Finite Group Databasebeta

← #27144357

Subgroups — #27144357

Order 12416.

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

normal subgroups only

This group has 600 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
1C2×#15690768C211yesmaximal, characteristic
2C2×#15690767C211yesmaximal, characteristic
3C2×C4⋊C4×D97C211yesmaximal, characteristic
4#20260630C214yesmaximal, complemented
5#20260606C214yesmaximal
6#20260604C211yesmaximal, characteristic, complemented
7#20260601C211yesmaximal, characteristic, complemented
8#20260600C211yesmaximal, characteristic, complemented
9#384×C97C211yesmaximal, characteristic, complemented
10C22×C4×D97C2212yes—
11C22×C4×D97C2211yescharacteristic
12#15690768—24no—
13#15690767C418yescomplemented
14C2×C4⋊Dic97C2211yescharacteristic
15C2×C4⋊Dic97C2211yescharacteristic
16C2×#13095490C2212yes—
17C2×#13095490C2214yes—
18C2×#13095490C2212yes—
19C4⋊C4×D97—24no—
20C2×#13095488C2212yes—
21C2×#13095488C2212yes—
22C2×C4×Dic97C2212yes—
23C2×C4×Dic97C2211yescharacteristic
24#15690717C2214yes—
25#15690717C2214yes—
26C2×C4⋊C4×C97C2211yescharacteristic
27C2×C4⋊C4×C97C2211yescharacteristic
28C2×C4⋊C4×C97C2214yes—
29C2×C42×C97C2211yescharacteristic
30C23×D97C2311yescharacteristic
31C22×Dic97C2312yes—
32C22×Dic97C2312yes—
33C22×Dic97C2312yes—
34C22×Dic97C2311yescharacteristic
35C2×C4×D97—24no—
36C2×C4×D97—44no—
37C2×C4×D97C2×C414yes—
38C2×C4×D97D414yes—
39C2×C4×D97Q814yes—
40C4⋊Dic97—22no—
41C4⋊Dic97—22no—
42#13095490—24no—
43#13095490—44no—
44#13095490C2×C418yescomplemented
45#13095488—24no—
46#13095488C2×C418yes—
47C4×Dic97—24no—
48C4×Dic97C2×C414yescomplemented
49C22×C4×C97C2312yes—
50C22×C4×C97C2312yes—
51C22×C4×C97C2312yes—
52C22×C4×C97C2311yescharacteristic
53C4⋊C4×C97—22no—
54C4⋊C4×C97—22no—
55C4⋊C4×C97—44no—
56C42×C97C2×C414yescomplemented
57C23×C97C2411yescharacteristic
58C22×D97—22no—
59C22×D97—22no—
60C22×D97C22×C412yes—
61C22×D97C2×D412yes—
62C22×D97C2×Q812yes—
63C2×C4×C97—22no—
64C2×C4×C97—22no—
65C2×C4×C97—22no—
66C2×C4×C97—22no—
67C2×C4×C97—24no—
68C2×C4×C97—24no—
69C2×C4×C97C22×C414yes—
70C2×C4×C97C22×C412yes—
71C2×C4×C97C2×D412yes—
72C2×C4×C97C2×Q812yes—
73C2×Dic97—22no—
74C2×Dic97—22no—
75C2×Dic97—22no—
76C2×Dic97—22no—
77C2×Dic97—24no—
78C2×Dic97—24no—
79C2×Dic97C22×C414yes—
80C2×Dic97C22×C412yes—
81C2×Dic97C2×D412yes—
82C2×Dic97C2×Q812yes—
83C4×D97—48no—
84C4×D97C4⋊C4116yescomplemented
85C22×C97C23×C411yescharacteristic
86C22×C97C22×D411yescharacteristic
87C22×C97C22×Q811yescharacteristic
88C22×C97H(5,𝔽2)12yes—
89C22×C972−1+412yes—
90C2×D97—24no—
91C2×D97—22no—
92C2×D97—22no—
93C2×D97—24no—
94C2×D97C2×C4⋊C414yes—
95C4×C97—24no—
96C4×C97—42no—
97C4×C97—42no—
98C4×C97C2×C4⋊C414yes—
99Dic97—24no—
100Dic97—42no—
101Dic97—42no—
102Dic97C2×C4⋊C414yes—
103C2×C97C22×C4⋊C411yescharacteristic
104C2×C97#52012yes—
105C2×C97#53511yescharacteristic
106C2×C97#53611yescharacteristic
107C2×C97#54412yes—
108D97—24no—
109#2317—971nomaximal, Sylow
110C97#231711yescharacteristic, complemented, Sylow
111C2×#117—971no—
112C2×#117—971no—
113C22×C4⋊C4—971no—
114#389—974no—
115#385—974no—
116#384—971no—
117#384D9711yescharacteristic, complemented
118#382—971no—
119#382—971no—
120C23×C4—971no—
121C23×C4—972no—
122#117—978no—
123#117—1944no—
124C2×C4⋊C4—972no—
125C2×C4⋊C4—974no—
126C2×C4⋊C4—972no—
127C2×C4⋊C4—971no—
128C2×C4⋊C4—971no—
129C2×C4⋊C4—1944no—
130C2×C4⋊C4C2×D9711yescharacteristic
131C2×C4⋊C4C2×D9711yescharacteristic
132C2×C4⋊C4C2×D9714yes—
133C2×C22⋊C4—972no—
134C2×C22⋊C4—972no—
135C2×C42—971no—
136C2×C42—972no—
137C2×C42C2×D9711yescharacteristic
138#95—974no—
139#95—974no—
140C24—971no—
141C22×C4—971no—
142C22×C4—974no—
143C22×C4—974no—
144C22×C4—974no—
145C22×C4—972no—
146C22×C4—972no—
147C22×C4—972no—
148C22×C4—3884no—
149C22×C4—1944no—
150C22×C4C22×D9712yes—
151C22×C4C22×D9712yes—
152C22×C4C22×D9712yes—
153C22×C4C22×D9711yescharacteristic
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—44no—
157C4⋊C4—978no—
158C4⋊C4—3884no—
159C4⋊C4—1944no—
160C4⋊C4—1942no—
161C4⋊C4—1942no—
162C22⋊C4—978no—
163C22⋊C4—1944no—
164C42—974no—
165C42—1944no—
166C42C4×D9714yescomplemented
167C23—972no—
168C23—972no—
169C23—972no—
170C23—1942no—
171C23—1942no—
172C23C23×D9711yescharacteristic
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—24no—
179C2×C4—972no—
180C2×C4—972no—
181C2×C4—972no—
182C2×C4—9716no—
183C2×C4—974no—
184C2×C4—3888no—
185C2×C4—1942no—
186C2×C4—1942no—
187C2×C4—1944no—
188C2×C4—1944no—
189C2×C4—1942no—
190C2×C4—1942no—
191C2×C4C2×C4×D9714yes—
192C2×C4C2×C4×D9712yes—
193C2×C4D4×D9712yes—
194C2×C4Q8×D9712yes—
195C22—974no—
196C22—1944no—
197C22—1942no—
198C22—1942no—
199C22—1944no—
200C22C22×C4×D9711yescharacteristic
201C22C2×D4×D9711yescharacteristic
202C22C2×Q8×D9711yescharacteristic
203C22#1569085512yes—
204C22#1569085812yes—
205C4—24no—
206C4—42no—
207C4—42no—
208C4—974no—
209C4—3882no—
210C4—3882no—
211C4—1944no—
212C4C4⋊C4×D9714yes—
213C2—1944no—
214C2C2×C4⋊C4×D9711yescharacteristic
215C2#2026121612yes—
216C2#2026129811yescharacteristic
217C2#2026131011yescharacteristic
218C2#2026136212yes—
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#384×C97C2
socle, cosocleC23×C97C24
prime radicalC23×D97C23
hypercenter#384D97
hypocenterC97#2317
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
4. #20260630C2semidirect
6. #20260604C2semidirect
7. #20260601C2semidirect
8. #20260600C2semidirect
9. #384×C97C2semidirect
13. #15690767C4semidirect
44. #13095490C2×C4semidirect
48. C4×Dic97C2×C4semidirect
56. C42×C97C2×C4semidirect
84. C4×D97C4⋊C4semidirect
110. C97#2317semidirect
117. #384D97semidirect
166. C42C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907682
C2×#156907672
C2×C4⋊C4×D972
#202606302
#202606062
#202606042
#202606012
#202606002
#384×C972
#231797