Finite Group Databasebeta

← #64093⋊C3 #11721424

Subgroups — #64093⋊C3 #11721424

Order 864.

363 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 75 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×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
4C2×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
5C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
6#3752⋊C3C211yesmaximal, characteristic, complemented
7#3750⋊C3C211yesmaximal, characteristic, complemented
8#64093C311yesmaximal, characteristic, complemented
9C3×#884—31nomaximal
10C23×H(3,𝔽3)C2211yescharacteristic
11C22×(C3⋊D3)⋊C3C2211yescharacteristic
12C22×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×C4×H(3,𝔽3)C2211yescharacteristic
14C2×C4×H(3,𝔽3)C2211yescharacteristic
15#638⋊C3—22no—
16#638⋊C3—22no—
17C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C4⋊((C3⋊D3)⋊C3)—22no—
20C4×(C3⋊D3)⋊C3—22no—
21C2×#638C2×C311yescharacteristic, complemented
22C2×#638C2×C311yescharacteristic, complemented
23C2×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
24C2×C4×C3⋊D3C2×C311yescharacteristic, complemented
25C2×C3×#67—31no—
26C2×C3×#67—31no—
27C2×C3×D12—31no—
28C2×C3×C4×D3—31no—
29C32×C22⋊C4—21no—
30C32×C22⋊C4D311yescharacteristic, complemented
31C32×C22⋊C4C2×C311yescharacteristic, complemented
32#3752C2×C311yescharacteristic, complemented
33#3750C2×C311yescharacteristic, complemented
34C3×(C2×D3)⋊C4—31no—
35C3×Dic3⋊C4—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
46C4×H(3,𝔽3)—21no—
47C4×H(3,𝔽3)—21no—
48(C3⋊Dic3)⋊C3—21no—
49(C3⋊Dic3)⋊C3D412yescomplemented
50C3×#121—91no—
51#884—31no—
52#884—91no—
53C23×C32—21no—
54C23×C32C2×D311yescharacteristic
55C23×C32C22×C311yescharacteristic
56C22×C3⋊D3C22×C311yescharacteristic
57C22×C3⋊D3C22×C311yescharacteristic
58C22×C3×D3—31no—
59C22×C3×D3—31no—
60C2×C32×C4—21no—
61C2×C32×C4—21no—
62C2×C32×C4C2×D311yescharacteristic
63C2×C32×C4C2×D311yescharacteristic
64C2×C32×C4C22×C311yescharacteristic
65C2×C32×C4C22×C311yescharacteristic
66#638—22no—
67#638—22no—
68C2×C3⋊Dic3C22×C311yescharacteristic
69C2×C3⋊Dic3C22×C311yescharacteristic
70C4⋊(C3⋊D3)—22no—
71C4×C3⋊D3—22no—
72C3×#67—62no—
73C3×#67—62no—
74C2×C3×Dic3—31no—
75C2×C3×Dic3—31no—
76C3×D12—62no—
77C3×C4×D3—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×C3×D4—91no—
87C22×C3×C4—91no—
88C2×#67—31no—
89C2×#67—31no—
90C2×#67—91no—
91C2×#67—91no—
92C2×D12—31no—
93C2×D12—91no—
94C2×C4×D3—31no—
95C2×C4×D3—91no—
96C3×C4⋊C4—91no—
97C3×C22⋊C4—31no—
98C3×C22⋊C4—31no—
99C3×C22⋊C4—61no—
100C3×C22⋊C4—91no—
101C3×C22⋊C4C3×D311yescharacteristic
102(C2×D3)⋊C4—31no—
103(C2×D3)⋊C4—91no—
104Dic3⋊C4—31no—
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×C32C23×C311yescharacteristic
118C2×C3⋊D3—21no—
119C2×C3⋊D3—21no—
120C2×C3⋊D3—21no—
121C2×C3⋊D3—21no—
122C2×C3⋊D3—21no—
123C2×C3⋊D3C3×D412yescomplemented
124C2×C3×D3—32no—
125C2×C3×D3—61no—
126C2×C3×D3—61no—
127C2×C3×D3—61no—
128C2×C3×D3—61no—
129C2×C3×D3—61no—
130C32×C4—21no—
131C32×C4—21no—
132C32×C4—21no—
133C32×C4—21no—
134C32×C4—41no—
135C32×C4—41no—
136C3⋊Dic3—21no—
137C3⋊Dic3C3×D412yescomplemented
138C3×Dic3—32no—
139C3×Dic3—61no—
140#121—91noSylow
141H(3,𝔽3)#12111yescharacteristic, complemented, Sylow
142C23×C3—31no—
143C23×C3—31no—
144C23×C3—61no—
145C23×C3—91no—
146C23×C3—91no—
147C23×C3C2×C3×D311yescharacteristic
148C22×D3—31no—
149C22×D3—31no—
150C22×D3—91no—
151C22×D3—91no—
152C3×D4—182no—
153C3×D4—182no—
154C3×D4—182no—
155C2×C3×C4—31no—
156C2×C3×C4—31no—
157C2×C3×C4—31no—
158C2×C3×C4—31no—
159C2×C3×C4—61no—
160C2×C3×C4—61no—
161C2×C3×C4—91no—
162C2×C3×C4—91no—
163C2×C3×C4—182no—
164C2×C3×C4C2×C3×D311yescharacteristic
165C2×C3×C4C2×C3×D311yescharacteristic
166#67—62no—
167#67—62no—
168#67—182no—
169#67—182no—
170C2×Dic3—31no—
171C2×Dic3—31no—
172C2×Dic3—91no—
173C2×Dic3—91no—
174D12—62no—
175D12—182no—
176C4×D3—62no—
177C4×D3—182no—
178C2×C32—21no—
179C2×C32—21no—
180C2×C32—21no—
181C2×C32—42no—
182C2×C32—41no—
183C2×C32—41no—
184C2×C32#23511yescharacteristic
185C2×C32D3×D411yescharacteristic
186C2×C32D3×D411yescharacteristic
187C2×C32C2×C3×D411yescharacteristic
188C2×C32C2×C3×D411yescharacteristic
189C2×C32C3×#4111yescharacteristic
190C3⋊D3—22no—
191C3⋊D3—41no—
192C3×D3—62no—
193C3×D3—121no—
194C2×D4—91no—
195C2×D4—91no—
196C2×D4—91no—
197C22×C4—91no—
198C4⋊C4—91no—
199C22⋊C4—91no—
200C22⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
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×C3C22×C3×D311yescharacteristic
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—21no—
236C3×C4—21no—
237C3×C4—61no—
238C3×C4—61no—
239C3×C4—61no—
240C3×C4—61no—
241C3×C4—92no—
242C3×C4—121no—
243C3×C4—121no—
244C3×C4—181no—
245Dic3—32no—
246Dic3—61no—
247Dic3—92no—
248Dic3—181no—
249C32—21no—
250C32#88411yescharacteristic, complemented
251C32C3×#12111yescharacteristic, complemented
252C23—91no—
253C23—91no—
254C23C2×(C3⋊D3)⋊C311yescharacteristic
255D4—182no—
256D4—182no—
257D4—182no—
258C2×C4—91no—
259C2×C4—91no—
260C2×C4—182no—
261C2×C4C2×(C3⋊D3)⋊C311yescharacteristic
262C2×C4C2×(C3⋊D3)⋊C311yescharacteristic
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×C3C3×#23511yescharacteristic
279C2×C3C3×D3×D411yescharacteristic
280C2×C3C3×D3×D411yescharacteristic
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—
294C22C22×(C3⋊D3)⋊C311yescharacteristic
295C4—21no—
296C4—21no—
297C4—92no—
298C4—181no—
299C3—31no—
300C3—31no—
301C3—61no—
302C3C3×#88411yescharacteristic
303C2—41no—
304C2—182no—
305C2—361no—
306C2#3828⋊C311yescharacteristic
307C2D4×(C3⋊D3)⋊C311yescharacteristic
308C2D4×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C23×C3
Frattini, socleC22×C3C22×C3×D3
FittingC22⋊C4×H(3,𝔽3)C2
hypercenterC22⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#121
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#638⋊C3C2semidirect
2. C2×#638⋊C3C2semidirect
3. C2×C4⋊((C3⋊D3)⋊C3)C2semidirect
4. C2×C4×(C3⋊D3)⋊C3C2semidirect
5. C22⋊C4×H(3,𝔽3)C2semidirect
6. #3752⋊C3C2semidirect
7. #3750⋊C3C2semidirect
8. #64093C3semidirect
21. C2×#638C2×C3semidirect
22. C2×#638C2×C3semidirect
23. C2×C4⋊(C3⋊D3)C2×C3semidirect
24. C2×C4×C3⋊D3C2×C3semidirect
30. C32×C22⋊C4D3semidirect
31. C32×C22⋊C4C2×C3semidirect
32. #3752C2×C3semidirect
33. #3750C2×C3semidirect
45. C2×(C3⋊D3)⋊C3D4semidirect
49. (C3⋊Dic3)⋊C3D4semidirect
123. C2×C3⋊D3C3×D4semidirect
137. C3⋊Dic3C3×D4semidirect
141. H(3,𝔽3)#121semidirect
200. C22⋊C4(C3⋊D3)⋊C3semidirect
250. C32#884semidirect
251. C32C3×#121semidirect

Maximal subgroups

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

subgroupindex
C2×#638⋊C32
C2×C4⋊((C3⋊D3)⋊C3)2
C2×C4×(C3⋊D3)⋊C32
C22⋊C4×H(3,𝔽3)2
#3752⋊C32
#3750⋊C32
#640933
C3×#8843