Finite Group Databasebeta

← #451×D97 #27150927

Subgroups — #451×D97 #27150927

Order 12416.

740 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 155 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
1C22×Q8×D97C211yesmaximal, characteristic
2C2×Q16×D97C212yesmaximal, complemented
3C22⋊Q8×D97C211yesmaximal, characteristic
4#20260893C211yesmaximal, characteristic, complemented
5#20260892C212yesmaximal, complemented
6#20260891C211yesmaximal, characteristic, complemented
7#20260890C211yesmaximal, characteristic, complemented
8#20260889C212yesmaximal, complemented
9#103×D97C212yesmaximal, complemented
10C22⋊C8×D97C211yesmaximal, characteristic
11#451×C97C211yesmaximal, characteristic, complemented
12C22×Dic194—21no—
13C22×Dic194C2211yescharacteristic
14C2×Q8×D97—28no—
15C2×Q8×D97—21no—
16C2×Q8×D97—21no—
17C2×Q8×D97C2212yes—
18C2×Q8×D97C2211yescharacteristic
19C2×Q8×D97C2212yes—
20C22×C4×D97—21no—
21C22×C4×D97C2211yescharacteristic
22C2×Dic388C2212yescomplemented
23C2×#13095501C2212yescomplemented
24C2×#13095501C2212yescomplemented
25Q16×D97—28no—
26C2×C8×D97C2212yes—
27#15690790C2211yescharacteristic
28#15690789C2211yescharacteristic
29#15690788—21no—
30#15690787—21no—
31#15690786C2211yescharacteristic
32C4⋊C4×D97—21no—
33C4⋊C4×D97C2211yescharacteristic
34C22⋊C4×D97—21no—
35#15690735C2212yescomplemented
36#15690734C2212yescomplemented
37#15690733C2212yescomplemented
38C22⋊#11715895C2211yescharacteristic
39#15690722C2212yes—
40C22×Q8×C97C2211yescharacteristic
41C2×Q16×C97C2212yescomplemented
42C22⋊Q8×C97C2211yescharacteristic
43#103×C97C2212yescomplemented
44C22⋊C8×C97C2211yescharacteristic
45C23×D97D411yescharacteristic
46C2×Dic194—28no—
47C2×Dic194—21no—
48C2×Dic194—22no—
49C2×Dic194—22no—
50C2×Dic194—24no—
51C2×Dic194—21no—
52C2×Dic194C2311yescharacteristic
53C2×Dic194C2312yes—
54Q8×D97—28no—
55Q8×D97—28no—
56Q8×D97—24no—
57Q8×D97—24no—
58Q8×D97—24no—
59Q8×D97—24no—
60Q8×D97D418yes—
61C22×Dic97—21no—
62C22×Dic97D411yescharacteristic
63C2×C4×D97—28no—
64C2×C4×D97—22no—
65C2×C4×D97—22no—
66C2×C4×D97—24no—
67C2×C4×D97—21no—
68C2×C4×D97—21no—
69C2×C4×D97D411yescharacteristic
70C2×C4×D97D411yescharacteristic
71C2×C4×D97C2311yescharacteristic
72C2×C4×D97C2311yescharacteristic
73Dic388—24no—
74#13095501—24no—
75#13095501—24no—
76C2×#11715895C2312yes—
77C8×D97—24no—
78C4⋊Dic97—21no—
79C4⋊Dic97C2311yescharacteristic
80#13095490—21no—
81#13095490—21no—
82#13095490—21no—
83C22⋊Dic97—21no—
84#13095488—21no—
85#13095488—21no—
86C2×Q8×C97—24no—
87C2×Q8×C97—21no—
88C2×Q8×C97C2311yescharacteristic
89C2×Q8×C97C2312yes—
90C22×C4×C97—21no—
91C22×C4×C97C2311yescharacteristic
92Q16×C97—24no—
93C2×C8×C97C2312yes—
94C4⋊C4×C97—21no—
95C4⋊C4×C97C2311yescharacteristic
96C22⋊C4×C97—21no—
97C23×C97C2×D411yescharacteristic
98C22×D97—22no—
99C22×D97—22no—
100C22×D97Q1614yescomplemented
101C22×D97C2×D411yescharacteristic
102C22×D97C2×D411yescharacteristic
103Q8×C97—24no—
104Q8×C97—22no—
105Q8×C97—22no—
106Q8×C97C2×D414yes—
107C2×C4×C97—24no—
108C2×C4×C97—22no—
109C2×C4×C97—22no—
110C2×C4×C97—21no—
111C2×C4×C97—21no—
112C2×C4×C97C2×D411yescharacteristic
113C2×C4×C97C2411yescharacteristic
114C2×Dic97—24no—
115C2×Dic97—22no—
116C2×Dic97—22no—
117C2×Dic97—21no—
118C2×Dic97—21no—
119C2×Dic97C2×D411yescharacteristic
120C2×Dic97C2×D411yescharacteristic
121C4×D97—28no—
122C4×D97—28no—
123C4×D97—22no—
124C4×D97—22no—
125C4×D97—22no—
126C4×D97—42no—
127C4×D97—42no—
128C4×D97C2×D414yes—
129Dic194—28no—
130Dic194—24no—
131Dic194—22no—
132Dic194—28no—
133Dic194—22no—
134Dic194—42no—
135Dic194C2×D414yes—
136C8×C97—22no—
137#11715895—22no—
138C22×C97—21no—
139C22×C97—21no—
140C22×C97C2×Q1612yescomplemented
141C22×C97C22×D411yescharacteristic
142C2×D97—21no—
143C2×D97—21no—
144C2×D97—28no—
145C2×D97#12012yes—
146C2×D97C2×Q1612yes—
147C2×D97C2×Q1612yescomplemented
148C2×D97#13712yes—
149C4×C97—24no—
150C4×C97—21no—
151C4×C97—41no—
152C4×C97—41no—
153C4×C97C22×D412yes—
154Dic97—21no—
155Dic97—24no—
156Dic97—41no—
157Dic97—41no—
158Dic97#12012yes—
159C2×C97—22no—
160C2×C97C2×#12011yescharacteristic
161C2×C97C22×Q1611yescharacteristic
162C2×C97C2×#13711yescharacteristic
163D97—22no—
164D97#45114yesdirect product factor
165C2×#451—971nomaximal, Sylow
166C97C2×#45111yescharacteristic, complemented, Sylow
167C23×Q8—971no—
168C22×Q16—972no—
169C2×C22⋊Q8—971no—
170#451—971no—
171#451—972no—
172#451—971no—
173#451—972no—
174#451—971no—
175#451D9711yescharacteristic, direct product factor
176C2×#103—972no—
177C2×C22⋊C8—971no—
178C22×Q8—972no—
179C22×Q8—971no—
180C22×Q8—972no—
181C22×Q8—971no—
182C22×Q8—1941no—
183C22×Q8—1941no—
184C22×Q8—1941no—
185C22×Q8—1948no—
186C22×Q8C2×D9711yescharacteristic
187C23×C4—971no—
188C23×C4—1941no—
189C2×Q16—972no—
190C2×Q16—972no—
191C2×Q16—972no—
192C2×Q16—1948no—
193C2×Q16C2×D9712yescomplemented
194C22×C8—972no—
195C22⋊Q8—971no—
196C22⋊Q8—971no—
197C22⋊Q8—971no—
198C22⋊Q8—1941no—
199C22⋊Q8—1941no—
200C22⋊Q8C2×D9711yescharacteristic
201C2×C4⋊C4—971no—
202C2×C4⋊C4—1941no—
203C2×C22⋊C4—1941no—
204#103—972no—
205#103—972no—
206#103—972no—
207#103C2×D9712yescomplemented
208C22⋊C8—971no—
209C22⋊C8—972no—
210C22⋊C8C2×D9711yescharacteristic
211C24—971no—
212C2×Q8—24no—
213C2×Q8—21no—
214C2×Q8—978no—
215C2×Q8—972no—
216C2×Q8—971no—
217C2×Q8—1941no—
218C2×Q8—1942no—
219C2×Q8—1942no—
220C2×Q8—1944no—
221C2×Q8—1948no—
222C2×Q8—1944no—
223C2×Q8—1944no—
224C2×Q8—1944no—
225C2×Q8—1948no—
226C2×Q8—1948no—
227C2×Q8—1944no—
228C2×Q8—1941no—
229C2×Q8C22×D9711yescharacteristic
230C2×Q8C22×D9712yes—
231C22×C4—21no—
232C22×C4—971no—
233C22×C4—971no—
234C22×C4—971no—
235C22×C4—971no—
236C22×C4—971no—
237C22×C4—1944no—
238C22×C4—1941no—
239C22×C4—1941no—
240C22×C4—1942no—
241C22×C4—1942no—
242C22×C4—1948no—
243C22×C4—1941no—
244C22×C4C22×D9711yescharacteristic
245Q16—24no—
246Q16—1944no—
247Q16—1944no—
248Q16—1944no—
249C2×C8—972no—
250C2×C8—1944no—
251C2×C8C22×D9712yes—
252C4⋊C4—21no—
253C4⋊C4—971no—
254C4⋊C4—1941no—
255C4⋊C4—1941no—
256C4⋊C4—1941no—
257C4⋊C4—1941no—
258C4⋊C4C22×D9711yescharacteristic
259C22⋊C4—21no—
260C22⋊C4—1941no—
261C22⋊C4—1941no—
262C22⋊C4—1941no—
263C23—971no—
264C23—974no—
265C23—971no—
266C23—1942no—
267C23—1942no—
268C23D4×D9711yescharacteristic
269Q8—24no—
270Q8—22no—
271Q8—22no—
272Q8—974no—
273Q8—3882no—
274Q8—1948no—
275Q8—1948no—
276Q8—1944no—
277Q8—1942no—
278Q8—1942no—
279Q8D4×D9714yes—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—971no—
286C2×C4—974no—
287C2×C4—971no—
288C2×C4—3882no—
289C2×C4—3882no—
290C2×C4—1942no—
291C2×C4—1942no—
292C2×C4—1948no—
293C2×C4—1948no—
294C2×C4—1942no—
295C2×C4—1941no—
296C2×C4—1941no—
297C2×C4—1944no—
298C2×C4—1942no—
299C2×C4—1942no—
300C2×C4D4×D9711yescharacteristic
301C2×C4C23×D9711yescharacteristic
302C8—22no—
303C8—1942no—
304C22—21no—
305C22—21no—
306C22—972no—
307C22—972no—
308C22—972no—
309C22—972no—
310C22—1948no—
311C22—1941no—
312C22—1941no—
313C22Q16×D9712yescomplemented
314C22C2×D4×D9711yescharacteristic
315C4—24no—
316C4—21no—
317C4—41no—
318C4—41no—
319C4—972no—
320C4—3881no—
321C4—3881no—
322C4—1941no—
323C4—1944no—
324C4C2×D4×D9712yes—
325C2—22no—
326C2—974no—
327C2—1942no—
328C2#120×D9711yescharacteristic
329C2C2×Q16×D9711yescharacteristic
330C2#137×D9711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#451×C97C2
socleC22×C97C22×D4
prime radicalC23×D97D4
hypercenter#451D97
hypocenterC97C2×#451
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
2. C2×Q16×D97C2semidirect
4. #20260893C2semidirect
5. #20260892C2semidirect
6. #20260891C2semidirect
7. #20260890C2semidirect
8. #20260889C2semidirect
9. #103×D97C2semidirect
11. #451×C97C2semidirect
22. C2×Dic388C22semidirect
23. C2×#13095501C22semidirect
24. C2×#13095501C22semidirect
35. #15690735C22semidirect
36. #15690734C22semidirect
37. #15690733C22semidirect
41. C2×Q16×C97C22semidirect
43. #103×C97C22semidirect
100. C22×D97Q16semidirect
140. C22×C97C2×Q16semidirect
147. C2×D97C2×Q16semidirect
164. D97#451direct (N × H)
166. C97C2×#451semidirect
175. #451D97direct (N × H)
193. C2×Q16C2×D97semidirect
207. #103C2×D97semidirect
313. C22Q16×D97semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D972
C2×Q16×D972
C22⋊Q8×D972
#202608932
#202608922
#202608912
#202608902
#202608892
#103×D972
C22⋊C8×D972
#451×C972
C2×#45197