Finite Group Databasebeta

← #27142056

Subgroups — #27142056

Order 12416.

412 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 105 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×#15690784C211yesmaximal, characteristic
2C2×#15690769C211yesmaximal, characteristic, complemented
3C2×#15690732C212yesmaximal, complemented
4#20260498C212yesmaximal, complemented
5C2×C22⋊C8×C97C211yesmaximal, characteristic, complemented
6C22×D388C2211yescharacteristic
7C22×C22⋊D97—21no—
8#15690784—22no—
9#15690784C414yescomplemented
10#15690769—24no—
11C2×C4⋊Dic97C2211yescharacteristic
12C2×C4⋊Dic97C2212yes—
13C2×C22⋊Dic97—21no—
14C2×#13095488—21no—
15C2×C4×Dic97—21no—
16#15690732—24no—
17C23×C4×C97C2211yescharacteristic
18C22×C8×C97C2212yes—
19C22⋊C8×C97—22no—
20C23×D97—21no—
21C2×D388—22no—
22C2×D388—22no—
23C2×D388—21no—
24C2×D388C2×C412yes—
25C2×C22⋊D97—24no—
26C2×C22⋊D97—44no—
27C22×Dic97—22no—
28C22×Dic97—21no—
29C4⋊Dic97—22no—
30C4⋊Dic97—21no—
31C4⋊Dic97C2×C412yes—
32C4⋊Dic97D414yescomplemented
33C22⋊Dic97—24no—
34#13095488—24no—
35C4×Dic97—24no—
36C24×C97D411yescharacteristic
37C22×C4×C97—22no—
38C22×C4×C97—22no—
39C22×C4×C97C2×C412yes—
40C22×C4×C97D411yescharacteristic
41C22×C4×C97D412yes—
42C22×C4×C97C2311yescharacteristic
43C2×C8×C97—22no—
44C2×C8×C97—22no—
45C2×C8×C97—22no—
46C23×C97—21no—
47C23×C97—21no—
48C23×C97—22no—
49C23×C97—22no—
50C23×C97C22⋊C412yes—
51C23×C97C2×D411yescharacteristic
52C22×D97—22no—
53C22×D97—41no—
54C22×D97—41no—
55C22×D97—42no—
56C22×D97—42no—
57C2×C4×C97—21no—
58C2×C4×C97—21no—
59C2×C4×C97—24no—
60C2×C4×C97—24no—
61C2×C4×C97C22⋊C412yes—
62C2×C4×C97C22×C411yescharacteristic
63C2×C4×C97C2×D412yes—
64C2×C4×C97C2×D411yescharacteristic
65C2×C4×C97#4112yes—
66C22⋊D97—48no—
67C2×Dic97—22no—
68C2×Dic97—24no—
69C2×Dic97—42no—
70C2×Dic97—42no—
71C2×Dic97—42no—
72D388—24no—
73D388—42no—
74C8×C97—42no—
75C22×C97—22no—
76C22×C97—22no—
77C22×C97—22no—
78C22×C97—42no—
79C22×C97—42no—
80C22×C97C2×C22⋊C411yescharacteristic
81C22×C97#12011yescharacteristic
82C22×C97#12311yescharacteristic
83C22×C97#13512yes—
84C22×C97AGL(1,ℤ/8)12yes—
85C2×D97—42no—
86C2×D97—42no—
87C2×D97—42no—
88C2×D97—82no—
89C2×D97—82no—
90C4×C97—22no—
91C4×C97C4×D412yes—
92C4×C97#12112yes—
93Dic97—42no—
94Dic97—44no—
95C2×C97—42no—
96C2×C97#38611yescharacteristic
97C2×C97#41611yescharacteristic
98C2×C97#41811yescharacteristic
99C2×C97#44912yes—
100C2×C97#48212yes—
101D97—82no—
102#1875—971nomaximal, Sylow
103C97#187511yescharacteristic, complemented, Sylow
104C2×#121—971no—
105C2×#117—971no—
106C2×#102—972no—
107C2×C22⋊C8D9711yescharacteristic, complemented
108#340—972no—
109C22×D4—971no—
110C22×D4—1941no—
111C23×C4C2×D9711yescharacteristic
112C22×C8C2×D9712yes—
113#121—974no—
114#121—1942no—
115#117—1944no—
116C2×C4⋊C4—972no—
117C2×C4⋊C4—971no—
118C2×C22⋊C4—1941no—
119C2×C22⋊C4—1941no—
120C2×C42—1941no—
121#102—1944no—
122C22⋊C8—22no—
123C24—1941no—
124C24D38811yescharacteristic
125C2×D4—972no—
126C2×D4—3884no—
127C2×D4—1944no—
128C2×D4—1942no—
129C2×D4—1941no—
130C2×D4—1942no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—1942no—
134C22×C4—1941no—
135C22×C4C4×D9712yes—
136C22×C4D38811yescharacteristic
137C22×C4C22⋊D9712yes—
138C22×C4C22×D9711yescharacteristic
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C4⋊C4—972no—
143C4⋊C4—974no—
144C4⋊C4—1942no—
145C4⋊C4—1941no—
146C22⋊C4—1944no—
147C22⋊C4—1944no—
148C42—1944no—
149C23—21no—
150C23—21no—
151C23—22no—
152C23—22no—
153C23—3881no—
154C23—3881no—
155C23—3882no—
156C23—3882no—
157C23—1942no—
158C23#1309548812yes—
159C23C2×D38811yescharacteristic
160D4—3888no—
161D4—3882no—
162D4—1944no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—24no—
166C2×C4—24no—
167C2×C4—3882no—
168C2×C4—3882no—
169C2×C4—3882no—
170C2×C4—1942no—
171C2×C4—1944no—
172C2×C4#1309548812yes—
173C2×C4C2×C4×D9711yescharacteristic
174C2×C4D4×D9712yes—
175C2×C4C2×C22⋊D9711yescharacteristic
176C2×C4#1309551012yes—
177C8—42no—
178C22—22no—
179C22—22no—
180C22—22no—
181C22—42no—
182C22—42no—
183C22—7762no—
184C22—7762no—
185C22—3882no—
186C22—3882no—
187C22—3882no—
188C22C2×#1309548811yescharacteristic
189C22#1569078011yescharacteristic
190C22#1569079411yescharacteristic
191C22#1569083212yes—
192C22#1569083812yes—
193C4—22no—
194C4—3884no—
195C4—3882no—
196C4#1569077012yes—
197C4#1569078212yes—
198C2—42no—
199C2—7762no—
200C2#2026061211yescharacteristic
201C2#2026073211yescharacteristic
202C2#2026074111yescharacteristic
203C2#2026088212yes—
204C2#2026106012yes—
subgroupG/H
centerC23C2×D388
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×C22⋊C8×C97C2
socleC23×C97C2×D4
prime radicalC2×#15690784C2
cosocleC22×C4×C97C23
hypercenterC2×C22⋊C8D97
hypocenterC97#1875
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#15690769C2semidirect
3. C2×#15690732C2semidirect
4. #20260498C2semidirect
5. C2×C22⋊C8×C97C2semidirect
9. #15690784C4semidirect
32. C4⋊Dic97D4semidirect
103. C97#1875semidirect
107. C2×C22⋊C8D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907842
C2×#156907692
C2×#156907322
#202604982
C2×C22⋊C8×C972
#187597