Finite Group Databasebeta

← #11721154

Subgroups — #11721154

Order 864.

363 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 68 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)⋊C3C211yesmaximal, characteristic, complemented
2C2×#6200C211yesmaximal, characteristic
3C2×#6199C211yesmaximal, characteristic, complemented
4#88542C211yesmaximal, characteristic, complemented
5#3750⋊C3C211yesmaximal, characteristic, complemented
6#88450C211yesmaximal, characteristic, complemented
7#88447C211yesmaximal, characteristic, complemented
8#63881—31nomaximal
9#63879—31nomaximal
10C22×#272C2211yescharacteristic
11C22×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×C4×H(3,𝔽3)C2211yescharacteristic
13C2×#1084C2211yescharacteristic
14C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C4×(C3⋊D3)⋊C3—22no—
18#6200C414yescomplemented
19#6199—22no—
20C2×C4×C3⋊D3D311yescharacteristic, complemented
21C2×C3×C4×D3—31no—
22C2×#626—31no—
23C2×#625—31no—
24C2×D3×Dic3—31no—
25C2×D3×Dic3—31no—
26#3750D311yescharacteristic, complemented
27C3×(C2×D3)⋊C4—31no—
28C3×(C2×D3)⋊C4—31no—
29C3×(C2×D3)⋊C4—61no—
30C3×Dic3⋊C4—31no—
31#3722—31no—
32(C2×D3)⋊Dic3—31no—
33Dic32—31no—
34Dic32—31no—
35C22×H(3,𝔽3)C2311yescharacteristic
36C2×#272—21no—
37C2×#272—21no—
38C2×#272C2×C412yescomplemented
39C2×(C3⋊D3)⋊C3—21no—
40C2×(C3⋊D3)⋊C3—21no—
41C2×(C3⋊D3)⋊C3C2×C412yescomplemented
42C4×H(3,𝔽3)—21no—
43#1084—21no—
44(C3⋊Dic3)⋊C3—21no—
45(C3⋊Dic3)⋊C3C2×C412yescomplemented
46(C3⋊Dic3)⋊C3D412yescomplemented
47C4×#67—91no—
48#893—91no—
49#67⋊C4—91no—
50C22×C3⋊D3C2×D311yescharacteristic
51C22×C3×D3—31no—
52C22×C3×D3—31no—
53C22×C3×D3—31no—
54C22×C3×D3—61no—
55C2×C32×C4—21no—
56C2×C32×C4C2×D311yescharacteristic
57C2×C32×C4C2×D311yescharacteristic
58C2×C3⋊Dic3C2×D311yescharacteristic
59C2×C3⋊Dic3C2×D311yescharacteristic
60C2×C3⋊Dic3C2×D311yescharacteristic
61C4×C3⋊D3—22no—
62C2×C3×Dic3—31no—
63C2×C3×Dic3—31no—
64C2×C3×Dic3—31no—
65C2×C3×Dic3—31no—
66C2×C3×Dic3—31no—
67C2×C3×Dic3—61no—
68C3×C4×D3—62no—
69#626—34no—
70#625—34no—
71D3×Dic3—62no—
72D3×Dic3—62no—
73C2×H(3,𝔽3)C22×C411yescharacteristic
74C2×H(3,𝔽3)C2×D411yescharacteristic
75C2×H(3,𝔽3)#4111yescharacteristic
76#272—22no—
77(C3⋊D3)⋊C3—22no—
78C22×C3×C4—91no—
79C2×#67—91no—
80C2×#67—91no—
81C22×Dic3—91no—
82C2×D12—91no—
83C2×C4×D3—31no—
84C2×C4×D3—91no—
85C2×C4×D3—91no—
86C2×C4×D3—91no—
87C3×C4⋊C4—91no—
88C3×C22⋊C4—91no—
89#217—91no—
90(C2×D3)⋊C4—31no—
91(C2×D3)⋊C4—31no—
92(C2×D3)⋊C4—61no—
93(C2×D3)⋊C4—91no—
94(C2×D3)⋊C4—91no—
95Dic3⋊C4—31no—
96Dic3⋊C4—181no—
97C4×Dic3—91no—
98C4×Dic3—91no—
99C4×Dic3—91no—
100C22×C32—21no—
101C22×C32C22×D311yescharacteristic
102C22×C32C22×D311yescharacteristic
103C2×C3⋊D3—21no—
104C2×C3⋊D3—21no—
105C2×C3⋊D3C4×D312yescomplemented
106C2×C3×D3—32no—
107C2×C3×D3—32no—
108C2×C3×D3—32no—
109C2×C3×D3—62no—
110C2×C3×D3—61no—
111C2×C3×D3—61no—
112C2×C3×D3—61no—
113C2×C3×D3—61no—
114C2×C3×D3—61no—
115C2×C3×D3—61no—
116C2×C3×D3—121no—
117C2×C3×D3—121no—
118C32×C4—21no—
119C32×C4—21no—
120C32×C4—41no—
121C3⋊Dic3—21no—
122C3⋊Dic3C4×D312yescomplemented
123C3⋊Dic3#6712yescomplemented
124C3×Dic3—32no—
125C3×Dic3—32no—
126C3×Dic3—61no—
127C3×Dic3—61no—
128C3×Dic3—61no—
129C3×Dic3—121no—
130C4×D4—271noSylow
131H(3,𝔽3)C4×D411yescharacteristic, complemented, Sylow
132C23×C3—91no—
133C23×C3—91no—
134C22×D3—31no—
135C22×D3—31no—
136C22×D3—31no—
137C22×D3—61no—
138C22×D3—91no—
139C2×C3×C4—31no—
140C2×C3×C4—61no—
141C2×C3×C4—91no—
142C2×C3×C4—91no—
143C2×C3×C4—91no—
144C2×C3×C4—91no—
145C2×C3×C4—121no—
146C2×C3×C4—182no—
147C2×C3×C4D3211yescharacteristic
148#67—94no—
149#67—94no—
150C2×Dic3—31no—
151C2×Dic3—31no—
152C2×Dic3—31no—
153C2×Dic3—31no—
154C2×Dic3—31no—
155C2×Dic3—61no—
156C2×Dic3—91no—
157C2×Dic3—181no—
158C2×Dic3—181no—
159C2×Dic3—182no—
160D12—94no—
161C4×D3—62no—
162C4×D3—182no—
163C4×D3—182no—
164C4×D3—182no—
165C2×C32—21no—
166C2×C32—21no—
167C2×C32—21no—
168C2×C32C2×C4×D311yescharacteristic
169C2×C32C2×C4×D311yescharacteristic
170C2×C32#23511yescharacteristic
171C2×C32D3×D411yescharacteristic
172C2×C32#23911yescharacteristic
173C2×C32C2×#6711yescharacteristic
174C3⋊D3—22no—
175C3×D3—62no—
176C3×D3—62no—
177C3×D3—62no—
178C3×D3—122no—
179C2×D4—271no—
180C22×C4—91no—
181C22×C4—271no—
182C4⋊C4—91no—
183C22⋊C4—91no—
184C22⋊C4—271no—
185C42—271no—
186C22×C3—31no—
187C22×C3—61no—
188C22×C3—92no—
189C22×C3—92no—
190C22×C3—121no—
191C22×C3—181no—
192C22×C3—181no—
193C22×C3—181no—
194C22×C3—181no—
195C22×C3C2×D3211yescharacteristic
196C2×D3—32no—
197C2×D3—32no—
198C2×D3—32no—
199C2×D3—62no—
200C2×D3—61no—
201C2×D3—61no—
202C2×D3—61no—
203C2×D3—61no—
204C2×D3—61no—
205C2×D3—61no—
206C2×D3—92no—
207C2×D3—121no—
208C2×D3—121no—
209C2×D3—181no—
210C2×D3—181no—
211C3×C4—21no—
212C3×C4—61no—
213C3×C4—92no—
214C3×C4—92no—
215C3×C4—121no—
216C3×C4—121no—
217C3×C4—181no—
218C3×C4—181no—
219Dic3—32no—
220Dic3—32no—
221Dic3—61no—
222Dic3—61no—
223Dic3—61no—
224Dic3—92no—
225Dic3—121no—
226Dic3—182no—
227Dic3—361no—
228C32—21no—
229C32#89311yescharacteristic, complemented
230C32C4×#6711yescharacteristic, complemented
231C23—91no—
232C23—91no—
233D4—274no—
234C2×C4—91no—
235C2×C4—91no—
236C2×C4—91no—
237C2×C4—91no—
238C2×C4—182no—
239C2×C4—542no—
240C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
241C2×C3—31no—
242C2×C3—31no—
243C2×C3—31no—
244C2×C3—61no—
245C2×C3—61no—
246C2×C3—61no—
247C2×C3—121no—
248C2×C3—121no—
249C2×C3—121no—
250C2×C3—182no—
251C2×C3—182no—
252C2×C3#379911yescharacteristic
253C2×C3C4×D3211yescharacteristic
254C2×C3D3×#6711yescharacteristic
255D3—62no—
256D3—62no—
257D3—62no—
258D3—122no—
259D3—182no—
260C22—92no—
261C22—92no—
262C22—181no—
263C22—181no—
264C22—181no—
265C22—181no—
266C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
267C4—21no—
268C4—92no—
269C4—92no—
270C4—181no—
271C4—181no—
272C3—31no—
273C3—61no—
274C3—121no—
275C3#6390411yescharacteristic
276C2—182no—
277C2—182no—
278C2#8865211yescharacteristic
279C2C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
280C2#8867611yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C22×C4
Frattini, socleC22×C3C2×D32
FittingC2×C4×H(3,𝔽3)C22
prime radicalC2×#6200C2
cosocleC22×H(3,𝔽3)C23
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4×D4
LCSG ▷ C2×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)⋊C3C2semidirect
3. C2×#6199C2semidirect
4. #88542C2semidirect
5. #3750⋊C3C2semidirect
6. #88450C2semidirect
7. #88447C2semidirect
18. #6200C4semidirect
20. C2×C4×C3⋊D3D3semidirect
26. #3750D3semidirect
38. C2×#272C2×C4semidirect
41. C2×(C3⋊D3)⋊C3C2×C4semidirect
45. (C3⋊Dic3)⋊C3C2×C4semidirect
46. (C3⋊Dic3)⋊C3D4semidirect
105. C2×C3⋊D3C4×D3semidirect
122. C3⋊Dic3C4×D3semidirect
123. C3⋊Dic3#67semidirect
131. H(3,𝔽3)C4×D4semidirect
229. C32#893semidirect
230. C32C4×#67semidirect

Maximal subgroups

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

subgroupindex
C2×C4×(C3⋊D3)⋊C32
C2×#62002
C2×#61992
#885422
#3750⋊C32
#884502
#884472
#638813
#638793