Finite Group Databasebeta

← #11721315

Subgroups — #11721315

Order 864.

383 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C22×#1084C211yesmaximal, characteristic, complemented
2C2×#638⋊C3C211yesmaximal, characteristic
3C2×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic, complemented
4#3757⋊C3C211yesmaximal, characteristic, complemented
5#88450C211yesmaximal, characteristic, complemented
6#88448C211yesmaximal, characteristic, complemented
7#88447C211yesmaximal, characteristic, complemented
8Dic3×#67—31nomaximal
9#63952—31nomaximal
10C23×H(3,𝔽3)C2211yescharacteristic
11C22×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×#1084—22no—
13C2×#1084C2211yescharacteristic
14C2×#1084C2211yescharacteristic
15#638⋊C3C414yescomplemented
16C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
19((C3⋊D3)⋊C3)⋊C4—22no—
20C2×#638D311yescharacteristic
21C2×C3×#67—31no—
22C22×C3×Dic3—31no—
23C22×C3×Dic3—31no—
24C22×C3×Dic3—61no—
25C2×#624—31no—
26C2×D3×Dic3—31no—
27#3757D311yescharacteristic, complemented
28C3×#217—31no—
29#3724—31no—
30Dic3⋊Dic3—31no—
31#3722—31no—
32(C2×D3)⋊Dic3—31no—
33Dic32—31no—
34Dic32—31no—
35C22×H(3,𝔽3)—21no—
36C22×H(3,𝔽3)—21no—
37C22×H(3,𝔽3)C2×C412yescomplemented
38C22×H(3,𝔽3)C2311yescharacteristic
39C2×(C3⋊D3)⋊C3—21no—
40C2×(C3⋊D3)⋊C3—21no—
41C2×(C3⋊D3)⋊C3C2×C412yescomplemented
42#1084—21no—
43#1084D412yescomplemented
44(C3⋊Dic3)⋊C3—21no—
45(C3⋊Dic3)⋊C3—21no—
46(C3⋊Dic3)⋊C3C2×C412yescomplemented
47D4×Dic3—91no—
48C4×#67—91no—
49#67⋊C4—91no—
50C23×C32—21no—
51C23×C32C2×D311yescharacteristic
52C23×C32C2×D311yescharacteristic
53C22×C3⋊D3C2×D311yescharacteristic
54C22×C3×D3—31no—
55#638Dic314yescomplemented
56C2×C3⋊Dic3C2×D311yescharacteristic
57C2×C3⋊Dic3C2×D311yescharacteristic
58C2×C3⋊Dic3C2×D311yescharacteristic
59C3×#67—34no—
60C2×C3×Dic3—31no—
61C2×C3×Dic3—31no—
62C2×C3×Dic3—31no—
63C2×C3×Dic3—31no—
64C2×C3×Dic3—31no—
65C2×C3×Dic3—31no—
66C2×C3×Dic3—31no—
67C2×C3×Dic3—61no—
68C2×C3×Dic3—61no—
69C2×C3×Dic3—62no—
70C2×C3×Dic3—62no—
71C2×C3×Dic3—64no—
72#624—62no—
73D3×Dic3—62no—
74C2×H(3,𝔽3)—22no—
75C2×H(3,𝔽3)C22×C411yescharacteristic
76C2×H(3,𝔽3)C2×D411yescharacteristic
77C2×H(3,𝔽3)#4111yescharacteristic
78(C3⋊D3)⋊C3—22no—
79C2×C3×D4—91no—
80C22×C3×C4—91no—
81C2×#67—31no—
82C2×#67—91no—
83C22×Dic3—31no—
84C22×Dic3—31no—
85C22×Dic3—61no—
86C22×Dic3—91no—
87C2×C4×D3—91no—
88C2×C4×D3—91no—
89C3×C22⋊C4—91no—
90#217—31no—
91#217—91no—
92#217—91no—
93(C2×D3)⋊C4—91no—
94(C2×D3)⋊C4—91no—
95C4⋊Dic3—91no—
96Dic3⋊C4—91no—
97Dic3⋊C4—91no—
98C4×Dic3—91no—
99C4×Dic3—91no—
100C4×Dic3—91no—
101C22×C32—22no—
102C22×C32—22no—
103C22×C32—21no—
104C22×C32—21no—
105C22×C32—21no—
106C22×C32—21no—
107C22×C32—22no—
108C22×C32—21no—
109C22×C32C4×D312yescomplemented
110C22×C32C2×Dic312yescomplemented
111C22×C32C22×D311yescharacteristic
112C22×C32C22×D311yescharacteristic
113C2×C3⋊D3—21no—
114C2×C3⋊D3—21no—
115C2×C3⋊D3C2×Dic312yescomplemented
116C2×C3×D3—32no—
117C2×C3×D3—61no—
118C2×C3×D3—61no—
119C3⋊Dic3—21no—
120C3⋊Dic3—21no—
121C3⋊Dic3C2×Dic312yescomplemented
122C3×Dic3—32no—
123C3×Dic3—32no—
124C3×Dic3—32no—
125C3×Dic3—62no—
126C3×Dic3—62no—
127C3×Dic3—61no—
128C3×Dic3—61no—
129C3×Dic3—61no—
130C3×Dic3—61no—
131C4×D4—271noSylow
132H(3,𝔽3)C4×D411yescharacteristic, complemented, Sylow
133C23×C3—31no—
134C23×C3—31no—
135C23×C3—61no—
136C23×C3—91no—
137C23×C3D3211yescharacteristic
138C22×D3—31no—
139C22×D3—91no—
140C3×D4—94no—
141C2×C3×C4—91no—
142C2×C3×C4—91no—
143C2×C3×C4—91no—
144C2×C3×C4—91no—
145C2×C3×C4—91no—
146C2×C3×C4—182no—
147#67—34no—
148#67—94no—
149C2×Dic3—31no—
150C2×Dic3—31no—
151C2×Dic3—31no—
152C2×Dic3—31no—
153C2×Dic3—31no—
154C2×Dic3—31no—
155C2×Dic3—31no—
156C2×Dic3—61no—
157C2×Dic3—61no—
158C2×Dic3—62no—
159C2×Dic3—62no—
160C2×Dic3—64no—
161C2×Dic3—91no—
162C2×Dic3—91no—
163C2×Dic3—91no—
164C2×Dic3—182no—
165C4×D3—182no—
166C4×D3—182no—
167C2×C32—24no—
168C2×C32—22no—
169C2×C32—22no—
170C2×C32—21no—
171C2×C32—21no—
172C2×C32—21no—
173C2×C32C2×C4×D311yescharacteristic
174C2×C32D3×D411yescharacteristic
175C2×C32D3×D411yescharacteristic
176C2×C32#23711yescharacteristic
177C2×C32#23711yescharacteristic
178C2×C32C22×Dic311yescharacteristic
179C3⋊D3—22no—
180C3×D3—62no—
181C2×D4—91no—
182C22×C4—91no—
183C22×C4—271no—
184C4⋊C4—271no—
185C22⋊C4—91no—
186C22⋊C4—271no—
187C42—271no—
188C22×C3—21no—
189C22×C3—21no—
190C22×C3—32no—
191C22×C3—32no—
192C22×C3—31no—
193C22×C3—31no—
194C22×C3—61no—
195C22×C3—61no—
196C22×C3—61no—
197C22×C3—61no—
198C22×C3—62no—
199C22×C3—62no—
200C22×C3—62no—
201C22×C3—61no—
202C22×C3—92no—
203C22×C3—181no—
204C22×C3—181no—
205C22×C3D3×Dic312yes—
206C22×C3C2×D3211yescharacteristic
207C2×D3—32no—
208C2×D3—61no—
209C2×D3—61no—
210C2×D3—92no—
211C2×D3—181no—
212C2×D3—181no—
213C3×C4—92no—
214C3×C4—92no—
215C3×C4—181no—
216C3×C4—181no—
217C3×C4—181no—
218Dic3—32no—
219Dic3—32no—
220Dic3—32no—
221Dic3—62no—
222Dic3—61no—
223Dic3—61no—
224Dic3—62no—
225Dic3—61no—
226Dic3—61no—
227Dic3—92no—
228Dic3—181no—
229Dic3—181no—
230C32—21no—
231C32#67⋊C411yescharacteristic, complemented
232C32D4×Dic311yescharacteristic, complemented
233C23—91no—
234C23((C3⋊D3)⋊C3)⋊C211yescharacteristic
235D4—94no—
236C2×C4—91no—
237C2×C4—91no—
238C2×C4—91no—
239C2×C4—91no—
240C2×C4—91no—
241C2×C4—182no—
242C2×C4—542no—
243C2×C3—22no—
244C2×C3—31no—
245C2×C3—31no—
246C2×C3—31no—
247C2×C3—31no—
248C2×C3—31no—
249C2×C3—31no—
250C2×C3—62no—
251C2×C3—62no—
252C2×C3—64no—
253C2×C3—61no—
254C2×C3—61no—
255C2×C3—61no—
256C2×C3—182no—
257C2×C3C2×D3×Dic311yescharacteristic
258C2×C3#380511yescharacteristic
259C2×C3#381111yescharacteristic
260D3—62no—
261D3—182no—
262C22—21no—
263C22—21no—
264C22—92no—
265C22—181no—
266C22—181no—
267C22((C3⋊D3)⋊C3)⋊C412yescomplemented
268C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
269C4—92no—
270C4—92no—
271C4—181no—
272C4—181no—
273C4—181no—
274C3—31no—
275C3—31no—
276C3—61no—
277C3#67⋊Dic311yescharacteristic
278C2—22no—
279C2—182no—
280C2C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
281C2#8867211yescharacteristic
282C2#8867711yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C22×C4
Frattini, socleC22×C3C2×D32
FittingC23×H(3,𝔽3)C22
prime radicalC2×#638⋊C3C2
cosocleC22×H(3,𝔽3)C23
hypercenterC23((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4×D4
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×#1084C2semidirect
3. C2×((C3⋊D3)⋊C3)⋊C4C2semidirect
4. #3757⋊C3C2semidirect
5. #88450C2semidirect
6. #88448C2semidirect
7. #88447C2semidirect
15. #638⋊C3C4semidirect
27. #3757D3semidirect
37. C22×H(3,𝔽3)C2×C4semidirect
41. C2×(C3⋊D3)⋊C3C2×C4semidirect
43. #1084D4semidirect
46. (C3⋊Dic3)⋊C3C2×C4semidirect
55. #638Dic3semidirect
109. C22×C32C4×D3semidirect
110. C22×C32C2×Dic3semidirect
115. C2×C3⋊D3C2×Dic3semidirect
121. C3⋊Dic3C2×Dic3semidirect
132. H(3,𝔽3)C4×D4semidirect
231. C32#67⋊C4semidirect
232. C32D4×Dic3semidirect
267. C22((C3⋊D3)⋊C3)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#10842
C2×#638⋊C32
C2×((C3⋊D3)⋊C3)⋊C42
#3757⋊C32
#884502
#884482
#884472
Dic3×#673
#639523