Finite Group Databasebeta

← #27155823

Subgroups — #27155823

Order 12416.

992 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261461C211yesmaximal, characteristic, complemented
2#20261460C213yesmaximal, complemented
3#20261456C213yesmaximal, complemented
4#20261427C213yesmaximal, complemented
5#20261425C213yesmaximal, complemented
6Q8×C22⋊D97C211yesmaximal, characteristic, complemented
7Q8×C22⋊D97C211yesmaximal, characteristic, complemented
8#20261346C213yesmaximal, complemented
9#20261345C213yesmaximal, complemented
10#20261287C213yesmaximal, complemented
11#20261265C213yesmaximal
12#20261213C211yesmaximal, characteristic, complemented
13#20261211C211yesmaximal, characteristic, complemented
14C2×Q8×Dic97C211yesmaximal, characteristic, complemented
15D4×Q8×C97C211yesmaximal, characteristic, complemented
16C2×Q8×D97C2211yescharacteristic
17#15690809C2213yes—
18#15690808C2213yes—
19#15690808C2213yes—
20#15690805C2216yes—
21#15690805C2213yes—
22#15690803C2216yes—
23#15690803C2213yes—
24#15690802C2213yes—
25#15690801C2216yes—
26#15690801C2213yes—
27#15690800C2216yes—
28#15690800C2213yes—
29#15690800C2216yes—
30#15690796C2213yes—
31#15690794C2213yes—
32#15690791C2213yes—
33#15690789C2213yes—
34#15690789C2213yes—
35#15690787C2213yes—
36#15690786C2213yes—
37#15690786C2213yes—
38#15690782C2211yescharacteristic
39Q8×Dic97C2211yescharacteristic
40Q8×Dic97C2211yescharacteristic
41Q8×Dic97C2213yes—
42Q8×Dic97C2213yes—
43Q8×Dic97C2211yescharacteristic
44Q8×Dic97C2213yes—
45Q8×Dic97C2213yes—
46C4×Dic194C2213yes—
47C4×Dic194C2213yes—
48#15690775C2216yes—
49#15690775C2213yes—
50C4×C22⋊D97C2213yes—
51C4×C22⋊D97C2213yes—
52#15690769C2213yes—
53#15690769C2213yes—
54#15690768C2213yes—
55#15690767C2213yes—
56#15690766C2216yes—
57C2×C4⋊Dic97C2213yes—
58C2×C4×Dic97C2213yes—
59C22×Q8×C97C2211yescharacteristic
60C22×Q8×C97C2211yescharacteristic
61C4⋊Q8×C97C2213yes—
62C22⋊Q8×C97C2213yes—
63C22⋊Q8×C97C2213yes—
64C4×Q8×C97C2211yescharacteristic
65C4×D4×C97C2213yes—
66C2×Dic194C2313yes—
67C2×Dic194C2313yes—
68Q8×D97—24no—
69C2×C22⋊D97C2311yescharacteristic
70C2×C22⋊D97C2311yescharacteristic
71C22×Dic97C2311yescharacteristic
72C22×Dic97C2313yes—
73C2×C4×D97C2313yes—
74C4⋊Dic97C2313yes—
75C4⋊Dic97C2313yes—
76C4⋊Dic97C2316yes—
77C4⋊Dic97C2313yes—
78C4⋊Dic97C2313yes—
79C4⋊Dic97C2313yes—
80C4⋊Dic97C2313yes—
81#13095490C2316yes—
82#13095490C2316yes—
83#13095490C2313yes—
84#13095490C2313yes—
85#13095490C2313yes—
86#13095490C2313yes—
87#13095490C2316yes—
88#13095490C2311yescharacteristic
89#13095490C2313yes—
90C22⋊Dic97C2313yes—
91C22⋊Dic97C2311yescharacteristic
92C22⋊Dic97C2313yes—
93C22⋊Dic97C2313yes—
94#13095488C2311yescharacteristic
95#13095488C2313yes—
96#13095488C2313yes—
97#13095488C2313yes—
98C4×Dic97C2313yes—
99C4×Dic97C2313yes—
100C4×Dic97C2313yes—
101C4×Dic97C2313yes—
102C4×Dic97C2316yes—
103C4×Dic97C2316yes—
104C4×Dic97C2313yes—
105C2×Q8×C97—24no—
106C2×Q8×C97—24no—
107C2×Q8×C97C2313yes—
108C2×Q8×C97C2313yes—
109C2×Q8×C97C2311yescharacteristic
110C2×D4×C97C2311yescharacteristic
111C22×C4×C97C2313yes—
112C22×C4×C97C2313yes—
113C4⋊C4×C97C2313yes—
114C4⋊C4×C97C2316yes—
115C4⋊C4×C97C2313yes—
116C22⋊C4×C97C2313yes—
117C22⋊C4×C97C2313yes—
118C42×C97C2313yes—
119C23×C97C2411yescharacteristic
120C23×C97C2411yescharacteristic
121C22×D97C2411yescharacteristic
122Q8×C97—26no—
123Q8×C97—26no—
124Q8×C97#4114yes—
125D4×C97—22no—
126C2×C4×C97—26no—
127C2×C4×C97—26no—
128C2×C4×C97C2411yescharacteristic
129C2×C4×C97C2413yes—
130C2×C4×C97C2413yes—
131C2×C4×C97C2413yes—
132C2×C4×C97C2413yes—
133C22⋊D97—22no—
134C22⋊D97—22no—
135C2×Dic97—26no—
136C2×Dic97—22no—
137C2×Dic97C2411yescharacteristic
138C2×Dic97C2413yes—
139C2×Dic97C2413yes—
140C2×Dic97C2411yescharacteristic
141C2×Dic97C2413yes—
142C2×Dic97C2413yes—
143C2×Dic97C2411yescharacteristic
144C4×D97—26no—
145Dic194—26no—
146Dic194—26no—
147C22×C97—21no—
148C22×C97—21no—
149C22×C97—21no—
150C22×C97—21no—
151C22×C97—21no—
152C22×C972−1+412yes—
153C22×C97C2511yescharacteristic
154C2×D97—21no—
155C2×D97—21no—
156C2×D97—21no—
157C4×C97—23no—
158C4×C97—23no—
159C4×C97—23no—
160C4×C97—21no—
161C4×C97C2×#4116yes—
162Dic97—23no—
163Dic97—23no—
164Dic97—21no—
165Dic97—23no—
166Dic97—23no—
167Dic97—21no—
168Dic972−1+412yes—
169C2×C97—22no—
170C2×C97—41no—
171C2×C97C22×#4111yescharacteristic
172C2×C97C2×2−1+411yescharacteristic
173C2×C97C2×2−1+411yescharacteristic
174D97—41no—
175#3523—971nomaximal, Sylow
176C97#352311yescharacteristic, complemented, Sylow
177#557—973no—
178#557—971no—
179#556—973no—
180#552—973no—
181#552—973no—
182D4×Q8—971no—
183D4×Q8—971no—
184D4×Q8D9711yescharacteristic, complemented
185#541—973no—
186#541—973no—
187#533—973no—
188#529—973no—
189#519—971no—
190#519—971no—
191C2×C4×Q8—971no—
192C22×Q8—971no—
193C22×Q8C2×D9711yescharacteristic
194C22×Q8C2×D9711yescharacteristic
195C4⋊Q8—973no—
196C4⋊Q8—973no—
197C4⋊Q8—973no—
198C4⋊Q8C2×D9713yes—
199#126—973no—
200#126—976no—
201#126—973no—
202#126—976no—
203#125—973no—
204#125—976no—
205#125—976no—
206#125—976no—
207#125—973no—
208#125—973no—
209#124—973no—
210#123—973no—
211#123—973no—
212C22⋊Q8—973no—
213C22⋊Q8—973no—
214C22⋊Q8—973no—
215C22⋊Q8—973no—
216C22⋊Q8—973no—
217C22⋊Q8C2×D9713yes—
218C22⋊Q8C2×D9713yes—
219#121—971no—
220C4×Q8—971no—
221C4×Q8—973no—
222C4×Q8—971no—
223C4×Q8—973no—
224C4×Q8—973no—
225C4×Q8—971no—
226C4×Q8—973no—
227C4×Q8—973no—
228C4×Q8—976no—
229C4×Q8—973no—
230C4×Q8—973no—
231C4×Q8C2×D9711yescharacteristic
232C4×D4—973no—
233C4×D4—973no—
234C4×D4C2×D9713yes—
235#117—973no—
236#117—973no—
237#117—973no—
238#117—976no—
239#117—973no—
240C2×C4⋊C4—973no—
241C2×C42—973no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—973no—
245C2×Q8—973no—
246C2×Q8—1944no—
247C2×Q8C22×D9713yes—
248C2×Q8C22×D9713yes—
249C2×Q8C22×D9711yescharacteristic
250C2×D4—971no—
251C2×D4—971no—
252C2×D4C22×D9711yescharacteristic
253C22×C4—971no—
254C22×C4—973no—
255C22×C4—973no—
256C22×C4C22×D9713yes—
257C22×C4C22×D9713yes—
258C4⋊C4—973no—
259C4⋊C4—973no—
260C4⋊C4—976no—
261C4⋊C4—976no—
262C4⋊C4—973no—
263C4⋊C4—973no—
264C4⋊C4—971no—
265C4⋊C4—976no—
266C4⋊C4—973no—
267C4⋊C4—973no—
268C4⋊C4—973no—
269C4⋊C4—976no—
270C4⋊C4—973no—
271C4⋊C4—973no—
272C4⋊C4—973no—
273C4⋊C4—973no—
274C4⋊C4C22×D9713yes—
275C4⋊C4C22×D9713yes—
276C4⋊C4C22×D9716yes—
277C22⋊C4—971no—
278C22⋊C4—973no—
279C22⋊C4—973no—
280C22⋊C4—973no—
281C22⋊C4—971no—
282C22⋊C4—973no—
283C22⋊C4—973no—
284C22⋊C4—973no—
285C22⋊C4C22×D9713yes—
286C22⋊C4C22×D9713yes—
287C42—973no—
288C42—973no—
289C42—973no—
290C42—973no—
291C42—976no—
292C42—976no—
293C42—973no—
294C42C22×D9713yes—
295C23—971no—
296C23C23×D9711yescharacteristic
297C23C23×D9711yescharacteristic
298Q8—26no—
299Q8—26no—
300Q8—1946no—
301Q8—1946no—
302Q8#1309551014yes—
303D4—22no—
304D4—1942no—
305D4—1942no—
306C2×C4—26no—
307C2×C4—26no—
308C2×C4—971no—
309C2×C4—971no—
310C2×C4—973no—
311C2×C4—973no—
312C2×C4—971no—
313C2×C4—973no—
314C2×C4—973no—
315C2×C4—1942no—
316C2×C4—1946no—
317C2×C4—1946no—
318C2×C4C23×D9713yes—
319C2×C4C23×D9713yes—
320C2×C4C23×D9713yes—
321C2×C4C23×D9711yescharacteristic
322C2×C4C23×D9713yes—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—1941no—
329C22—1941no—
330C22—1941no—
331C22#1569085812yes—
332C22C24×D9711yescharacteristic
333C4—23no—
334C4—23no—
335C4—23no—
336C4—21no—
337C4—972no—
338C4—1943no—
339C4—1941no—
340C4—1943no—
341C4—1943no—
342C4—1943no—
343C4—1941no—
344C4C2×#1309551016yes—
345C2—22no—
346C2—41no—
347C2—3881no—
348C2C22×#1309551011yescharacteristic
349C22−1+4×D9711yescharacteristic
350C2C2×#1569085811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D97
commutator, socle, cosocleC22×C97C25
FittingD4×Q8×C97C2
prime radical#15690782C22
hypercenterD4×Q8D97
hypocenterC97#3523
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. #20261461C2semidirect
2. #20261460C2semidirect
3. #20261456C2semidirect
4. #20261427C2semidirect
5. #20261425C2semidirect
6. Q8×C22⋊D97C2semidirect
7. Q8×C22⋊D97C2semidirect
8. #20261346C2semidirect
9. #20261345C2semidirect
10. #20261287C2semidirect
12. #20261213C2semidirect
13. #20261211C2semidirect
14. C2×Q8×Dic97C2semidirect
15. D4×Q8×C97C2semidirect
176. C97#3523semidirect
184. D4×Q8D97semidirect

Maximal subgroups

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

subgroupindex
#202614612
#202614602
#202614562
#202614272
#202614252
Q8×C22⋊D972
#202613462
#202613452
#202612872
#202612652
#202612132
#202612112
C2×Q8×Dic972
D4×Q8×C972
#352397