Finite Group Databasebeta

← #27157738

Subgroups — #27157738

Order 12416.

332 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 66 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#20261815C211yesmaximal, characteristic, complemented
2#20261734C212yesmaximal, complemented
3#20261716C211yesmaximal, characteristic, complemented
4#20261606C212yesmaximal, complemented
5#20261286C211yesmaximal, characteristic
6#15690898—24no—
7D97⋊M16—22no—
8D97⋊M16C2212yes—
9#15690889C2212yes—
10#15690875—24no—
11#15690869—22no—
12C2×#13095512C2211yescharacteristic
13C2×#13095512C2211yescharacteristic
14#15690807C411yescharacteristic, complemented
15#15690799C411yescharacteristic, complemented
16#15690799C411yescharacteristic, complemented
17#15690785—22no—
18#15690772—22no—
19C42×D97C2211yescharacteristic
20C4⋊Q8×C97C411yescharacteristic, complemented
21#13095521—24no—
22#13095521D414yescomplemented
23#13095520—22no—
24#13095519—24no—
25#13095519—22no—
26C4⋊#87604—22no—
27C22⋊#87604—22no—
28C4×#87604—22no—
29#13095512—22no—
30#13095512—22no—
31#13095512—22no—
32#13095512—22no—
33C2×D388—22no—
34C2×D388—21no—
35C2×D388C2×C411yescharacteristic
36C2×D388C2×C411yescharacteristic
37C2×D388D412yes—
38C2×C4×D97—21no—
39C2×C4×D97—21no—
40C2×C4×D97D412yes—
41C2×C4×D97C2311yescharacteristic
42#13095488—22no—
43#13095488—22no—
44C4×Dic97—21no—
45C4×Dic97C2×C411yescharacteristic
46C2×Q8×C97C2×C411yescharacteristic
47C2×Q8×C97C2×C411yescharacteristic
48C4⋊C4×C97—22no—
49C42×C97C2×C411yescharacteristic
50C22×D97—21no—
51C22×D97C22⋊C412yes—
52C22×D97C2×D411yescharacteristic
53C2×#87604—42no—
54Q8×C97—22no—
55Q8×C97—22no—
56C2×C4×C97—21no—
57C2×C4×C97—21no—
58C2×C4×C97C22⋊C412yes—
59C2×C4×C97C22×C411yescharacteristic
60C2×Dic97—21no—
61C2×Dic97C2×D411yescharacteristic
62D388—24no—
63D388—22no—
64D388—22no—
65D388—44no—
66D388—42no—
67C4×D97—22no—
68C4×D97—22no—
69C4×D97—42no—
70C4×D97—41no—
71C4×D97—41no—
72C4×D97C2×D412yes—
73C4×D97#4112yes—
74#11715897—24no—
75C22×C97C2×C22⋊C411yescharacteristic
76C2×D97—22no—
77C2×D97—41no—
78C2×D97—42no—
79C2×D97—42no—
80C2×D97—41no—
81C2×D97#12112yes—
82#87604—82no—
83C4×C97—22no—
84C4×C97—22no—
85C4×C97—41no—
86C4×C97C4×D412yes—
87Dic97—41no—
88Dic97#12011yescharacteristic
89Dic97#12311yescharacteristic
90C2×C97—21no—
91C2×C97#38611yescharacteristic
92D97—22no—
93D97—42no—
94D97—81no—
95#1884—971nomaximal, Sylow
96C97#188411yescharacteristic, complemented, Sylow
97#568—971no—
98#532—971no—
99#421—972no—
100#410—971no—
101#339—972no—
102C2×#41—971no—
103C2×#41—971no—
104#131—1944no—
105C2×M16—972no—
106C2×M16—1942no—
107C4⋊Q8#8760411yescharacteristic, complemented
108#127—971no—
109#124—971no—
110#124—971no—
111#121—1942no—
112C4×D4—1942no—
113#117—972no—
114C2×C42—971no—
115#104—1944no—
116#99—1942no—
117#41—1942no—
118#41—1942no—
119#41—1942no—
120#41—1942no—
121C2×Q8C2×#8760411yescharacteristic
122C2×Q8C2×#8760411yescharacteristic
123C2×D4—972no—
124C2×D4—971no—
125C2×D4—971no—
126C2×D4—1942no—
127C2×D4—1941no—
128C22×C4—971no—
129C22×C4—972no—
130C22×C4—1941no—
131C22×C4—1941no—
132M16—974no—
133M16—1942no—
134M16—1944no—
135C2×C8—1942no—
136C2×C8—1944no—
137C4⋊C4—22no—
138C4⋊C4—1942no—
139C22⋊C4—1942no—
140C22⋊C4—1942no—
141C22⋊C4—1942no—
142C42—971no—
143C42—1941no—
144C42—1942no—
145C42C2×#8760411yescharacteristic
146C23—971no—
147C23—972no—
148C23—1941no—
149Q8—22no—
150Q8—22no—
151D4—3884no—
152D4—3882no—
153D4—1944no—
154D4—1942no—
155D4—1942no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—971no—
159C2×C4—972no—
160C2×C4—972no—
161C2×C4—3882no—
162C2×C4—3881no—
163C2×C4—3881no—
164C2×C4—3882no—
165C2×C4—1942no—
166C2×C4—1942no—
167C2×C4—1941no—
168C2×C4C22⋊#8760412yes—
169C2×C4C22×#8760411yescharacteristic
170C8—1944no—
171C22—972no—
172C22—3882no—
173C22—3882no—
174C22—3881no—
175C22—3881no—
176C22—1942no—
177C22C2×C22⋊#8760411yescharacteristic
178C4—22no—
179C4—22no—
180C4—41no—
181C4—7762no—
182C4—971no—
183C4—971no—
184C4—3881no—
185C4D4×#8760412yes—
186C2—21no—
187C2—7761no—
188C2—3882no—
189C2—1942no—
190C2#2026168311yescharacteristic
subgroupG/H
centerC2#20261683
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4⋊Q8×C97C4
socleC2×C97#386
prime radical#20261286C2
cosocleC2×C4×D97C23
hypercenterC4⋊Q8#87604
hypocenterC97#1884
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #20261815C2semidirect
2. #20261734C2semidirect
3. #20261716C2semidirect
4. #20261606C2semidirect
14. #15690807C4semidirect
15. #15690799C4semidirect
16. #15690799C4semidirect
20. C4⋊Q8×C97C4semidirect
22. #13095521D4semidirect
96. C97#1884semidirect
107. C4⋊Q8#87604semidirect

Maximal subgroups

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

subgroupindex
#202618152
#202617342
#202617162
#202616062
#202612862
#188497