Finite Group Databasebeta

← #27158262

Subgroups — #27158262

Order 12416.

576 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×#87604C211yesmaximal, characteristic, complemented
2C2×#15690889C211yesmaximal, characteristic
3(C2×#87604)⋊Q8C211yesmaximal, characteristic, complemented
4#20261686C211yesmaximal, characteristic, complemented
5#20261685C212yesmaximal
6#20261685C212yesmaximal
7#20261684C212yesmaximal, complemented
8C4⋊C4×#87604C212yesmaximal, complemented
9C4⋊C4×#87604C211yesmaximal, characteristic, complemented
10C4×C22⋊#87604C211yesmaximal, characteristic
11C22⋊Q8×D97C211yesmaximal, characteristic
12Q8×#87604—24no—
13#15690889—24no—
14C2×C4⋊#87604C2212yes—
15C2×C4⋊#87604C2212yes—
16C2×C4⋊#87604C2211yescharacteristic
17C2×C4⋊#87604C2211yescharacteristic
18C2×C22⋊#87604C2211yescharacteristic
19C2×C22⋊#87604C2211yescharacteristic
20C2×C22⋊#87604C2212yes—
21C2×C4×#87604C2211yescharacteristic
22C2×C4×#87604C2211yescharacteristic
23C2×C4×#87604C2212yes—
24C2×C4×#87604C2212yes—
25C2×C4×#87604C2212yes—
26(C2×#87604)⋊C4C2211yescharacteristic
27(C2×#87604)⋊C4C2211yescharacteristic
28(C2×#87604)⋊C4C2212yes—
29(C2×#87604)⋊C4C2212yes—
30(C2×#87604)⋊C4C2212yes—
31(C2×#87604)⋊C4C2212yes—
32C2×Q8×D97C2211yescharacteristic
33C22×C4×D97C2211yescharacteristic
34#15690790C411yescharacteristic, complemented
35#15690789C411yescharacteristic, complemented
36#15690788C412yescomplemented
37#15690787C412yescomplemented
38#15690786C411yescharacteristic, complemented
39C4⋊C4×D97C2212yes—
40C4⋊C4×D97C2211yescharacteristic
41C22⋊C4×D97C2212yes—
42C22⋊Q8×C97C411yescharacteristic, complemented
43C22×#87604C2311yescharacteristic
44C22×#87604C2312yes—
45C22×#87604C2311yescharacteristic
46C22×#87604C2312yes—
47C22×#87604C2311yescharacteristic
48C22×#87604C2311yescharacteristic
49C4⋊#87604—24no—
50C4⋊#87604—24no—
51C4⋊#87604—22no—
52C4⋊#87604D414yes—
53C22⋊#87604—24no—
54C22⋊#87604—22no—
55C22⋊#87604—22no—
56C4×#87604—24no—
57C4×#87604—24no—
58C4×#87604—24no—
59C4×#87604—22no—
60C4×#87604—22no—
61C23×D97C2311yescharacteristic
62C2×Dic194C2×C412yes—
63C2×Dic194C2×C411yescharacteristic
64Q8×D97—22no—
65Q8×D97—22no—
66C22×Dic97C2×C411yescharacteristic
67C2×C4×D97—22no—
68C2×C4×D97—22no—
69C2×C4×D97C2×C411yescharacteristic
70C2×C4×D97C2×C411yescharacteristic
71C2×C4×D97C2312yes—
72C2×C4×D97C2312yes—
73C2×C4×D97C2311yescharacteristic
74C2×C4×D97C2311yescharacteristic
75C4⋊Dic97C2×C411yescharacteristic
76C4⋊Dic97C2×C412yes—
77#13095490C2×C412yes—
78#13095490C2×C412yes—
79#13095490C2×C412yes—
80C22⋊Dic97C2×C412yes—
81#13095488C2×C412yes—
82#13095488C2×C412yes—
83C2×Q8×C97C2×C411yescharacteristic
84C22×C4×C97C2×C411yescharacteristic
85C4⋊C4×C97C2×C412yes—
86C4⋊C4×C97C2×C411yescharacteristic
87C22⋊C4×C97C2×C412yes—
88C23×C97C22×C411yescharacteristic
89C22×D97—21no—
90C22×D97—21no—
91C22×D97—21no—
92C22×D97—21no—
93C22×D97—21no—
94C22×D97—21no—
95C22×D97C22×C411yescharacteristic
96C22×D97C2411yescharacteristic
97C2×#87604—21no—
98C2×#87604—22no—
99C2×#87604—21no—
100C2×#87604—22no—
101C2×#87604—21no—
102C2×#87604—21no—
103C2×#87604—22no—
104C2×#87604—21no—
105C2×#87604—21no—
106C2×#87604—21no—
107C2×#87604—21no—
108C2×#87604—22no—
109C2×#87604—21no—
110C2×#87604—22no—
111C2×#87604—21no—
112C2×#87604C2×D414yes—
113C2×#87604#4112yes—
114C2×#87604#4112yes—
115Q8×C97—22no—
116C2×C4×C97—22no—
117C2×C4×C97C22×C412yes—
118C2×C4×C97C22×C412yes—
119C2×C4×C97C22×C411yescharacteristic
120C2×C4×C97C22×C411yescharacteristic
121C2×Dic97—22no—
122C2×Dic97C22×C412yes—
123C2×Dic97C22×C412yes—
124C2×Dic97C22×C411yescharacteristic
125C2×Dic97C22×C411yescharacteristic
126C4×D97—21no—
127C4×D97—22no—
128C4×D97—21no—
129C4×D97—22no—
130C4×D97—22no—
131C4×D97—22no—
132C4×D97—22no—
133C4×D97—22no—
134C4×D97C2×D412yes—
135C4×D97#4112yes—
136Dic194—22no—
137Dic194—24no—
138C22×C97—21no—
139C22×C97—21no—
140C22×C97—21no—
141C22×C97C23×C411yescharacteristic
142C2×D97—21no—
143C2×D97—21no—
144C2×D97—21no—
145C2×D97—41no—
146C2×D97—41no—
147C2×D97—41no—
148C2×D97—41no—
149C2×D97C22×D411yescharacteristic
150C2×D97C2×#4111yescharacteristic
151C2×D97C2×#4111yescharacteristic
152C2×D97C2×#4111yescharacteristic
153C2×D972−1+411yescharacteristic
154C2×D972−1+411yescharacteristic
155#87604—24no—
156#87604—22no—
157#87604—22no—
158#87604—41no—
159#87604—42no—
160#87604—41no—
161C4×C97—22no—
162C4×C97—21no—
163C4×C97—22no—
164C4×C97C4×D412yes—
165Dic97—22no—
166Dic97—21no—
167Dic97—22no—
168Dic97C4×D412yes—
169C2×C97—41no—
170C2×C97C2×C4×D411yescharacteristic
171C2×C97C4×#4111yescharacteristic
172C2×C97#51911yescharacteristic
173D97—41no—
174D97#54711yescharacteristic
175D97#54811yescharacteristic
176D97#55211yescharacteristic
177D97#55611yescharacteristic
178#2362—971nomaximal, Sylow
179C97#236211yescharacteristic, complemented, Sylow
180C2×C22⋊Q8—971no—
181C2×C4×Q8—971no—
182C2×#117—971no—
183#391—971no—
184#389—971no—
185#388—972no—
186#388—972no—
187#387—972no—
188C4×C4⋊C4—972no—
189C4×C4⋊C4—971no—
190C4×C22⋊C4—971no—
191C22×Q8—971no—
192C23×C4—971no—
193C22⋊Q8—971no—
194C22⋊Q8—971no—
195C22⋊Q8—972no—
196C22⋊Q8—971no—
197C22⋊Q8—972no—
198C22⋊Q8#8760411yescharacteristic, complemented
199C4×Q8—1944no—
200#117—1944no—
201C2×C4⋊C4—971no—
202C2×C4⋊C4—972no—
203C2×C4⋊C4—971no—
204C2×C4⋊C4—971no—
205C2×C4⋊C4—972no—
206C2×C4⋊C4—972no—
207C2×C22⋊C4—972no—
208C2×C22⋊C4—972no—
209C2×C22⋊C4—971no—
210C2×C22⋊C4—971no—
211C2×C42—972no—
212C2×C42—971no—
213C2×C42—972no—
214C2×C42—971no—
215C2×C42—972no—
216#95—971no—
217#95—971no—
218#95—972no—
219#95—972no—
220#95—972no—
221#95—972no—
222C24—971no—
223C2×Q8—972no—
224C2×Q8—971no—
225C2×Q8—1942no—
226C2×Q8—1942no—
227C2×Q8C2×#8760411yescharacteristic
228C22×C4—971no—
229C22×C4—971no—
230C22×C4—972no—
231C22×C4—971no—
232C22×C4—971no—
233C22×C4—971no—
234C22×C4—971no—
235C22×C4—971no—
236C22×C4—972no—
237C22×C4—972no—
238C22×C4—971no—
239C22×C4—972no—
240C22×C4—971no—
241C22×C4—1942no—
242C22×C4—1942no—
243C22×C4C2×#8760411yescharacteristic
244C4⋊C4—974no—
245C4⋊C4—972no—
246C4⋊C4—972no—
247C4⋊C4—972no—
248C4⋊C4—971no—
249C4⋊C4—972no—
250C4⋊C4—1944no—
251C4⋊C4—1942no—
252C4⋊C4—1944no—
253C4⋊C4C2×#8760411yescharacteristic
254C4⋊C4C2×#8760412yes—
255C22⋊C4—972no—
256C22⋊C4—972no—
257C22⋊C4—972no—
258C22⋊C4—1942no—
259C22⋊C4—1942no—
260C22⋊C4—1944no—
261C22⋊C4C2×#8760412yes—
262C42—1942no—
263C42—1944no—
264C42—1944no—
265C42—1942no—
266C42—1944no—
267C23—971no—
268C23—971no—
269C23—1941no—
270C23—1941no—
271C23—1941no—
272C23—1941no—
273C23—1941no—
274C23—1941no—
275C23C22×#8760411yescharacteristic
276Q8—22no—
277Q8—1944no—
278Q8—1942no—
279C2×C4—22no—
280C2×C4—971no—
281C2×C4—972no—
282C2×C4—972no—
283C2×C4—974no—
284C2×C4—972no—
285C2×C4—972no—
286C2×C4—972no—
287C2×C4—972no—
288C2×C4—971no—
289C2×C4—1942no—
290C2×C4—1941no—
291C2×C4—1942no—
292C2×C4—1942no—
293C2×C4—1942no—
294C2×C4—1941no—
295C2×C4—1942no—
296C2×C4—1942no—
297C2×C4—1942no—
298C2×C4—1941no—
299C2×C4—1941no—
300C2×C4—1941no—
301C2×C4—1941no—
302C2×C4—1942no—
303C2×C4—1941no—
304C2×C4—1942no—
305C2×C4—1941no—
306C2×C4—1941no—
307C2×C4—1941no—
308C2×C4—1942no—
309C2×C4—1941no—
310C2×C4—1941no—
311C2×C4—1942no—
312C2×C4—1942no—
313C2×C4C22×#8760412yes—
314C2×C4C22×#8760412yes—
315C2×C4C22×#8760411yescharacteristic
316C2×C4C22×#8760411yescharacteristic
317C22—21no—
318C22—21no—
319C22—21no—
320C22—971no—
321C22—971no—
322C22—971no—
323C22—971no—
324C22—971no—
325C22—971no—
326C22—3881no—
327C22—3881no—
328C22—3881no—
329C22—3881no—
330C22—1941no—
331C22—1941no—
332C22—1941no—
333C22C23×#8760411yescharacteristic
334C4—21no—
335C4—22no—
336C4—22no—
337C4—972no—
338C4—3882no—
339C4—3881no—
340C4—3881no—
341C4—1942no—
342C4—1944no—
343C4—1942no—
344C4—1941no—
345C4—1942no—
346C4—1942no—
347C4D4×#8760412yes—
348C2—41no—
349C2—971no—
350C2—971no—
351C2—971no—
352C2—971no—
353C2—3881no—
354C2C2×D4×#8760411yescharacteristic
355C2#41×#8760411yescharacteristic
356C2#2026180311yescharacteristic
subgroupG/H
center, FrattiniC22C23×#87604
commutator, socleC22×C97C23×C4
FittingC22⋊Q8×C97C4
prime radicalC23×D97C23
cosocleC22×D97C24
hypercenterC22⋊Q8#87604
hypocenterC97#2362
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×Q8×#87604C2semidirect
3. (C2×#87604)⋊Q8C2semidirect
4. #20261686C2semidirect
7. #20261684C2semidirect
8. C4⋊C4×#87604C2semidirect
9. C4⋊C4×#87604C2semidirect
34. #15690790C4semidirect
35. #15690789C4semidirect
36. #15690788C4semidirect
37. #15690787C4semidirect
38. #15690786C4semidirect
42. C22⋊Q8×C97C4semidirect
179. C97#2362semidirect
198. C22⋊Q8#87604semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×#876042
C2×#156908892
(C2×#87604)⋊Q82
#202616862
#202616852
#202616842
C4⋊C4×#876042
C4×C22⋊#876042
C22⋊Q8×D972
#236297