Finite Group Databasebeta

← #11721765

Subgroups — #11721765

Order 864.

407 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 407 conjugacy classes of subgroups — too many to draw. Only its 90 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
2C2×#6229C211yesmaximal, characteristic, complemented
3C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4#88542C212yesmaximal
5C4⋊#1084C212yesmaximal, complemented
6C3×#897—31nomaximal
7C3×#897—33nomaximal
8C22×#272C2211yescharacteristic
9C2×C4×H(3,𝔽3)C2212yes—
10C2×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#1084C2212yes—
12C2×#1084C2211yescharacteristic
13C4×#272—22no—
14#6229—22no—
15C2×C3×C4×D3—31no—
16C2×C3×C4×D3—33no—
17C2×C3×Dic6—31no—
18C2×C3×Dic6—33no—
19C32×C4⋊C4D311yescharacteristic, complemented
20C32×C4⋊C4D313yescomplemented
21C3×(C2×D3)⋊C4—32no—
22C3×(C2×D3)⋊C4—36no—
23C3×C4⋊Dic3—32no—
24C3×C4⋊Dic3—36no—
25C22×H(3,𝔽3)C2311yescharacteristic
26C2×#272—21no—
27C2×#272—21no—
28C2×#272Q812yescomplemented
29C4×H(3,𝔽3)—22no—
30C4×H(3,𝔽3)D412yes—
31#1084—22no—
32#1084—21no—
33C3×C22⋊Q8—91no—
34#897—31no—
35#897—33no—
36C22×C3×D3—31no—
37C22×C3×D3—33no—
38C2×C32×C4C2×D312yes—
39C2×C32×C4C2×D311yescharacteristic
40C2×C32×C4C2×D316yes—
41C2×C32×C4C2×D313yes—
42C2×C3×Dic3—31no—
43C2×C3×Dic3—33no—
44C2×C3×Dic3—32no—
45C2×C3×Dic3—36no—
46C3×C4×D3—62no—
47C3×C4×D3—66no—
48C3×Dic6—62no—
49C3×Dic6—66no—
50C2×H(3,𝔽3)C2×D411yescharacteristic
51C2×H(3,𝔽3)C2×Q811yescharacteristic
52C2×H(3,𝔽3)#4111yescharacteristic
53#272—22no—
54C2×C3×Q8—91no—
55C22×C3×C4—91no—
56C2×C4×D3—31no—
57C2×C4×D3—33no—
58C2×Dic6—31no—
59C2×Dic6—33no—
60C3×C4⋊C4—31no—
61C3×C4⋊C4—63no—
62C3×C4⋊C4—92no—
63C3×C4⋊C4C3⋊D311yescharacteristic
64C3×C22⋊C4—92no—
65(C2×D3)⋊C4—32no—
66(C2×D3)⋊C4—36no—
67C4⋊Dic3—32no—
68C4⋊Dic3—36no—
69C22×C32C22×D311yescharacteristic
70C22×C32C22×D313yes—
71C2×C3×D3—32no—
72C2×C3×D3—36no—
73C2×C3×D3—61no—
74C2×C3×D3—63no—
75C2×C3×D3—61no—
76C2×C3×D3—63no—
77C32×C4—22no—
78C32×C4—26no—
79C32×C4D1212yes—
80C32×C4D1216yes—
81C3×Dic3—62no—
82C3×Dic3—66no—
83C3×Dic3—61no—
84C3×Dic3—63no—
85C22⋊Q8—91noSylow
86H(3,𝔽3)C22⋊Q811yescharacteristic, complemented, Sylow
87C23×C3—91no—
88C22×D3—31no—
89C22×D3—33no—
90C3×Q8—182no—
91C2×C3×C4—31no—
92C2×C3×C4—32no—
93C2×C3×C4—63no—
94C2×C3×C4—66no—
95C2×C3×C4—92no—
96C2×C3×C4—91no—
97C2×C3×C4—182no—
98C2×C3×C4C2×C3⋊D312yes—
99C2×C3×C4C2×C3⋊D311yescharacteristic
100C2×Dic3—31no—
101C2×Dic3—32no—
102C2×Dic3—33no—
103C2×Dic3—36no—
104C4×D3—62no—
105C4×D3—66no—
106Dic6—62no—
107Dic6—66no—
108C2×C32C2×D1211yescharacteristic
109C2×C32C2×D1213yes—
110C2×C32#23711yescharacteristic
111C2×C32#23713yes—
112C2×C32D3×Q811yescharacteristic
113C2×C32D3×Q813yes—
114C3×D3—62no—
115C3×D3—66no—
116C2×Q8—91no—
117C22×C4—91no—
118C4⋊C4—92no—
119C4⋊C4#27211yescharacteristic, complemented
120C22⋊C4—92no—
121C22×C3—31no—
122C22×C3—63no—
123C22×C3—92no—
124C22×C3—181no—
125C22×C3—181no—
126C22×C3C22×C3⋊D311yescharacteristic
127C2×D3—32no—
128C2×D3—36no—
129C2×D3—61no—
130C2×D3—61no—
131C2×D3—63no—
132C2×D3—63no—
133C3×C4—22no—
134C3×C4—32no—
135C3×C4—62no—
136C3×C4—66no—
137C3×C4—126no—
138C3×C4—181no—
139C3×C4—182no—
140C3×C4C4⋊(C3⋊D3)12yes—
141Dic3—62no—
142Dic3—61no—
143Dic3—66no—
144Dic3—63no—
145C32#89711yescharacteristic, complemented
146C32#89713yescomplemented
147C23—91no—
148Q8—182no—
149C2×C4—92no—
150C2×C4—91no—
151C2×C4—182no—
152C2×C4C2×#27212yes—
153C2×C4C2×#27211yescharacteristic
154C2×C3—31no—
155C2×C3—31no—
156C2×C3—31no—
157C2×C3—63no—
158C2×C3—63no—
159C2×C3—63no—
160C2×C3—182no—
161C2×C3C2×C4⋊(C3⋊D3)11yescharacteristic
162C2×C3#383011yescharacteristic
163C2×C3Q8×C3⋊D311yescharacteristic
164D3—62no—
165D3—66no—
166C22—92no—
167C22—181no—
168C22—181no—
169C22C22×#27211yescharacteristic
170C4—22no—
171C4—181no—
172C4—182no—
173C4C4⋊#27212yes—
174C3—31no—
175C3—63no—
176C3#6410611yescharacteristic
177C2—182no—
178C2C2×C4⋊#27211yescharacteristic
179C2#8874411yescharacteristic
180C2Q8×#27211yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C23
FittingC4⋊C4×H(3,𝔽3)C2
prime radicalC22×#272C22
hypercenterC3×C4⋊C4C3⋊D3
hypocenterH(3,𝔽3)C22⋊Q8
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#6229C2semidirect
3. C4⋊C4×H(3,𝔽3)C2semidirect
5. C4⋊#1084C2semidirect
19. C32×C4⋊C4D3semidirect
20. C32×C4⋊C4D3semidirect
28. C2×#272Q8semidirect
86. H(3,𝔽3)C22⋊Q8semidirect
119. C4⋊C4#272semidirect
145. C32#897semidirect
146. C32#897semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2722
C2×#62292
C4⋊C4×H(3,𝔽3)2
#885422
C4⋊#10842
C3×#8973