Finite Group Databasebeta

← #11721763

Subgroups — #11721763

Order 864.

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

normal subgroups only

This group has 517 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⋊#272C212yesmaximal, complemented
2C2×C4⋊#272C211yesmaximal, characteristic, complemented
3C2×C4×#272C211yesmaximal, characteristic, complemented
4C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
5#88542C212yesmaximal, complemented
6C3×#895—31nomaximal
7C3×#895—33nomaximal
8C22×#272C2212yes—
9C22×#272C2211yescharacteristic
10C2×C4×H(3,𝔽3)C2212yes—
11C2×C4×H(3,𝔽3)C2211yescharacteristic
12C2×#1084C2211yescharacteristic
13C4⋊#272—24no—
14C4⋊#272—22no—
15C4×#272—22no—
16C2×C3×D12—32no—
17C2×C3×D12—36no—
18C2×C3×D12—31no—
19C2×C3×D12—33no—
20C2×C3×C4×D3—31no—
21C2×C3×C4×D3—33no—
22C32×C4⋊C4D311yescharacteristic, complemented
23C32×C4⋊C4D313yescomplemented
24C3×(C2×D3)⋊C4—32no—
25C3×(C2×D3)⋊C4—36no—
26C22×H(3,𝔽3)C2311yescharacteristic
27C2×#272—21no—
28C2×#272—21no—
29C2×#272—22no—
30C2×#272—22no—
31C2×#272—22no—
32C2×#272D412yescomplemented
33C4×H(3,𝔽3)—22no—
34C4×H(3,𝔽3)D412yescomplemented
35#1084—21no—
36C3×#121—91no—
37#895—31no—
38#895—33no—
39C22×C3×D3—31no—
40C22×C3×D3—33no—
41C22×C3×D3—32no—
42C22×C3×D3—36no—
43C2×C32×C4C2×D312yes—
44C2×C32×C4C2×D311yescharacteristic
45C2×C32×C4C2×D316yes—
46C2×C32×C4C2×D313yes—
47C2×C3×Dic3—31no—
48C2×C3×Dic3—33no—
49C3×D12—64no—
50C3×D12—612no—
51C3×D12—62no—
52C3×D12—66no—
53C3×C4×D3—62no—
54C3×C4×D3—66no—
55C2×H(3,𝔽3)C2×D411yescharacteristic
56C2×H(3,𝔽3)C2×D411yescharacteristic
57C2×H(3,𝔽3)#4111yescharacteristic
58#272—22no—
59#272—42no—
60C2×C3×D4—92no—
61C2×C3×D4—91no—
62C22×C3×C4—91no—
63C2×D12—31no—
64C2×D12—32no—
65C2×D12—33no—
66C2×D12—36no—
67C2×C4×D3—31no—
68C2×C4×D3—33no—
69C3×C4⋊C4—31no—
70C3×C4⋊C4—63no—
71C3×C4⋊C4C3⋊D311yescharacteristic
72C3×C22⋊C4—92no—
73(C2×D3)⋊C4—32no—
74(C2×D3)⋊C4—36no—
75C22×C32C22×D311yescharacteristic
76C22×C32C22×D313yes—
77C2×C3×D3—32no—
78C2×C3×D3—36no—
79C2×C3×D3—61no—
80C2×C3×D3—63no—
81C2×C3×D3—61no—
82C2×C3×D3—63no—
83C2×C3×D3—62no—
84C2×C3×D3—66no—
85C2×C3×D3—62no—
86C2×C3×D3—66no—
87C2×C3×D3—62no—
88C2×C3×D3—66no—
89C32×C4—22no—
90C32×C4—26no—
91C32×C4D1212yescomplemented
92C32×C4D1216yescomplemented
93C3×Dic3—61no—
94C3×Dic3—63no—
95#121—91noSylow
96H(3,𝔽3)#12111yescharacteristic, complemented, Sylow
97C23×C3—91no—
98C23×C3—92no—
99C22×D3—31no—
100C22×D3—33no—
101C22×D3—32no—
102C22×D3—36no—
103C3×D4—184no—
104C3×D4—182no—
105C2×C3×C4—31no—
106C2×C3×C4—32no—
107C2×C3×C4—63no—
108C2×C3×C4—66no—
109C2×C3×C4—91no—
110C2×C3×C4—182no—
111C2×C3×C4C2×C3⋊D312yes—
112C2×C3×C4C2×C3⋊D311yescharacteristic
113C2×Dic3—31no—
114C2×Dic3—33no—
115D12—62no—
116D12—64no—
117D12—66no—
118D12—612no—
119C4×D3—62no—
120C4×D3—66no—
121C2×C32C2×D1211yescharacteristic
122C2×C32C2×D1213yes—
123C2×C32D3×D411yescharacteristic
124C2×C32D3×D413yes—
125C2×C32#23911yescharacteristic
126C2×C32#23913yes—
127C3×D3—62no—
128C3×D3—66no—
129C3×D3—122no—
130C3×D3—126no—
131C2×D4—92no—
132C2×D4—91no—
133C22×C4—91no—
134C4⋊C4#27211yescharacteristic, complemented
135C22⋊C4—92no—
136C22×C3—31no—
137C22×C3—63no—
138C22×C3—92no—
139C22×C3—181no—
140C22×C3—181no—
141C22×C3—182no—
142C22×C3—182no—
143C22×C3—182no—
144C22×C3C22×C3⋊D311yescharacteristic
145C2×D3—32no—
146C2×D3—36no—
147C2×D3—61no—
148C2×D3—61no—
149C2×D3—63no—
150C2×D3—63no—
151C2×D3—62no—
152C2×D3—62no—
153C2×D3—62no—
154C2×D3—66no—
155C2×D3—66no—
156C2×D3—66no—
157C3×C4—22no—
158C3×C4—32no—
159C3×C4—62no—
160C3×C4—66no—
161C3×C4—126no—
162C3×C4—181no—
163C3×C4C4⋊(C3⋊D3)12yes—
164Dic3—61no—
165Dic3—63no—
166C32#89511yescharacteristic, complemented
167C32#89513yescomplemented
168C23—91no—
169C23—92no—
170D4—184no—
171D4—182no—
172C2×C4—91no—
173C2×C4—182no—
174C2×C4C2×#27212yes—
175C2×C4C2×#27211yescharacteristic
176C2×C3—31no—
177C2×C3—31no—
178C2×C3—31no—
179C2×C3—63no—
180C2×C3—63no—
181C2×C3—63no—
182C2×C3—182no—
183C2×C3—362no—
184C2×C3C2×C4⋊(C3⋊D3)11yescharacteristic
185C2×C3D4×C3⋊D311yescharacteristic
186C2×C3#383211yescharacteristic
187D3—62no—
188D3—66no—
189D3—122no—
190D3—126no—
191C22—92no—
192C22—181no—
193C22—181no—
194C22—182no—
195C22—182no—
196C22—182no—
197C22C22×#27211yescharacteristic
198C4—22no—
199C4—181no—
200C4C4⋊#27212yescomplemented
201C3—31no—
202C3—63no—
203C3#6410411yescharacteristic
204C2—182no—
205C2—362no—
206C2C2×C4⋊#27211yescharacteristic
207C2D4×#27211yescharacteristic
208C2#8874811yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C23
FittingC4⋊C4×H(3,𝔽3)C2
hypercenterC3×C4⋊C4C3⋊D3
hypocenterH(3,𝔽3)#121
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
1. C2×C4⋊#272C2semidirect
2. C2×C4⋊#272C2semidirect
3. C2×C4×#272C2semidirect
4. C4⋊C4×H(3,𝔽3)C2semidirect
5. #88542C2semidirect
22. C32×C4⋊C4D3semidirect
23. C32×C4⋊C4D3semidirect
32. C2×#272D4semidirect
34. C4×H(3,𝔽3)D4semidirect
91. C32×C4D12semidirect
92. C32×C4D12semidirect
96. H(3,𝔽3)#121semidirect
134. C4⋊C4#272semidirect
166. C32#895semidirect
167. C32#895semidirect
200. C4C4⋊#272semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#2722
C2×C4×#2722
C4⋊C4×H(3,𝔽3)2
#885422
C3×#8953