Finite Group Databasebeta

← #11721127

Subgroups — #11721127

Order 864.

363 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 58 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×#6200C211yesmaximal, characteristic, complemented
3C2×#6200C211yesmaximal, characteristic, complemented
4C2×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic, complemented
5#88542C211yesmaximal, characteristic, complemented
6#3750⋊C3C211yesmaximal, characteristic, complemented
7#88450C211yesmaximal, characteristic, complemented
8#63910—31nomaximal
9#63885—31nomaximal
10C22×#272C2211yescharacteristic
11C22×(C3⋊D3)⋊C3C2211yescharacteristic
12C22×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×C4×H(3,𝔽3)C2211yescharacteristic
14C2×#1084C2211yescharacteristic
15C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C4⋊((C3⋊D3)⋊C3)—22no—
18#6200—22no—
19#6200—22no—
20((C3⋊D3)⋊C3)⋊C4—22no—
21C2×C4⋊(C3⋊D3)D311yescharacteristic, complemented
22C2×C3×D12—31no—
23C2×#626—31no—
24C2×#626—31no—
25C2×#625—31no—
26C2×#625—31no—
27C2×#624—31no—
28C2×D3×Dic3—31no—
29#3750D311yescharacteristic, complemented
30C3×(C2×D3)⋊C4—31no—
31C3×(C2×D3)⋊C4—31no—
32C3×(C2×D3)⋊C4—61no—
33C3×Dic3⋊C4—31no—
34#3722—31no—
35(C2×D3)⋊Dic3—31no—
36C22×H(3,𝔽3)C2311yescharacteristic
37C2×#272—21no—
38C2×#272—21no—
39C2×#272—21no—
40C2×(C3⋊D3)⋊C3—21no—
41C2×(C3⋊D3)⋊C3—21no—
42C2×(C3⋊D3)⋊C3—21no—
43C2×(C3⋊D3)⋊C3—21no—
44C2×(C3⋊D3)⋊C3—21no—
45C2×(C3⋊D3)⋊C3D412yescomplemented
46C4×H(3,𝔽3)—21no—
47#1084—21no—
48(C3⋊Dic3)⋊C3—21no—
49(C3⋊Dic3)⋊C3D412yescomplemented
50#939—91no—
51#895—91no—
52#884—91no—
53C22×C3⋊D3C2×D311yescharacteristic
54C22×C3⋊D3C2×D311yescharacteristic
55C22×C3×D3—31no—
56C22×C3×D3—31no—
57C22×C3×D3—31no—
58C22×C3×D3—31no—
59C22×C3×D3—61no—
60C2×C32×C4—21no—
61C2×C32×C4C2×D311yescharacteristic
62C2×C32×C4C2×D311yescharacteristic
63C2×C3⋊Dic3C2×D311yescharacteristic
64C2×C3⋊Dic3C2×D311yescharacteristic
65C4⋊(C3⋊D3)—22no—
66C2×C3×Dic3—31no—
67C2×C3×Dic3—31no—
68C2×C3×Dic3—31no—
69C2×C3×Dic3—31no—
70C2×C3×Dic3—61no—
71C3×D12—62no—
72#626—62no—
73#626—62no—
74#625—62no—
75#625—62no—
76#624—62no—
77D3×Dic3—62no—
78C2×H(3,𝔽3)C2×D411yescharacteristic
79C2×H(3,𝔽3)C2×D411yescharacteristic
80C2×H(3,𝔽3)#4111yescharacteristic
81#272—41no—
82(C3⋊D3)⋊C3—22no—
83(C3⋊D3)⋊C3—41no—
84C2×C3×D4—91no—
85C2×#67—91no—
86C2×#67—91no—
87C2×#67—91no—
88C2×#67—91no—
89C22×Dic3—91no—
90C2×D12—31no—
91C2×D12—91no—
92C2×D12—91no—
93C2×D12—181no—
94C2×C4×D3—91no—
95C2×C4×D3—91no—
96C3×C4⋊C4—91no—
97C3×C22⋊C4—91no—
98#217—91no—
99(C2×D3)⋊C4—31no—
100(C2×D3)⋊C4—31no—
101(C2×D3)⋊C4—61no—
102(C2×D3)⋊C4—91no—
103(C2×D3)⋊C4—91no—
104Dic3⋊C4—31no—
105Dic3⋊C4—181no—
106C22×C32—21no—
107C22×C32C22×D311yescharacteristic
108C22×C32C22×D311yescharacteristic
109C2×C3⋊D3—21no—
110C2×C3⋊D3—21no—
111C2×C3⋊D3—21no—
112C2×C3⋊D3—21no—
113C2×C3⋊D3—21no—
114C2×C3⋊D3#6712yescomplemented
115C2×C3×D3—32no—
116C2×C3×D3—61no—
117C2×C3×D3—61no—
118C2×C3×D3—62no—
119C2×C3×D3—61no—
120C2×C3×D3—61no—
121C2×C3×D3—61no—
122C2×C3×D3—61no—
123C2×C3×D3—61no—
124C2×C3×D3—61no—
125C2×C3×D3—61no—
126C2×C3×D3—61no—
127C2×C3×D3—61no—
128C2×C3×D3—121no—
129C2×C3×D3—121no—
130C32×C4—21no—
131C32×C4—21no—
132C32×C4—41no—
133C3⋊Dic3—21no—
134C3⋊Dic3D1212yescomplemented
135C3×Dic3—32no—
136C3×Dic3—61no—
137C3×Dic3—61no—
138C3×Dic3—61no—
139C3×Dic3—121no—
140#121—271noSylow
141H(3,𝔽3)#12111yescharacteristic, complemented, Sylow
142C23×C3—91no—
143C23×C3—91no—
144C23×C3—91no—
145C22×D3—31no—
146C22×D3—31no—
147C22×D3—31no—
148C22×D3—31no—
149C22×D3—61no—
150C22×D3—181no—
151C22×D3—181no—
152C3×D4—182no—
153C2×C3×C4—61no—
154C2×C3×C4—61no—
155C2×C3×C4—91no—
156C2×C3×C4—91no—
157C2×C3×C4—91no—
158C2×C3×C4—121no—
159C2×C3×C4D3211yescharacteristic
160#67—182no—
161#67—182no—
162#67—182no—
163#67—182no—
164C2×Dic3—31no—
165C2×Dic3—31no—
166C2×Dic3—31no—
167C2×Dic3—31no—
168C2×Dic3—61no—
169C2×Dic3—181no—
170C2×Dic3—181no—
171C2×Dic3—182no—
172D12—62no—
173D12—182no—
174D12—182no—
175D12—182no—
176C4×D3—182no—
177C4×D3—182no—
178C2×C32—21no—
179C2×C32—21no—
180C2×C32—21no—
181C2×C32C2×D1211yescharacteristic
182C2×C32D3×D411yescharacteristic
183C2×C32D3×D411yescharacteristic
184C2×C32#23711yescharacteristic
185C2×C32#23911yescharacteristic
186C2×C32C2×#6711yescharacteristic
187C3⋊D3—22no—
188C3⋊D3—41no—
189C3×D3—62no—
190C3×D3—122no—
191C3×D3—121no—
192C3×D3—121no—
193C3×D3—121no—
194C2×D4—91no—
195C2×D4—271no—
196C2×D4—271no—
197C22×C4—271no—
198C4⋊C4—91no—
199C22⋊C4—91no—
200C22⋊C4—271no—
201C22×C3—61no—
202C22×C3—61no—
203C22×C3—92no—
204C22×C3—121no—
205C22×C3—181no—
206C22×C3—181no—
207C22×C3—181no—
208C22×C3—181no—
209C22×C3—181no—
210C22×C3—181no—
211C22×C3—181no—
212C22×C3—181no—
213C22×C3C2×D3211yescharacteristic
214C2×D3—32no—
215C2×D3—61no—
216C2×D3—61no—
217C2×D3—61no—
218C2×D3—61no—
219C2×D3—61no—
220C2×D3—61no—
221C2×D3—61no—
222C2×D3—61no—
223C2×D3—62no—
224C2×D3—61no—
225C2×D3—61no—
226C2×D3—61no—
227C2×D3—121no—
228C2×D3—121no—
229C2×D3—182no—
230C2×D3—181no—
231C2×D3—361no—
232C2×D3—361no—
233C2×D3—361no—
234C2×D3—361no—
235C3×C4—21no—
236C3×C4—61no—
237C3×C4—92no—
238C3×C4—121no—
239C3×C4—121no—
240C3×C4—181no—
241C3×C4—181no—
242Dic3—32no—
243Dic3—61no—
244Dic3—61no—
245Dic3—61no—
246Dic3—121no—
247Dic3—182no—
248Dic3—361no—
249C32—21no—
250C32#89511yescharacteristic, complemented
251C32#93911yescharacteristic, complemented
252C23—91no—
253C23—91no—
254C23—91no—
255D4—182no—
256D4—542no—
257D4—542no—
258C2×C4—91no—
259C2×C4—91no—
260C2×C4—91no—
261C2×C4—542no—
262C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
263C2×C3—61no—
264C2×C3—61no—
265C2×C3—61no—
266C2×C3—61no—
267C2×C3—61no—
268C2×C3—61no—
269C2×C3—121no—
270C2×C3—121no—
271C2×C3—121no—
272C2×C3—182no—
273C2×C3—361no—
274C2×C3—361no—
275C2×C3#379611yescharacteristic
276C2×C3D3×D1211yescharacteristic
277C2×C3D3×#6711yescharacteristic
278D3—62no—
279D3—121no—
280D3—121no—
281D3—122no—
282D3—121no—
283D3—362no—
284D3—361no—
285C22—92no—
286C22—181no—
287C22—181no—
288C22—181no—
289C22—181no—
290C22—181no—
291C22—181no—
292C22—181no—
293C22—181no—
294C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
295C4—21no—
296C4—92no—
297C4—181no—
298C4—181no—
299C3—61no—
300C3—61no—
301C3—121no—
302C3#6388611yescharacteristic
303C2—182no—
304C2—361no—
305C2—361no—
306C2#8864911yescharacteristic
307C2#8865411yescharacteristic
308C2#8867611yescharacteristic
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×#6200C2semidirect
3. C2×#6200C2semidirect
4. C2×((C3⋊D3)⋊C3)⋊C4C2semidirect
5. #88542C2semidirect
6. #3750⋊C3C2semidirect
7. #88450C2semidirect
21. C2×C4⋊(C3⋊D3)D3semidirect
29. #3750D3semidirect
45. C2×(C3⋊D3)⋊C3D4semidirect
49. (C3⋊Dic3)⋊C3D4semidirect
114. C2×C3⋊D3#67semidirect
134. C3⋊Dic3D12semidirect
141. H(3,𝔽3)#121semidirect
250. C32#895semidirect
251. C32#939semidirect

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×#62002
C2×((C3⋊D3)⋊C3)⋊C42
#885422
#3750⋊C32
#884502
#639103
#638853