Finite Group Databasebeta

← #11721762

Subgroups — #11721762

Order 864.

429 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 429 conjugacy classes of subgroups — too many to draw. Only its 76 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×C4×#272C211yesmaximal, characteristic, complemented
3C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4#88542C211yesmaximal, characteristic, complemented
5#88542C211yesmaximal, characteristic, complemented
6#88542C211yesmaximal, characteristic, complemented
7#88540C211yesmaximal, characteristic, complemented
8C3×#894—32nomaximal
9C3×#894—32nomaximal
10C22×#272C2211yescharacteristic
11C22×#272C2211yescharacteristic
12C2×C4×H(3,𝔽3)C2211yescharacteristic
13C2×C4×H(3,𝔽3)C2211yescharacteristic
14C2×C4×H(3,𝔽3)C2211yescharacteristic
15C2×#1084C2211yescharacteristic
16C2×#1084C2211yescharacteristic
17C4⋊#272—22no—
18C4×#272—22no—
19C2×C3×D12—32no—
20C2×C3×D12—32no—
21C2×C3×C4×D3—32no—
22C2×C3×C4×D3—32no—
23C32×C4⋊C4D312yescomplemented
24C32×C4⋊C4D312yescomplemented
25C3×(C2×D3)⋊C4—32no—
26C3×(C2×D3)⋊C4—32no—
27C3×(C2×D3)⋊C4—32no—
28C3×(C2×D3)⋊C4—32no—
29C3×(C2×D3)⋊C4—32no—
30C3×(C2×D3)⋊C4—32no—
31C3×Dic3⋊C4—32no—
32C3×Dic3⋊C4—32no—
33C22×H(3,𝔽3)C2311yescharacteristic
34C2×#272—21no—
35C2×#272—21no—
36C2×#272—21no—
37C2×#272—21no—
38C2×#272—21no—
39C2×#272D412yes—
40C4×H(3,𝔽3)—21no—
41C4×H(3,𝔽3)—21no—
42C4×H(3,𝔽3)—21no—
43#1084—21no—
44#1084—21no—
45C3×#123—91no—
46#894—32no—
47#894—32no—
48C22×C3×D3—32no—
49C22×C3×D3—32no—
50C22×C3×D3—32no—
51C22×C3×D3—32no—
52C2×C32×C4C2×D312yes—
53C2×C32×C4C2×D312yes—
54C2×C32×C4C2×D312yes—
55C2×C32×C4C2×D312yes—
56C2×C32×C4C2×D312yes—
57C2×C32×C4C2×D312yes—
58C2×C3×Dic3—32no—
59C2×C3×Dic3—32no—
60C2×C3×Dic3—32no—
61C2×C3×Dic3—32no—
62C3×D12—64no—
63C3×D12—64no—
64C3×C4×D3—64no—
65C3×C4×D3—64no—
66C2×H(3,𝔽3)C2×D411yescharacteristic
67C2×H(3,𝔽3)#4111yescharacteristic
68C2×H(3,𝔽3)#4111yescharacteristic
69#272—22no—
70#272—41no—
71C2×C3×D4—91no—
72C22×C3×C4—91no—
73C2×D12—32no—
74C2×D12—32no—
75C2×C4×D3—32no—
76C2×C4×D3—32no—
77C3×C4⋊C4—32no—
78C3×C4⋊C4—62no—
79C3×C4⋊C4—91no—
80C3×C4⋊C4C3⋊D311yescharacteristic
81C3×C22⋊C4—91no—
82C3×C22⋊C4—91no—
83C3×C22⋊C4—91no—
84(C2×D3)⋊C4—32no—
85(C2×D3)⋊C4—32no—
86(C2×D3)⋊C4—32no—
87(C2×D3)⋊C4—32no—
88(C2×D3)⋊C4—32no—
89(C2×D3)⋊C4—32no—
90Dic3⋊C4—32no—
91Dic3⋊C4—32no—
92C22×C32C22×D312yes—
93C22×C32C22×D312yes—
94C2×C3×D3—34no—
95C2×C3×D3—34no—
96C2×C3×D3—62no—
97C2×C3×D3—62no—
98C2×C3×D3—62no—
99C2×C3×D3—62no—
100C2×C3×D3—62no—
101C2×C3×D3—62no—
102C2×C3×D3—62no—
103C2×C3×D3—62no—
104C2×C3×D3—62no—
105C2×C3×D3—62no—
106C32×C4—22no—
107C32×C4—22no—
108C32×C4—22no—
109C32×C4—22no—
110C32×C4—22no—
111C32×C4—22no—
112C3×Dic3—62no—
113C3×Dic3—62no—
114C3×Dic3—62no—
115C3×Dic3—62no—
116#123—91noSylow
117H(3,𝔽3)#12311yescharacteristic, complemented, Sylow
118C23×C3—91no—
119C23×C3—91no—
120C22×D3—32no—
121C22×D3—32no—
122C22×D3—32no—
123C22×D3—32no—
124C3×D4—182no—
125C2×C3×C4—32no—
126C2×C3×C4—32no—
127C2×C3×C4—32no—
128C2×C3×C4—62no—
129C2×C3×C4—62no—
130C2×C3×C4—62no—
131C2×C3×C4—91no—
132C2×C3×C4—91no—
133C2×C3×C4—182no—
134C2×C3×C4C2×C3⋊D311yescharacteristic
135C2×C3×C4C2×C3⋊D311yescharacteristic
136C2×C3×C4C2×C3⋊D311yescharacteristic
137C2×Dic3—32no—
138C2×Dic3—32no—
139C2×Dic3—32no—
140C2×Dic3—32no—
141D12—64no—
142D12—64no—
143C4×D3—64no—
144C4×D3—64no—
145C2×C32#23512yes—
146C2×C32#23512yes—
147C2×C32D3×D412yes—
148C2×C32D3×D412yes—
149C2×C32#23912yes—
150C2×C32#23912yes—
151C3×D3—64no—
152C3×D3—64no—
153C3×D3—122no—
154C3×D3—122no—
155C2×D4—91no—
156C22×C4—91no—
157C4⋊C4—91no—
158C4⋊C4#27211yescharacteristic, complemented
159C22⋊C4—91no—
160C22⋊C4—91no—
161C22⋊C4—91no—
162C22×C3—32no—
163C22×C3—62no—
164C22×C3—92no—
165C22×C3—181no—
166C22×C3—181no—
167C22×C3—181no—
168C22×C3—181no—
169C22×C3—181no—
170C22×C3C22×C3⋊D311yescharacteristic
171C2×D3—34no—
172C2×D3—34no—
173C2×D3—62no—
174C2×D3—62no—
175C2×D3—62no—
176C2×D3—62no—
177C2×D3—62no—
178C2×D3—62no—
179C2×D3—62no—
180C2×D3—62no—
181C2×D3—62no—
182C2×D3—62no—
183C3×C4—21no—
184C3×C4—21no—
185C3×C4—21no—
186C3×C4—62no—
187C3×C4—62no—
188C3×C4—62no—
189C3×C4—62no—
190C3×C4—122no—
191C3×C4—122no—
192C3×C4—181no—
193C3×C4—181no—
194Dic3—62no—
195Dic3—62no—
196Dic3—62no—
197Dic3—62no—
198C32#89412yescomplemented
199C32#89412yescomplemented
200C23—91no—
201C23—91no—
202D4—182no—
203C2×C4—91no—
204C2×C4—91no—
205C2×C4—182no—
206C2×C4C2×#27211yescharacteristic
207C2×C4C2×#27211yescharacteristic
208C2×C4C2×#27211yescharacteristic
209C2×C3—32no—
210C2×C3—32no—
211C2×C3—32no—
212C2×C3—62no—
213C2×C3—62no—
214C2×C3—62no—
215C2×C3—182no—
216C2×C3—361no—
217C2×C3#382811yescharacteristic
218C2×C3D4×C3⋊D311yescharacteristic
219C2×C3#383211yescharacteristic
220D3—64no—
221D3—64no—
222D3—122no—
223D3—122no—
224C22—92no—
225C22—181no—
226C22—181no—
227C22—181no—
228C22—181no—
229C22—181no—
230C22C22×#27211yescharacteristic
231C4—21no—
232C4—21no—
233C4—21no—
234C4—181no—
235C4—181no—
236C3—32no—
237C3—62no—
238C3#6410311yescharacteristic
239C2—182no—
240C2—361no—
241C2#8874011yescharacteristic
242C2D4×#27211yescharacteristic
243C2#8874811yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C23
FittingC4⋊C4×H(3,𝔽3)C2
prime radicalC2×C4⋊#272C2
hypercenterC3×C4⋊C4C3⋊D3
hypocenterH(3,𝔽3)#123
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×C4×#272C2semidirect
3. C4⋊C4×H(3,𝔽3)C2semidirect
4. #88542C2semidirect
5. #88542C2semidirect
6. #88542C2semidirect
7. #88540C2semidirect
23. C32×C4⋊C4D3semidirect
24. C32×C4⋊C4D3semidirect
117. H(3,𝔽3)#123semidirect
158. C4⋊C4#272semidirect
198. C32#894semidirect
199. C32#894semidirect

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
#885402
C3×#8943