Finite Group Databasebeta

← #27140830

Subgroups — #27140830

Order 12416.

210 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 55 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#20261463C211yesmaximal, characteristic, complemented
2#20261081C211yesmaximal, characteristic
3#20260932C211yesmaximal, characteristic
4#20260482C211yesmaximal, characteristic, complemented
5Q8⋊#11715895C211yesmaximal, characteristic, complemented
6#20260465C211yesmaximal, characteristic, complemented
7#460×C97C211yesmaximal, characteristic
8C2×#13095499—21no—
9#15690810C2211yescharacteristic
10#15690808—21no—
11#15690802—21no—
12#15690800—22no—
13C4×Dic194—21no—
14C4×Dic194C2211yescharacteristic
15#15690775—21no—
16C4⋊#11715895C2211yescharacteristic
17#15690731—21no—
18#15690731—22no—
19C4×#11715895C2211yescharacteristic
20C2×SD16×C97—21no—
21#127×C97C2211yescharacteristic
22C4×Q8×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24#102×C97—21no—
25C2×Dic194—21no—
26C2×Dic194D411yescharacteristic
27#13095499—42no—
28C2×#11715895—21no—
29C2×#11715895—21no—
30C4⋊Dic97—21no—
31C4⋊Dic97D411yescharacteristic
32C4⋊Dic97D412yes—
33#13095490—22no—
34#13095490—22no—
35#13095490—22no—
36#13095490—21no—
37C4×Dic97—22no—
38C4×Dic97—21no—
39C2×Q8×C97D411yescharacteristic
40C2×D4×C97—21no—
41C2×D4×C97—21no—
42C2×D4×C97—21no—
43SD16×C97—42no—
44C2×C8×C97—21no—
45C4⋊C4×C97—21no—
46C4⋊C4×C97D411yescharacteristic
47C42×C97—21no—
48C42×C97C2311yescharacteristic
49C23×C97—41no—
50Q8×C97—21no—
51Q8×C97—21no—
52D4×C97—42no—
53D4×C97—41no—
54D4×C97—41no—
55D4×C97—41no—
56D4×C97—41no—
57C2×C4×C97—21no—
58C2×C4×C97—21no—
59C2×C4×C97C2×D411yescharacteristic
60C2×C4×C97C2×D411yescharacteristic
61C2×C4×C97C2×D411yescharacteristic
62C2×Dic97—21no—
63C2×Dic97—21no—
64C2×Dic97—22no—
65Dic194—21no—
66Dic194—41no—
67Dic194—41no—
68Dic194SD1612yescomplemented
69C8×C97—41no—
70#11715895—41no—
71#11715895—41no—
72C22×C97—81no—
73C22×C97—81no—
74C22×C97—81no—
75C22×C97#12011yescharacteristic
76C4×C97—21no—
77C4×C97—41no—
78C4×C97—41no—
79C4×C97C2×SD1611yescharacteristic
80C4×C97#13511yescharacteristic
81C4×C97AGL(1,ℤ/8)11yescharacteristic
82C4×C97#13711yescharacteristic
83Dic97—22no—
84Dic97—42no—
85Dic97—41no—
86C2×C97—161no—
87C2×C97#44811yescharacteristic
88C2×C97#44911yescharacteristic
89C2×C97#45411yescharacteristic
90#1626—971nomaximal, Sylow
91C97#162611yescharacteristic, complemented, Sylow
92#557—971no—
93#486—971no—
94#460—971no—
95#460D9711yescharacteristic
96#332—971no—
97Q8⋊C8—971no—
98Q8⋊C8—971no—
99C2×SD16—21no—
100C2×SD16—1941no—
101C4⋊Q8—971no—
102C4⋊Q8—1941no—
103#127C2×D9711yescharacteristic
104#125—1941no—
105#125—1942no—
106C4×Q8—971no—
107C4×Q8—1941no—
108C4×Q8—1941no—
109C4×Q8C2×D9711yescharacteristic
110C4⋊C8—971no—
111C4⋊C8C2×D9711yescharacteristic
112#102—21no—
113#102—1941no—
114#102—1942no—
115C4×C8—971no—
116C2×Q8—971no—
117C2×Q8—1941no—
118C2×Q8C22⋊D9711yescharacteristic
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—21no—
122SD16—42no—
123SD16—3882no—
124C2×C8—21no—
125C2×C8—1941no—
126C2×C8—1941no—
127C4⋊C4—21no—
128C4⋊C4—971no—
129C4⋊C4—972no—
130C4⋊C4—1942no—
131C4⋊C4—1942no—
132C4⋊C4—1942no—
133C4⋊C4—1941no—
134C4⋊C4—1941no—
135C4⋊C4C22⋊D9711yescharacteristic
136C42—21no—
137C42—1942no—
138C42—1941no—
139C42C22×D9711yescharacteristic
140C23—41no—
141Q8—21no—
142Q8—21no—
143Q8—972no—
144Q8—3881no—
145Q8—3881no—
146Q8—1941no—
147D4—42no—
148D4—41no—
149D4—41no—
150D4—41no—
151D4—41no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—1941no—
155C2×C4—1942no—
156C2×C4—1941no—
157C2×C4D4×D9711yescharacteristic
158C2×C4D4×D9711yescharacteristic
159C2×C4C2×C22⋊D9711yescharacteristic
160C8—41no—
161C8—3881no—
162C8—3881no—
163C22—81no—
164C22—81no—
165C22—81no—
166C22#1569077811yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—3882no—
171C4—3881no—
172C4—1942no—
173C4SD16×D9711yescharacteristic
174C4#1569083011yescharacteristic
175C4#1569083611yescharacteristic
176C4#1569083911yescharacteristic
177C2—161no—
178C2#2026087211yescharacteristic
179C2#2026087811yescharacteristic
180C2#2026090611yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#460×C97C2
socleC22×C97#120
prime radical#127×C97C22
hypercenter#460D97
hypocenterC97#1626
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #20261463C2semidirect
4. #20260482C2semidirect
5. Q8⋊#11715895C2semidirect
6. #20260465C2semidirect
68. Dic194SD16semidirect
91. C97#1626semidirect

Maximal subgroups

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

subgroupindex
#202614632
#202610812
#202609322
#202604822
Q8⋊#117158952
#202604652
#460×C972
#162697