Finite Group Databasebeta

← #27145046

Subgroups — #27145046

Order 12416.

1564 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 1564 conjugacy classes of subgroups — too many to draw. Only its 213 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊D97C211yesmaximal, characteristic, complemented
2C2×#15690779C212yesmaximal, complemented
3C2×#15690778C212yesmaximal, complemented
4#20260643C214yesmaximal, complemented
5#20260615C212yesmaximal, complemented
6#20260614C212yesmaximal, complemented
7#20260591C211yesmaximal, characteristic, complemented
8C23×D4×C97C211yesmaximal, characteristic, complemented
9C24×D97C2211yescharacteristic, complemented
10C22×C22⋊D97—28no—
11C22×C22⋊D97—24no—
12C22×C22⋊D97C2212yes—
13C22×C22⋊D97C2212yescomplemented
14C22×C22⋊D97C2214yescomplemented
15C23×Dic97C2211yescharacteristic, complemented
16#15690779—28no—
17#15690778—28no—
18C2×C22⋊Dic97C2212yescomplemented
19C2×C22⋊Dic97C2214yescomplemented
20C2×C22⋊Dic97C2212yescomplemented
21C2×C22⋊Dic97C2212yescomplemented
22C2×#13095488C2214yescomplemented
23#15690717C2214yescomplemented
24C25×C97C2211yescharacteristic
25C25×C97C2211yescharacteristic, complemented
26C22×D4×C97—24no—
27C22×D4×C97—28no—
28C22×D4×C97C2212yes—
29C22×D4×C97C2212yescomplemented
30C23×C4×C97C2211yescharacteristic, complemented
31C23×D97—22no—
32C23×D97—24no—
33C23×D97—24no—
34C23×D97—42no—
35C23×D97C2312yes—
36C2×C22⋊D97—232no—
37C2×C22⋊D97—24no—
38C2×C22⋊D97—24no—
39C2×C22⋊D97—24no—
40C2×C22⋊D97—24no—
41C2×C22⋊D97—28no—
42C2×C22⋊D97—28no—
43C2×C22⋊D97—24no—
44C2×C22⋊D97—44no—
45C2×C22⋊D97D418yescomplemented
46C22×Dic97—22no—
47C22×Dic97—24no—
48C22×Dic97C2314yes—
49C22×Dic97C2312yes—
50C22⋊Dic97—24no—
51C22⋊Dic97—24no—
52C22⋊Dic97—28no—
53C22⋊Dic97—42no—
54#13095488—28no—
55C24×C97—24no—
56C24×C97—24no—
57C24×C97—22no—
58C24×C97—24no—
59C24×C97—24no—
60C24×C97—42no—
61C24×C97—42no—
62C24×C97D414yescomplemented
63C24×C97C2311yescharacteristic
64C24×C97C2312yes—
65C24×C97C2311yescharacteristic
66C24×C97C2311yescharacteristic
67C2×D4×C97—232no—
68C2×D4×C97—24no—
69C2×D4×C97—24no—
70C2×D4×C97—24no—
71C2×D4×C97—24no—
72C2×D4×C97—24no—
73C2×D4×C97D418yescomplemented
74C22×C4×C97—22no—
75C22×C4×C97—24no—
76C22×C4×C97C2312yes—
77C23×C97—22no—
78C23×C97—22no—
79C23×C97—22no—
80C23×C97—22no—
81C23×C97—28no—
82C23×C97—28no—
83C23×C97—28no—
84C23×C97—22no—
85C23×C97—22no—
86C23×C97—22no—
87C23×C97—24no—
88C23×C97—22no—
89C23×C97—22no—
90C23×C97—24no—
91C23×C97—44no—
92C23×C97—416no—
93C23×C97—416no—
94C23×C97—44no—
95C23×C97—44no—
96C23×C97C2×D414yes—
97C23×C97C2×D412yes—
98C23×C97C2×D412yes—
99C23×C97C2×D412yes—
100C23×C97C2×D414yes—
101C23×C97C2×D414yescomplemented
102C23×C97C2411yescharacteristic
103C22×D97—28no—
104C22×D97—24no—
105C22×D97—22no—
106C22×D97—22no—
107C22×D97—24no—
108C22×D97—416no—
109C22×D97—44no—
110C22×D97—44no—
111C22×D97C2×D414yescomplemented
112D4×C97—216no—
113D4×C97—216no—
114C2×C4×C97—28no—
115C2×C4×C97—22no—
116C2×C4×C97—22no—
117C2×C4×C97C2×D414yescomplemented
118C22⋊D97—216no—
119C22⋊D97—216no—
120C22⋊D97—416no—
121C2×Dic97—28no—
122C2×Dic97—22no—
123C2×Dic97—22no—
124C2×Dic97—24no—
125C2×Dic97—24no—
126C2×Dic97—24no—
127C2×Dic97C2×D414yescomplemented
128C22×C97—22no—
129C22×C97—22no—
130C22×C97—22no—
131C22×C97—22no—
132C22×C97—28no—
133C22×C97—28no—
134C22×C97—22no—
135C22×C97—22no—
136C22×C97—24no—
137C22×C97—22no—
138C22×C97—28no—
139C22×C97—416no—
140C22×C97—416no—
141C22×C97—42no—
142C22×C97—42no—
143C22×C97—44no—
144C22×C97—44no—
145C22×C97#12014yescomplemented
146C22×C97#12014yescomplemented
147C22×C97C22×D412yes—
148C22×C97C22×D411yescharacteristic
149C22×C97C22×D412yes—
150C22×C97H(5,𝔽2)12yes—
151C2×D97—24no—
152C2×D97—28no—
153C2×D97—416no—
154C2×D97—42no—
155C2×D97—42no—
156C2×D97—44no—
157C4×C97—24no—
158Dic97—24no—
159Dic97—44no—
160C2×C97—24no—
161C2×C97—24no—
162C2×C97—42no—
163C2×C97—44no—
164C2×C97C2×#12011yescharacteristic
165C2×C97C2×#12011yescharacteristic
166C2×C97#53411yescharacteristic
167C2×C97D4214yes—
168D97—44no—
169#2403—971nomaximal, Sylow
170C97#240311yescharacteristic, complemented, Sylow
171C23×D4—971no—
172C23×D4D9711yescharacteristic, complemented
173C2×#120—972no—
174C2×#120—972no—
175#392—974no—
176#386—972no—
177#386—972no—
178#379—971no—
179C25—971no—
180C25C2×D9711yescharacteristic
181C25C2×D9711yescharacteristic, complemented
182C22×D4—24no—
183C22×D4—28no—
184C22×D4—972no—
185C22×D4—974no—
186C22×D4—972no—
187C22×D4—1948no—
188C22×D4—1944no—
189C22×D4C2×D9712yes—
190C22×D4C2×D9712yescomplemented
191C23×C4—971no—
192C23×C4C2×D9711yescharacteristic, complemented
193#120—1948no—
194#120—1948no—
195C2×C22⋊C4—972no—
196C2×C22⋊C4—972no—
197C2×C22⋊C4—972no—
198C2×C22⋊C4—974no—
199C2×C22⋊C4—974no—
200#95—974no—
201C24—24no—
202C24—24no—
203C24—22no—
204C24—24no—
205C24—24no—
206C24—42no—
207C24—42no—
208C24—972no—
209C24—3882no—
210C24—1944no—
211C24—1944no—
212C24—1942no—
213C24C22⋊D9714yescomplemented
214C24C22×D9711yescharacteristic
215C24C22×D9712yes—
216C24C22×D9711yescharacteristic
217C24C22×D9711yescharacteristic
218C2×D4—232no—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—24no—
224C2×D4—978no—
225C2×D4—3884no—
226C2×D4—1944no—
227C2×D4—1948no—
228C2×D4—1948no—
229C2×D4—1944no—
230C2×D4—19432no—
231C2×D4—1944no—
232C2×D4—1944no—
233C2×D4—1944no—
234C2×D4C22⋊D9718yescomplemented
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—972no—
238C22×C4—974no—
239C22×C4—1944no—
240C22×C4—1942no—
241C22×C4C22×D9712yes—
242C22⋊C4—3882no—
243C22⋊C4—1948no—
244C22⋊C4—1944no—
245C22⋊C4—1944no—
246C22⋊C4—1948no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—28no—
252C23—28no—
253C23—28no—
254C23—24no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—24no—
261C23—44no—
262C23—416no—
263C23—416no—
264C23—44no—
265C23—44no—
266C23—974no—
267C23—3884no—
268C23—38816no—
269C23—3884no—
270C23—1948no—
271C23—1942no—
272C23—1944no—
273C23—1942no—
274C23—1944no—
275C23D4×D9714yes—
276C23C2×C22⋊D9712yes—
277C23C2×C22⋊D9712yes—
278C23C2×C22⋊D9714yes—
279C23C2×C22⋊D9712yes—
280C23C2×C22⋊D9714yescomplemented
281C23C23×D9711yescharacteristic
282D4—216no—
283D4—216no—
284D4—38816no—
285D4—19416no—
286D4—19416no—
287C2×C4—28no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—974no—
291C2×C4—1944no—
292C2×C4—1944no—
293C2×C4—1944no—
294C2×C4—1948no—
295C2×C4—1942no—
296C2×C4—1942no—
297C2×C4C2×C22⋊D9714yescomplemented
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—28no—
303C22—28no—
304C22—22no—
305C22—22no—
306C22—24no—
307C22—22no—
308C22—28no—
309C22—416no—
310C22—416no—
311C22—42no—
312C22—42no—
313C22—44no—
314C22—44no—
315C22—38816no—
316C22—3882no—
317C22—3882no—
318C22—3884no—
319C22—1944no—
320C22—1948no—
321C22#1569077814yescomplemented
322C22#1569077914yescomplemented
323C22C2×D4×D9712yes—
324C22C22×C22⋊D9711yescharacteristic
325C22C22×C22⋊D9712yes—
326C22#1569085512yes—
327C4—24no—
328C4—3884no—
329C4—1944no—
330C2—24no—
331C2—24no—
332C2—42no—
333C2—44no—
334C2—3884no—
335C2C2×#1569077811yescharacteristic
336C2C2×#1569077911yescharacteristic
337C2#2026129411yescharacteristic
338C2D4×C22⋊D9714yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
FittingC23×D4×C97C2
hypercenterC23×D4D97
hypocenterC97#2403
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D97C2semidirect
2. C2×#15690779C2semidirect
3. C2×#15690778C2semidirect
4. #20260643C2semidirect
5. #20260615C2semidirect
6. #20260614C2semidirect
7. #20260591C2semidirect
8. C23×D4×C97C2semidirect
9. C24×D97C22semidirect
13. C22×C22⋊D97C22semidirect
14. C22×C22⋊D97C22semidirect
15. C23×Dic97C22semidirect
18. C2×C22⋊Dic97C22semidirect
19. C2×C22⋊Dic97C22semidirect
20. C2×C22⋊Dic97C22semidirect
21. C2×C22⋊Dic97C22semidirect
22. C2×#13095488C22semidirect
23. #15690717C22semidirect
25. C25×C97C22semidirect
29. C22×D4×C97C22semidirect
30. C23×C4×C97C22semidirect
45. C2×C22⋊D97D4semidirect
62. C24×C97D4semidirect
73. C2×D4×C97D4semidirect
101. C23×C97C2×D4semidirect
111. C22×D97C2×D4semidirect
117. C2×C4×C97C2×D4semidirect
127. C2×Dic97C2×D4semidirect
145. C22×C97#120semidirect
146. C22×C97#120semidirect
170. C97#2403semidirect
172. C23×D4D97semidirect
181. C25C2×D97semidirect
190. C22×D4C2×D97semidirect
192. C23×C4C2×D97semidirect
213. C24C22⋊D97semidirect
234. C2×D4C22⋊D97semidirect
280. C23C2×C22⋊D97semidirect
297. C2×C4C2×C22⋊D97semidirect
321. C22#15690778semidirect
322. C22#15690779semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D972
C2×#156907792
C2×#156907782
#202606432
#202606152
#202606142
#202605912
C23×D4×C972
#240397