Finite Group Databasebeta

← #11721141

Subgroups — #11721141

Order 864.

355 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 64 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⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
2C2×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3C2×#6198C212yesmaximal, complemented
4C4⋊#1084C211yesmaximal, characteristic, complemented
5#88450C212yesmaximal, complemented
6#63916—31nomaximal
7#63909—31nomaximal
8C22×(C3⋊D3)⋊C3C2212yes—
9C22×(C3⋊D3)⋊C3C2211yescharacteristic
10C2×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#1084C2212yes—
12C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C4⋊((C3⋊D3)⋊C3)—22no—
14C4×(C3⋊D3)⋊C3—22no—
15#6198—24no—
16C2×C4⋊(C3⋊D3)D311yescharacteristic, complemented
17C2×C4×C3⋊D3D311yescharacteristic, complemented
18C2×C3×D12—31no—
19C2×C3×C4×D3—31no—
20C2×#626—32no—
21C2×#626—32no—
22C3×C4⋊Dic3—31no—
23C3×C4⋊Dic3—31no—
24C3×C4⋊Dic3—61no—
25#3722—32no—
26(C2×D3)⋊Dic3—32no—
27C22×H(3,𝔽3)C2311yescharacteristic
28C2×(C3⋊D3)⋊C3—21no—
29C2×(C3⋊D3)⋊C3—21no—
30C2×(C3⋊D3)⋊C3—22no—
31C2×(C3⋊D3)⋊C3—22no—
32C2×(C3⋊D3)⋊C3—22no—
33C2×(C3⋊D3)⋊C3D412yescomplemented
34C4×H(3,𝔽3)D412yescomplemented
35#1084—22no—
36(C3⋊Dic3)⋊C3—21no—
37#938—91no—
38#930—91no—
39#895—91no—
40C22×C3⋊D3C2×D312yes—
41C22×C3⋊D3C2×D311yescharacteristic
42C22×C3×D3—31no—
43C22×C3×D3—32no—
44C2×C32×C4—21no—
45C2×C32×C4C2×D311yescharacteristic
46C2×C32×C4C2×D311yescharacteristic
47C2×C3⋊Dic3C2×D311yescharacteristic
48C4⋊(C3⋊D3)—22no—
49C4×C3⋊D3—22no—
50C2×C3×Dic3—31no—
51C2×C3×Dic3—32no—
52C2×C3×Dic3—32no—
53C2×C3×Dic3—62no—
54C3×D12—62no—
55C3×C4×D3—62no—
56#626—64no—
57#626—64no—
58C2×H(3,𝔽3)C2×D411yescharacteristic
59C2×H(3,𝔽3)C2×D411yescharacteristic
60C2×H(3,𝔽3)#4111yescharacteristic
61(C3⋊D3)⋊C3—22no—
62(C3⋊D3)⋊C3—42no—
63C2×C3×D4—91no—
64C22×C3×C4—91no—
65C2×#67—92no—
66C2×#67—92no—
67C2×D12—31no—
68C2×D12—92no—
69C2×D12—181no—
70C2×C4×D3—31no—
71C2×C4×D3—181no—
72C3×C4⋊C4—91no—
73#217—92no—
74(C2×D3)⋊C4—92no—
75(C2×D3)⋊C4—92no—
76C4⋊Dic3—31no—
77C4⋊Dic3—31no—
78C4⋊Dic3—61no—
79C22×C32—21no—
80C22×C32C22×D311yescharacteristic
81C22×C32C22×D311yescharacteristic
82C2×C3⋊D3—21no—
83C2×C3⋊D3—21no—
84C2×C3⋊D3—22no—
85C2×C3⋊D3—22no—
86C2×C3⋊D3—22no—
87C2×C3⋊D3D1212yescomplemented
88C2×C3×D3—32no—
89C2×C3×D3—61no—
90C2×C3×D3—61no—
91C2×C3×D3—62no—
92C2×C3×D3—62no—
93C2×C3×D3—62no—
94C32×C4—22no—
95C32×C4#6712yescomplemented
96C32×C4#6712yescomplemented
97C3⋊Dic3—21no—
98C3×Dic3—62no—
99C3×Dic3—62no—
100C3×Dic3—61no—
101C3×Dic3—122no—
102#121—271noSylow
103H(3,𝔽3)#12111yescharacteristic, complemented, Sylow
104C23×C3—91no—
105C23×C3—92no—
106C22×D3—31no—
107C22×D3—32no—
108C22×D3—181no—
109C22×D3—182no—
110C3×D4—182no—
111C2×C3×C4—61no—
112C2×C3×C4—61no—
113C2×C3×C4—91no—
114C2×C3×C4—92no—
115C2×C3×C4—121no—
116C2×C3×C4—182no—
117C2×C3×C4D3211yescharacteristic
118#67—184no—
119#67—184no—
120C2×Dic3—32no—
121C2×Dic3—32no—
122C2×Dic3—31no—
123C2×Dic3—62no—
124C2×Dic3—181no—
125D12—62no—
126D12—182no—
127D12—184no—
128C4×D3—62no—
129C4×D3—182no—
130C2×C32—21no—
131C2×C32—21no—
132C2×C32—21no—
133C2×C32C2×D1211yescharacteristic
134C2×C32#23511yescharacteristic
135C2×C32D3×D411yescharacteristic
136C2×C32#23711yescharacteristic
137C2×C32C2×#6711yescharacteristic
138C2×C32C2×#6711yescharacteristic
139C3⋊D3—22no—
140C3⋊D3—42no—
141C3×D3—62no—
142C3×D3—122no—
143C2×D4—91no—
144C2×D4—272no—
145C22×C4—91no—
146C4⋊C4—91no—
147C22⋊C4—272no—
148C22×C3—61no—
149C22×C3—61no—
150C22×C3—92no—
151C22×C3—121no—
152C22×C3—181no—
153C22×C3—181no—
154C22×C3—182no—
155C22×C3—182no—
156C22×C3—182no—
157C22×C3C2×D3211yescharacteristic
158C2×D3—32no—
159C2×D3—61no—
160C2×D3—61no—
161C2×D3—62no—
162C2×D3—62no—
163C2×D3—62no—
164C2×D3—182no—
165C2×D3—181no—
166C2×D3—181no—
167C2×D3—182no—
168C2×D3—362no—
169C2×D3—362no—
170C3×C4—62no—
171C3×C4—62no—
172C3×C4—122no—
173C3×C4—181no—
174C3×C4—182no—
175C3×C4#62512yes—
176Dic3—62no—
177Dic3—62no—
178Dic3—61no—
179Dic3—122no—
180Dic3—181no—
181C32—21no—
182C32#93011yescharacteristic, complemented
183C32#93811yescharacteristic, complemented
184C23—91no—
185C23—92no—
186D4—182no—
187D4—544no—
188C2×C4—91no—
189C2×C4—92no—
190C2×C4—182no—
191C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
192C2×C3—61no—
193C2×C3—61no—
194C2×C3—61no—
195C2×C3—61no—
196C2×C3—61no—
197C2×C3—61no—
198C2×C3—121no—
199C2×C3—121no—
200C2×C3—121no—
201C2×C3—182no—
202C2×C3—362no—
203C2×C3#379511yescharacteristic
204C2×C3D3×D1211yescharacteristic
205C2×C3C2×#62511yescharacteristic
206D3—62no—
207D3—122no—
208D3—182no—
209D3—362no—
210C22—92no—
211C22—181no—
212C22—181no—
213C22—182no—
214C22—182no—
215C22—182no—
216C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
217C4—182no—
218C4—181no—
219C4#619812yescomplemented
220C3—61no—
221C3—61no—
222C3—121no—
223C3#6390811yescharacteristic
224C2—182no—
225C2—362no—
226C2#8864811yescharacteristic
227C2#8865411yescharacteristic
228C2C2×#619811yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C23
Frattini, socleC22×C3C2×D32
FittingC2×C4×H(3,𝔽3)C22
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#121
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
1. C2×C4⋊((C3⋊D3)⋊C3)C2semidirect
2. C2×C4×(C3⋊D3)⋊C3C2semidirect
3. C2×#6198C2semidirect
4. C4⋊#1084C2semidirect
5. #88450C2semidirect
16. C2×C4⋊(C3⋊D3)D3semidirect
17. C2×C4×C3⋊D3D3semidirect
33. C2×(C3⋊D3)⋊C3D4semidirect
34. C4×H(3,𝔽3)D4semidirect
87. C2×C3⋊D3D12semidirect
95. C32×C4#67semidirect
96. C32×C4#67semidirect
103. H(3,𝔽3)#121semidirect
182. C32#930semidirect
183. C32#938semidirect
219. C4#6198semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊((C3⋊D3)⋊C3)2
C2×C4×(C3⋊D3)⋊C32
C2×#61982
C4⋊#10842
#884502
#639163
#639093