C22⋊C4—971no—
270C22⋊C4—971no—
271C22⋊C4—971no—
272C22⋊C4—971no—
273C22⋊C4—971no—
274C22⋊C4—971no—
275C42—971no—
276C42—971no—
277C42—971no—
278C42—971no—
279C23—22no—
280C23—22no—
281C23—41no—
282C23—971no—
283C23—971no—
284C23—3881no—
285C23—1942no—
286C23—1942no—
287C23C2×C4×D9711yescharacteristic
288C23C2×C4×D9711yescharacteristic
289C23C2×C4×D9711yescharacteristic
290D4—974no—
291D4—1942no—
292D4—1942no—
293D4—1942no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—971no—
302C2×C4—971no—
303C2×C4—971no—
304C2×C4—971no—
305C2×C4—971no—
306C2×C4—971no—
307C2×C4—3881no—
308C2×C4—1942no—
309C2×C4—1942no—
310C2×C4—1942no—
311C2×C4—1942no—
312C2×C4—1941no—
313C2×C4—1941no—
314C2×C4—1941no—
315C2×C4—1941no—
316C2×C4C2×C4×D9711yescharacteristic
317C2×C4C2×C4×D9711yescharacteristic
318C2×C4C2×C4×D9711yescharacteristic
319C2×C4D4×D9712yes—
320C2×C4#1309551012yes—
321C2×C4C23×D9711yescharacteristic
322C8—22no—
323C8—22no—
324C8—974no—
325C8—1942no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—44no—
334C22—972no—
335C22—3884no—
336C22—1941no—
337C22—1941no—
338C22—1941no—
339C22—1941no—
340C22—1941no—
341C22#1569077012yes—
342C22M16×D9712yescomplemented
343C22C22×C4×D9711yescharacteristic
344C4—21no—
345C4—21no—
346C4—972no—
347C4—1941no—
348C4—1941no—
349C4—1941no—
350C4—1941no—
351C4—1941no—
352C4#1569077012yes—
353C4C2×D4×D9711yescharacteristic
354C4C2×#1309551011yescharacteristic
355C2—22no—
356C2—22no—
357C2—41no—
358C2—3881no—
359C2—1942no—
360C2C2×#1569077011yescharacteristic
361C2C2×M16×D9711yescharacteristic
362C2#2026150111yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D97
commutator, socleC22×C97C23×C4
Fitting#407×C97C2
prime radicalC22×C22⋊D97C4
cosocleC2×C4×C97C24
hypercenter#407D97
hypocenterC97D4×M16
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C22×#13095495C2semidirect
2. C2×M16×D97C2semidirect
4. #20260810C2semidirect
5. #20260809C2semidirect
6. #20260803C2semidirect
7. #20260798C2semidirect
8. #20260758C2semidirect
9. #20260698C2semidirect
10. M16×Dic97C2semidirect
11. #407×C97C2semidirect
17. C2×#13095495C22semidirect
18. C2×#13095494C22semidirect
20. C22×#11715895C22semidirect
21. C2×C8×D97C22semidirect
34. #15690739C22semidirect
35. #15690739C22semidirect
36. #15690722C22semidirect
37. #15690722C22semidirect
38. #15690720C22semidirect
39. C8×Dic97C22semidirect
43. C22⋊C8×C97C22semidirect
44. C22⋊C8×C97C22semidirect
62. #13095495D4semidirect
124. C22⋊D97M16semidirect
143. #11715895C2×D4semidirect
153. C22×C97C2×M16semidirect
161. C2×D97C2×M16semidirect
172. Dic97C2×M16semidirect
182. C97D4×M16semidirect
192. #407D97semidirect
223. C22⋊C8C2×D97semidirect
224. C22⋊C8C2×D97semidirect
342. C22M16×D97semidirect

Maximal subgroups

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

subgroupindex
C22×#130954952
C2×M16×D972
C2×C4×C22⋊D972
#202608102
#202608092
#202608032
#202607982
#202607582
#202606982
M16×Dic972
#407×C972
D4×M1697