Finite Group Databasebeta

← #27150006

Subgroups — #27150006

Order 12416.

348 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#20260676C212yesmaximal
2#20260669C212yesmaximal
3#20260667C212yesmaximal
4#20260665C212yesmaximal
5#20260656C212yesmaximal
6#20260655C211yesmaximal, characteristic
7#20260604C211yesmaximal, characteristic
8#20260602C212yesmaximal
9#400×C97C211yesmaximal, characteristic
10C22×Dic194C2211yescharacteristic
11C2×C4⋊Dic97C2212yes—
12C2×C4⋊Dic97C2212yes—
13C2×C4⋊Dic97C2211yescharacteristic
14C2×#13095490C2212yes—
15C2×#13095490C2212yes—
16C2×#13095490C2212yes—
17C2×#13095490C2212yes—
18C2×C4×Dic97C2211yescharacteristic
19#15690717C2212yes—
20#15690717C2211yescharacteristic
21#15690717C2212yes—
22#15690717C2212yes—
23#15690717C2212yes—
24#15690717C2212yes—
25#15690717C2212yes—
26C2×C4⋊C4×C97C2211yescharacteristic
27C2×C4⋊C4×C97C2211yescharacteristic
28#95×C97C2212yes—
29#95×C97C2212yes—
30#95×C97C2211yescharacteristic
31C2×Dic194—42no—
32C2×Dic194—41no—
33C22×Dic97C2312yes—
34C22×Dic97C2312yes—
35C22×Dic97C2312yes—
36C22×Dic97C2312yes—
37C4⋊Dic97—42no—
38C4⋊Dic97—41no—
39C4⋊Dic97—42no—
40#13095490—42no—
41#13095490—42no—
42#13095490—42no—
43#13095490—42no—
44C4×Dic97Q814yescomplemented
45C22×C4×C97C2311yescharacteristic
46C22×C4×C97C2311yescharacteristic
47C22×C4×C97C2312yes—
48C22×C4×C97C2312yes—
49C22×C4×C97C2311yescharacteristic
50C4⋊C4×C97—41no—
51C4⋊C4×C97—41no—
52C23×C97C2411yescharacteristic
53C2×C4×C97—21no—
54C2×C4×C97—21no—
55C2×C4×C97—21no—
56C2×C4×C97—22no—
57C2×C4×C97—21no—
58C2×C4×C97—22no—
59C2×C4×C97—22no—
60C2×C4×C97—22no—
61C2×C4×C97—22no—
62C2×C4×C97—22no—
63C2×C4×C97—24no—
64C2×C4×C97C2×Q812yes—
65C2×Dic97—22no—
66C2×Dic97—22no—
67C2×Dic97—22no—
68C2×Dic97—22no—
69C2×Dic97—22no—
70C2×Dic97—22no—
71C2×Dic97—22no—
72C2×Dic97—22no—
73C2×Dic97—22no—
74C2×Dic97—22no—
75C2×Dic97—22no—
76C2×Dic97C2×Q814yes—
77Dic194—44no—
78C22×C97C22×Q811yescharacteristic
79C22×C97H(5,𝔽2)11yescharacteristic
80C22×C97H(5,𝔽2)12yes—
81C22×C972−1+412yes—
82C22×C972−1+411yescharacteristic
83C4×C97—22no—
84C4×C97—41no—
85C4×C97—41no—
86C4×C97—42no—
87C4×C97—42no—
88Dic97—24no—
89Dic97—42no—
90Dic97—42no—
91Dic97—42no—
92C2×C97#54411yescharacteristic
93C2×C97#54412yes—
94C2×C97#56011yescharacteristic
95C2×C97#56311yescharacteristic
96C2×C97#56312yes—
97#2843—971nomaximal, Sylow
98C97#284311yescharacteristic, complemented, Sylow
99#400—972no—
100#400D9711yescharacteristic
101#398—972no—
102#398—972no—
103#398—972no—
104#395—971no—
105#395—972no—
106#384—971no—
107#383—972no—
108C22×Q8—971no—
109C2×C4⋊C4—972no—
110C2×C4⋊C4—972no—
111C2×C4⋊C4—972no—
112C2×C4⋊C4—972no—
113C2×C4⋊C4—971no—
114C2×C4⋊C4—972no—
115C2×C4⋊C4—972no—
116C2×C4⋊C4C2×D9711yescharacteristic
117C2×C4⋊C4C2×D9711yescharacteristic
118C2×C42—971no—
119#95—972no—
120#95—972no—
121#95—972no—
122#95—972no—
123#95—972no—
124#95—971no—
125#95—972no—
126#95C2×D9712yes—
127#95C2×D9712yes—
128#95C2×D9711yescharacteristic
129C2×Q8—3881no—
130C2×Q8—3882no—
131C22×C4—972no—
132C22×C4—972no—
133C22×C4—972no—
134C22×C4—972no—
135C22×C4C22×D9712yes—
136C22×C4C22×D9712yes—
137C22×C4C22×D9711yescharacteristic
138C22×C4C22×D9711yescharacteristic
139C22×C4C22×D9711yescharacteristic
140C4⋊C4—41no—
141C4⋊C4—41no—
142C4⋊C4—3882no—
143C4⋊C4—3882no—
144C4⋊C4—3882no—
145C4⋊C4—3882no—
146C4⋊C4—3881no—
147C4⋊C4—3882no—
148C4⋊C4—3882no—
149C42—974no—
150C23C23×D9711yescharacteristic
151Q8—3884no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—24no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—974no—
164C2×C4—1942no—
165C2×C4—1942no—
166C2×C4—1942no—
167C2×C4—1942no—
168C2×C4—1942no—
169C2×C4—1942no—
170C2×C4—1942no—
171C2×C4—1942no—
172C2×C4—1942no—
173C2×C4—1942no—
174C2×C4—1942no—
175C2×C4Q8×D9712yes—
176C22C2×Q8×D9711yescharacteristic
177C22#1569085511yescharacteristic
178C22#1569085612yes—
179C22#1569085712yes—
180C22#1569085711yescharacteristic
181C4—22no—
182C4—41no—
183C4—41no—
184C4—42no—
185C4—42no—
186C4—3882no—
187C4—3882no—
188C4—3882no—
189C4—1944no—
190C2#2026136411yescharacteristic
191C2#2026136512yes—
192C2#2026147511yescharacteristic
193C2#2026148511yescharacteristic
194C2#2026148712yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, prime radical, cosocleC23×C97C24
Fitting#400×C97C2
hypercenter#400D97
hypocenterC97#2843
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
44. C4×Dic97Q8semidirect
98. C97#2843semidirect

Maximal subgroups

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

subgroupindex
#202606762
#202606692
#202606672
#202606652
#202606562
#202606552
#202606042
#202606022
#400×C972
#284397