Finite Group Databasebeta

← #11722448

Subgroups — #11722448

Order 864.

222 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 42 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×C4×#272C211yesmaximal, characteristic
2C22⋊#1088C212yesmaximal
3C4⋊#1088C212yesmaximal, complemented
4C4×#1088C212yesmaximal, complemented
5C22×#272C2211yescharacteristic
6C2×#1088C2214yes—
7C2×C4×H(3,𝔽3)C411yescharacteristic, complemented
8C2×#1084C411yescharacteristic, complemented
9C4×#272C411yescharacteristic, complemented
10C4×#272C411yescharacteristic, complemented
11C4×#272C2211yescharacteristic
12C4×#272C2211yescharacteristic
13C2×C3×C4×D3—61no—
14C2×C3×C4×D3—61no—
15C22×H(3,𝔽3)C2×C411yescharacteristic
16C2×#272—22no—
17C2×#272C2×C411yescharacteristic
18C2×#272C2311yescharacteristic
19#1088—24no—
20C4×H(3,𝔽3)C2×C411yescharacteristic
21C4×H(3,𝔽3)C2×C411yescharacteristic
22#1084C2×C411yescharacteristic
23#1084C2×C411yescharacteristic
24C3×#117—91nomaximal
25C22×C3×D3—61no—
26C22×C3×D3—61no—
27C2×C32×C4—21no—
28C2×C32×C4—21no—
29C2×C3×Dic3—61no—
30C2×C3×Dic3—61no—
31C3×C4×D3—61no—
32C3×C4×D3—61no—
33C3×C4×D3—61no—
34C3×C4×D3—61no—
35C3×C4×D3—61no—
36C3×C4×D3—61no—
37C3×C4×D3—61no—
38C3×C4×D3—61no—
39C2×H(3,𝔽3)—21no—
40C2×H(3,𝔽3)C22×C411yescharacteristic
41#272—21no—
42#272#4112yes—
43C22×C3×C4—91no—
44C2×C4×D3—61no—
45C2×C4×D3—61no—
46C3×C4⋊C4—92no—
47C3×C22⋊C4—92no—
48C3×C42—92no—
49C22×C32—21no—
50C22×C32—21no—
51C2×C3×D3—61no—
52C2×C3×D3—61no—
53C2×C3×D3—61no—
54C2×C3×D3—61no—
55C2×C3×D3—62no—
56C2×C3×D3—62no—
57C2×C3×D3—64no—
58C32×C4—21no—
59C32×C4—21no—
60C32×C4—21no—
61C32×C4—21no—
62C3×Dic3—61no—
63C3×Dic3—61no—
64C3×Dic3—61no—
65C3×Dic3—61no—
66#117—91noSylow
67H(3,𝔽3)#11711yescharacteristic, complemented, Sylow
68C23×C3—91no—
69C22×D3—61no—
70C22×D3—61no—
71C2×C3×C4—61no—
72C2×C3×C4—91no—
73C2×C3×C4—91no—
74C2×C3×C4—91no—
75C2×C3×C4—91no—
76C2×C3×C4—91no—
77C2×C3×C4—94no—
78C2×C3×C4—121no—
79C2×C3×C4#15711yescharacteristic
80C2×Dic3—61no—
81C2×Dic3—61no—
82C4×D3—61no—
83C4×D3—61no—
84C4×D3—61no—
85C4×D3—61no—
86C4×D3—61no—
87C4×D3—61no—
88C4×D3—61no—
89C4×D3—61no—
90C2×C32—21no—
91C2×C32—21no—
92C2×C32—22no—
93C2×C32—21no—
94C2×C32—21no—
95C3×D3—62no—
96C3×D3—62no—
97C3×D3—62no—
98C3×D3—62no—
99C22×C4—91no—
100C4⋊C4—92no—
101C22⋊C4—92no—
102C42—92no—
103C22×C3—61no—
104C22×C3—91no—
105C22×C3—91no—
106C22×C3—121no—
107C22×C3—182no—
108C22×C3C2×#15711yescharacteristic
109C2×D3—61no—
110C2×D3—61no—
111C2×D3—61no—
112C2×D3—61no—
113C2×D3—62no—
114C2×D3—62no—
115C2×D3—64no—
116C3×C4—61no—
117C3×C4—61no—
118C3×C4—91no—
119C3×C4—91no—
120C3×C4—121no—
121C3×C4—121no—
122C3×C4—184no—
123C3×C4C2×#15711yescharacteristic
124C3×C4C2×#15711yescharacteristic
125Dic3—61no—
126Dic3—61no—
127Dic3—61no—
128Dic3—61no—
129C32—21no—
130C32—21no—
131C23—91no—
132C2×C4—91no—
133C2×C4—91no—
134C2×C4—91no—
135C2×C4—91no—
136C2×C4—91no—
137C2×C4—94no—
138C2×C4#108811yescharacteristic, complemented
139C2×C3—21no—
140C2×C3—61no—
141C2×C3—61no—
142C2×C3—92no—
143C2×C3—121no—
144C2×C3—122no—
145C2×C3—121no—
146C2×C3—181no—
147C2×C3C22×#15711yescharacteristic
148D3—62no—
149D3—62no—
150D3—62no—
151D3—62no—
152C22—91no—
153C22—91no—
154C22—182no—
155C22C2×#108811yescharacteristic
156C4—91no—
157C4—91no—
158C4—184no—
159C4C2×#108811yescharacteristic
160C4C2×#108811yescharacteristic
161C3—61no—
162C3—121no—
163C3#6428611yescharacteristic
164C2—21no—
165C2—92no—
166C2—181no—
167C2C22×#108811yescharacteristic
subgroupG/H
centerC3×C4C2×#157
commutatorC2×H(3,𝔽3)C22×C4
Frattini, socleC2×C3C22×#157
FittingC2×C4×H(3,𝔽3)C4
prime radicalC22×#272C22
cosocleC2×#272C23
hypercenterC2×C3×C4#157
hypocenterH(3,𝔽3)#117
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C3×C4 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
3. C4⋊#1088C2semidirect
4. C4×#1088C2semidirect
7. C2×C4×H(3,𝔽3)C4semidirect
8. C2×#1084C4semidirect
9. C4×#272C4semidirect
10. C4×#272C4semidirect
67. H(3,𝔽3)#117semidirect
138. C2×C4#1088semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2722
C22⋊#10882
C4⋊#10882
C4×#10882
C3×#1179