Finite Group Databasebeta

← #64105⋊C3 #11721470

Subgroups — #64105⋊C3 #11721470

Order 864.

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

normal subgroups only

This group has 285 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×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×#634⋊C3C211yesmaximal, characteristic, complemented
3C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4#3752⋊C3C211yesmaximal, characteristic
5#3752⋊C3C211yesmaximal, characteristic
6#3750⋊C3C211yesmaximal, characteristic, complemented
7#3750⋊C3C211yesmaximal, characteristic, complemented
8#64105C311yesmaximal, characteristic, complemented
9C3×#896—31nomaximal
10C22×(C3⋊D3)⋊C3C2211yescharacteristic
11C2×C4×H(3,𝔽3)C2211yescharacteristic
12C2×C4×H(3,𝔽3)C2211yescharacteristic
13C2×C4×H(3,𝔽3)C2211yescharacteristic
14C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C4×(C3⋊D3)⋊C3—22no—
18#634⋊C3—22no—
19C2×C4×C3⋊D3C2×C311yescharacteristic
20C2×#634C2×C311yescharacteristic, complemented
21C2×C3×C4×D3—31no—
22C2×C3×Dic6—31no—
23C32×C4⋊C4—21no—
24C32×C4⋊C4D311yescharacteristic, complemented
25C32×C4⋊C4C2×C311yescharacteristic, complemented
26#3752C2×C311yescharacteristic
27#3752C2×C311yescharacteristic
28#3750C2×C311yescharacteristic, complemented
29#3750C2×C311yescharacteristic, complemented
30C3×(C2×D3)⋊C4—31no—
31C3×(C2×D3)⋊C4—31no—
32C3×Dic3⋊C4—31no—
33C3×Dic3⋊C4—31no—
34C22×H(3,𝔽3)C2311yescharacteristic
35C2×(C3⋊D3)⋊C3—21no—
36C2×(C3⋊D3)⋊C3—21no—
37C2×(C3⋊D3)⋊C3Q812yescomplemented
38C4×H(3,𝔽3)—21no—
39C4×H(3,𝔽3)—21no—
40C4×H(3,𝔽3)—21no—
41(C3⋊Dic3)⋊C3—21no—
42(C3⋊Dic3)⋊C3—21no—
43(C3⋊Dic3)⋊C3D412yes—
44C3×C22⋊Q8—91no—
45#896—31no—
46#896—91no—
47C22×C3⋊D3C22×C311yescharacteristic
48C22×C3×D3—31no—
49C2×C32×C4—21no—
50C2×C32×C4—21no—
51C2×C32×C4—21no—
52C2×C32×C4C2×D311yescharacteristic
53C2×C32×C4C2×D311yescharacteristic
54C2×C32×C4C2×D311yescharacteristic
55C2×C32×C4C22×C311yescharacteristic
56C2×C32×C4C22×C311yescharacteristic
57C2×C32×C4C22×C311yescharacteristic
58C2×C3⋊Dic3C22×C311yescharacteristic
59C2×C3⋊Dic3C22×C311yescharacteristic
60C2×C3⋊Dic3C22×C311yescharacteristic
61C4×C3⋊D3—22no—
62#634—22no—
63C2×C3×Dic3—31no—
64C2×C3×Dic3—31no—
65C2×C3×Dic3—31no—
66C3×C4×D3—62no—
67C3×Dic6—62no—
68C2×H(3,𝔽3)C2×D411yescharacteristic
69C2×H(3,𝔽3)C2×Q811yescharacteristic
70C2×H(3,𝔽3)#4111yescharacteristic
71(C3⋊D3)⋊C3—22no—
72C2×C3×Q8—91no—
73C22×C3×C4—91no—
74C2×C4×D3—31no—
75C2×C4×D3—91no—
76C2×Dic6—31no—
77C2×Dic6—91no—
78C3×C4⋊C4—31no—
79C3×C4⋊C4—61no—
80C3×C4⋊C4—61no—
81C3×C4⋊C4—91no—
82C3×C4⋊C4—91no—
83C3×C4⋊C4C3×D311yescharacteristic
84C3×C22⋊C4—91no—
85C3×C22⋊C4—91no—
86(C2×D3)⋊C4—31no—
87(C2×D3)⋊C4—31no—
88(C2×D3)⋊C4—91no—
89(C2×D3)⋊C4—91no—
90Dic3⋊C4—31no—
91Dic3⋊C4—31no—
92Dic3⋊C4—91no—
93Dic3⋊C4—91no—
94C22×C32—21no—
95C22×C32C22×D311yescharacteristic
96C22×C32C23×C311yescharacteristic
97C2×C3⋊D3—21no—
98C2×C3⋊D3—21no—
99C2×C3⋊D3C3×Q812yescomplemented
100C2×C3×D3—32no—
101C2×C3×D3—61no—
102C2×C3×D3—61no—
103C32×C4—22no—
104C32×C4—21no—
105C32×C4—21no—
106C32×C4—21no—
107C32×C4—21no—
108C32×C4—21no—
109C32×C4—21no—
110C32×C4—41no—
111C32×C4—41no—
112C3⋊Dic3—21no—
113C3⋊Dic3—21no—
114C3⋊Dic3C3×D412yes—
115C3×Dic3—32no—
116C3×Dic3—61no—
117C3×Dic3—61no—
118C22⋊Q8—91noSylow
119H(3,𝔽3)C22⋊Q811yescharacteristic, complemented, Sylow
120C23×C3—91no—
121C22×D3—31no—
122C22×D3—91no—
123C3×Q8—182no—
124C2×C3×C4—31no—
125C2×C3×C4—31no—
126C2×C3×C4—31no—
127C2×C3×C4—61no—
128C2×C3×C4—61no—
129C2×C3×C4—61no—
130C2×C3×C4—61no—
131C2×C3×C4—61no—
132C2×C3×C4—61no—
133C2×C3×C4—91no—
134C2×C3×C4—91no—
135C2×C3×C4—91no—
136C2×C3×C4—182no—
137C2×C3×C4C2×C3×D311yescharacteristic
138C2×C3×C4C2×C3×D311yescharacteristic
139C2×C3×C4C2×C3×D311yescharacteristic
140C2×Dic3—31no—
141C2×Dic3—31no—
142C2×Dic3—31no—
143C2×Dic3—91no—
144C2×Dic3—91no—
145C2×Dic3—91no—
146C4×D3—62no—
147C4×D3—182no—
148Dic6—62no—
149Dic6—182no—
150C2×C32—21no—
151C2×C32—21no—
152C2×C32—21no—
153C2×C32#23511yescharacteristic
154C2×C32D3×D411yescharacteristic
155C2×C32D3×Q811yescharacteristic
156C2×C32C2×C3×D411yescharacteristic
157C2×C32C2×C3×Q811yescharacteristic
158C2×C32C3×#4111yescharacteristic
159C3⋊D3—22no—
160C3×D3—62no—
161C2×Q8—91no—
162C22×C4—91no—
163C4⋊C4—91no—
164C4⋊C4—91no—
165C4⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
166C22⋊C4—91no—
167C22⋊C4—91no—
168C22×C3—31no—
169C22×C3—61no—
170C22×C3—61no—
171C22×C3—92no—
172C22×C3—181no—
173C22×C3—181no—
174C22×C3C22×C3×D311yescharacteristic
175C2×D3—32no—
176C2×D3—61no—
177C2×D3—61no—
178C2×D3—92no—
179C2×D3—181no—
180C2×D3—181no—
181C3×C4—21no—
182C3×C4—21no—
183C3×C4—21no—
184C3×C4—61no—
185C3×C4—61no—
186C3×C4—61no—
187C3×C4—62no—
188C3×C4—61no—
189C3×C4—92no—
190C3×C4—121no—
191C3×C4—121no—
192C3×C4—121no—
193C3×C4—121no—
194C3×C4—181no—
195C3×C4—181no—
196Dic3—32no—
197Dic3—61no—
198Dic3—61no—
199Dic3—92no—
200Dic3—181no—
201Dic3—181no—
202C32—21no—
203C32#89611yescharacteristic, complemented
204C32C3×C22⋊Q811yescharacteristic, complemented
205C23—91no—
206Q8—182no—
207C2×C4—91no—
208C2×C4—91no—
209C2×C4—91no—
210C2×C4—182no—
211C2×C4C2×(C3⋊D3)⋊C311yescharacteristic
212C2×C4C2×(C3⋊D3)⋊C311yescharacteristic
213C2×C4C2×(C3⋊D3)⋊C311yescharacteristic
214C2×C3—31no—
215C2×C3—31no—
216C2×C3—31no—
217C2×C3—61no—
218C2×C3—61no—
219C2×C3—61no—
220C2×C3—61no—
221C2×C3—61no—
222C2×C3—61no—
223C2×C3—182no—
224C2×C3C3×#23511yescharacteristic
225C2×C3C3×D3×D411yescharacteristic
226C2×C3C3×D3×Q811yescharacteristic
227D3—62no—
228D3—182no—
229C22—92no—
230C22—181no—
231C22—181no—
232C22C22×(C3⋊D3)⋊C311yescharacteristic
233C4—21no—
234C4—21no—
235C4—21no—
236C4—92no—
237C4—181no—
238C4—181no—
239C3—31no—
240C3—61no—
241C3—61no—
242C3C3×#89611yescharacteristic
243C2—182no—
244C2#3828⋊C311yescharacteristic
245C2D4×(C3⋊D3)⋊C311yescharacteristic
246C2Q8×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C23×C3
Frattini, socleC22×C3C22×C3×D3
FittingC4⋊C4×H(3,𝔽3)C2
prime radicalC22×(C3⋊D3)⋊C3C22
hypercenterC4⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×C22⋊Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#634⋊C3C2semidirect
3. C4⋊C4×H(3,𝔽3)C2semidirect
6. #3750⋊C3C2semidirect
7. #3750⋊C3C2semidirect
8. #64105C3semidirect
20. C2×#634C2×C3semidirect
24. C32×C4⋊C4D3semidirect
25. C32×C4⋊C4C2×C3semidirect
28. #3750C2×C3semidirect
29. #3750C2×C3semidirect
37. C2×(C3⋊D3)⋊C3Q8semidirect
99. C2×C3⋊D3C3×Q8semidirect
119. H(3,𝔽3)C22⋊Q8semidirect
165. C4⋊C4(C3⋊D3)⋊C3semidirect
203. C32#896semidirect
204. C32C3×C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×(C3⋊D3)⋊C32
C2×#634⋊C32
C4⋊C4×H(3,𝔽3)2
#3752⋊C32
#3750⋊C32
#641053
C3×#8963