Finite Group Databasebeta

← #27140891

Subgroups — #27140891

Order 12416.

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

normal subgroups only

This group has 188 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#20261460C211yesmaximal, characteristic
2#20261027C211yesmaximal, characteristic
3#20261027C211yesmaximal, characteristic
4#20260481C211yesmaximal, characteristic
5#20260467C211yesmaximal, characteristic
6#20260467C211yesmaximal, characteristic
7#487×C97C211yesmaximal, characteristic
8#15690808—21no—
9#15690801—21no—
10#15690800—22no—
11Q8×Dic97—21no—
12C4×Dic194C2211yescharacteristic
13C4×Dic194C2211yescharacteristic
14#15690775—21no—
15#15690743—21no—
16#15690743—21no—
17C4⋊#11715895C2211yescharacteristic
18C4⋊#11715895C2211yescharacteristic
19#15690733—21no—
20#15690733—21no—
21C4⋊Q8×C97C2211yescharacteristic
22C4⋊Q8×C97C2211yescharacteristic
23#103×C97—22no—
24C4×C8×C97C2211yescharacteristic
25C2×Dic194D411yescharacteristic
26C2×Dic194D411yescharacteristic
27C2×#11715895—21no—
28C2×#11715895—21no—
29C4⋊Dic97—22no—
30C4⋊Dic97D411yescharacteristic
31C4⋊Dic97D411yescharacteristic
32#13095490—22no—
33#13095490—22no—
34#13095490—21no—
35#13095490—21no—
36C4×Dic97—22no—
37C4×Dic97—21no—
38C4×Dic97—21no—
39C2×Q8×C97—21no—
40C2×Q8×C97—21no—
41C2×C8×C97—21no—
42C4⋊C4×C97—21no—
43C4⋊C4×C97—21no—
44C4⋊C4×C97—21no—
45C4⋊C4×C97D412yes—
46C42×C97C2311yescharacteristic
47Q8×C97—42no—
48Q8×C97—41no—
49Q8×C97—41no—
50C2×C4×C97—22no—
51C2×C4×C97—41no—
52C2×C4×C97C2×D411yescharacteristic
53C2×C4×C97C2×D411yescharacteristic
54C2×C4×C97C2×D411yescharacteristic
55C2×Dic97—21no—
56C2×Dic97—21no—
57C2×Dic97—21no—
58C2×Dic97—21no—
59Dic194—21no—
60Dic194—21no—
61Dic194—21no—
62Dic194Q1612yes—
63C8×C97—41no—
64#11715895—41no—
65#11715895—41no—
66C22×C97#12011yescharacteristic
67C4×C97—21no—
68C4×C97—42no—
69C4×C97—81no—
70C4×C97C2×Q1611yescharacteristic
71C4×C97#13511yescharacteristic
72C4×C97#13711yescharacteristic
73C4×C97#13711yescharacteristic
74Dic97—22no—
75Dic97—41no—
76Dic97—41no—
77Dic97—41no—
78C2×C97#45111yescharacteristic
79C2×C97#45211yescharacteristic
80C2×C97#45411yescharacteristic
81#1636—971nomaximal, Sylow
82C97#163611yescharacteristic, complemented, Sylow
83#557—971no—
84#487D9711yescharacteristic
85#477—971no—
86#477—971no—
87#332—971no—
88Q8⋊C8—971no—
89Q8⋊C8—971no—
90C4⋊Q8—1941no—
91C4⋊Q8C2×D9711yescharacteristic
92C4⋊Q8C2×D9711yescharacteristic
93#125—1941no—
94#125—1942no—
95C4×Q8—971no—
96C4×Q8—971no—
97C4×Q8—1941no—
98C4×Q8—1941no—
99#107—1941no—
100#107—1941no—
101C4⋊C8—971no—
102C4⋊C8—971no—
103#103—22no—
104#103—1941no—
105#103—1941no—
106C4×C8C2×D9711yescharacteristic
107C2×Q8—21no—
108C2×Q8—21no—
109C2×Q8—971no—
110C2×Q8—971no—
111C2×C8—21no—
112C2×C8—1941no—
113C2×C8—1941no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—971no—
118C4⋊C4—971no—
119C4⋊C4—1942no—
120C4⋊C4—1942no—
121C4⋊C4—1942no—
122C4⋊C4—1941no—
123C4⋊C4—1941no—
124C4⋊C4C22⋊D9712yes—
125C42—1942no—
126C42—1941no—
127C42—1941no—
128C42C22×D9711yescharacteristic
129Q8—42no—
130Q8—41no—
131Q8—41no—
132Q8—972no—
133Q8—1941no—
134Q8—1941no—
135Q8—1941no—
136C2×C4—22no—
137C2×C4—41no—
138C2×C4—1941no—
139C2×C4—1941no—
140C2×C4—1941no—
141C2×C4—1941no—
142C2×C4D4×D9711yescharacteristic
143C2×C4D4×D9711yescharacteristic
144C2×C4C2×C22⋊D9711yescharacteristic
145C8—41no—
146C8—3881no—
147C8—3881no—
148C22#1569077811yescharacteristic
149C4—21no—
150C4—42no—
151C4—81no—
152C4—3881no—
153C4—3881no—
154C4—3881no—
155C4—1942no—
156C4Q16×D9711yescharacteristic
157C4#1569083111yescharacteristic
158C4#1569083911yescharacteristic
159C4#1569084211yescharacteristic
160C2#2026088911yescharacteristic
161C2#2026089411yescharacteristic
162C2#2026090711yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#487×C97C2
socle, prime radicalC22×C97#120
hypercenter#487D97
hypocenterC97#1636
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
82. C97#1636semidirect

Maximal subgroups

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

subgroupindex
#202614602
#202610272
#202604812
#202604672
#487×C972
#163697