Finite Group Databasebeta

← #27152030

Subgroups — #27152030

Order 12416.

358 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261345C211yesmaximal, characteristic
2#20261213C211yesmaximal, characteristic
3#20261077C211yesmaximal, characteristic
4#20261073C211yesmaximal, characteristic
5#20261044C211yesmaximal, characteristic, complemented
6#20261040C211yesmaximal, characteristic, complemented
7#20260943C211yesmaximal, characteristic, complemented
8#20260943C211yesmaximal, characteristic, complemented
9#20260891C211yesmaximal, characteristic
10#20260842C211yesmaximal, characteristic, complemented
11#20260839C211yesmaximal, characteristic, complemented
12Q16×Dic97C211yesmaximal, characteristic, complemented
13#20260827C211yesmaximal, characteristic, complemented
14#20260793C211yesmaximal, characteristic
15#451×C97C211yesmaximal, characteristic
16C2×#13095501C2211yescharacteristic
17C2×#13095501C2211yescharacteristic
18#15690808C2211yescharacteristic
19#15690803—21no—
20#15690803—21no—
21#15690801—21no—
22#15690801C2211yescharacteristic
23#15690800—21no—
24#15690789C2211yescharacteristic
25Q8×Dic97—21no—
26Q8×Dic97—21no—
27Q8×Dic97C2211yescharacteristic
28Q8×Dic97C2211yescharacteristic
29Q8×Dic97C2211yescharacteristic
30Q8×Dic97C2211yescharacteristic
31Q8×Dic97C2211yescharacteristic
32#15690775C2211yescharacteristic
33#15690769—21no—
34#15690769—21no—
35#15690769C2211yescharacteristic
36#15690769C2211yescharacteristic
37#15690744C2211yescharacteristic
38#15690743C2211yescharacteristic
39#15690742C2211yescharacteristic
40#15690741C2211yescharacteristic
41#15690739C2211yescharacteristic
42#15690739C2211yescharacteristic
43#15690735C2211yescharacteristic
44#15690735C2211yescharacteristic
45#15690734C2211yescharacteristic
46#15690734C2211yescharacteristic
47#15690734C2211yescharacteristic
48#15690734C2211yescharacteristic
49C22⋊#11715895C2211yescharacteristic
50#15690720C2211yescharacteristic
51C8×Dic97C2211yescharacteristic
52C22×Q8×C97C2211yescharacteristic
53C2×Q16×C97C2211yescharacteristic
54C2×Q16×C97C2211yescharacteristic
55C22⋊Q8×C97C2211yescharacteristic
56#103×C97C2211yescharacteristic
57#103×C97C2211yescharacteristic
58C22⋊C8×C97C2211yescharacteristic
59C2×Dic194C2311yescharacteristic
60#13095501—22no—
61#13095501—22no—
62C2×#11715895C2311yescharacteristic
63C2×#11715895C2311yescharacteristic
64C4⋊Dic97—21no—
65C4⋊Dic97—21no—
66C4⋊Dic97—21no—
67C4⋊Dic97—21no—
68C4⋊Dic97—21no—
69C4⋊Dic97—21no—
70C4⋊Dic97D411yescharacteristic
71C4⋊Dic97D411yescharacteristic
72C4⋊Dic97C2311yescharacteristic
73C4⋊Dic97C2311yescharacteristic
74C4⋊Dic97C2311yescharacteristic
75#13095490—21no—
76#13095490—21no—
77#13095490—21no—
78#13095490—21no—
79#13095490—21no—
80C22⋊Dic97—21no—
81C22⋊Dic97—21no—
82C22⋊Dic97D411yescharacteristic
83C22⋊Dic97D411yescharacteristic
84C4×Dic97—21no—
85C4×Dic97—21no—
86C4×Dic97—21no—
87C4×Dic97—21no—
88C4×Dic97—21no—
89C4×Dic97—21no—
90C4×Dic97—21no—
91C4×Dic97C2311yescharacteristic
92C4×Dic97C2311yescharacteristic
93C2×Q8×C97—22no—
94C2×Q8×C97—22no—
95C2×Q8×C97—21no—
96C2×Q8×C97C2311yescharacteristic
97C2×Q8×C97C2311yescharacteristic
98C2×Q8×C97C2311yescharacteristic
99C22×C4×C97—21no—
100C22×C4×C97C2311yescharacteristic
101Q16×C97—22no—
102Q16×C97—22no—
103C2×C8×C97C2311yescharacteristic
104C2×C8×C97C2311yescharacteristic
105C4⋊C4×C97—21no—
106C4⋊C4×C97C2311yescharacteristic
107C22⋊C4×C97—21no—
108C23×C97C2×D411yescharacteristic
109Q8×C97—24no—
110Q8×C97—21no—
111Q8×C97—21no—
112Q8×C97—21no—
113Q8×C97—21no—
114Q8×C97#4112yes—
115Q8×C97#4112yes—
116C2×C4×C97—24no—
117C2×C4×C97—21no—
118C2×C4×C97—21no—
119C2×C4×C97—21no—
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97C2×D411yescharacteristic
124C2×C4×C97C2411yescharacteristic
125C2×Dic97—21no—
126C2×Dic97—21no—
127C2×Dic97—21no—
128C2×Dic97—21no—
129C2×Dic97C2×D411yescharacteristic
130C2×Dic97C2×D411yescharacteristic
131C2×Dic97C2×D411yescharacteristic
132C2×Dic97C2×D411yescharacteristic
133Dic194—21no—
134Dic194—21no—
135C8×C97—21no—
136C8×C97—21no—
137#11715895—21no—
138#11715895—21no—
139C22×C97—21no—
140C22×C97—21no—
141C22×C97—21no—
142C22×C97C22×D411yescharacteristic
143C4×C97—22no—
144C4×C97—22no—
145C4×C97—21no—
146C4×C97—41no—
147C4×C97—41no—
148C4×C97C2×#4111yescharacteristic
149C4×C97C2×#4111yescharacteristic
150Dic97—21no—
151Dic97—21no—
152Dic97—21no—
153Dic97—21no—
154Dic97—41no—
155Dic97—41no—
156Dic97—41no—
157Dic97—41no—
158C2×C97—41no—
159C2×C97#52511yescharacteristic
160C2×C97#57811yescharacteristic
161C2×C97#57811yescharacteristic
162#3127—971nomaximal, Sylow
163C97#312711yescharacteristic, complemented, Sylow
164#541—971no—
165#519—971no—
166#485—971no—
167#485—971no—
168#479—971no—
169#479—971no—
170#462—971no—
171#462—971no—
172#451—971no—
173#451D9711yescharacteristic
174#441—971no—
175#441—971no—
176C4×Q16—971no—
177C4×Q16—971no—
178#433—971no—
179C22×Q8C2×D9711yescharacteristic
180C2×Q16—971no—
181C2×Q16—971no—
182C2×Q16C2×D9711yescharacteristic
183C2×Q16C2×D9711yescharacteristic
184C4⋊Q8—971no—
185#126—1941no—
186#126—1941no—
187#125—971no—
188#125—1941no—
189#125—1941no—
190C22⋊Q8—971no—
191C22⋊Q8C2×D9711yescharacteristic
192C4×Q8—971no—
193C4×Q8—971no—
194C4×Q8—971no—
195C4×Q8—971no—
196C4×Q8—971no—
197C4×Q8—971no—
198C4×Q8—1941no—
199C4×Q8—1941no—
200#117—971no—
201#117—971no—
202#117—1941no—
203#117—1941no—
204#107—971no—
205#107—971no—
206#106—971no—
207#106—971no—
208C4⋊C8—971no—
209C4⋊C8—971no—
210#103—971no—
211#103—971no—
212#103—971no—
213#103—971no—
214#103—971no—
215#103—971no—
216#103C2×D9711yescharacteristic
217#103C2×D9711yescharacteristic
218C22⋊C8—971no—
219C22⋊C8C2×D9711yescharacteristic
220#97—971no—
221C4×C8—971no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—21no—
225C2×Q8—971no—
226C2×Q8C22×D9711yescharacteristic
227C2×Q8C22×D9711yescharacteristic
228C2×Q8C22×D9711yescharacteristic
229C22×C4—21no—
230C22×C4C22×D9711yescharacteristic
231Q16—22no—
232Q16—22no—
233Q16—1942no—
234Q16—1942no—
235C2×C8—971no—
236C2×C8—971no—
237C2×C8C22×D9711yescharacteristic
238C2×C8C22×D9711yescharacteristic
239C4⋊C4—21no—
240C4⋊C4—971no—
241C4⋊C4—971no—
242C4⋊C4—971no—
243C4⋊C4—971no—
244C4⋊C4—971no—
245C4⋊C4—1941no—
246C4⋊C4—1941no—
247C4⋊C4—1941no—
248C4⋊C4—1941no—
249C4⋊C4—1941no—
250C4⋊C4—1941no—
251C4⋊C4—1941no—
252C4⋊C4—1941no—
253C4⋊C4—1941no—
254C4⋊C4—1941no—
255C4⋊C4—1941no—
256C4⋊C4C22×D9711yescharacteristic
257C22⋊C4—21no—
258C22⋊C4—971no—
259C22⋊C4—971no—
260C22⋊C4—1941no—
261C22⋊C4—1941no—
262C42—971no—
263C42—971no—
264C42—1941no—
265C42—1941no—
266C42—1941no—
267C42—1941no—
268C42—1941no—
269C42—1941no—
270C42—1941no—
271C23D4×D9711yescharacteristic
272Q8—24no—
273Q8—21no—
274Q8—21no—
275Q8—21no—
276Q8—21no—
277Q8—1941no—
278Q8—1941no—
279Q8#1309551012yes—
280Q8#1309551012yes—
281C2×C4—24no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—971no—
289C2×C4—971no—
290C2×C4—971no—
291C2×C4—971no—
292C2×C4—1941no—
293C2×C4—1941no—
294C2×C4—1941no—
295C2×C4—1941no—
296C2×C4D4×D9711yescharacteristic
297C2×C4C23×D9711yescharacteristic
298C8—21no—
299C8—21no—
300C8—1941no—
301C8—1941no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22C2×D4×D9711yescharacteristic
306C4—22no—
307C4—22no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—3881no—
312C4—3881no—
313C4—3881no—
314C4—3881no—
315C4—1941no—
316C4—1941no—
317C4—1941no—
318C4—1941no—
319C4C2×#1309551011yescharacteristic
320C4C2×#1309551011yescharacteristic
321C2—41no—
322C2#2026124111yescharacteristic
323C2#2026154811yescharacteristic
324C2#2026155111yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#451×C97C2
socleC22×C97C22×D4
prime radicalC23×C97C2×D4
hypercenter#451D97
hypocenterC97#3127
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
5. #20261044C2semidirect
6. #20261040C2semidirect
7. #20260943C2semidirect
8. #20260943C2semidirect
10. #20260842C2semidirect
11. #20260839C2semidirect
12. Q16×Dic97C2semidirect
13. #20260827C2semidirect
163. C97#3127semidirect

Maximal subgroups

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

subgroupindex
#202613452
#202612132
#202610772
#202610732
#202610442
#202610402
#202609432
#202608912
#202608422
#202608392
Q16×Dic972
#202608272
#202607932
#451×C972
#312797