Finite Group Databasebeta

← #11721313

Subgroups — #11721313

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×#638⋊C3C211yesmaximal, characteristic, complemented
2C2×#638⋊C3C211yesmaximal, characteristic, complemented
3C2×#6198C211yesmaximal, characteristic, complemented
4C2×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic, complemented
5#88552C211yesmaximal, characteristic, complemented
6#88450C211yesmaximal, characteristic, complemented
7#88448C211yesmaximal, characteristic, complemented
8#63971—31nomaximal
9#63958—31nomaximal
10C23×H(3,𝔽3)C2211yescharacteristic
11C22×(C3⋊D3)⋊C3C2211yescharacteristic
12C22×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×#1084C2211yescharacteristic
14C2×#1084C2211yescharacteristic
15#638⋊C3—22no—
16#638⋊C3—22no—
17C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
19#6198—22no—
20((C3⋊D3)⋊C3)⋊C4—22no—
21C2×#638D311yescharacteristic, complemented
22C2×#638D311yescharacteristic, complemented
23C2×C3×#67—31no—
24C2×C3×#67—31no—
25C2×#626—31no—
26C2×#626—31no—
27C2×#624—31no—
28C2×D3×Dic3—31no—
29C3×#217—31no—
30C3×#217—31no—
31C3×#217—61no—
32#3724—31no—
33Dic3⋊Dic3—31no—
34#3722—31no—
35(C2×D3)⋊Dic3—31no—
36C22×H(3,𝔽3)—21no—
37C22×H(3,𝔽3)—21no—
38C22×H(3,𝔽3)—21no—
39C22×H(3,𝔽3)C2311yescharacteristic
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
46#1084—21no—
47#1084—21no—
48(C3⋊Dic3)⋊C3—21no—
49(C3⋊Dic3)⋊C3D412yescomplemented
50#939—91no—
51#938—91no—
52#884—91no—
53C23×C32—21no—
54C23×C32C2×D311yescharacteristic
55C23×C32C2×D311yescharacteristic
56C22×C3⋊D3C2×D311yescharacteristic
57C22×C3⋊D3C2×D311yescharacteristic
58C22×C3×D3—31no—
59C22×C3×D3—31no—
60#638—22no—
61#638—22no—
62C2×C3⋊Dic3C2×D311yescharacteristic
63C2×C3⋊Dic3C2×D311yescharacteristic
64C3×#67—62no—
65C3×#67—62no—
66C2×C3×Dic3—31no—
67C2×C3×Dic3—31no—
68C2×C3×Dic3—31no—
69C2×C3×Dic3—31no—
70C2×C3×Dic3—31no—
71C2×C3×Dic3—31no—
72C2×C3×Dic3—61no—
73C2×C3×Dic3—61no—
74#626—62no—
75#626—62no—
76#624—62no—
77D3×Dic3—62no—
78C2×H(3,𝔽3)—41no—
79C2×H(3,𝔽3)C2×D411yescharacteristic
80C2×H(3,𝔽3)C2×D411yescharacteristic
81C2×H(3,𝔽3)#4111yescharacteristic
82(C3⋊D3)⋊C3—22no—
83(C3⋊D3)⋊C3—41no—
84C2×C3×D4—91no—
85C2×C3×D4—91no—
86C2×#67—31no—
87C2×#67—31no—
88C2×#67—91no—
89C2×#67—91no—
90C2×#67—91no—
91C2×#67—91no—
92C22×Dic3—91no—
93C2×D12—91no—
94C2×C4×D3—91no—
95C2×C4×D3—91no—
96C3×C22⋊C4—91no—
97#217—31no—
98#217—31no—
99#217—61no—
100#217—91no—
101(C2×D3)⋊C4—91no—
102(C2×D3)⋊C4—91no—
103C4⋊Dic3—91no—
104Dic3⋊C4—91no—
105Dic3⋊C4—91no—
106C22×C32—22no—
107C22×C32—21no—
108C22×C32—21no—
109C22×C32—21no—
110C22×C32—21no—
111C22×C32—21no—
112C22×C32—21no—
113C22×C32—21no—
114C22×C32—41no—
115C22×C32—41no—
116C22×C32C22×D311yescharacteristic
117C22×C32C22×D311yescharacteristic
118C2×C3⋊D3—21no—
119C2×C3⋊D3—21no—
120C2×C3⋊D3—21no—
121C2×C3⋊D3—21no—
122C2×C3⋊D3—21no—
123C2×C3⋊D3#6712yescomplemented
124C2×C3×D3—32no—
125C2×C3×D3—61no—
126C2×C3×D3—61no—
127C2×C3×D3—61no—
128C2×C3×D3—61no—
129C2×C3×D3—61no—
130C3⋊Dic3—21no—
131C3⋊Dic3#6712yescomplemented
132C3×Dic3—32no—
133C3×Dic3—61no—
134C3×Dic3—61no—
135C3×Dic3—61no—
136C3×Dic3—61no—
137C3×Dic3—61no—
138C3×Dic3—121no—
139C3×Dic3—121no—
140#121—271noSylow
141H(3,𝔽3)#12111yescharacteristic, complemented, Sylow
142C23×C3—31no—
143C23×C3—31no—
144C23×C3—61no—
145C23×C3—91no—
146C23×C3—91no—
147C23×C3D3211yescharacteristic
148C22×D3—31no—
149C22×D3—31no—
150C22×D3—91no—
151C22×D3—91no—
152C3×D4—182no—
153C3×D4—182no—
154C2×C3×C4—91no—
155C2×C3×C4—91no—
156C2×C3×C4—91no—
157C2×C3×C4—91no—
158#67—62no—
159#67—62no—
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—31no—
169C2×Dic3—31no—
170C2×Dic3—61no—
171C2×Dic3—61no—
172C2×Dic3—91no—
173C2×Dic3—91no—
174C2×Dic3—182no—
175D12—182no—
176C4×D3—182no—
177C4×D3—182no—
178C2×C32—21no—
179C2×C32—21no—
180C2×C32—21no—
181C2×C32—42no—
182C2×C32—41no—
183C2×C32—41no—
184C2×C32D3×D411yescharacteristic
185C2×C32D3×D411yescharacteristic
186C2×C32#23711yescharacteristic
187C2×C32#23711yescharacteristic
188C2×C32C2×#6711yescharacteristic
189C2×C32C2×#6711yescharacteristic
190C3⋊D3—22no—
191C3⋊D3—41no—
192C3×D3—62no—
193C3×D3—121no—
194C2×D4—91no—
195C2×D4—91no—
196C2×D4—271no—
197C22×C4—271no—
198C4⋊C4—271no—
199C22⋊C4—91no—
200C22⋊C4—271no—
201C22×C3—21no—
202C22×C3—21no—
203C22×C3—21no—
204C22×C3—31no—
205C22×C3—31no—
206C22×C3—61no—
207C22×C3—61no—
208C22×C3—61no—
209C22×C3—61no—
210C22×C3—61no—
211C22×C3—61no—
212C22×C3—62no—
213C22×C3—61no—
214C22×C3—92no—
215C22×C3—121no—
216C22×C3—121no—
217C22×C3—181no—
218C22×C3—181no—
219C22×C3—181no—
220C22×C3—181no—
221C22×C3—181no—
222C22×C3C2×D3211yescharacteristic
223C2×D3—32no—
224C2×D3—61no—
225C2×D3—61no—
226C2×D3—61no—
227C2×D3—61no—
228C2×D3—61no—
229C2×D3—92no—
230C2×D3—181no—
231C2×D3—181no—
232C2×D3—181no—
233C2×D3—181no—
234C2×D3—181no—
235C3×C4—92no—
236C3×C4—181no—
237C3×C4—181no—
238C3×C4—181no—
239Dic3—32no—
240Dic3—61no—
241Dic3—61no—
242Dic3—61no—
243Dic3—61no—
244Dic3—61no—
245Dic3—92no—
246Dic3—121no—
247Dic3—121no—
248Dic3—181no—
249C32—21no—
250C32#93811yescharacteristic, complemented
251C32#93911yescharacteristic, complemented
252C23—91no—
253C23—91no—
254C23((C3⋊D3)⋊C3)⋊C211yescharacteristic
255D4—182no—
256D4—182no—
257D4—542no—
258C2×C4—91no—
259C2×C4—91no—
260C2×C4—91no—
261C2×C4—91no—
262C2×C4—542no—
263C2×C3—31no—
264C2×C3—31no—
265C2×C3—31no—
266C2×C3—31no—
267C2×C3—31no—
268C2×C3—31no—
269C2×C3—41no—
270C2×C3—61no—
271C2×C3—61no—
272C2×C3—61no—
273C2×C3—121no—
274C2×C3—121no—
275C2×C3—122no—
276C2×C3—182no—
277C2×C3—361no—
278C2×C3#380511yescharacteristic
279C2×C3D3×#6711yescharacteristic
280C2×C3D3×#6711yescharacteristic
281D3—62no—
282D3—121no—
283D3—182no—
284D3—361no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—92no—
289C22—181no—
290C22—181no—
291C22—181no—
292C22—181no—
293C22—181no—
294C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
295C4—92no—
296C4—181no—
297C4—181no—
298C4—181no—
299C3—31no—
300C3—31no—
301C3—61no—
302C3#6397011yescharacteristic
303C2—41no—
304C2—182no—
305C2—361no—
306C2#8867211yescharacteristic
307C2#8867611yescharacteristic
308C2#8867611yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C23
Frattini, socleC22×C3C2×D32
FittingC23×H(3,𝔽3)C22
hypercenterC23((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#121
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C2×#638⋊C3C2semidirect
2. C2×#638⋊C3C2semidirect
3. C2×#6198C2semidirect
4. C2×((C3⋊D3)⋊C3)⋊C4C2semidirect
5. #88552C2semidirect
6. #88450C2semidirect
7. #88448C2semidirect
21. C2×#638D3semidirect
22. C2×#638D3semidirect
45. C2×(C3⋊D3)⋊C3D4semidirect
49. (C3⋊Dic3)⋊C3D4semidirect
123. C2×C3⋊D3#67semidirect
131. C3⋊Dic3#67semidirect
141. H(3,𝔽3)#121semidirect
250. C32#938semidirect
251. C32#939semidirect

Maximal subgroups

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

subgroupindex
C2×#638⋊C32
C2×#61982
C2×((C3⋊D3)⋊C3)⋊C42
#885522
#884502
#884482
#639713
#639583