Finite Group Databasebeta

← #27157914

Subgroups — #27157914

Order 12416.

292 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 88 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#20261736C211yesmaximal, characteristic
2C4×C4⋊#87604C211yesmaximal, characteristic
3#20261608C214yesmaximal
4C4⋊Q8×D97C211yesmaximal, characteristic
5C2×#13095519C2212yes—
6C2×C4⋊#87604C2212yes—
7C2×C4×#87604—21no—
8#15690877—21no—
9C4⋊#11715897—21no—
10(C2×#87604)⋊C4—21no—
11C2×Q8×D97—21no—
12#15690810C411yescharacteristic, complemented
13#15690809—21no—
14#15690809C412yescomplemented
15#15690808—21no—
16C4⋊C4×D97—21no—
17C4⋊C4×D97C2212yes—
18C42×D97C2211yescharacteristic
19C4⋊Q8×C97C411yescharacteristic, complemented
20C22×#87604—22no—
21#13095519—24no—
22C2×#11715897—22no—
23C4⋊#87604—22no—
24C4⋊#87604Q814yescomplemented
25C4×#87604—42no—
26C2×Dic194—21no—
27C2×Dic194—22no—
28Q8×D97—44no—
29C2×C4×D97—22no—
30C2×C4×D97D411yescharacteristic
31C2×C4×D97D411yescharacteristic
32C2×C4×D97C2311yescharacteristic
33C4⋊Dic97—21no—
34C4⋊Dic97C2×C412yes—
35#13095490—22no—
36#13095490—22no—
37C4×Dic97C2×C411yescharacteristic
38C4×Dic97D411yescharacteristic
39C4×Dic97D411yescharacteristic
40C2×Q8×C97—21no—
41C4⋊C4×C97—21no—
42C4⋊C4×C97C2×C412yes—
43C42×C97C2×C411yescharacteristic
44C22×D97C2×D411yescharacteristic
45C2×#87604—24no—
46C2×#87604—42no—
47C2×#87604—42no—
48Q8×C97—42no—
49C2×C4×C97—22no—
50C2×C4×C97C22⋊C411yescharacteristic
51C2×C4×C97C22⋊C411yescharacteristic
52C2×C4×C97C22×C411yescharacteristic
53C2×Dic97—22no—
54C2×Dic97C22⋊C411yescharacteristic
55C2×Dic97C22⋊C411yescharacteristic
56C2×Dic97C2×D411yescharacteristic
57C4×D97—22no—
58C4×D97—44no—
59C4×D97SD1612yes—
60C4×D97Q1612yes—
61C4×D97C2×Q812yes—
62C4×D97#4112yes—
63Dic194—44no—
64Dic194—42no—
65#11715897—42no—
66C22×C97C2×C22⋊C411yescharacteristic
67C2×D97C22⋊Q812yes—
68C2×D97C2×SD1611yescharacteristic
69C2×D97C2×Q1611yescharacteristic
70C2×D97AGL(1,ℤ/8)12yes—
71#87604—44no—
72C4×C97—21no—
73C4×C97—42no—
74C4×C97#10312yes—
75C4×C97C4×Q812yes—
76Dic97—21no—
77Dic97—42no—
78Dic97#10312yes—
79Dic97#12411yescharacteristic
80Dic97C4⋊Q811yescharacteristic
81C2×C97#39111yescharacteristic
82C2×C97C2×#10311yescharacteristic
83C2×C97#41811yescharacteristic
84D97#47612yes—
85D97#47712yes—
86#1987—971nomaximal, Sylow
87C97#198711yescharacteristic, complemented, Sylow
88C2×C4⋊Q8—971no—
89C2×C4⋊C8—971no—
90C4×C4⋊C4—971no—
91#340—974no—
92C22×Q8—1941no—
93C22×C8—972no—
94C4⋊Q8—972no—
95C4⋊Q8—971no—
96C4⋊Q8—1941no—
97C4⋊Q8—1941no—
98C4⋊Q8#8760411yescharacteristic, complemented
99C2×C4⋊C4—972no—
100C2×C4⋊C4—972no—
101C2×C4⋊C4—1941no—
102C2×C42—971no—
103C2×C42—1941no—
104C4⋊C8—1941no—
105C4⋊C8—1941no—
106#95—1941no—
107C2×Q8—21no—
108C2×Q8—3884no—
109C2×Q8—1942no—
110C2×Q8—1941no—
111C22×C4—971no—
112C22×C4—971no—
113C22×C4—971no—
114C22×C4—1942no—
115C22×C4—1942no—
116C2×C8—1942no—
117C2×C8—1944no—
118C4⋊C4—21no—
119C4⋊C4—974no—
120C4⋊C4—972no—
121C4⋊C4—1942no—
122C4⋊C4—1942no—
123C4⋊C4—1942no—
124C4⋊C4—1941no—
125C4⋊C4C2×#8760412yes—
126C42—971no—
127C42—971no—
128C42—971no—
129C42—3882no—
130C42C2×#8760411yescharacteristic
131C23—971no—
132Q8—42no—
133Q8—3882no—
134Q8—3884no—
135C2×C4—22no—
136C2×C4—971no—
137C2×C4—971no—
138C2×C4—972no—
139C2×C4—972no—
140C2×C4—972no—
141C2×C4—972no—
142C2×C4—971no—
143C2×C4—3884no—
144C2×C4—3882no—
145C2×C4—3882no—
146C2×C4—1942no—
147C2×C4—1944no—
148C2×C4—1942no—
149C2×C4C22⋊#8760411yescharacteristic
150C2×C4C22⋊#8760411yescharacteristic
151C2×C4C22×#8760411yescharacteristic
152C8—3882no—
153C22—972no—
154C22—971no—
155C22—971no—
156C22—972no—
157C22C2×C22⋊#8760411yescharacteristic
158C4—21no—
159C4—42no—
160C4—971no—
161C4—971no—
162C4—972no—
163C4—3884no—
164C4—3882no—
165C4—1941no—
166C4#1569087312yes—
167C4Q8×#8760412yes—
168C2—972no—
169C2—972no—
170C2(C2×#87604)⋊Q811yescharacteristic
171C2C2×#1569087311yescharacteristic
172C2#2026172711yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4⋊Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C2×D4
cosocleC2×C4×D97C23
hypercenterC4⋊Q8#87604
hypocenterC97#1987
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
12. #15690810C4semidirect
14. #15690809C4semidirect
19. C4⋊Q8×C97C4semidirect
24. C4⋊#87604Q8semidirect
87. C97#1987semidirect
98. C4⋊Q8#87604semidirect

Maximal subgroups

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

subgroupindex
#202617362
C4×C4⋊#876042
#202616082
C4⋊Q8×D972
#198797