Finite Group Databasebeta

← #27157920

Subgroups — #27157920

Order 12416.

276 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 78 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#20261694C211yesmaximal, characteristic
2#20261679C211yesmaximal, characteristic
3#20261608C214yesmaximal
4C4⋊Q8×D97C211yesmaximal, characteristic
5C2×#13095519C2212yes—
6C2×C4⋊#87604C2212yes—
7#15690865—21no—
8#15690865—21no—
9(C2×#87604)⋊C4—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—22no—
22#13095519Q814yescomplemented
23C2×#11715897—22no—
24C4⋊#87604—24no—
25C2×Dic194—21no—
26C2×Dic194—22no—
27Q8×D97—44no—
28C2×C4×D97—22no—
29C2×C4×D97D411yescharacteristic
30C2×C4×D97D411yescharacteristic
31C2×C4×D97C2311yescharacteristic
32C4⋊Dic97—21no—
33C4⋊Dic97C2×C412yes—
34#13095490—22no—
35#13095490—22no—
36C4×Dic97C2×C411yescharacteristic
37C4×Dic97D411yescharacteristic
38C4×Dic97D411yescharacteristic
39C2×Q8×C97—21no—
40C4⋊C4×C97—21no—
41C4⋊C4×C97C2×C412yes—
42C42×C97C2×C411yescharacteristic
43C22×D97C2×D411yescharacteristic
44C2×#87604—42no—
45C2×#87604—42no—
46C2×#87604—42no—
47Q8×C97—42no—
48C2×C4×C97—22no—
49C2×C4×C97C22⋊C411yescharacteristic
50C2×C4×C97C22⋊C411yescharacteristic
51C2×C4×C97C22×C411yescharacteristic
52C2×Dic97—22no—
53C2×Dic97C22⋊C411yescharacteristic
54C2×Dic97C22⋊C411yescharacteristic
55C2×Dic97C2×D411yescharacteristic
56C4×D97—22no—
57C4×D97—22no—
58C4×D97—44no—
59C4×D97C2×Q812yes—
60C4×D97#4112yes—
61Dic194—44no—
62Dic194—42no—
63#11715897—24no—
64C22×C97C2×C22⋊C411yescharacteristic
65C2×D97#12411yescharacteristic
66C2×D97C4⋊Q811yescharacteristic
67C2×D97AGL(1,ℤ/8)12yes—
68C2×D97#13712yes—
69#87604—82no—
70C4×C97—21no—
71C4×C97—21no—
72C4×C97—42no—
73C4×C97C4×Q812yes—
74Dic97—21no—
75Dic97—21no—
76Dic97—42no—
77Dic97C22⋊Q812yes—
78C2×C97#39111yescharacteristic
79C2×C97#41811yescharacteristic
80C2×C97#41911yescharacteristic
81D97#48912yes—
82D97#50112yes—
83#1993—971nomaximal, Sylow
84C97#199311yescharacteristic, complemented, Sylow
85C2×C4⋊Q8—971no—
86C2×#97—971no—
87#382—971no—
88#340—974no—
89C22×Q8—1941no—
90C22×C8—972no—
91C4⋊Q8—972no—
92C4⋊Q8—971no—
93C4⋊Q8—1941no—
94C4⋊Q8—1941no—
95C4⋊Q8#8760411yescharacteristic, complemented
96C2×C4⋊C4—972no—
97C2×C4⋊C4—972no—
98C2×C4⋊C4—1941no—
99C2×C42—971no—
100#97—1941no—
101#97—1941no—
102#95—1941no—
103#95—1941no—
104C2×Q8—21no—
105C2×Q8—3884no—
106C2×Q8—1942no—
107C2×Q8—1941no—
108C22×C4—971no—
109C22×C4—971no—
110C22×C4—971no—
111C22×C4—1942no—
112C22×C4—1942no—
113C2×C8—974no—
114C2×C8—1942no—
115C2×C8—1942no—
116C4⋊C4—21no—
117C4⋊C4—972no—
118C4⋊C4—1942no—
119C4⋊C4—1942no—
120C4⋊C4—1944no—
121C4⋊C4—1941no—
122C4⋊C4C2×#8760412yes—
123C42—971no—
124C42—971no—
125C42—971no—
126C42C2×#8760411yescharacteristic
127C23—971no—
128Q8—42no—
129Q8—3882no—
130Q8—3884no—
131C2×C4—22no—
132C2×C4—971no—
133C2×C4—972no—
134C2×C4—972no—
135C2×C4—971no—
136C2×C4—971no—
137C2×C4—3884no—
138C2×C4—3882no—
139C2×C4—3882no—
140C2×C4—3882no—
141C2×C4—1942no—
142C2×C4—1942no—
143C2×C4—1942no—
144C2×C4C22⋊#8760411yescharacteristic
145C2×C4C22⋊#8760411yescharacteristic
146C2×C4C22×#8760411yescharacteristic
147C8—1944no—
148C22—971no—
149C22—971no—
150C22—972no—
151C22—972no—
152C22C2×C22⋊#8760411yescharacteristic
153C4—21no—
154C4—21no—
155C4—42no—
156C4—7762no—
157C4—972no—
158C4—3882no—
159C4—1941no—
160C4—1941no—
161C4Q8×#8760412yes—
162C2—972no—
163C2—972no—
164C2(C2×#87604)⋊Q811yescharacteristic
165C2#2026172711yescharacteristic
166C2#2026172811yescharacteristic
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#1993
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
22. #13095519Q8semidirect
84. C97#1993semidirect
95. C4⋊Q8#87604semidirect

Maximal subgroups

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

subgroupindex
#202616942
#202616792
#202616082
C4⋊Q8×D972
#199397