Finite Group Databasebeta

← #27151115

Subgroups — #27151115

Order 12416.

704 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 161 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
1C2×#15690840C212yesmaximal, complemented
2#20261306C211yesmaximal, characteristic, complemented
3#20260918C214yesmaximal, complemented
4#20260913C214yesmaximal, complemented
5#20260752C212yesmaximal, complemented
6#20260719C211yesmaximal, characteristic, complemented
7#585×C97C211yesmaximal, characteristic
8C22×Dic194C2211yescharacteristic, complemented
9#15690840—24no—
10#15690840C2218yescomplemented
11C2×#13095501—22no—
12C2×#13095499—22no—
13C2×#13095495C2212yescomplemented
14#15690808C2212yescomplemented
15#15690797C2212yescomplemented
16#15690791—22no—
17#15690789—22no—
18#15690769C2211yescharacteristic, complemented
19#15690737C2218yescomplemented
20#15690729C2212yescomplemented
21#15690727C2212yescomplemented
222−1+4×C97—22no—
232−1+4×C97C2212yes—
24H(5,𝔽2)×C97—22no—
25H(5,𝔽2)×C97—22no—
26H(5,𝔽2)×C97C2212yes—
27C2×#41×C97—24no—
28C2×#41×C97—22no—
29C2×#41×C97C2212yes—
30C2×#41×C97C2211yescharacteristic
31C2×Dic194—22no—
32C2×Dic194—22no—
33C2×Dic194—21no—
34C2×Dic194C2312yes—
35C22×Dic97—21no—
36#13095501—28no—
37#13095499—28no—
38#13095495—22no—
39#13095495C2314yes—
40C2×#11715895—22no—
41C4⋊Dic97—21no—
42#13095490—24no—
43C22⋊Dic97—21no—
44C22⋊Dic97—24no—
45C4×Dic97C2312yes—
46#41×C97—24no—
47#41×C97—28no—
48#41×C97—24no—
49#41×C97—24no—
50#41×C97—24no—
51#41×C97—22no—
52#41×C97—22no—
53#41×C97—24no—
54#41×C97—22no—
55#41×C97—22no—
56#41×C97D414yes—
57#41×C97C2314yes—
58C2×Q8×C97—22no—
59C2×Q8×C97—24no—
60C2×Q8×C97D412yes—
61C2×Q8×C97C2311yescharacteristic
62C2×D4×C97—28no—
63C2×D4×C97—22no—
64C2×D4×C97—24no—
65C2×D4×C97—22no—
66C2×D4×C97—24no—
67C2×D4×C97D412yes—
68C2×D4×C97D412yes—
69C2×D4×C97C2311yescharacteristic
70C22×C4×C97—24no—
71C22×C4×C97—22no—
72C22×C4×C97D412yes—
73C22×C4×C97C2311yescharacteristic
74C23×C97—24no—
75C23×C97—22no—
76C23×C97C2×D412yes—
77C23×C97C2×D411yescharacteristic
78Q8×C97—24no—
79Q8×C97—22no—
80Q8×C97—22no—
81Q8×C97C2×D414yes—
82D4×C97—28no—
83D4×C97—24no—
84D4×C97—24no—
85D4×C97—24no—
86D4×C97—24no—
87D4×C97—22no—
88D4×C97—22no—
89D4×C97C2×D414yes—
90C2×C4×C97—24no—
91C2×C4×C97—28no—
92C2×C4×C97—22no—
93C2×C4×C97—24no—
94C2×C4×C97—22no—
95C2×C4×C97—22no—
96C2×C4×C97—24no—
97C2×C4×C97C2×D412yes—
98C2×C4×C97C2×D411yescharacteristic
99C2×C4×C97C2×D412yes—
100C2×C4×C97C2×D412yes—
101C2×C4×C97C2411yescharacteristic
102C2×Dic97—22no—
103C2×Dic97—22no—
104C2×Dic97—42no—
105Dic194—24no—
106Dic194—41no—
107Dic194—41no—
108#11715895—24no—
109C22×C97—22no—
110C22×C97—24no—
111C22×C97—41no—
112C22×C97—42no—
113C22×C97—82no—
114C22×C97—84no—
115C22×C97#12012yes—
116C22×C97C22×D411yescharacteristic
117C4×C97—24no—
118C4×C97—22no—
119C4×C97#12011yescharacteristic
120C4×C97#12011yescharacteristic
121C4×C97C22×D412yes—
122Dic97—42no—
123Dic97—42no—
124C2×C97—21no—
125C2×C97—22no—
126C2×C97—42no—
127C2×C97—44no—
128C2×C97C2×#12011yescharacteristic
129#3022—971nomaximal, Sylow
130C97#302211yescharacteristic, complemented, Sylow
131#585D9711yescharacteristic
132C2×#137—972no—
133#536—971no—
134#456—974no—
135#455—974no—
136#421—972no—
137#413—971no—
1382−1+4—22no—
1392−1+4C2×D9712yes—
140H(5,𝔽2)—22no—
141H(5,𝔽2)—22no—
142H(5,𝔽2)C2×D9712yes—
143C2×#41—24no—
144C2×#41—22no—
145C2×#41C2×D9712yes—
146C2×#41C2×D9711yescharacteristic
147C22×Q8—971no—
148#137—978no—
149#137—1944no—
150C2×Q16—1942no—
151C2×SD16—1942no—
152C2×M16—972no—
153C4⋊Q8—972no—
154#124—972no—
155#123—1942no—
156C22⋊Q8—1942no—
157#117—971no—
158#104—978no—
159#101—972no—
160#100—972no—
161#41—24no—
162#41—28no—
163#41—24no—
164#41—24no—
165#41—24no—
166#41—22no—
167#41—22no—
168#41—24no—
169#41—22no—
170#41—22no—
171#41C22⋊D9714yes—
172#41C22×D9714yes—
173C2×Q8—22no—
174C2×Q8—24no—
175C2×Q8—972no—
176C2×Q8—1942no—
177C2×Q8—1942no—
178C2×Q8—1941no—
179C2×Q8C22⋊D9712yes—
180C2×Q8C22×D9711yescharacteristic
181C2×D4—28no—
182C2×D4—22no—
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—24no—
186C2×D4C22⋊D9712yes—
187C2×D4C22⋊D9712yes—
188C2×D4C22×D9711yescharacteristic
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—1941no—
192C22×C4C22⋊D9712yes—
193C22×C4C22×D9711yescharacteristic
194Q16—1948no—
195SD16—1948no—
196M16—974no—
197M16—1942no—
198C2×C8—1942no—
199C4⋊C4—1944no—
200C4⋊C4—1941no—
201C22⋊C4—1941no—
202C22⋊C4—1944no—
203C42—972no—
204C23—24no—
205C23—22no—
206C23C2×C22⋊D9712yes—
207C23C2×C22⋊D9711yescharacteristic
208Q8—24no—
209Q8—22no—
210Q8—22no—
211Q8—3881no—
212Q8—3881no—
213Q8—1944no—
214Q8C2×C22⋊D9714yes—
215D4—28no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—24no—
220D4—22no—
221D4—22no—
222D4C2×C22⋊D9714yes—
223C2×C4—24no—
224C2×C4—28no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—3882no—
231C2×C4—1942no—
232C2×C4—1942no—
233C2×C4C2×C22⋊D9712yes—
234C2×C4C2×C22⋊D9711yescharacteristic
235C2×C4C2×C22⋊D9712yes—
236C2×C4C2×C22⋊D9712yes—
237C2×C4C23×D9711yescharacteristic
238C8—1944no—
239C22—22no—
240C22—24no—
241C22—41no—
242C22—42no—
243C22—82no—
244C22—84no—
245C22#1569077912yes—
246C22C22×C22⋊D9711yescharacteristic
247C4—24no—
248C4—22no—
249C4—3882no—
250C4—3882no—
251C4#1569077911yescharacteristic
252C4#1569077911yescharacteristic
253C4C22×C22⋊D9712yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—44no—
258C2C2×#1569077911yescharacteristic
subgroupG/H
centerC2C2×#15690779
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting, prime radical#585×C97C2
socleC2×C97C2×#120
hypercenter#585D97
hypocenterC97#3022
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. C2×#15690840C2semidirect
2. #20261306C2semidirect
3. #20260918C2semidirect
4. #20260913C2semidirect
5. #20260752C2semidirect
6. #20260719C2semidirect
8. C22×Dic194C22semidirect
10. #15690840C22semidirect
13. C2×#13095495C22semidirect
14. #15690808C22semidirect
15. #15690797C22semidirect
18. #15690769C22semidirect
19. #15690737C22semidirect
20. #15690729C22semidirect
21. #15690727C22semidirect
130. C97#3022semidirect

Maximal subgroups

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

subgroupindex
C2×#156908402
#202613062
#202609182
#202609132
#202607522
#202607192
#585×C972
#302297