Finite Group Databasebeta

← #11721138

Subgroups — #11721138

Order 864.

359 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 359 conjugacy classes of subgroups — too many to draw. Only its 66 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
1C2×#6198C211yesmaximal, characteristic
2C2×((C3⋊D3)⋊C3)⋊C4C212yesmaximal, complemented
3C4×#1084C211yesmaximal, characteristic, complemented
4#3752⋊C3C212yesmaximal, complemented
5#88449C211yesmaximal, characteristic, complemented
6#63880—32nomaximal
7C22×(C3⋊D3)⋊C3C2212yes—
8C2×C4×H(3,𝔽3)C2211yescharacteristic
9C2×#1084C2211yescharacteristic
10C2×#1084C2211yescharacteristic
11C2×(C3⋊Dic3)⋊C3C2212yes—
12#6198C414yescomplemented
13((C3⋊D3)⋊C3)⋊C4—24no—
14C2×#626—32no—
15C2×#624—32no—
16C2×D3×Dic3—32no—
17#3752D312yescomplemented
18C3×(C2×D3)⋊C4—32no—
19C3×C4×Dic3—32no—
20C3×C4×Dic3—61no—
21Dic3⋊Dic3—32no—
22C22×H(3,𝔽3)C2311yescharacteristic
23C2×(C3⋊D3)⋊C3—22no—
24C2×(C3⋊D3)⋊C3—22no—
25C2×(C3⋊D3)⋊C3C2×C414yescomplemented
26C4×H(3,𝔽3)—21no—
27#1084C2×C412yescomplemented
28#1084D412yescomplemented
29(C3⋊Dic3)⋊C3—22no—
30#67⋊C4—92no—
31C4×D12—91no—
32C22×C3⋊D3C2×D312yes—
33C22×C3×D3—32no—
34C2×C32×C4—21no—
35C2×C32×C4C2×D312yes—
36C2×C3⋊Dic3C2×D312yes—
37C2×C3×Dic3—32no—
38C2×C3×Dic3—32no—
39C2×C3×Dic3—32no—
40C2×C3×Dic3—61no—
41C2×C3×Dic3—61no—
42#626—38no—
43#624—64no—
44D3×Dic3—64no—
45C2×H(3,𝔽3)C22×C411yescharacteristic
46C2×H(3,𝔽3)C2×D411yescharacteristic
47C2×H(3,𝔽3)#4111yescharacteristic
48(C3⋊D3)⋊C3—24no—
49C2×#67—92no—
50C22×Dic3—92no—
51C2×D12—91no—
52C2×C4×D3—92no—
53C2×C4×D3—92no—
54C3×C22⋊C4—92no—
55C3×C42—91no—
56(C2×D3)⋊C4—32no—
57(C2×D3)⋊C4—182no—
58C4⋊Dic3—91no—
59Dic3⋊C4—92no—
60C4×Dic3—32no—
61C4×Dic3—61no—
62C22×C32—21no—
63C22×C32C22×D312yes—
64C2×C3⋊D3—22no—
65C2×C3⋊D3—22no—
66C2×C3⋊D3C4×D314yescomplemented
67C2×C3×D3—34no—
68C2×C3×D3—62no—
69C2×C3×D3—62no—
70C32×C4—21no—
71C32×C4—21no—
72C32×C4—22no—
73C3⋊Dic3—22no—
74C3×Dic3—34no—
75C3×Dic3—34no—
76C3×Dic3—62no—
77C3×Dic3—62no—
78C3×Dic3—62no—
79C4×D4—271noSylow
80H(3,𝔽3)C4×D411yescharacteristic, complemented, Sylow
81C23×C3—92no—
82C22×D3—32no—
83C22×D3—182no—
84C2×C3×C4—61no—
85C2×C3×C4—62no—
86C2×C3×C4—91no—
87C2×C3×C4—91no—
88C2×C3×C4—92no—
89C2×C3×C4D3211yescharacteristic
90#67—98no—
91C2×Dic3—32no—
92C2×Dic3—32no—
93C2×Dic3—32no—
94C2×Dic3—61no—
95C2×Dic3—61no—
96C2×Dic3—182no—
97C2×Dic3—184no—
98D12—94no—
99C4×D3—184no—
100C4×D3—184no—
101C2×C32—21no—
102C2×C32—21no—
103C2×C32—21no—
104C2×C32C2×C4×D312yes—
105C2×C32D3×D412yes—
106C2×C32#23712yes—
107C3⋊D3—24no—
108C3×D3—64no—
109C2×D4—271no—
110C22×C4—272no—
111C4⋊C4—271no—
112C22⋊C4—92no—
113C42—91no—
114C22×C3—62no—
115C22×C3—61no—
116C22×C3—94no—
117C22×C3—182no—
118C22×C3—182no—
119C22×C3C2×D3211yescharacteristic
120C2×D3—34no—
121C2×D3—62no—
122C2×D3—62no—
123C2×D3—184no—
124C2×D3—362no—
125C2×D3—362no—
126C3×C4—21no—
127C3×C4—61no—
128C3×C4—92no—
129C3×C4—92no—
130C3×C4—121no—
131C3×C4—122no—
132C3×C4—182no—
133Dic3—34no—
134Dic3—34no—
135Dic3—62no—
136Dic3—62no—
137Dic3—62no—
138Dic3—362no—
139C32—21no—
140C32#67⋊C412yescomplemented
141C23—92no—
142D4—274no—
143C2×C4—91no—
144C2×C4—91no—
145C2×C4—92no—
146C2×C4—544no—
147C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
148C2×C3—62no—
149C2×C3—61no—
150C2×C3—62no—
151C2×C3—61no—
152C2×C3—61no—
153C2×C3—62no—
154C2×C3—184no—
155C2×C3C4×D3211yescharacteristic
156C2×C3#380211yescharacteristic
157C2×C3#380511yescharacteristic
158D3—64no—
159D3—364no—
160C22—94no—
161C22—182no—
162C22—182no—
163C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
164C4—21no—
165C4—92no—
166C4—92no—
167C4—182no—
168C3—62no—
169C3—61no—
170C3#6390211yescharacteristic
171C2—184no—
172C2C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
173C2#8865511yescharacteristic
174C2#8867211yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C22×C4
Frattini, socleC22×C3C2×D32
FittingC2×C4×H(3,𝔽3)C22
prime radicalC2×#6198C2
cosocleC22×H(3,𝔽3)C23
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4×D4
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×((C3⋊D3)⋊C3)⋊C4C2semidirect
3. C4×#1084C2semidirect
4. #3752⋊C3C2semidirect
5. #88449C2semidirect
12. #6198C4semidirect
17. #3752D3semidirect
25. C2×(C3⋊D3)⋊C3C2×C4semidirect
27. #1084C2×C4semidirect
28. #1084D4semidirect
66. C2×C3⋊D3C4×D3semidirect
80. H(3,𝔽3)C4×D4semidirect
140. C32#67⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#61982
C2×((C3⋊D3)⋊C3)⋊C42
C4×#10842
#3752⋊C32
#884492
#638803