Finite Group Databasebeta

← #11794297

Subgroups — #11794297

Order 1152.

284 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 53 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#10596543C211yesmaximal, characteristic
2#10593736C211yesmaximal, characteristic
3#10593544C212yesmaximal
4#10590151C212yesmaximal
5#10590094C211yesmaximal, characteristic
6#64286C2212yes—
7C2×C4⋊#157C2211yescharacteristic
8C2×C4⋊#157C2211yescharacteristic
9#64283C2212yes—
10C22⋊PSU(3,𝔽2)—22no—
11C4⋊PSU(3,𝔽2)—22no—
12#64243—22no—
13C42×C3⋊D3C2211yescharacteristic
14#63781—22no—
15#63746—22no—
16#63745—22no—
17C22×#157—21no—
18C22×#157D412yes—
19C2×PSU(3,𝔽2)—22no—
20C2×C4×C3⋊D3—21no—
21C2×C4×C3⋊D3C2311yescharacteristic
22C22⋊#157—22no—
23#3792—22no—
24C4⋊#157—22no—
25C4⋊#157—22no—
26C4⋊#157D412yes—
27C4⋊#157D412yes—
28C4×#157—22no—
29#3788Q814yescomplemented
30#3787—22no—
31#3777—22no—
32#3777—24no—
33C32×C42Q811yescharacteristic, complemented
34C4×C3⋊Dic3—21no—
35C4×C3⋊Dic3Q811yescharacteristic, complemented
36#2061—91nomaximal, Sylow
37C42×D3—121no—
38C22×C3⋊D3C2×D411yescharacteristic
39C2×#157—22no—
40C2×#157—22no—
41C2×#157—22no—
42C2×#157—42no—
43C2×#157—42no—
44C2×#157—41no—
45PSU(3,𝔽2)—82no—
46C2×C32×C4—21no—
47C2×C32×C4C2×Q811yescharacteristic
48C2×C3⋊Dic3—21no—
49C2×C3⋊Dic3#4111yescharacteristic
50C4×C3⋊D3—41no—
51C4×C3⋊D3—41no—
52C4×C3⋊D3C2×D412yes—
53C4×C3⋊D3C2×Q812yes—
54#622—24no—
55#544—91no—
56#428—92no—
57#382—91no—
58#343—91no—
59#339—92no—
60C2×C4×D3—121no—
61C2×C4×D3—122no—
62C3×C42—41no—
63C4×Dic3—121no—
64C4×Dic3—122no—
65C22×C32C22⋊Q811yescharacteristic
66C2×C3⋊D3—21no—
67C2×C3⋊D3—21no—
68C2×C3⋊D3#12011yescharacteristic
69C2×C3⋊D3C4⋊Q811yescharacteristic
70#157—42no—
71#157—42no—
72#157—42no—
73#157—81no—
74C32×C4—41no—
75C32×C4C4⋊Q812yes—
76C3⋊Dic3—41no—
77C3⋊Dic3C22⋊Q812yes—
78C2×M16—92no—
79C4⋊Q8—182no—
80#125—182no—
81C22⋊Q8—182no—
82#117—92no—
83C2×C4⋊C4—91no—
84C2×C4⋊C4—91no—
85C2×C42—91no—
86#107—182no—
87#106—182no—
88#95—182no—
89C22×D3—121no—
90C2×C3×C4—42no—
91C2×C3×C4—41no—
92C2×Dic3—121no—
93C2×Dic3—122no—
94C4×D3—122no—
95C4×D3—122no—
96C4×D3—124no—
97C4×D3—124no—
98C2×C32—21no—
99C2×C32#39511yescharacteristic
100C3⋊D3—21no—
101C3⋊D3#45411yescharacteristic
102C3⋊D3#45511yescharacteristic
103C2×Q8—182no—
104C22×C4—91no—
105C22×C4—92no—
106C22×C4—181no—
107C22×C4—181no—
108M16—94no—
109M16—182no—
110C2×C8—182no—
111C4⋊C4—92no—
112C4⋊C4—92no—
113C4⋊C4—182no—
114C4⋊C4—182no—
115C4⋊C4—182no—
116C4⋊C4—184no—
117C22⋊C4—182no—
118C42—91no—
119C42—181no—
120C42—182no—
121C42PSU(3,𝔽2)11yescharacteristic, complemented
122C22×C3—41no—
123C2×D3—121no—
124C2×D3—121no—
125C2×D3—122no—
126C2×D3—122no—
127C3×C4—44no—
128C3×C4—42no—
129Dic3—122no—
130Dic3—124no—
131C32#206111yescharacteristic, complemented, Sylow
132C23—91no—
133Q8—722no—
134C2×C4—21no—
135C2×C4—91no—
136C2×C4—92no—
137C2×C4—92no—
138C2×C4—182no—
139C2×C4—182no—
140C2×C4—182no—
141C2×C4—181no—
142C2×C4—361no—
143C2×C4—361no—
144C2×C4—362no—
145C2×C4—361no—
146C2×C4—362no—
147C2×C4C2×PSU(3,𝔽2)11yescharacteristic
148C8—184no—
149C2×C3—42no—
150C2×C3—41no—
151D3—121no—
152D3—121no—
153D3—122no—
154C22—91no—
155C22—91no—
156C22—181no—
157C22—181no—
158C22C22⋊PSU(3,𝔽2)11yescharacteristic
159C4—41no—
160C4—92no—
161C4—361no—
162C4—362no—
163C4—362no—
164C4—362no—
165C4—721no—
166C4C4⋊PSU(3,𝔽2)12yes—
167C3—41no—
168C2—21no—
169C2—91no—
170C2—91no—
171C2—181no—
172C2#1059355611yescharacteristic
subgroupG/H
centerC2#10593556
commutator, cosocleC2×C4×C3⋊D3C23
FrattiniC2×C4C2×PSU(3,𝔽2)
FittingC32×C42Q8
socleC2×C32#395
prime radicalC22×C3⋊D3C2×D4
hypercenterC42PSU(3,𝔽2)
hypocenterC32#2061
LCSG ▷ C2×C4×C3⋊D3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
29. #3788Q8semidirect
33. C32×C42Q8semidirect
35. C4×C3⋊Dic3Q8semidirect
121. C42PSU(3,𝔽2)semidirect
131. C32#2061semidirect

Maximal subgroups

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

subgroupindex
#105965432
#105937362
#105935442
#105901512
#105900942
#20619