Finite Group Databasebeta

← #27141070

Subgroups — #27141070

Order 12416.

184 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 49 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#20261145C211yesmaximal, characteristic, complemented
2#20261105C211yesmaximal, characteristic, complemented
3#20261094C211yesmaximal, characteristic
4#20260477C211yesmaximal, characteristic, complemented
5#20260477C211yesmaximal, characteristic, complemented
6#20260475C211yesmaximal, characteristic
7#563×C97C211yesmaximal, characteristic
8#15690810C2211yescharacteristic
9#15690743—21no—
10#15690741—21no—
11#15690734—21no—
12#15690733—21no—
13#15690733—21no—
14#15690731—21no—
15#15690721C2211yescharacteristic
16#15690721C2211yescharacteristic
17#15690721C2211yescharacteristic
18C4⋊Q8×C97—21no—
19#126×C97—21no—
20#126×C97—21no—
21#125×C97C2211yescharacteristic
22#125×C97C2211yescharacteristic
23#124×C97C2211yescharacteristic
24#123×C97—21no—
25C22⋊Q8×C97—21no—
26C22⋊Q8×C97—21no—
27C2×Dic194—21no—
28C2×#11715895—21no—
29C2×#11715895—21no—
30C2×#11715895—21no—
31C4⋊Dic97—21no—
32C4⋊Dic97—21no—
33C2×Q8×C97—21no—
34C2×Q8×C97D411yescharacteristic
35C2×D4×C97D411yescharacteristic
36C22×C4×C97—21no—
37C4⋊C4×C97—21no—
38C4⋊C4×C97—21no—
39C4⋊C4×C97—21no—
40C4⋊C4×C97—21no—
41C4⋊C4×C97—21no—
42C4⋊C4×C97—21no—
43C4⋊C4×C97D411yescharacteristic
44C4⋊C4×C97D411yescharacteristic
45C4⋊C4×C97D411yescharacteristic
46C4⋊C4×C97D411yescharacteristic
47C22⋊C4×C97—21no—
48C22⋊C4×C97—21no—
49C22⋊C4×C97—21no—
50C22⋊C4×C97—21no—
51C22⋊C4×C97—21no—
52C42×C97—21no—
53C42×C97C2311yescharacteristic
54C23×C97—21no—
55Q8×C97—41no—
56Q8×C97—81no—
57D4×C97—41no—
58C2×C4×C97—21no—
59C2×C4×C97—21no—
60C2×C4×C97—21no—
61C2×C4×C97—21no—
62C2×C4×C97—21no—
63C2×C4×C97—81no—
64C2×C4×C97C2×D411yescharacteristic
65C2×C4×C97C2×D411yescharacteristic
66C2×C4×C97C2×D411yescharacteristic
67C2×Dic97—41no—
68Dic194—42no—
69#11715895—22no—
70#11715895—41no—
71#11715895—41no—
72C22×C97—41no—
73C22×C97—41no—
74C22×C97—41no—
75C22×C97#12011yescharacteristic
76C4×C97—21no—
77C4×C97—21no—
78C4×C97—21no—
79C4×C97—41no—
80C4×C97—41no—
81C4×C97—41no—
82C4×C97—41no—
83C4×C97—41no—
84Dic97—81no—
85C2×C97—81no—
86C2×C97#45411yescharacteristic
87C2×C97#45511yescharacteristic
88C2×C97#45611yescharacteristic
89#1664—971nomaximal, Sylow
90C97#166411yescharacteristic, complemented, Sylow
91#563D9711yescharacteristic
92#501—971no—
93#491—971no—
94#489—971no—
95#330—971no—
96#330—971no—
97#329—971no—
98C4⋊Q8—21no—
99C4⋊Q8—971no—
100#126—21no—
101#126—21no—
102#125C2×D9711yescharacteristic
103#125C2×D9711yescharacteristic
104#124C2×D9711yescharacteristic
105#123—21no—
106C22⋊Q8—21no—
107C22⋊Q8—21no—
108#107—1941no—
109#106—1941no—
110#103—1941no—
111#103—1941no—
112#103—1941no—
113#102—1941no—
114#97—971no—
115#97—971no—
116#97—971no—
117C2×Q8—21no—
118C2×Q8—1941no—
119C2×Q8C22⋊D9711yescharacteristic
120C2×D4C22⋊D9711yescharacteristic
121C22×C4—21no—
122C2×C8—1941no—
123C2×C8—1941no—
124C2×C8—1941no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—1941no—
132C4⋊C4—1941no—
133C4⋊C4C22⋊D9711yescharacteristic
134C4⋊C4C22⋊D9711yescharacteristic
135C4⋊C4C22⋊D9711yescharacteristic
136C4⋊C4C22⋊D9711yescharacteristic
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C22⋊C4—21no—
141C22⋊C4—21no—
142C42—21no—
143C42C22×D9711yescharacteristic
144C23—21no—
145Q8—41no—
146Q8—81no—
147Q8—3882no—
148D4—41no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—81no—
155C2×C4—3881no—
156C2×C4C2×C22⋊D9711yescharacteristic
157C2×C4C2×C22⋊D9711yescharacteristic
158C2×C4C2×C22⋊D9711yescharacteristic
159C8—3881no—
160C8—3881no—
161C8—1942no—
162C22—41no—
163C22—41no—
164C22—41no—
165C22#1569077911yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—41no—
170C4—41no—
171C4—41no—
172C4—41no—
173C4—41no—
174C4—7761no—
175C2—81no—
176C2#2026090811yescharacteristic
177C2#2026091311yescharacteristic
178C2#2026091811yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#563×C97C2
socleC22×C97#120
prime radicalC2×D4×C97D4
hypercenter#563D97
hypocenterC97#1664
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #20261145C2semidirect
2. #20261105C2semidirect
4. #20260477C2semidirect
5. #20260477C2semidirect
90. C97#1664semidirect

Maximal subgroups

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

subgroupindex
#202611452
#202611052
#202610942
#202604772
#202604752
#563×C972
#166497