Finite Group Databasebeta

← #27158223

Subgroups — #27158223

Order 12416.

556 conjugacy classes of subgroups.

The subgroup lattice has 360 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#20261686C212yesmaximal, complemented
6#20261685C212yesmaximal
7#20261685C212yesmaximal
8#20261684C212yesmaximal, complemented
9#20261677C211yesmaximal, characteristic
10C4⋊C4×#87604C211yesmaximal, characteristic, complemented
11C22⋊Q8×D97C211yesmaximal, characteristic
12Q8×#87604—24no—
13#15690889—24no—
14C2×C4⋊#87604C2211yescharacteristic
15C2×C4⋊#87604C2211yescharacteristic
16C2×C4⋊#87604C2211yescharacteristic
17C2×C4⋊#87604C2212yes—
18C2×C4⋊#87604C2211yescharacteristic
19C2×C4⋊#87604C2212yes—
20C2×C22⋊#87604C2211yescharacteristic
21C2×C22⋊#87604C2211yescharacteristic
22C2×C22⋊#87604C2212yes—
23C2×C4×#87604C2212yes—
24C2×C4×#87604C2211yescharacteristic
25C2×C4×#87604C2212yes—
26C2×C4×#87604C2211yescharacteristic
27(C2×#87604)⋊C4C2212yes—
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×D97C2211yescharacteristic
40C4⋊C4×D97C2212yes—
41C22⋊C4×D97C2212yes—
42C22⋊Q8×C97C411yescharacteristic, complemented
43C22×#87604C2311yescharacteristic
44C22×#87604C2312yes—
45C22×#87604C2311yescharacteristic
46C22×#87604C2312yes—
47C22×#87604C2311yescharacteristic
48C22×#87604C2311yescharacteristic
49C4⋊#87604—22no—
50C4⋊#87604—22no—
51C4⋊#87604—24no—
52C4⋊#87604—22no—
53C4⋊#87604—22no—
54C4⋊#87604—42no—
55C22⋊#87604—22no—
56C22⋊#87604—22no—
57C22⋊#87604—42no—
58C4×#87604—24no—
59C4×#87604—22no—
60C4×#87604—42no—
61C4×#87604D414yes—
62C23×D97C2311yescharacteristic
63C2×Dic194C2×C412yes—
64C2×Dic194C2×C411yescharacteristic
65Q8×D97—22no—
66Q8×D97—22no—
67C22×Dic97C2×C411yescharacteristic
68C2×C4×D97—22no—
69C2×C4×D97—22no—
70C2×C4×D97C2×C411yescharacteristic
71C2×C4×D97C2×C411yescharacteristic
72C2×C4×D97C2312yes—
73C2×C4×D97C2312yes—
74C2×C4×D97C2311yescharacteristic
75C2×C4×D97C2311yescharacteristic
76C4⋊Dic97C2×C411yescharacteristic
77C4⋊Dic97C2×C412yes—
78#13095490C2×C412yes—
79#13095490C2×C412yes—
80#13095490C2×C412yes—
81C22⋊Dic97C2×C412yes—
82#13095488C2×C412yes—
83#13095488C2×C412yes—
84C2×Q8×C97C2×C411yescharacteristic
85C22×C4×C97C2×C411yescharacteristic
86C4⋊C4×C97C2×C412yes—
87C4⋊C4×C97C2×C411yescharacteristic
88C22⋊C4×C97C2×C412yes—
89C23×C97C22×C411yescharacteristic
90C22×D97—21no—
91C22×D97—21no—
92C22×D97—21no—
93C22×D97—21no—
94C22×D97—21no—
95C22×D97—21no—
96C22×D97C22×C411yescharacteristic
97C22×D97C2411yescharacteristic
98C2×#87604—21no—
99C2×#87604—21no—
100C2×#87604—21no—
101C2×#87604—21no—
102C2×#87604—21no—
103C2×#87604—21no—
104C2×#87604—22no—
105C2×#87604—22no—
106C2×#87604—22no—
107C2×#87604—22no—
108C2×#87604—21no—
109C2×#87604—21no—
110C2×#87604—21no—
111C2×#87604—21no—
112C2×#87604—22no—
113C2×#87604—22no—
114C2×#87604C2×D412yes—
115C2×#87604C2×D412yes—
116Q8×C97—22no—
117C2×C4×C97—22no—
118C2×C4×C97C22×C412yes—
119C2×C4×C97C22×C412yes—
120C2×C4×C97C22×C411yescharacteristic
121C2×C4×C97C22×C411yescharacteristic
122C2×Dic97—22no—
123C2×Dic97C22×C412yes—
124C2×Dic97C22×C412yes—
125C2×Dic97C22×C411yescharacteristic
126C2×Dic97C22×C411yescharacteristic
127C4×D97—21no—
128C4×D97—22no—
129C4×D97—21no—
130C4×D97—22no—
131C4×D97—22no—
132C4×D97—22no—
133C4×D97—22no—
134C4×D97—22no—
135C4×D97C2×D412yes—
136C4×D97#4112yes—
137Dic194—22no—
138Dic194—24no—
139C22×C97—21no—
140C22×C97—21no—
141C22×C97—21no—
142C22×C97C23×C411yescharacteristic
143C2×D97—21no—
144C2×D97—21no—
145C2×D97—21no—
146C2×D97—41no—
147C2×D97—41no—
148C2×D97—41no—
149C2×D97—41no—
150C2×D97C22×D411yescharacteristic
151C2×D97C2×#4111yescharacteristic
152C2×D97H(5,𝔽2)11yescharacteristic
153C2×D972−1+411yescharacteristic
154C2×D972−1+411yescharacteristic
155C2×D972−1+411yescharacteristic
156#87604—22no—
157#87604—22no—
158#87604—41no—
159#87604—41no—
160#87604—42no—
161#87604—42no—
162C4×C97—22no—
163C4×C97—21no—
164C4×C97—22no—
165C4×C97C4×D412yes—
166Dic97—22no—
167Dic97—21no—
168Dic97—22no—
169Dic97C4×D412yes—
170C2×C97—41no—
171C2×C97C2×C4×D411yescharacteristic
172C2×C97#51911yescharacteristic
173C2×C97#52011yescharacteristic
174D97—41no—
175D97#53611yescharacteristic
176D97#53711yescharacteristic
177D97#54111yescharacteristic
178D97#54211yescharacteristic
179#2323—971nomaximal, Sylow
180C97#232311yescharacteristic, complemented, Sylow
181C2×C22⋊Q8—971no—
182C2×C4×Q8—971no—
183C2×#117—971no—
184#391—971no—
185#389—972no—
186#389—971no—
187#388—972no—
188#388—972no—
189#387—972no—
190#380—971no—
191C4×C4⋊C4—971no—
192C22×Q8—971no—
193C23×C4—971no—
194C22⋊Q8—971no—
195C22⋊Q8—971no—
196C22⋊Q8—972no—
197C22⋊Q8—971no—
198C22⋊Q8—972no—
199C22⋊Q8#8760411yescharacteristic, complemented
200C4×Q8—1944no—
201#117—1944no—
202C2×C4⋊C4—971no—
203C2×C4⋊C4—972no—
204C2×C4⋊C4—972no—
205C2×C4⋊C4—971no—
206C2×C4⋊C4—971no—
207C2×C4⋊C4—971no—
208C2×C4⋊C4—972no—
209C2×C4⋊C4—971no—
210C2×C22⋊C4—972no—
211C2×C22⋊C4—972no—
212C2×C22⋊C4—971no—
213C2×C22⋊C4—971no—
214C2×C42—971no—
215C2×C42—972no—
216C2×C42—971no—
217C2×C42—972no—
218#95—972no—
219#95—972no—
220#95—972no—
221#95—972no—
222#95—972no—
223C24—971no—
224C2×Q8—972no—
225C2×Q8—971no—
226C2×Q8—1942no—
227C2×Q8—1942no—
228C2×Q8C2×#8760411yescharacteristic
229C22×C4—971no—
230C22×C4—971no—
231C22×C4—971no—
232C22×C4—971no—
233C22×C4—971no—
234C22×C4—971no—
235C22×C4—972no—
236C22×C4—972no—
237C22×C4—971no—
238C22×C4—972no—
239C22×C4—971no—
240C22×C4—971no—
241C22×C4—972no—
242C22×C4—1942no—
243C22×C4—1942no—
244C22×C4C2×#8760411yescharacteristic
245C4⋊C4—972no—
246C4⋊C4—972no—
247C4⋊C4—972no—
248C4⋊C4—971no—
249C4⋊C4—972no—
250C4⋊C4—3882no—
251C4⋊C4—1942no—
252C4⋊C4—1942no—
253C4⋊C4—1942no—
254C4⋊C4—1942no—
255C4⋊C4—1944no—
256C4⋊C4C2×#8760411yescharacteristic
257C4⋊C4C2×#8760412yes—
258C22⋊C4—972no—
259C22⋊C4—972no—
260C22⋊C4—972no—
261C22⋊C4—3882no—
262C22⋊C4—1942no—
263C22⋊C4—1942no—
264C22⋊C4C2×#8760412yes—
265C42—974no—
266C42—3882no—
267C42—1942no—
268C42—1944no—
269C23—971no—
270C23—971no—
271C23—1941no—
272C23—1941no—
273C23—1941no—
274C23—1941no—
275C23—1941no—
276C23—1941no—
277C23C22×#8760411yescharacteristic
278Q8—22no—
279Q8—1944no—
280Q8—1942no—
281C2×C4—22no—
282C2×C4—971no—
283C2×C4—972no—
284C2×C4—972no—
285C2×C4—972no—
286C2×C4—972no—
287C2×C4—972no—
288C2×C4—972no—
289C2×C4—971no—
290C2×C4—1942no—
291C2×C4—1942no—
292C2×C4—1942no—
293C2×C4—1941no—
294C2×C4—1941no—
295C2×C4—1942no—
296C2×C4—1942no—
297C2×C4—1942no—
298C2×C4—1942no—
299C2×C4—1941no—
300C2×C4—1941no—
301C2×C4—1942no—
302C2×C4—1942no—
303C2×C4—1941no—
304C2×C4—1942no—
305C2×C4—1942no—
306C2×C4—1941no—
307C2×C4—1941no—
308C2×C4—1942no—
309C2×C4—1941no—
310C2×C4—1941no—
311C2×C4—1941no—
312C2×C4—1941no—
313C2×C4—1941no—
314C2×C4—1942no—
315C2×C4C22×#8760412yes—
316C2×C4C22×#8760412yes—
317C2×C4C22×#8760411yescharacteristic
318C2×C4C22×#8760411yescharacteristic
319C22—21no—
320C22—21no—
321C22—21no—
322C22—971no—
323C22—971no—
324C22—971no—
325C22—971no—
326C22—971no—
327C22—971no—
328C22—3881no—
329C22—3881no—
330C22—3881no—
331C22—3881no—
332C22—1941no—
333C22—1941no—
334C22—1941no—
335C22C23×#8760411yescharacteristic
336C4—21no—
337C4—22no—
338C4—22no—
339C4—972no—
340C4—3881no—
341C4—3882no—
342C4—3882no—
343C4—3881no—
344C4—1942no—
345C4—1942no—
346C4—1941no—
347C4—1942no—
348C4—1942no—
349C4D4×#8760412yes—
350C2—41no—
351C2—971no—
352C2—971no—
353C2—971no—
354C2—971no—
355C2—3881no—
356C2C2×D4×#8760411yescharacteristic
357C2#2026180311yescharacteristic
358C2#2026180411yescharacteristic
subgroupG/H
center, FrattiniC22C23×#87604
commutator, socleC22×C97C23×C4
FittingC22⋊Q8×C97C4
prime radicalC23×D97C23
cosocleC22×D97C24
hypercenterC22⋊Q8#87604
hypocenterC97#2323
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
5. #20261686C2semidirect
8. #20261684C2semidirect
10. C4⋊C4×#87604C2semidirect
34. #15690790C4semidirect
35. #15690789C4semidirect
36. #15690788C4semidirect
37. #15690787C4semidirect
38. #15690786C4semidirect
42. C22⋊Q8×C97C4semidirect
180. C97#2323semidirect
199. 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
#202616772
C4⋊C4×#876042
C22⋊Q8×D972
#232397