Finite Group Databasebeta

← #11721145

Subgroups — #11721145

Order 864.

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

normal subgroups only

This group has 296 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
1C2×#6198C211yesmaximal, characteristic
2C2×#6196C211yesmaximal, characteristic, complemented
3C4×#1084C211yesmaximal, characteristic, complemented
4#3752⋊C3C212yesmaximal, complemented
5#88450C212yesmaximal, complemented
6#63852—32nomaximal
7C22×(C3⋊D3)⋊C3C2212yes—
8C2×C4×H(3,𝔽3)C2211yescharacteristic
9C2×#1084C2211yescharacteristic
10C2×#1084C2211yescharacteristic
11C2×(C3⋊Dic3)⋊C3C2212yes—
12#6198—22no—
13#6196—22no—
14C2×#627—32no—
15C2×#626—32no—
16#3752D312yescomplemented
17C3×(C2×D3)⋊C4—32no—
18C3×C4×Dic3—32no—
19C3×C4×Dic3—61no—
20#3722—32no—
21(C2×D3)⋊Dic3—32no—
22C22×H(3,𝔽3)C2311yescharacteristic
23C2×(C3⋊D3)⋊C3—22no—
24C2×(C3⋊D3)⋊C3—22no—
25C2×(C3⋊D3)⋊C3—22no—
26C4×H(3,𝔽3)—21no—
27#1084—21no—
28#1084D412yes—
29(C3⋊Dic3)⋊C3—22no—
30#885—92no—
31#875—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#627—64no—
43#626—64no—
44C2×H(3,𝔽3)C2×D411yescharacteristic
45C2×H(3,𝔽3)#4111yescharacteristic
46C2×H(3,𝔽3)#4111yescharacteristic
47(C3⋊D3)⋊C3—42no—
48C2×#67—92no—
49C2×D12—91no—
50C2×Dic6—91no—
51C2×Dic6—92no—
52C3×C22⋊C4—92no—
53C3×C42—91no—
54#217—92no—
55(C2×D3)⋊C4—32no—
56(C2×D3)⋊C4—92no—
57(C2×D3)⋊C4—92no—
58(C2×D3)⋊C4—182no—
59C4×Dic3—32no—
60C4×Dic3—61no—
61C22×C32—21no—
62C22×C32C22×D312yes—
63C2×C3⋊D3—22no—
64C2×C3⋊D3—22no—
65C2×C3⋊D3—22no—
66C2×C3×D3—62no—
67C2×C3×D3—62no—
68C2×C3×D3—62no—
69C32×C4—22no—
70C32×C4—22no—
71C3⋊Dic3—22no—
72C3×Dic3—34no—
73C3×Dic3—62no—
74C3×Dic3—62no—
75C3×Dic3—62no—
76C3×Dic3—62no—
77#124—271noSylow
78H(3,𝔽3)#12411yescharacteristic, complemented, Sylow
79C23×C3—92no—
80C22×D3—32no—
81C22×D3—182no—
82C2×C3×C4—61no—
83C2×C3×C4—62no—
84C2×C3×C4—91no—
85C2×C3×C4—92no—
86C2×C3×C4—91no—
87C2×C3×C4D3211yescharacteristic
88#67—184no—
89C2×Dic3—32no—
90C2×Dic3—32no—
91C2×Dic3—32no—
92C2×Dic3—61no—
93C2×Dic3—61no—
94C2×Dic3—182no—
95D12—182no—
96Dic6—184no—
97Dic6—182no—
98C2×C32—21no—
99C2×C32—21no—
100C2×C32—21no—
101C2×C32#23512yes—
102C2×C32D3×D412yes—
103C2×C32#23712yes—
104C3⋊D3—42no—
105C3×D3—122no—
106C2×Q8—271no—
107C2×D4—271no—
108C22⋊C4—92no—
109C22⋊C4—272no—
110C42—91no—
111C22×C3—62no—
112C22×C3—61no—
113C22×C3—182no—
114C22×C3—182no—
115C22×C3—182no—
116C22×C3C2×D3211yescharacteristic
117C2×D3—62no—
118C2×D3—62no—
119C2×D3—62no—
120C2×D3—182no—
121C2×D3—362no—
122C2×D3—362no—
123C3×C4—21no—
124C3×C4—62no—
125C3×C4—92no—
126C3×C4—122no—
127C3×C4—182no—
128C3×C4—181no—
129Dic3—34no—
130Dic3—62no—
131Dic3—62no—
132Dic3—62no—
133Dic3—62no—
134Dic3—362no—
135C32—21no—
136C32#88512yescomplemented
137C23—92no—
138Q8—542no—
139D4—542no—
140C2×C4—91no—
141C2×C4—91no—
142C2×C4—92no—
143C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
144C2×C3—61no—
145C2×C3—62no—
146C2×C3—62no—
147C2×C3—61no—
148C2×C3—61no—
149C2×C3—62no—
150C2×C3—362no—
151C2×C3#379811yescharacteristic
152C2×C3#380211yescharacteristic
153C2×C3#380511yescharacteristic
154D3—122no—
155D3—362no—
156C22—182no—
157C22—182no—
158C22—182no—
159C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
160C4—21no—
161C4—92no—
162C4—181no—
163C4—182no—
164C3—62no—
165C3—61no—
166C3#6391511yescharacteristic
167C2—362no—
168C2#8865011yescharacteristic
169C2#8865511yescharacteristic
170C2#8867211yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C23
Frattini, socleC22×C3C2×D32
FittingC2×C4×H(3,𝔽3)C22
prime radicalC2×#6198C2
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#124
LCSG ▷ C22×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×#6196C2semidirect
3. C4×#1084C2semidirect
4. #3752⋊C3C2semidirect
5. #88450C2semidirect
16. #3752D3semidirect
78. H(3,𝔽3)#124semidirect
136. C32#885semidirect

Maximal subgroups

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

subgroupindex
C2×#61982
C2×#61962
C4×#10842
#3752⋊C32
#884502
#638523