Finite Group Databasebeta

← #27150696

Subgroups — #27150696

Order 12416.

412 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13095510C211yesmaximal, characteristic
2#20260864C211yesmaximal, characteristic, complemented
3#20260863C211yesmaximal, characteristic, complemented
4#20260811C211yesmaximal, characteristic, complemented
5#20260810C211yesmaximal, characteristic, complemented
6#20260807C211yesmaximal, characteristic, complemented
7#20260806C211yesmaximal, characteristic, complemented
8#20260803C211yesmaximal, characteristic, complemented
9#20260787C211yesmaximal, characteristic, complemented
10#20260785C211yesmaximal, characteristic, complemented
11#20260785C211yesmaximal, characteristic, complemented
12#20260687C211yesmaximal, characteristic, complemented
13M16×Dic97C211yesmaximal, characteristic, complemented
14C2×C8×Dic97C211yesmaximal, characteristic, complemented
15#435×C97C211yesmaximal, characteristic, complemented
16C2×#13095510C411yescharacteristic
17C2×#13095495C2211yescharacteristic
18C2×#13095494C2211yescharacteristic
19C2×#13095494C2211yescharacteristic
20C22×#11715895C2211yescharacteristic
21C4×Dic194C2211yescharacteristic
22#15690775C411yescharacteristic
23D4×Dic97C411yescharacteristic
24C4×C22⋊D97C2211yescharacteristic
25C4×C22⋊D97C2211yescharacteristic
26#15690770C411yescharacteristic
27#15690770C411yescharacteristic
28#15690768C411yescharacteristic
29#15690766C411yescharacteristic
30#15690766C411yescharacteristic
31C2×C4×Dic97C2211yescharacteristic
32C2×C4×Dic97C2211yescharacteristic
33C42×D97C2211yescharacteristic
34C4⋊#11715895C2211yescharacteristic
35#15690739C2211yescharacteristic
36#15690739C2211yescharacteristic
37#15690739C2211yescharacteristic
38#15690739C2211yescharacteristic
39C22⋊#11715895C2211yescharacteristic
40C22⋊#11715895C2211yescharacteristic
41#15690722C2211yescharacteristic
42#15690722C2211yescharacteristic
43#15690721C2211yescharacteristic
44#15690720C2211yescharacteristic
45#15690720C2211yescharacteristic
46C8×Dic97C2211yescharacteristic
47C8×Dic97C2211yescharacteristic
48C8×Dic97C2211yescharacteristic
49C8×Dic97C2211yescharacteristic
50C8×Dic97C2211yescharacteristic
51C8×Dic97C2211yescharacteristic
52C2×M16×C97C2211yescharacteristic
53C22×C8×C97C2211yescharacteristic
54C4×D4×C97C2211yescharacteristic
55C4⋊C8×C97C2211yescharacteristic
56C22⋊C8×C97C2211yescharacteristic
57C22⋊C8×C97C2211yescharacteristic
58#97×C97C2211yescharacteristic
59#13095510—24no—
60C2×Dic194C2×C411yescharacteristic
61C2×C22⋊D97C2×C411yescharacteristic
62C2×C22⋊D97C2×C411yescharacteristic
63C22×Dic97C2×C411yescharacteristic
64C22×Dic97C2×C411yescharacteristic
65C22×Dic97C2×C411yescharacteristic
66C22×Dic97C2×C411yescharacteristic
67C2×C4×D97C2×C411yescharacteristic
68C2×C4×D97C2×C411yescharacteristic
69C2×C4×D97C2311yescharacteristic
70#13095495—22no—
71#13095494—22no—
72#13095494—22no—
73C2×#11715895—22no—
74C2×#11715895C2311yescharacteristic
75C2×#11715895C2311yescharacteristic
76C2×#11715895C2311yescharacteristic
77C2×#11715895C2311yescharacteristic
78C4⋊Dic97C2×C411yescharacteristic
79#13095490C2×C411yescharacteristic
80#13095490C2×C411yescharacteristic
81#13095490C2×C411yescharacteristic
82#13095490C2×C411yescharacteristic
83C22⋊Dic97C2×C411yescharacteristic
84C22⋊Dic97C2×C411yescharacteristic
85#13095488C2×C411yescharacteristic
86#13095488C2×C411yescharacteristic
87C4×Dic97C2×C411yescharacteristic
88C4×Dic97C2×C411yescharacteristic
89C4×Dic97C2×C411yescharacteristic
90C4×Dic97C2×C411yescharacteristic
91C4×Dic97C2×C411yescharacteristic
92C4×Dic97C2×C411yescharacteristic
93C4×Dic97C2311yescharacteristic
94C4×Dic97C2311yescharacteristic
95C4×Dic97C2311yescharacteristic
96C2×D4×C97C2×C411yescharacteristic
97C22×C4×C97C2311yescharacteristic
98C22×C4×C97C2311yescharacteristic
99M16×C97—22no—
100C2×C8×C97—22no—
101C2×C8×C97C2311yescharacteristic
102C2×C8×C97C2311yescharacteristic
103C2×C8×C97C2311yescharacteristic
104C2×C8×C97C2311yescharacteristic
105C4⋊C4×C97C2×C411yescharacteristic
106C22⋊C4×C97C2×C411yescharacteristic
107C22⋊C4×C97C2×C411yescharacteristic
108C42×C97C2311yescharacteristic
109C23×C97C22×C411yescharacteristic
110C23×C97C22×C411yescharacteristic
111C22×D97C22×C411yescharacteristic
112D4×C97—22no—
113C2×C4×C97—21no—
114C2×C4×C97—21no—
115C2×C4×C97—21no—
116C2×C4×C97—21no—
117C2×C4×C97C22×C411yescharacteristic
118C2×C4×C97C22×C411yescharacteristic
119C2×C4×C97C22×C411yescharacteristic
120C2×C4×C97C22×C411yescharacteristic
121C2×C4×C97C2411yescharacteristic
122C22⋊D97—22no—
123C22⋊D97—22no—
124C2×Dic97—22no—
125C2×Dic97—22no—
126C2×Dic97—22no—
127C2×Dic97M1614yescomplemented
128C2×Dic97C22×C411yescharacteristic
129C2×Dic97C22×C411yescharacteristic
130C2×Dic97C22×C411yescharacteristic
131C2×Dic97C22×C411yescharacteristic
132C2×Dic97C22×C411yescharacteristic
133C2×Dic97C22×C411yescharacteristic
134C2×Dic97C22×C411yescharacteristic
135C4×D97—21no—
136C4×D97—21no—
137C4×D97—22no—
138C4×D97—22no—
139Dic194—22no—
140C8×C97—21no—
141C8×C97—21no—
142C8×C97—21no—
143C8×C97#4112yescomplemented
144#11715895—21no—
145#11715895—21no—
146#11715895—21no—
147#11715895#4112yescomplemented
148C22×C97—21no—
149C22×C97—21no—
150C22×C97—21no—
151C22×C97—21no—
152C22×C97—21no—
153C22×C97C2×M1612yes—
154C22×C97C23×C411yescharacteristic
155C2×D97—21no—
156C2×D97—21no—
157C2×D97—21no—
158C4×C97—21no—
159C4×C97—21no—
160C4×C97—21no—
161C4×C97—21no—
162C4×C97C2×#4111yescharacteristic
163C4×C97C2×#4111yescharacteristic
164Dic97—21no—
165Dic97—21no—
166Dic97—21no—
167Dic97—21no—
168Dic97C2×M1612yes—
169Dic97C2×M1612yes—
170Dic97#13112yes—
171C2×C97—22no—
172C2×C97—41no—
173C2×C97C4×#4111yescharacteristic
174C2×C97C22×M1611yescharacteristic
175C2×C97C2×#13111yescharacteristic
176D97—41no—
177#2965—971nomaximal, Sylow
178C97#296511yescharacteristic, complemented, Sylow
179C4×#41—971no—
180#446—971no—
181#446—971no—
182#436—971no—
183#436—971no—
184#435—971no—
185#435—971no—
186#435—971no—
187#435D9711yescharacteristic, complemented
188#431—971no—
189#431—971no—
190#431—971no—
191C4×M16—971no—
192C4×M16—971no—
193C2×C4×C8—971no—
194C2×#41—971no—
195C2×M16—971no—
196C2×M16—971no—
197C2×M16—971no—
198C2×M16C2×D9711yescharacteristic
199C22×C8—971no—
200C22×C8C2×D9711yescharacteristic
201C4×Q8—971no—
202C4×Q8—971no—
203C4×D4—971no—
204C4×D4—971no—
205C4×D4—971no—
206C4×D4—971no—
207C4×D4—971no—
208C4×D4C2×D9711yescharacteristic
209#117—971no—
210#117—971no—
211#117—971no—
212C2×C42—971no—
213C2×C42—971no—
214C2×C42—971no—
215C4⋊C8—971no—
216C4⋊C8—971no—
217C4⋊C8—971no—
218C4⋊C8—971no—
219C4⋊C8—971no—
220C4⋊C8C2×D9711yescharacteristic
221C22⋊C8—971no—
222C22⋊C8—971no—
223C22⋊C8—971no—
224C22⋊C8—971no—
225C22⋊C8C2×D9711yescharacteristic
226C22⋊C8C2×D9711yescharacteristic
227#97—971no—
228#97—971no—
229#97—971no—
230#97C2×D9711yescharacteristic
231C4×C8—971no—
232C4×C8—971no—
233C4×C8—971no—
234C4×C8—971no—
235C4×C8—971no—
236C4×C8—971no—
237#41—1944no—
238C2×Q8—971no—
239C2×D4—971no—
240C2×D4—971no—
241C2×D4C4×D9711yescharacteristic
242C22×C4—971no—
243C22×C4—971no—
244C22×C4—971no—
245C22×C4—971no—
246C22×C4—971no—
247C22×C4—971no—
248C22×C4—971no—
249C22×C4C22×D9711yescharacteristic
250C22×C4C22×D9711yescharacteristic
251M16—22no—
252M16—1942no—
253M16—1942no—
254M16—1942no—
255C2×C8—22no—
256C2×C8—971no—
257C2×C8—971no—
258C2×C8—971no—
259C2×C8—971no—
260C2×C8—1942no—
261C2×C8C22×D9711yescharacteristic
262C2×C8C22×D9711yescharacteristic
263C2×C8C22×D9711yescharacteristic
264C2×C8C22×D9711yescharacteristic
265C4⋊C4—971no—
266C4⋊C4—971no—
267C4⋊C4—971no—
268C4⋊C4—971no—
269C4⋊C4—971no—
270C4⋊C4C4×D9711yescharacteristic
271C22⋊C4—971no—
272C22⋊C4—971no—
273C22⋊C4—971no—
274C22⋊C4—971no—
275C22⋊C4C4×D9711yescharacteristic
276C22⋊C4C4×D9711yescharacteristic
277C42—971no—
278C42—971no—
279C42—971no—
280C42—971no—
281C42—971no—
282C42—971no—
283C42—971no—
284C42—971no—
285C42—971no—
286C42C22×D9711yescharacteristic
287C23—971no—
288C23C2×C4×D9711yescharacteristic
289C23C2×C4×D9711yescharacteristic
290Q8—1942no—
291D4—22no—
292D4—1942no—
293D4—1942no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—974no—
299C2×C4—971no—
300C2×C4—971no—
301C2×C4—971no—
302C2×C4—971no—
303C2×C4—971no—
304C2×C4—971no—
305C2×C4—971no—
306C2×C4—1942no—
307C2×C4—1942no—
308C2×C4—1942no—
309C2×C4—1942no—
310C2×C4—1942no—
311C2×C4—1941no—
312C2×C4—1941no—
313C2×C4C2×C4×D9711yescharacteristic
314C2×C4C2×C4×D9711yescharacteristic
315C2×C4C2×C4×D9711yescharacteristic
316C2×C4C2×C4×D9711yescharacteristic
317C2×C4C23×D9711yescharacteristic
318C8—21no—
319C8—21no—
320C8—21no—
321C8—972no—
322C8—1941no—
323C8—1941no—
324C8—1941no—
325C8#1309551012yescomplemented
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—1941no—
332C22—1941no—
333C22—1941no—
334C22M16×D9712yes—
335C22C22×C4×D9711yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—972no—
341C4—972no—
342C4—972no—
343C4—1941no—
344C4—1941no—
345C4—1941no—
346C4—1941no—
347C4#41×D9711yescharacteristic
348C4C2×#1309551011yescharacteristic
349C2—22no—
350C2—41no—
351C2—3881no—
352C2C4×#1309551011yescharacteristic
353C2C2×M16×D9711yescharacteristic
354C2#131×D9711yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D97
commutator, socleC22×C97C23×C4
Fitting#435×C97C2
prime radicalC2×#13095510C4
cosocleC2×C4×C97C24
hypercenter#435D97
hypocenterC97#2965
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ #435

Complements

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

NH ≅ G/Ntype
2. #20260864C2semidirect
3. #20260863C2semidirect
4. #20260811C2semidirect
5. #20260810C2semidirect
6. #20260807C2semidirect
7. #20260806C2semidirect
8. #20260803C2semidirect
9. #20260787C2semidirect
10. #20260785C2semidirect
11. #20260785C2semidirect
12. #20260687C2semidirect
13. M16×Dic97C2semidirect
14. C2×C8×Dic97C2semidirect
15. #435×C97C2semidirect
127. C2×Dic97M16semidirect
143. C8×C97#41semidirect
147. #11715895#41semidirect
178. C97#2965semidirect
187. #435D97semidirect
325. C8#13095510semidirect

Maximal subgroups

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

subgroupindex
C4×#130955102
#202608642
#202608632
#202608112
#202608102
#202608072
#202608062
#202608032
#202607872
#202607852
#202606872
M16×Dic972
C2×C8×Dic972
#435×C972
#296597