Finite Group Databasebeta

← #27144319

Subgroups — #27144319

Order 12416.

784 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 229 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×#15690768C211yesmaximal, characteristic
3C2×#15690767C211yesmaximal, characteristic
4#20260642C211yesmaximal, characteristic, complemented
5#20260641C211yesmaximal, characteristic, complemented
6#20260640C211yesmaximal, characteristic, complemented
7#20260639C214yesmaximal, complemented
8#20260597C214yesmaximal
9#391×C97C211yesmaximal, characteristic, complemented
10C22×Dic194C2212yes—
11C22×Dic194C2211yescharacteristic
12C2×Q8×D97—24no—
13C2×Q8×D97—24no—
14C2×Q8×D97C418yescomplemented
15C22×C4×D97C2212yes—
16C22×C4×D97C2211yescharacteristic
17#15690768—24no—
18#15690767—24no—
19C2×C4⋊Dic97C2211yescharacteristic
20C2×#13095490C2212yes—
21C2×#13095488C2212yes—
22C2×#13095488C2212yes—
23C2×C4×Dic97C2212yes—
24C2×C4×Dic97C2211yescharacteristic
25#15690717C2214yes—
26#15690717C2214yes—
27#15690717C2214yes—
28C22×Q8×C97C2211yescharacteristic
29C2×C4⋊C4×C97C2211yescharacteristic
30C2×C42×C97C2211yescharacteristic
31#95×C97C2214yes—
32C23×D97C2311yescharacteristic
33C2×Dic194—22no—
34C2×Dic194—22no—
35C2×Dic194—24no—
36C2×Dic194—24no—
37C2×Dic194C2×C414yes—
38C2×Dic194C2×C418yes—
39Q8×D97—232no—
40C22×Dic97C2312yes—
41C22×Dic97C2312yes—
42C22×Dic97C2312yes—
43C22×Dic97C2311yescharacteristic
44C2×C4×D97—24no—
45C2×C4×D97—24no—
46C2×C4×D97C2×C418yes—
47C2×C4×D97C2×C414yes—
48C2×C4×D97D414yes—
49C2×C4×D97D414yes—
50C4⋊Dic97—22no—
51#13095490—24no—
52#13095488—24no—
53#13095488—24no—
54C4×Dic97—22no—
55C4×Dic97—24no—
56C2×Q8×C97—22no—
57C2×Q8×C97—22no—
58C2×Q8×C97C2×C414yes—
59C22×C4×C97C2312yes—
60C22×C4×C97C2312yes—
61C22×C4×C97C2312yes—
62C22×C4×C97C2311yescharacteristic
63C4⋊C4×C97—22no—
64C42×C97—22no—
65C23×C97C2411yescharacteristic
66C22×D97—22no—
67C22×D97—22no—
68C22×D97C22×C412yes—
69C22×D97C2×D412yes—
70C22×D97C2×D412yes—
71Q8×C97—28no—
72C2×C4×C97—22no—
73C2×C4×C97—22no—
74C2×C4×C97—22no—
75C2×C4×C97—22no—
76C2×C4×C97—24no—
77C2×C4×C97—22no—
78C2×C4×C97—22no—
79C2×C4×C97C22×C414yes—
80C2×C4×C97C22×C412yes—
81C2×C4×C97C2×D412yes—
82C2×C4×C97C2×D412yes—
83C2×Dic97—22no—
84C2×Dic97—22no—
85C2×Dic97—22no—
86C2×Dic97—22no—
87C2×Dic97—22no—
88C2×Dic97—22no—
89C2×Dic97—24no—
90C2×Dic97C22×C414yes—
91C2×Dic97C22×C412yes—
92C2×Dic97C2×D412yes—
93C2×Dic97C2×D412yes—
94C4×D97—216no—
95C4×D97C22⋊C4116yes—
96Dic194—28no—
97Dic194—216no—
98C22×C97C23×C411yescharacteristic
99C22×C97C22×D411yescharacteristic
100C22×C97C22×D411yescharacteristic
101C22×C972−1+412yes—
102C22×C972−1+412yes—
103C2×D97—24no—
104C2×D97—22no—
105C2×D97—24no—
106C2×D97—22no—
107C2×D97C2×C22⋊C414yes—
108C4×C97—24no—
109C4×C97—42no—
110C4×C97—42no—
111C4×C97C2×C22⋊C414yes—
112Dic97—24no—
113Dic97—42no—
114Dic97—42no—
115Dic97C2×C22⋊C414yes—
116C2×C97C22×C22⋊C411yescharacteristic
117C2×C97#51912yes—
118C2×C97#53611yescharacteristic
119C2×C97#53612yes—
120C2×C97#53611yescharacteristic
121D97—24no—
122#2310—971nomaximal, Sylow
123C97#231011yescharacteristic, complemented, Sylow
124C23×Q8—971no—
125C2×#117—971no—
126C2×#117—971no—
127#391—971no—
128#391—971no—
129#391—971no—
130#391—974no—
131#391D9711yescharacteristic, complemented
132#381—974no—
133C22×Q8—978no—
134C22×Q8—972no—
135C22×Q8—971no—
136C22×Q8—1944no—
137C22×Q8—1944no—
138C22×Q8C2×D9711yescharacteristic
139C23×C4—971no—
140C23×C4—972no—
141#117—1944no—
142#117—1944no—
143C2×C4⋊C4—972no—
144C2×C4⋊C4—971no—
145C2×C4⋊C4C2×D9711yescharacteristic
146C2×C22⋊C4—972no—
147C2×C22⋊C4—972no—
148C2×C42—972no—
149C2×C42—971no—
150C2×C42C2×D9711yescharacteristic
151#95—974no—
152#95—974no—
153#95—974no—
154#95C2×D9714yes—
155C24—971no—
156C2×Q8—22no—
157C2×Q8—22no—
158C2×Q8—978no—
159C2×Q8—974no—
160C2×Q8—1944no—
161C2×Q8—1944no—
162C2×Q8—19432no—
163C2×Q8—1942no—
164C2×Q8—1942no—
165C2×Q8C4×D9714yes—
166C22×C4—971no—
167C22×C4—974no—
168C22×C4—974no—
169C22×C4—974no—
170C22×C4—972no—
171C22×C4—978no—
172C22×C4—972no—
173C22×C4—972no—
174C22×C4—1944no—
175C22×C4—1944no—
176C22×C4C22×D9712yes—
177C22×C4C22×D9712yes—
178C22×C4C22×D9712yes—
179C22×C4C22×D9711yescharacteristic
180C4⋊C4—22no—
181C4⋊C4—1944no—
182C4⋊C4—1942no—
183C22⋊C4—1944no—
184C22⋊C4—1944no—
185C42—22no—
186C42—1944no—
187C42—1942no—
188C23—972no—
189C23—972no—
190C23—972no—
191C23—1942no—
192C23—1942no—
193C23C23×D9711yescharacteristic
194Q8—28no—
195Q8—19416no—
196Q8—1948no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—972no—
205C2×C4—972no—
206C2×C4—972no—
207C2×C4—9716no—
208C2×C4—974no—
209C2×C4—19416no—
210C2×C4—1942no—
211C2×C4—1942no—
212C2×C4—1942no—
213C2×C4—1942no—
214C2×C4—1944no—
215C2×C4—1942no—
216C2×C4—1942no—
217C2×C4C2×C4×D9714yes—
218C2×C4C2×C4×D9712yes—
219C2×C4D4×D9712yes—
220C2×C4D4×D9712yes—
221C22—974no—
222C22—1944no—
223C22—1942no—
224C22—1944no—
225C22—1942no—
226C22C22×C4×D9711yescharacteristic
227C22C2×D4×D9711yescharacteristic
228C22C2×D4×D9711yescharacteristic
229C22#1569085712yes—
230C22#1569085812yes—
231C4—24no—
232C4—42no—
233C4—42no—
234C4—974no—
235C4—3882no—
236C4—3882no—
237C4—1944no—
238C4C22⋊C4×D9714yes—
239C2—1944no—
240C2C2×C22⋊C4×D9711yescharacteristic
241C2#2026121112yes—
242C2#2026130511yescharacteristic
243C2#2026130812yes—
244C2#2026131011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#391×C97C2
socle, cosocleC23×C97C24
prime radicalC23×D97C23
hypercenter#391D97
hypocenterC97#2310
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
4. #20260642C2semidirect
5. #20260641C2semidirect
6. #20260640C2semidirect
7. #20260639C2semidirect
9. #391×C97C2semidirect
14. C2×Q8×D97C4semidirect
123. C97#2310semidirect
131. #391D97semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D972
C2×#156907682
C2×#156907672
#202606422
#202606412
#202606402
#202606392
#202605972
#391×C972
#231097