Finite Group Databasebeta

← #27140222

Subgroups — #27140222

Order 12416.

238 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 81 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#20260788C211yesmaximal, characteristic, complemented
2#20260581C211yesmaximal, characteristic, complemented
3#20260472C212yesmaximal, complemented
4#20260469C212yesmaximal
5#529×C97C211yesmaximal, characteristic
6C2×C4×Dic97—21no—
7C4⋊#11715895—21no—
8C22⋊#11715895C2212yes—
9C4×#11715895—21no—
10#15690717—21no—
11#15690717C2212yes—
12#126×C97—21no—
13#125×C97C411yescharacteristic, complemented
14#124×C97C411yescharacteristic, complemented
15#123×C97—21no—
16C22⋊Q8×C97C2211yescharacteristic
17#121×C97C2211yescharacteristic
18C4×Q8×C97C411yescharacteristic, complemented
19C4×D4×C97—21no—
20C4×D4×C97C411yescharacteristic, complemented
21#117×C97—21no—
22C2×C42×C97C2211yescharacteristic
23C22×Dic97—22no—
24C2×#11715895—22no—
25C4×Dic97—24no—
26C2×Q8×C97C2×C411yescharacteristic
27C2×D4×C97—21no—
28C2×D4×C97C2×C411yescharacteristic
29C22×C4×C97—21no—
30C22×C4×C97D411yescharacteristic
31C22×C4×C97D411yescharacteristic
32C22×C4×C97C2311yescharacteristic
33C4⋊C4×C97—21no—
34C4⋊C4×C97—21no—
35C4⋊C4×C97—21no—
36C4⋊C4×C97—21no—
37C4⋊C4×C97C2×C411yescharacteristic
38C4⋊C4×C97C2×C411yescharacteristic
39C22⋊C4×C97—21no—
40C22⋊C4×C97—21no—
41C22⋊C4×C97—21no—
42C22⋊C4×C97—21no—
43C22⋊C4×C97—21no—
44C42×C97—21no—
45C42×C97C2×C411yescharacteristic
46C42×C97C2×C411yescharacteristic
47C42×C97D411yescharacteristic
48C42×C97D411yescharacteristic
49C23×C97—21no—
50C23×C97C2×D411yescharacteristic
51Q8×C97—41no—
52D4×C97—41no—
53D4×C97—42no—
54C2×C4×C97—22no—
55C2×C4×C97—22no—
56C2×C4×C97—21no—
57C2×C4×C97—21no—
58C2×C4×C97—21no—
59C2×C4×C97—41no—
60C2×C4×C97—42no—
61C2×C4×C97C22⋊C411yescharacteristic
62C2×C4×C97C22⋊C411yescharacteristic
63C2×C4×C97C22⋊C411yescharacteristic
64C2×C4×C97C22⋊C411yescharacteristic
65C2×C4×C97C22×C411yescharacteristic
66C2×C4×C97C2×D411yescharacteristic
67C2×Dic97—24no—
68C2×Dic97—42no—
69C2×Dic97—42no—
70#11715895—42no—
71C22×C97—21no—
72C22×C97—21no—
73C22×C97—41no—
74C22×C97—41no—
75C22×C97—41no—
76C22×C97C2×C22⋊C411yescharacteristic
77C22×C97#13512yes—
78C4×C97—21no—
79C4×C97—21no—
80C4×C97—21no—
81C4×C97—21no—
82C4×C97—21no—
83C4×C97—41no—
84C4×C97—41no—
85C4×C97—41no—
86C4×C97#10412yes—
87Dic97—44no—
88C2×C97—22no—
89C2×C97—81no—
90C2×C97#41011yescharacteristic
91C2×C97#41611yescharacteristic
92C2×C97C2×#10411yescharacteristic
93#1500—971nomaximal, Sylow
94C97#150011yescharacteristic, complemented, Sylow
95#529D9711yescharacteristic
96#431—971no—
97#376—971no—
98#328—972no—
99#327—972no—
100#126—21no—
101#125Dic9711yescharacteristic, complemented
102#124Dic9711yescharacteristic, complemented
103#123—21no—
104C22⋊Q8C2×D9711yescharacteristic
105#121C2×D9711yescharacteristic
106C4×Q8Dic9711yescharacteristic, complemented
107C4×D4—21no—
108C4×D4Dic9711yescharacteristic, complemented
109#117—21no—
110C2×C42—1941no—
111C2×C42C2×D9711yescharacteristic
112C4⋊C8—1941no—
113C22⋊C8—972no—
114C4×C8—1941no—
115#95—972no—
116#95—1941no—
117C2×Q8C2×Dic9711yescharacteristic
118C2×D4—21no—
119C2×D4C2×Dic9711yescharacteristic
120C22×C4—21no—
121C22×C4—1942no—
122C22×C4C22⋊D9711yescharacteristic
123C22×C4C22⋊D9711yescharacteristic
124C22×C4C22×D9711yescharacteristic
125C2×C8—1942no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4C2×Dic9711yescharacteristic
131C4⋊C4C2×Dic9711yescharacteristic
132C22⋊C4—21no—
133C22⋊C4—21no—
134C22⋊C4—21no—
135C22⋊C4—21no—
136C22⋊C4—21no—
137C42—21no—
138C42—1944no—
139C42C2×Dic9711yescharacteristic
140C42C2×Dic9711yescharacteristic
141C42C22⋊D9711yescharacteristic
142C42C22⋊D9711yescharacteristic
143C23—21no—
144C23C2×C22⋊D9711yescharacteristic
145Q8—41no—
146D4—41no—
147D4—42no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—42no—
155C2×C4—3882no—
156C2×C4—3882no—
157C2×C4—1944no—
158C2×C4C22⋊Dic9711yescharacteristic
159C2×C4C22⋊Dic9711yescharacteristic
160C2×C4C22⋊Dic9711yescharacteristic
161C2×C4C22⋊Dic9711yescharacteristic
162C2×C4C22×Dic9711yescharacteristic
163C2×C4C2×C22⋊D9711yescharacteristic
164C8—3882no—
165C22—21no—
166C22—21no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22C2×C22⋊Dic9711yescharacteristic
171C22#1569083012yes—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—41no—
178C4—41no—
179C4—41no—
180C4—3884no—
181C4#1569073712yes—
182C2—22no—
183C2—81no—
184C2#2026070911yescharacteristic
185C2#2026073111yescharacteristic
186C2C2×#1569073711yescharacteristic
subgroupG/H
centerC2×C4C22⋊Dic97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#529×C97C2
socleC22×C97C2×C22⋊C4
prime radical#121×C97C22
cosocleC22×C4×C97C23
hypercenter#529D97
hypocenterC97#1500
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #20260788C2semidirect
2. #20260581C2semidirect
3. #20260472C2semidirect
13. #125×C97C4semidirect
14. #124×C97C4semidirect
18. C4×Q8×C97C4semidirect
20. C4×D4×C97C4semidirect
94. C97#1500semidirect
101. #125Dic97semidirect
102. #124Dic97semidirect
106. C4×Q8Dic97semidirect
108. C4×D4Dic97semidirect

Maximal subgroups

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

subgroupindex
#202607882
#202605812
#202604722
#202604692
#529×C972
#150097