Finite Group Databasebeta

← #27141012

Subgroups — #27141012

Order 12416.

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

normal subgroups only

This group has 194 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
2#20261131C211yesmaximal, characteristic
3#20260942C211yesmaximal, characteristic
4#20260482C211yesmaximal, characteristic
5Q8⋊#11715895C211yesmaximal, characteristic
6#20260465C211yesmaximal, characteristic
7#475×C97C211yesmaximal, characteristic
8C2×#13095501—21no—
9#15690810C2211yescharacteristic
10#15690808—21no—
11#15690802—21no—
12#15690800—22no—
13C4×Dic194—21no—
14C4×Dic194C2211yescharacteristic
15#15690775—21no—
16#15690741—22no—
17C4⋊#11715895C2211yescharacteristic
18#15690734—21no—
19C4×#11715895C2211yescharacteristic
20C4⋊Q8×C97C2211yescharacteristic
21C4×Q8×C97C2211yescharacteristic
22#106×C97—21no—
23C4⋊C8×C97C2211yescharacteristic
24#103×C97—21no—
25C2×Dic194—21no—
26C2×Dic194D411yescharacteristic
27#13095501—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×C97—21no—
40C2×Q8×C97D411yescharacteristic
41C2×C8×C97—21no—
42C4⋊C4×C97—21no—
43C4⋊C4×C97—21no—
44C4⋊C4×C97—21no—
45C4⋊C4×C97D411yescharacteristic
46C42×C97—21no—
47C42×C97C2311yescharacteristic
48Q8×C97—21no—
49Q8×C97—21no—
50Q8×C97—41no—
51Q8×C97—41no—
52C2×C4×C97—21no—
53C2×C4×C97—21no—
54C2×C4×C97—41no—
55C2×C4×C97C2×D411yescharacteristic
56C2×C4×C97C2×D411yescharacteristic
57C2×C4×C97C2×D411yescharacteristic
58C2×Dic97—21no—
59C2×Dic97—21no—
60C2×Dic97—22no—
61Dic194—21no—
62Dic194—41no—
63Dic194—41no—
64Dic194SD1612yes—
65C8×C97—41no—
66#11715895—22no—
67#11715895—41no—
68C22×C97#12011yescharacteristic
69C4×C97—21no—
70C4×C97—41no—
71C4×C97—41no—
72C4×C97—81no—
73C4×C97C2×SD1611yescharacteristic
74C4×C97#13511yescharacteristic
75C4×C97#13711yescharacteristic
76C4×C97#13711yescharacteristic
77Dic97—22no—
78Dic97—42no—
79Dic97—41no—
80C2×C97#44811yescharacteristic
81C2×C97#45211yescharacteristic
82C2×C97#45611yescharacteristic
83#1653—971nomaximal, Sylow
84C97#165311yescharacteristic, complemented, Sylow
85#557—971no—
86#498—971no—
87#475D9711yescharacteristic
88#462—971no—
89#332—971no—
90Q8⋊C8—971no—
91Q8⋊C8—971no—
92C2×Q16—1941no—
93C4⋊Q8—971no—
94C4⋊Q8—1941no—
95C4⋊Q8C2×D9711yescharacteristic
96#125—1941no—
97#125—1942no—
98C4×Q8—971no—
99C4×Q8—1941no—
100C4×Q8—1941no—
101C4×Q8C2×D9711yescharacteristic
102#106—21no—
103#106—1942no—
104C4⋊C8—971no—
105C4⋊C8C2×D9711yescharacteristic
106#103—21no—
107#103—1941no—
108C4×C8—971no—
109C2×Q8—21no—
110C2×Q8—971no—
111C2×Q8—1941no—
112C2×Q8C22⋊D9711yescharacteristic
113Q16—3882no—
114C2×C8—21no—
115C2×C8—1941no—
116C2×C8—1941no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—971no—
121C4⋊C4—972no—
122C4⋊C4—1942no—
123C4⋊C4—1942no—
124C4⋊C4—1942no—
125C4⋊C4—1941no—
126C4⋊C4—1941no—
127C4⋊C4C22⋊D9711yescharacteristic
128C42—21no—
129C42—1942no—
130C42—1941no—
131C42C22×D9711yescharacteristic
132Q8—21no—
133Q8—21no—
134Q8—41no—
135Q8—41no—
136Q8—972no—
137Q8—3881no—
138Q8—3881no—
139Q8—1941no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—41no—
143C2×C4—1941no—
144C2×C4—1942no—
145C2×C4—1941no—
146C2×C4D4×D9711yescharacteristic
147C2×C4D4×D9711yescharacteristic
148C2×C4C2×C22⋊D9711yescharacteristic
149C8—41no—
150C8—3881no—
151C8—1942no—
152C22#1569077811yescharacteristic
153C4—21no—
154C4—41no—
155C4—41no—
156C4—81no—
157C4—3882no—
158C4—3881no—
159C4—1942no—
160C4SD16×D9711yescharacteristic
161C4#1569083011yescharacteristic
162C4#1569083911yescharacteristic
163C4#1569084111yescharacteristic
164C2#2026087211yescharacteristic
165C2#2026089511yescharacteristic
166C2#2026091611yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#475×C97C2
socle, prime radicalC22×C97#120
hypercenter#475D97
hypocenterC97#1653
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
84. C97#1653semidirect

Maximal subgroups

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

subgroupindex
#202614632
#202611312
#202609422
#202604822
Q8⋊#117158952
#202604652
#475×C972
#165397