Finite Group Databasebeta

← #11721237

Subgroups — #11721237

Order 864.

228 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 44 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
1#88722C211yesmaximal, characteristic
2((C3⋊D3)⋊C3)⋊Q8C211yesmaximal, characteristic, complemented
3#88550C211yesmaximal, characteristic, complemented
4#3756⋊C3C211yesmaximal, characteristic, complemented
5#88438C211yesmaximal, characteristic, complemented
6#88431C211yesmaximal, characteristic, complemented
7#88430C211yesmaximal, characteristic, complemented
8#63945—31nomaximal
9#63943—31nomaximal
10Q8×H(3,𝔽3)C2211yescharacteristic
11#6229C2211yescharacteristic
12C4⋊((C3⋊D3)⋊C3)—21no—
13C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C4×(C3⋊D3)⋊C3—21no—
15C4×(C3⋊D3)⋊C3C2211yescharacteristic
16#634⋊C3C2211yescharacteristic
17((C3⋊D3)⋊C3)⋊C4—21no—
18#6196—21no—
19#6180C2211yescharacteristic
20(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#3832D311yescharacteristic
22C3×#239—31no—
23#3797—31no—
24D3×Dic6—31no—
25#3756D311yescharacteristic, complemented
26C3×#216—31no—
27C3×#216—31no—
28C3×#216—31no—
29C3×#216—61no—
30#3717—31no—
31#3717—31no—
32#3716—31no—
33#3715—31no—
34#3712—31no—
35#3711—31no—
36C2×(C3⋊D3)⋊C3—21no—
37C2×(C3⋊D3)⋊C3D411yescharacteristic
38C4×H(3,𝔽3)—21no—
39C4×H(3,𝔽3)C2311yescharacteristic
40#1084—21no—
41(C3⋊Dic3)⋊C3—21no—
42(C3⋊Dic3)⋊C3D411yescharacteristic
43#951—91no—
44#918—91no—
45#918—91no—
46C32×Q8—21no—
47C32×Q8C2×D311yescharacteristic
48C32×Q8C2×D311yescharacteristic
49C4⋊(C3⋊D3)—21no—
50C4⋊(C3⋊D3)C2×D311yescharacteristic
51C4×C3⋊D3—21no—
52C4×C3⋊D3C2×D311yescharacteristic
53#634C2×D311yescharacteristic
54C3×D12—31no—
55C3×D12—61no—
56C3×C4×D3—31no—
57C3×C4×D3—61no—
58C3×Dic6—31no—
59C3×Dic6—31no—
60C3×Dic6—31no—
61C3×Dic6—61no—
62#627—61no—
63#627—61no—
64#624—61no—
65D3×Dic3—61no—
66C3⋊(C3⋊C8)C2×D311yescharacteristic
67C3×C3⋊C8—31no—
68C3×C3⋊C8—31no—
69C3×C3⋊C8—31no—
70C3×C3⋊C8—61no—
71C2×H(3,𝔽3)C2×D411yescharacteristic
72(C3⋊D3)⋊C3—21no—
73(C3⋊D3)⋊C3—41no—
74C3×#41—91no—
75#239—31no—
76#239—91no—
77D3×Q8—91no—
78D3×Q8—91no—
79C2×Dic6—91no—
80C3×Q16—91no—
81C3×Q16—91no—
82#216—31no—
83#216—31no—
84#216—31no—
85#216—61no—
86#216—91no—
87#215—91no—
88#215—91no—
89#214—91no—
90#214—91no—
91#208—91no—
92#204—91no—
93#204—91no—
94ΓL(1,𝔽25)—91no—
95ΓL(1,𝔽25)—91no—
96C2×C3⋊D3—21no—
97C2×C3⋊D3#6711yescharacteristic
98C2×C3×D3—31no—
99C2×C3×D3—61no—
100C32×C4—21no—
101C32×C4—21no—
102C32×C4—21no—
103C32×C4—41no—
104C32×C4C22×D311yescharacteristic
105C32×C4C22×D311yescharacteristic
106C3⋊Dic3—21no—
107C3⋊Dic3#6711yescharacteristic
108C3×Dic3—31no—
109C3×Dic3—61no—
110C3×Dic3—61no—
111C3×Dic3—61no—
112C3×Dic3—121no—
113#137—271noSylow
114H(3,𝔽3)#13711yescharacteristic, complemented, Sylow
115C3×Q8—31no—
116C3×Q8—31no—
117C3×Q8—61no—
118C3×Q8—91no—
119C3×Q8—91no—
120C3×Q8D3211yescharacteristic
121C3×D4—91no—
122C3×D4—181no—
123C2×C3×C4—91no—
124C2×C3×C4—181no—
125C2×Dic3—181no—
126D12—31no—
127D12—61no—
128D12—91no—
129D12—181no—
130C4×D3—31no—
131C4×D3—61no—
132C4×D3—91no—
133C4×D3—181no—
134C4×D3—181no—
135C4×D3—181no—
136Dic6—31no—
137Dic6—31no—
138Dic6—31no—
139Dic6—61no—
140Dic6—91no—
141Dic6—181no—
142Dic6—181no—
143Dic6—181no—
144C3×C8—91no—
145C3×C8—91no—
146C3⋊C8—31no—
147C3⋊C8—31no—
148C3⋊C8—31no—
149C3⋊C8—61no—
150C3⋊C8—91no—
151C2×C32—21no—
152C2×C32D3×D411yescharacteristic
153C2×C32C2×#6711yescharacteristic
154C3⋊D3—21no—
155C3⋊D3—41no—
156C3×D3—61no—
157C3×D3—121no—
158#41—91no—
159C2×Q8—271no—
160Q16—91no—
161Q16—91no—
162SD16—271no—
163SD16—271no—
164M16—271no—
165C22×C3—91no—
166C22×C3—181no—
167C2×D3—31no—
168C2×D3—61no—
169C2×D3—91no—
170C2×D3—181no—
171C3×C4—21no—
172C3×C4—31no—
173C3×C4—31no—
174C3×C4—61no—
175C3×C4—61no—
176C3×C4—61no—
177C3×C4—91no—
178C3×C4—121no—
179C3×C4—181no—
180C3×C4—181no—
181C3×C4C2×D3211yescharacteristic
182Dic3—31no—
183Dic3—61no—
184Dic3—61no—
185Dic3—61no—
186Dic3—91no—
187Dic3—121no—
188Dic3—181no—
189C32—21no—
190C32#91811yescharacteristic, complemented
191C32#95111yescharacteristic, complemented
192Q8—91no—
193Q8—91no—
194Q8—541no—
195Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
196D4—91no—
197D4—181no—
198C2×C4—91no—
199C2×C4—181no—
200C2×C4—541no—
201C8—91no—
202C8—91no—
203C2×C3—31no—
204C2×C3—31no—
205C2×C3—61no—
206C2×C3—181no—
207C2×C3—361no—
208C2×C3D3×#6711yescharacteristic
209D3—61no—
210D3—121no—
211D3—181no—
212D3—361no—
213C22—91no—
214C22—181no—
215C4—21no—
216C4—91no—
217C4—181no—
218C4—181no—
219C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
220C3—31no—
221C3—31no—
222C3—61no—
223C3#6394311yescharacteristic
224C2—181no—
225C2—361no—
226C2#8867611yescharacteristic
subgroupG/H
centerC2#88676
commutator, cosocleC4×H(3,𝔽3)C23
FrattiniC3×C4C2×D32
FittingQ8×H(3,𝔽3)C22
socleC2×C3D3×#67
prime radical#88722C2
hypercenterQ8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#137
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
2. ((C3⋊D3)⋊C3)⋊Q8C2semidirect
3. #88550C2semidirect
4. #3756⋊C3C2semidirect
5. #88438C2semidirect
6. #88431C2semidirect
7. #88430C2semidirect
25. #3756D3semidirect
114. H(3,𝔽3)#137semidirect
190. C32#918semidirect
191. C32#951semidirect

Maximal subgroups

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

subgroupindex
#887222
((C3⋊D3)⋊C3)⋊Q82
#885502
#3756⋊C32
#884382
#884312
#884302
#639453
#639433