Finite Group Databasebeta

← #11786160

Subgroups — #11786160

Order 1152.

304 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 52 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#10592406C211yesmaximal, characteristic
2#10590186C211yesmaximal, characteristic
3#10590184C211yesmaximal, characteristic
4#10590183C212yesmaximal
5#10590151C212yesmaximal
6#64286C2212yes—
7#64283C2212yes—
8C2×#3743C2211yescharacteristic
9C2×#3743C2211yescharacteristic
10C42×C3⋊D3C2211yescharacteristic
11C32×#97C411yescharacteristic, complemented
12C8×C3⋊Dic3C411yescharacteristic, complemented
13C8×C3⋊Dic3C411yescharacteristic, complemented
14#63630C411yescharacteristic, complemented
15#4707—62no—
16C2×C4×C3⋊D3—21no—
17C2×C4×C3⋊D3C2311yescharacteristic
18C22⋊#157—22no—
19#3792—22no—
20C4⋊#157—22no—
21C4×#157—22no—
22#3788—22no—
23#3787—22no—
24C2×C32×C8C2×C411yescharacteristic
25C2×C32×C8C2×C411yescharacteristic
26C32×C42C2×C411yescharacteristic
27C4×C3⋊Dic3—21no—
28C4×C3⋊Dic3C2×C411yescharacteristic
29C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
30C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
31#3743—22no—
32#3743—22no—
33#1841—91nomaximal, Sylow
34C2×ΓL(1,𝔽25)—62no—
35C2×ΓL(1,𝔽25)—62no—
36C42×D3—62no—
37C3×#97—22no—
38C8×Dic3—62no—
39C8×Dic3—62no—
40#803—62no—
41C22×C3⋊D3#4111yescharacteristic
42C2×#157—42no—
43C2×C32×C4—21no—
44C2×C32×C4C22×C411yescharacteristic
45C2×C3⋊Dic3—21no—
46C2×C3⋊Dic3#4111yescharacteristic
47C4×C3⋊D3—42no—
48C4×C3⋊D3#4112yes—
49C4×C3⋊D3#4112yes—
50#622—42no—
51C32×C8—21no—
52C32×C8—21no—
53C3⋊(C3⋊C8)—21no—
54C3⋊(C3⋊C8)—21no—
55C4×M16—91no—
56#344—92no—
57#339—92no—
58#338—91no—
59H(3,ℤ/4)—91no—
60C2×C4×D3—62no—
61C2×C4×D3—64no—
62C2×C3×C8—22no—
63C2×C3×C8—22no—
64C3×C42—22no—
65C4×Dic3—62no—
66C4×Dic3—64no—
67C2×C3⋊C8—62no—
68C2×C3⋊C8—62no—
69ΓL(1,𝔽25)—68no—
70ΓL(1,𝔽25)—68no—
71C22×C32#11711yescharacteristic
72C2×C3⋊D3—41no—
73C2×C3⋊D3#12612yes—
74#157—82no—
75C32×C4—41no—
76C32×C4#11711yescharacteristic
77C32×C4#11711yescharacteristic
78C3⋊Dic3—22no—
79C3⋊Dic3#12612yes—
80C2×M16—91no—
81C2×M16—91no—
82C2×M16—92no—
83#117—92no—
84C2×C42—91no—
85#97—91no—
86#97#15711yescharacteristic, complemented
87C4×C8—91no—
88C4×C8—91no—
89C22×D3—62no—
90C2×C3×C4—24no—
91C2×C3×C4—22no—
92C2×Dic3—62no—
93C2×Dic3—64no—
94C4×D3—64no—
95C4×D3—64no—
96C4×D3—128no—
97C3×C8—24no—
98C3×C8—24no—
99C3⋊C8—64no—
100C3⋊C8—64no—
101C2×C32—21no—
102C2×C32#38311yescharacteristic
103C3⋊D3—22no—
104C22×C4—91no—
105C22×C4—181no—
106M16—182no—
107M16—182no—
108M16—182no—
109M16—182no—
110C2×C8—91no—
111C2×C8—91no—
112C2×C8—182no—
113C2×C8C2×#15711yescharacteristic
114C2×C8C2×#15711yescharacteristic
115C4⋊C4—182no—
116C22⋊C4—182no—
117C42—91no—
118C42—181no—
119C42—182no—
120C42C2×#15711yescharacteristic
121C22×C3—22no—
122C2×D3—64no—
123C2×D3—124no—
124C3×C4—22no—
125C3×C4—22no—
126C3×C4—44no—
127Dic3—64no—
128Dic3—68no—
129C32#184111yescharacteristic, complemented, Sylow
130C23—91no—
131C2×C4—21no—
132C2×C4—91no—
133C2×C4—92no—
134C2×C4—92no—
135C2×C4—181no—
136C2×C4—362no—
137C2×C4—362no—
138C2×C4C22×#15711yescharacteristic
139C8—21no—
140C8—21no—
141C8—181no—
142C8—181no—
143C8—362no—
144C2×C3—24no—
145C2×C3—22no—
146D3—124no—
147C22—92no—
148C22—361no—
149C22#6428611yescharacteristic
150C4—41no—
151C4—92no—
152C4—182no—
153C4—722no—
154C4#6428611yescharacteristic
155C4#6428611yescharacteristic
156C3—22no—
157C2—21no—
158C2—182no—
159C2#1059373711yescharacteristic
subgroupG/H
centerC4#64286
commutatorC2×C32×C4C22×C4
FrattiniC2×C4C22×#157
FittingC32×#97C4
socleC2×C32#383
prime radicalC22×C3⋊D3#41
cosocleC2×C4×C3⋊D3C23
hypercenter#97#157
hypocenterC32#1841
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C2×C8 ◁ #97

Complements

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

NH ≅ G/Ntype
11. C32×#97C4semidirect
12. C8×C3⋊Dic3C4semidirect
13. C8×C3⋊Dic3C4semidirect
14. #63630C4semidirect
86. #97#157semidirect
129. C32#1841semidirect

Maximal subgroups

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

subgroupindex
#105924062
#105901862
#105901842
#105901832
#105901512
#18419