Finite Group Databasebeta

← #27144752

Subgroups — #27144752

Order 12416.

788 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690784C211yesmaximal, characteristic
2C2×#15690772C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D97C212yesmaximal, complemented
4#20260636C211yesmaximal, characteristic, complemented
5#20260631C211yesmaximal, characteristic, complemented
6#20260625C212yesmaximal, complemented
7#20260614C212yesmaximal, complemented
8#20260606C212yesmaximal, complemented
9C4⋊C4×Dic97C211yesmaximal, characteristic, complemented
10C4×C22⋊Dic97C211yesmaximal, characteristic, complemented
11C22×C4⋊C4×C97C211yesmaximal, characteristic, complemented
12C22×D388C2211yescharacteristic
13C22×C22⋊D97C2212yes—
14C22×C4×D97C2212yes—
15#15690784C418yescomplemented
16#15690772—24no—
17C4×C22⋊D97—28no—
18C2×C4⋊Dic97C2211yescharacteristic
19C2×#13095490C2212yes—
20C2×C22⋊Dic97C2211yescharacteristic
21C2×C22⋊Dic97C2211yescharacteristic
22C2×#13095488C2212yes—
23C2×#13095488C2212yes—
24C2×#13095488C2212yes—
25C2×#13095488C2212yes—
26C2×C4×Dic97C2212yes—
27C2×C4×Dic97C2211yescharacteristic
28C2×C4×Dic97C2211yescharacteristic
29#15690717C2212yes—
30#15690717C2212yes—
31#15690717C2211yescharacteristic
32#15690717C2211yescharacteristic
33C23×C4×C97C2212yes—
34C23×C4×C97C2211yescharacteristic
35C2×C4⋊C4×C97—24no—
36C2×C4⋊C4×C97C2212yes—
37C2×C4⋊C4×C97C2211yescharacteristic
38C2×C4⋊C4×C97C2211yescharacteristic
39C23×D97C2312yes—
40C2×D388—22no—
41C2×D388—22no—
42C2×D388C2×C414yescomplemented
43C2×C22⋊D97—24no—
44C2×C22⋊D97—24no—
45C2×C22⋊D97C2×C418yescomplemented
46C22×Dic97C2311yescharacteristic
47C22×Dic97C2311yescharacteristic
48C22×Dic97C2312yes—
49C22×Dic97C2311yescharacteristic
50C22×Dic97C2311yescharacteristic
51C2×C4×D97—24no—
52C2×C4×D97—24no—
53C2×C4×D97—42no—
54C4⋊Dic97C2×C414yescomplemented
55#13095490—24no—
56C22⋊Dic97—22no—
57C22⋊Dic97D414yescomplemented
58#13095488—24no—
59#13095488—24no—
60#13095488—42no—
61#13095488C2×C418yescomplemented
62C4×Dic97—22no—
63C4×Dic97—22no—
64C4×Dic97—24no—
65C24×C97C2311yescharacteristic
66C22×C4×C97—22no—
67C22×C4×C97—22no—
68C22×C4×C97—24no—
69C22×C4×C97—24no—
70C22×C4×C97—42no—
71C22×C4×C97C2×C414yescomplemented
72C22×C4×C97C2312yes—
73C22×C4×C97C2311yescharacteristic
74C22×C4×C97C2312yes—
75C22×C4×C97C2311yescharacteristic
76C4⋊C4×C97—24no—
77C4⋊C4×C97—22no—
78C4⋊C4×C97D414yescomplemented
79C23×C97—21no—
80C23×C97—21no—
81C23×C97—21no—
82C23×C97—21no—
83C23×C97C22×C412yes—
84C23×C97C2×D412yes—
85C23×C97#4112yes—
86C23×C97C2411yescharacteristic
87C22×D97—22no—
88C22×D97—22no—
89C22×D97—22no—
90C22×D97—22no—
91C22×D97—22no—
92C22×D97—22no—
93C22×D97C22×C414yes—
94C2×C4×C97—21no—
95C2×C4×C97—22no—
96C2×C4×C97—21no—
97C2×C4×C97—22no—
98C2×C4×C97—28no—
99C2×C4×C97—22no—
100C2×C4×C97—22no—
101C2×C4×C97—22no—
102C2×C4×C97—48no—
103C2×C4×C97C22×C412yes—
104C2×C4×C97C22×C412yes—
105C2×C4×C97C2×D414yes—
106C2×C4×C97C2×D412yes—
107C2×C4×C97#4112yes—
108C22⋊D97—216no—
109C2×Dic97—22no—
110C2×Dic97—21no—
111C2×Dic97—21no—
112C2×Dic97—22no—
113C2×Dic97—21no—
114C2×Dic97—21no—
115C2×Dic97—21no—
116C2×Dic97—21no—
117C2×Dic97—21no—
118C2×Dic97—21no—
119C2×Dic97—21no—
120C2×Dic97—21no—
121C2×Dic97C22×C414yes—
122C2×Dic97C2×D412yes—
123C2×Dic97C2×D412yes—
124D388—28no—
125C4×D97—48no—
126C22×C97—22no—
127C22×C97—22no—
128C22×C97—22no—
129C22×C97—22no—
130C22×C97—22no—
131C22×C97—22no—
132C22×C97C4×D414yescomplemented
133C22×C97C23×C411yescharacteristic
134C22×C97C22×D411yescharacteristic
135C22×C97C22×D411yescharacteristic
136C22×C97C2×#4111yescharacteristic
137C22×C97C2×#4111yescharacteristic
138C22×C97H(5,𝔽2)11yescharacteristic
139C22×C972−1+411yescharacteristic
140C2×D97—24no—
141C2×D97—24no—
142C2×D97—24no—
143C2×D97—42no—
144C2×D97—42no—
145C2×D97—42no—
146C2×D97—42no—
147C4×C97—22no—
148C4×C97—24no—
149C4×C97—42no—
150C4×C97C4×D414yescomplemented
151Dic97—24no—
152Dic97—22no—
153Dic97—22no—
154Dic97—41no—
155Dic97—41no—
156C2×C97—24no—
157C2×C97C2×C4×D411yescharacteristic
158C2×C97C2×C4×D411yescharacteristic
159C2×C97#52011yescharacteristic
160C2×C97D4211yescharacteristic
161C2×C97#54711yescharacteristic
162C2×C97#54811yescharacteristic
163C2×C97#55311yescharacteristic
164D97—44no—
165#2361—971nomaximal, Sylow
166C97#236111yescharacteristic, complemented, Sylow
167C2×#121—971no—
168C2×C4×D4—972no—
169C2×C4×D4—971no—
170C22×C4⋊C4D9711yescharacteristic, complemented
171#390—971no—
172#389—971no—
173#388—972no—
174#386—972no—
175#385—972no—
176C4×C4⋊C4—971no—
177C4×C22⋊C4—971no—
178C22×D4—972no—
179C22×D4—971no—
180C23×C4—972no—
181C23×C4C2×D9712yes—
182C23×C4C2×D9711yescharacteristic
183#121—978no—
184C4×D4—1948no—
185C4×D4—1944no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—971no—
188C2×C4⋊C4—972no—
189C2×C4⋊C4C2×D9712yes—
190C2×C4⋊C4C2×D9711yescharacteristic
191C2×C4⋊C4C2×D9711yescharacteristic
192C2×C22⋊C4—971no—
193C2×C22⋊C4—972no—
194C2×C22⋊C4—972no—
195C2×C22⋊C4—971no—
196C2×C22⋊C4—972no—
197C2×C22⋊C4—972no—
198C2×C42—971no—
199C2×C42—972no—
200C2×C42—971no—
201#95—971no—
202#95—971no—
203#95—972no—
204#95—972no—
205C24—972no—
206C24C22×D9711yescharacteristic
207C2×D4—978no—
208C2×D4—974no—
209C2×D4—1944no—
210C2×D4—1944no—
211C2×D4—1942no—
212C2×D4—1942no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—42no—
218C22×C4—972no—
219C22×C4—971no—
220C22×C4—971no—
221C22×C4—971no—
222C22×C4—971no—
223C22×C4—3882no—
224C22×C4—1944no—
225C22×C4—1944no—
226C22×C4C4×D9714yescomplemented
227C22×C4C22×D9712yes—
228C22×C4C22×D9711yescharacteristic
229C22×C4C22×D9712yes—
230C22×C4C22×D9711yescharacteristic
231C4⋊C4—24no—
232C4⋊C4—22no—
233C4⋊C4—974no—
234C4⋊C4—1944no—
235C4⋊C4C22⋊D9714yescomplemented
236C22⋊C4—974no—
237C22⋊C4—978no—
238C22⋊C4—3882no—
239C22⋊C4—1944no—
240C22⋊C4—1942no—
241C22⋊C4—1944no—
242C42—1944no—
243C42—1942no—
244C42—1942no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—974no—
250C23—1942no—
251C23—1942no—
252C23—1942no—
253C23—1942no—
254C23—1942no—
255C23—1942no—
256C23C2×C4×D9712yes—
257C23D4×D9712yes—
258C23#1309551212yes—
259C23C23×D9711yescharacteristic
260D4—19416no—
261D4—1948no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—28no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—48no—
271C2×C4—974no—
272C2×C4—972no—
273C2×C4—972no—
274C2×C4—3888no—
275C2×C4—1941no—
276C2×C4—1941no—
277C2×C4—1941no—
278C2×C4—1941no—
279C2×C4—1941no—
280C2×C4—1941no—
281C2×C4—1942no—
282C2×C4—1941no—
283C2×C4—1941no—
284C2×C4—1942no—
285C2×C4—1941no—
286C2×C4—1941no—
287C2×C4C2×C4×D9712yes—
288C2×C4C2×C4×D9712yes—
289C2×C4C2×C22⋊D9714yes—
290C2×C4C2×C22⋊D9712yes—
291C2×C4#1309551112yes—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—3882no—
299C22—3882no—
300C22—3882no—
301C22—3882no—
302C22—1944no—
303C22—1944no—
304C22—1944no—
305C22#1569077214yescomplemented
306C22C22×C4×D9711yescharacteristic
307C22C2×D4×D9711yescharacteristic
308C22C22×C22⋊D9711yescharacteristic
309C22C2×#1309551111yescharacteristic
310C22C2×#1309551211yescharacteristic
311C22#1569085511yescharacteristic
312C22#1569085811yescharacteristic
313C4—22no—
314C4—24no—
315C4—42no—
316C4—3881no—
317C4—3881no—
318C4—1942no—
319C4—1944no—
320C4—1942no—
321C4C4×C22⋊D9714yescomplemented
322C2—24no—
323C2—3884no—
324C2C2×C4×C22⋊D9711yescharacteristic
325C2C2×#1569077211yescharacteristic
326C2#2026121611yescharacteristic
327C2D4×C22⋊D9711yescharacteristic
328C2#2026138611yescharacteristic
329C2#2026139611yescharacteristic
330C2#2026143311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
FittingC22×C4⋊C4×C97C2
socle, cosocleC23×C97C24
prime radicalC2×#15690784C2
hypercenterC22×C4⋊C4D97
hypocenterC97#2361
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#15690772C2semidirect
3. C2×C4×C22⋊D97C2semidirect
4. #20260636C2semidirect
5. #20260631C2semidirect
6. #20260625C2semidirect
7. #20260614C2semidirect
8. #20260606C2semidirect
9. C4⋊C4×Dic97C2semidirect
10. C4×C22⋊Dic97C2semidirect
11. C22×C4⋊C4×C97C2semidirect
15. #15690784C4semidirect
42. C2×D388C2×C4semidirect
45. C2×C22⋊D97C2×C4semidirect
54. C4⋊Dic97C2×C4semidirect
57. C22⋊Dic97D4semidirect
61. #13095488C2×C4semidirect
71. C22×C4×C97C2×C4semidirect
78. C4⋊C4×C97D4semidirect
132. C22×C97C4×D4semidirect
150. C4×C97C4×D4semidirect
166. C97#2361semidirect
170. C22×C4⋊C4D97semidirect
226. C22×C4C4×D97semidirect
235. C4⋊C4C22⋊D97semidirect
305. C22#15690772semidirect
321. C4C4×C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907842
C2×#156907722
C2×C4×C22⋊D972
#202606362
#202606312
#202606252
#202606142
#202606062
C4⋊C4×Dic972
C4×C22⋊Dic972
C22×C4⋊C4×C972
#236197