Finite Group Databasebeta

← #27141653

Subgroups — #27141653

Order 12416.

612 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 101 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×#13095511C211yesmaximal, characteristic
2C2×#15690724C214yesmaximal, complemented
3#20260598C211yesmaximal, characteristic, complemented
4#381×C97C211yesmaximal, characteristic, complemented
5C2×#13095511—28no—
6C2×#13095511—22no—
7C2×#13095511—22no—
8C22×Dic194C411yescharacteristic, complemented
9C22×D388C411yescharacteristic, complemented
10C22×C22⋊D97C2212yes—
11C22×C4×D97C412yescomplemented
12C2×C22⋊Dic97C2212yes—
13#15690724—28no—
14#15690717—22no—
15C23×C4×C97C2211yescharacteristic
16C2×C22⋊C4×C97C2212yes—
17#95×C97—22no—
18C23×D97C2×C412yes—
19#13095511—216no—
20#13095511—216no—
21C2×Dic194—22no—
22C2×Dic194—24no—
23C2×D388—24no—
24C2×D388—22no—
25C2×C22⋊D97—24no—
26C2×C22⋊D97—22no—
27C2×C22⋊D97—22no—
28C2×C22⋊D97D418yes—
29C22×Dic97—22no—
30C22×Dic97C2×C412yes—
31C2×C4×D97—24no—
32C2×C4×D97—24no—
33C2×C4×D97—24no—
34C22⋊Dic97—24no—
35C24×C97C2311yescharacteristic
36C22×C4×C97—24no—
37C22×C4×C97—22no—
38C22×C4×C97C2×C411yescharacteristic
39C22×C4×C97C2×C411yescharacteristic
40C22×C4×C97D412yes—
41C22×C4×C97D412yes—
42C22⋊C4×C97—24no—
43C23×C97—42no—
44C23×C97C22×C411yescharacteristic
45C23×C97C2×D414yes—
46C23×C97C2×D411yescharacteristic
47C23×C97C2×D411yescharacteristic
48C22×D97—22no—
49C22×D97—22no—
50C22×D97—44no—
51C22×D97C22⋊C414yes—
52C2×C4×C97—22no—
53C2×C4×C97—22no—
54C2×C4×C97—24no—
55C2×C4×C97—24no—
56C2×C4×C97—44no—
57C2×C4×C97—42no—
58C2×C4×C97C22⋊C412yes—
59C2×C4×C97C22⋊C412yes—
60C22⋊D97—28no—
61C22⋊D97—28no—
62C2×Dic97—22no—
63C2×Dic97—22no—
64C2×Dic97—44no—
65C2×Dic97—42no—
66C2×Dic97C22⋊C414yes—
67D388—28no—
68C4×D97—216no—
69Dic194—28no—
70C22×C97—22no—
71C22×C97—22no—
72C22×C97—22no—
73C22×C97—44no—
74C22×C97C2×C22⋊C412yes—
75C22×C97C2×C22⋊C411yescharacteristic
76C22×C97#12012yes—
77C22×C97#12012yes—
78C2×D97—24no—
79C2×D97—44no—
80C2×D97—44no—
81C2×D97—44no—
82C4×C97—22no—
83C4×C97—22no—
84C4×C97—82no—
85Dic97—24no—
86Dic97—82no—
87C2×C97—22no—
88C2×C97—22no—
89C2×C97—22no—
90C2×C97#37911yescharacteristic
91C2×C97#41012yes—
92D97—44no—
93#1782—971nomaximal, Sylow
94C97#178211yescharacteristic, complemented, Sylow
95C22×#41—971no—
96C2×#99—974no—
97#381—971no—
98#381D9711yescharacteristic, complemented
99C2×#41—1948no—
100C2×#41—1942no—
101C2×#41—1942no—
102C22×Q8—971no—
103C22×D4—972no—
104C22×D4—971no—
105C23×C4—972no—
106C23×C4C2×D9711yescharacteristic
107C2×C22⋊C4—972no—
108C2×C22⋊C4C2×D9712yes—
109#99—1948no—
110#95—22no—
111#95—1942no—
112C24—972no—
113C24C22×D9711yescharacteristic
114#41—19416no—
115#41—19416no—
116C2×Q8—1944no—
117C2×Q8—1942no—
118C2×D4—978no—
119C2×D4—1944no—
120C2×D4—1944no—
121C2×D4—1942no—
122C2×D4—1942no—
123C2×D4—1942no—
124C22×C4—24no—
125C22×C4—22no—
126C22×C4—972no—
127C22×C4—1944no—
128C22×C4—1944no—
129C22×C4—1942no—
130C22×C4—1944no—
131C22×C4C4×D9711yescharacteristic
132C22×C4C4×D9711yescharacteristic
133C22×C4D38812yes—
134C22×C4C22⋊D9712yes—
135C22⋊C4—24no—
136C22⋊C4—1944no—
137C23—42no—
138C23—974no—
139C23—3884no—
140C23—1942no—
141C23—1942no—
142C23C2×C4×D9711yescharacteristic
143C23D4×D9714yes—
144C23C2×C22⋊D9711yescharacteristic
145C23C2×D38811yescharacteristic
146Q8—1948no—
147D4—1948no—
148D4—1948no—
149D4—1948no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—24no—
153C2×C4—24no—
154C2×C4—44no—
155C2×C4—42no—
156C2×C4—974no—
157C2×C4—3884no—
158C2×C4—3882no—
159C2×C4—19416no—
160C2×C4—1942no—
161C2×C4—1942no—
162C2×C4#1309548812yes—
163C2×C4#1309548812yes—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—44no—
168C22—3884no—
169C22—3884no—
170C22—3884no—
171C22—1944no—
172C22C22⋊C4×D9712yes—
173C22C2×#1309548811yescharacteristic
174C22#1569077812yes—
175C22#1569078012yes—
176C4—22no—
177C4—22no—
178C4—82no—
179C4—7762no—
180C4—1944no—
181C2—22no—
182C2—22no—
183C2—22no—
184C2—3884no—
185C2#2026059011yescharacteristic
186C2#2026070812yes—
subgroupG/H
centerC22C2×#13095488
commutatorC23×C97C22×C4
FrattiniC24C22×D97
Fitting#381×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC22×#13095511C2
cosocleC24×C97C23
hypercenter#381D97
hypocenterC97#1782
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×#15690724C2semidirect
3. #20260598C2semidirect
4. #381×C97C2semidirect
8. C22×Dic194C4semidirect
9. C22×D388C4semidirect
11. C22×C4×D97C4semidirect
94. C97#1782semidirect
98. #381D97semidirect

Maximal subgroups

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

subgroupindex
C22×#130955112
C2×#156907242
#202605982
#381×C972
#178297