Finite Group Databasebeta

← #27157913

Subgroups — #27157913

Order 12416.

260 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 72 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
1C4×#13095521C211yesmaximal, characteristic
2C4×C4⋊#87604C211yesmaximal, characteristic
3#20261605C212yesmaximal
4#20261605C212yesmaximal
5#20261291C211yesmaximal, characteristic
6C2×#13095521C2212yes—
7C2×C4⋊#87604—21no—
8C2×C4×#87604C2212yes—
9#15690865—21no—
10C4×#11715897—21no—
11(C2×#87604)⋊C4—21no—
12#15690809C411yescharacteristic, complemented
13#15690809C411yescharacteristic, complemented
14#15690802C411yescharacteristic, complemented
15#15690788—21no—
16#15690788—21no—
17#15690775—21no—
18#15690775—21no—
19#15690768C2211yescharacteristic
20#15690768C2211yescharacteristic
21C42×D97C2211yescharacteristic
22#125×C97C411yescharacteristic, complemented
23C22×#87604—22no—
24#13095521—22no—
25#13095521Q814yescomplemented
26C2×#11715897—22no—
27C4⋊#87604—24no—
28C4×#87604—24no—
29C2×Dic194—21no—
30C2×Dic194—21no—
31C2×C4×D97D411yescharacteristic
32C2×C4×D97D411yescharacteristic
33C2×C4×D97C2311yescharacteristic
34C4⋊Dic97C2×C411yescharacteristic
35C4⋊Dic97C2×C411yescharacteristic
36#13095490—21no—
37#13095490—21no—
38#13095490—21no—
39#13095490—21no—
40#13095488—21no—
41#13095488—21no—
42C4×Dic97—21no—
43C4×Dic97—21no—
44C4×Dic97C2×C411yescharacteristic
45C4×Dic97D411yescharacteristic
46C4×Dic97D411yescharacteristic
47C4⋊C4×C97—21no—
48C4⋊C4×C97—21no—
49C4⋊C4×C97C2×C411yescharacteristic
50C4⋊C4×C97C2×C411yescharacteristic
51C42×C97C2×C411yescharacteristic
52C22×D97C2×D411yescharacteristic
53C2×#87604—24no—
54C2×#87604—42no—
55C2×#87604—42no—
56C2×C4×C97—21no—
57C2×C4×C97—21no—
58C2×C4×C97C22⋊C411yescharacteristic
59C2×C4×C97C22⋊C411yescharacteristic
60C2×C4×C97C22×C411yescharacteristic
61C2×Dic97—21no—
62C2×Dic97—21no—
63C2×Dic97C22⋊C411yescharacteristic
64C2×Dic97C22⋊C411yescharacteristic
65C2×Dic97C2×D411yescharacteristic
66C4×D97—22no—
67C4×D97—41no—
68C4×D97C2×Q812yes—
69C4×D97#4112yes—
70Dic194—42no—
71Dic194—42no—
72#11715897—24no—
73C22×C97C2×C22⋊C411yescharacteristic
74C2×D97—21no—
75C2×D97—21no—
76C2×D97#12411yescharacteristic
77C2×D97C4⋊Q811yescharacteristic
78#87604—44no—
79C4×C97—21no—
80C4×C97—21no—
81C4×C97—41no—
82C4×C97—41no—
83C4×C97C4×Q812yes—
84Dic97—21no—
85Dic97—41no—
86Dic97—41no—
87Dic97#10412yes—
88Dic97C22⋊Q812yes—
89C2×C97#39111yescharacteristic
90C2×C97C2×#10411yescharacteristic
91C2×C97#42111yescharacteristic
92D97—21no—
93D97—21no—
94#1986—971nomaximal, Sylow
95C97#198611yescharacteristic, complemented, Sylow
96#533—971no—
97C4×M16—971no—
98C4×C4⋊C4—971no—
99#339—972no—
100#339—972no—
101C2×M16—972no—
102C4⋊Q8—971no—
103C4⋊Q8—971no—
104#125—971no—
105#125#8760411yescharacteristic, complemented
106C22⋊Q8—1941no—
107C22⋊Q8—1941no—
108C4×Q8—1941no—
109C4×Q8—1941no—
110#117—971no—
111#117—971no—
112C2×C4⋊C4—1941no—
113C2×C42—971no—
114C2×C42—972no—
115#97—1941no—
116C4×C8—1941no—
117#95—1941no—
118C2×Q8—1941no—
119C2×Q8—1941no—
120C22×C4—971no—
121C22×C4—971no—
122C22×C4—971no—
123C22×C4—1942no—
124M16—974no—
125M16—1942no—
126C2×C8—1942no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—971no—
130C4⋊C4—971no—
131C4⋊C4—1941no—
132C4⋊C4—1941no—
133C4⋊C4—1944no—
134C4⋊C4—1941no—
135C4⋊C4—1941no—
136C4⋊C4C2×#8760411yescharacteristic
137C4⋊C4C2×#8760411yescharacteristic
138C22⋊C4—1941no—
139C22⋊C4—1941no—
140C42—971no—
141C42—971no—
142C42—971no—
143C42—1941no—
144C42—1941no—
145C42—1944no—
146C42C2×#8760411yescharacteristic
147C23—971no—
148Q8—3882no—
149Q8—3882no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—971no—
153C2×C4—971no—
154C2×C4—972no—
155C2×C4—972no—
156C2×C4—971no—
157C2×C4—3881no—
158C2×C4—3882no—
159C2×C4—3882no—
160C2×C4—1942no—
161C2×C4—1944no—
162C2×C4—1941no—
163C2×C4—1941no—
164C2×C4C22⋊#8760411yescharacteristic
165C2×C4C22⋊#8760411yescharacteristic
166C2×C4C22×#8760411yescharacteristic
167C8—1944no—
168C22—971no—
169C22—971no—
170C22—1941no—
171C22—1941no—
172C22C2×C22⋊#8760411yescharacteristic
173C4—21no—
174C4—21no—
175C4—41no—
176C4—41no—
177C4—972no—
178C4—972no—
179C4—3884no—
180C4—3881no—
181C4—3881no—
182C4—1941no—
183C4Q8×#8760412yes—
184C2—1941no—
185C2—1941no—
186C2(C2×#87604)⋊Q811yescharacteristic
187C2D97⋊#10411yescharacteristic
188C2#2026173211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#125×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C2×D4
cosocleC2×C4×D97C23
hypercenter#125#87604
hypocenterC97#1986
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
12. #15690809C4semidirect
13. #15690809C4semidirect
14. #15690802C4semidirect
22. #125×C97C4semidirect
25. #13095521Q8semidirect
95. C97#1986semidirect
105. #125#87604semidirect

Maximal subgroups

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

subgroupindex
C4×#130955212
C4×C4⋊#876042
#202616052
#202612912
#198697