Finite Group Databasebeta

← #64103⋊C3 #11721468

Subgroups — #64103⋊C3 #11721468

Order 864.

299 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 71 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
2C2×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4#3752⋊C3C211yesmaximal, characteristic, complemented
5#3752⋊C3C211yesmaximal, characteristic, complemented
6#3752⋊C3C211yesmaximal, characteristic, complemented
7#3750⋊C3C211yesmaximal, characteristic, complemented
8#64103C311yesmaximal, characteristic, complemented
9C3×#894—31nomaximal
10C22×(C3⋊D3)⋊C3C2211yescharacteristic
11C22×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×C4×H(3,𝔽3)C2211yescharacteristic
13C2×C4×H(3,𝔽3)C2211yescharacteristic
14C2×C4×H(3,𝔽3)C2211yescharacteristic
15C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C4⋊((C3⋊D3)⋊C3)—22no—
18C4×(C3⋊D3)⋊C3—22no—
19C2×C4⋊(C3⋊D3)C2×C311yescharacteristic
20C2×C4×C3⋊D3C2×C311yescharacteristic, complemented
21C2×C3×D12—31no—
22C2×C3×C4×D3—31no—
23C32×C4⋊C4—21no—
24C32×C4⋊C4D311yescharacteristic, complemented
25C32×C4⋊C4C2×C311yescharacteristic, complemented
26#3752C2×C311yescharacteristic, complemented
27#3752C2×C311yescharacteristic, complemented
28#3752C2×C311yescharacteristic, complemented
29#3750C2×C311yescharacteristic, complemented
30C3×(C2×D3)⋊C4—31no—
31C3×(C2×D3)⋊C4—31no—
32C3×(C2×D3)⋊C4—31no—
33C3×Dic3⋊C4—31no—
34C22×H(3,𝔽3)C2311yescharacteristic
35C2×(C3⋊D3)⋊C3—21no—
36C2×(C3⋊D3)⋊C3—21no—
37C2×(C3⋊D3)⋊C3—21no—
38C2×(C3⋊D3)⋊C3—21no—
39C2×(C3⋊D3)⋊C3—21no—
40C2×(C3⋊D3)⋊C3D412yes—
41C4×H(3,𝔽3)—21no—
42C4×H(3,𝔽3)—21no—
43C4×H(3,𝔽3)—21no—
44(C3⋊Dic3)⋊C3—21no—
45(C3⋊Dic3)⋊C3—21no—
46C3×#123—91no—
47#894—31no—
48#894—91no—
49C22×C3⋊D3C22×C311yescharacteristic
50C22×C3⋊D3C22×C311yescharacteristic
51C22×C3×D3—31no—
52C22×C3×D3—31no—
53C2×C32×C4—21no—
54C2×C32×C4—21no—
55C2×C32×C4—21no—
56C2×C32×C4C2×D311yescharacteristic
57C2×C32×C4C2×D311yescharacteristic
58C2×C32×C4C2×D311yescharacteristic
59C2×C32×C4C22×C311yescharacteristic
60C2×C32×C4C22×C311yescharacteristic
61C2×C32×C4C22×C311yescharacteristic
62C2×C3⋊Dic3C22×C311yescharacteristic
63C2×C3⋊Dic3C22×C311yescharacteristic
64C4⋊(C3⋊D3)—22no—
65C4×C3⋊D3—22no—
66C2×C3×Dic3—31no—
67C2×C3×Dic3—31no—
68C3×D12—62no—
69C3×C4×D3—62no—
70C2×H(3,𝔽3)C2×D411yescharacteristic
71C2×H(3,𝔽3)#4111yescharacteristic
72C2×H(3,𝔽3)#4111yescharacteristic
73(C3⋊D3)⋊C3—22no—
74(C3⋊D3)⋊C3—41no—
75C2×C3×D4—91no—
76C22×C3×C4—91no—
77C2×D12—31no—
78C2×D12—91no—
79C2×C4×D3—31no—
80C2×C4×D3—91no—
81C3×C4⋊C4—31no—
82C3×C4⋊C4—61no—
83C3×C4⋊C4—61no—
84C3×C4⋊C4—91no—
85C3×C4⋊C4C3×D311yescharacteristic
86C3×C22⋊C4—91no—
87C3×C22⋊C4—91no—
88C3×C22⋊C4—91no—
89(C2×D3)⋊C4—31no—
90(C2×D3)⋊C4—31no—
91(C2×D3)⋊C4—31no—
92(C2×D3)⋊C4—91no—
93(C2×D3)⋊C4—91no—
94(C2×D3)⋊C4—91no—
95Dic3⋊C4—31no—
96Dic3⋊C4—91no—
97C22×C32—21no—
98C22×C32C22×D311yescharacteristic
99C22×C32C23×C311yescharacteristic
100C2×C3⋊D3—21no—
101C2×C3⋊D3—21no—
102C2×C3⋊D3—21no—
103C2×C3⋊D3—21no—
104C2×C3⋊D3—21no—
105C2×C3⋊D3C3×D412yes—
106C2×C3×D3—32no—
107C2×C3×D3—61no—
108C2×C3×D3—61no—
109C2×C3×D3—61no—
110C2×C3×D3—61no—
111C2×C3×D3—61no—
112C32×C4—22no—
113C32×C4—21no—
114C32×C4—21no—
115C32×C4—21no—
116C32×C4—21no—
117C32×C4—21no—
118C32×C4—21no—
119C32×C4—41no—
120C32×C4—41no—
121C3⋊Dic3—21no—
122C3⋊Dic3—21no—
123C3×Dic3—61no—
124C3×Dic3—61no—
125#123—91noSylow
126H(3,𝔽3)#12311yescharacteristic, complemented, Sylow
127C23×C3—91no—
128C23×C3—91no—
129C22×D3—31no—
130C22×D3—31no—
131C22×D3—91no—
132C22×D3—91no—
133C3×D4—182no—
134C2×C3×C4—31no—
135C2×C3×C4—31no—
136C2×C3×C4—31no—
137C2×C3×C4—61no—
138C2×C3×C4—61no—
139C2×C3×C4—61no—
140C2×C3×C4—61no—
141C2×C3×C4—61no—
142C2×C3×C4—61no—
143C2×C3×C4—91no—
144C2×C3×C4—91no—
145C2×C3×C4—182no—
146C2×C3×C4C2×C3×D311yescharacteristic
147C2×C3×C4C2×C3×D311yescharacteristic
148C2×C3×C4C2×C3×D311yescharacteristic
149C2×Dic3—31no—
150C2×Dic3—31no—
151C2×Dic3—91no—
152C2×Dic3—91no—
153D12—62no—
154D12—182no—
155C4×D3—62no—
156C4×D3—182no—
157C2×C32—21no—
158C2×C32—21no—
159C2×C32—21no—
160C2×C32#23511yescharacteristic
161C2×C32D3×D411yescharacteristic
162C2×C32#23911yescharacteristic
163C2×C32C2×C3×D411yescharacteristic
164C2×C32C3×#4111yescharacteristic
165C2×C32C3×#4111yescharacteristic
166C3⋊D3—22no—
167C3⋊D3—41no—
168C3×D3—62no—
169C3×D3—121no—
170C2×D4—91no—
171C22×C4—91no—
172C4⋊C4—91no—
173C4⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
174C22⋊C4—91no—
175C22⋊C4—91no—
176C22⋊C4—91no—
177C22×C3—31no—
178C22×C3—61no—
179C22×C3—61no—
180C22×C3—92no—
181C22×C3—181no—
182C22×C3—181no—
183C22×C3—181no—
184C22×C3—181no—
185C22×C3—181no—
186C22×C3C22×C3×D311yescharacteristic
187C2×D3—32no—
188C2×D3—61no—
189C2×D3—61no—
190C2×D3—61no—
191C2×D3—61no—
192C2×D3—61no—
193C2×D3—92no—
194C2×D3—181no—
195C2×D3—181no—
196C2×D3—181no—
197C2×D3—181no—
198C2×D3—181no—
199C3×C4—21no—
200C3×C4—21no—
201C3×C4—21no—
202C3×C4—61no—
203C3×C4—61no—
204C3×C4—61no—
205C3×C4—62no—
206C3×C4—61no—
207C3×C4—121no—
208C3×C4—121no—
209C3×C4—121no—
210C3×C4—121no—
211C3×C4—181no—
212C3×C4—181no—
213Dic3—61no—
214Dic3—61no—
215Dic3—181no—
216Dic3—181no—
217C32—21no—
218C32#89411yescharacteristic, complemented
219C32C3×#12311yescharacteristic, complemented
220C23—91no—
221C23—91no—
222D4—182no—
223C2×C4—91no—
224C2×C4—91no—
225C2×C4—182no—
226C2×C4C2×(C3⋊D3)⋊C311yescharacteristic
227C2×C4C2×(C3⋊D3)⋊C311yescharacteristic
228C2×C4C2×(C3⋊D3)⋊C311yescharacteristic
229C2×C3—31no—
230C2×C3—31no—
231C2×C3—31no—
232C2×C3—61no—
233C2×C3—61no—
234C2×C3—61no—
235C2×C3—61no—
236C2×C3—61no—
237C2×C3—61no—
238C2×C3—182no—
239C2×C3—361no—
240C2×C3C3×#23511yescharacteristic
241C2×C3C3×D3×D411yescharacteristic
242C2×C3C3×#23911yescharacteristic
243D3—62no—
244D3—121no—
245D3—182no—
246D3—361no—
247C22—92no—
248C22—181no—
249C22—181no—
250C22—181no—
251C22—181no—
252C22—181no—
253C22C22×(C3⋊D3)⋊C311yescharacteristic
254C4—21no—
255C4—21no—
256C4—21no—
257C4—181no—
258C4—181no—
259C3—31no—
260C3—61no—
261C3—61no—
262C3C3×#89411yescharacteristic
263C2—182no—
264C2—361no—
265C2#3828⋊C311yescharacteristic
266C2D4×(C3⋊D3)⋊C311yescharacteristic
267C2#8872211yescharacteristic
subgroupG/H
centerC22C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C23×C3
Frattini, socleC22×C3C22×C3×D3
FittingC4⋊C4×H(3,𝔽3)C2
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C2
hypercenterC4⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#123
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×(C3⋊D3)⋊C3C2semidirect
3. C4⋊C4×H(3,𝔽3)C2semidirect
4. #3752⋊C3C2semidirect
5. #3752⋊C3C2semidirect
6. #3752⋊C3C2semidirect
7. #3750⋊C3C2semidirect
8. #64103C3semidirect
20. C2×C4×C3⋊D3C2×C3semidirect
24. C32×C4⋊C4D3semidirect
25. C32×C4⋊C4C2×C3semidirect
26. #3752C2×C3semidirect
27. #3752C2×C3semidirect
28. #3752C2×C3semidirect
29. #3750C2×C3semidirect
126. H(3,𝔽3)#123semidirect
173. C4⋊C4(C3⋊D3)⋊C3semidirect
218. C32#894semidirect
219. C32C3×#123semidirect

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×C4×(C3⋊D3)⋊C32
C4⋊C4×H(3,𝔽3)2
#3752⋊C32
#3750⋊C32
#641033
C3×#8943