Finite Group Databasebeta

← #11721239

Subgroups — #11721239

Order 864.

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

normal subgroups only

This group has 235 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, complemented
2#88648C211yesmaximal, characteristic, complemented
3#88550C211yesmaximal, characteristic, complemented
4#3755⋊C3C211yesmaximal, characteristic, complemented
5#88438C211yesmaximal, characteristic, complemented
6#88432C211yesmaximal, characteristic, complemented
7((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic, complemented
8#63950—31nomaximal
9#63949—31nomaximal
10Q8×H(3,𝔽3)C2211yescharacteristic
11#6229C2211yescharacteristic
12C4⋊((C3⋊D3)⋊C3)—21no—
13C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C4×(C3⋊D3)⋊C3—21no—
16C4×(C3⋊D3)⋊C3C2211yescharacteristic
17#6198—21no—
18((C3⋊D3)⋊C3)⋊C4—21no—
19#6180C2211yescharacteristic
20(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#3832D311yescharacteristic, complemented
22C3×#239—31no—
23#3799—31no—
24#3796—31no—
25#3755D311yescharacteristic, complemented
26C3×#216—31no—
27C3×#216—31no—
28C3×#216—61no—
29C3×#215—31no—
30#3717—31no—
31#3715—31no—
32#3714—31no—
33#3714—31no—
34#3710—31no—
35D3×C3⋊C8—31no—
36C2×(C3⋊D3)⋊C3—21no—
37C2×(C3⋊D3)⋊C3—21no—
38C2×(C3⋊D3)⋊C3D411yescharacteristic
39C4×H(3,𝔽3)—21no—
40C4×H(3,𝔽3)C2311yescharacteristic
41#1084—21no—
42(C3⋊Dic3)⋊C3D411yescharacteristic
43#950—91no—
44#919—91no—
45#916—91no—
46C32×Q8—21no—
47C32×Q8C2×D311yescharacteristic
48C32×Q8C2×D311yescharacteristic
49C4⋊(C3⋊D3)—21no—
50C4⋊(C3⋊D3)C2×D311yescharacteristic
51C4⋊(C3⋊D3)C2×D311yescharacteristic
52C4×C3⋊D3—21no—
53C4×C3⋊D3C2×D311yescharacteristic
54C3×D12—31no—
55C3×D12—31no—
56C3×D12—61no—
57C3×C4×D3—31no—
58C3×C4×D3—61no—
59C3×Dic6—31no—
60C3×Dic6—31no—
61C3×Dic6—61no—
62#626—61no—
63#626—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—
74(C3⋊D3)⋊C3—41no—
75C3×#41—91no—
76#239—31no—
77#239—91no—
78#239—91no—
79#237—91no—
80#235—91no—
81C3×Q16—91no—
82C3×SD16—91no—
83#216—31no—
84#216—31no—
85#216—61no—
86#215—31no—
87#215—91no—
88#215—91no—
89#214—91no—
90#213—91no—
91#213—91no—
92C2×C3⋊C8—91no—
93D24—91no—
94#204—91no—
95D3×C8—91no—
96D3×C8—91no—
97C2×C3⋊D3—21no—
98C2×C3⋊D3—21no—
99C2×C3⋊D3#6711yescharacteristic
100C2×C3×D3—31no—
101C2×C3×D3—61no—
102C2×C3×D3—61no—
103C32×C4—21no—
104C32×C4—21no—
105C32×C4—21no—
106C32×C4—41no—
107C32×C4C22×D311yescharacteristic
108C32×C4C22×D311yescharacteristic
109C3⋊Dic3#6711yescharacteristic
110C3×Dic3—31no—
111C3×Dic3—61no—
112C3×Dic3—61no—
113C3×Dic3—121no—
114#135—271noSylow
115H(3,𝔽3)#13511yescharacteristic, complemented, Sylow
116C3×Q8—31no—
117C3×Q8—31no—
118C3×Q8—61no—
119C3×Q8—91no—
120C3×Q8D3211yescharacteristic
121C3×D4—91no—
122C3×D4—91no—
123C3×D4—181no—
124C2×C3×C4—91no—
125C2×C3×C4—181no—
126#67—181no—
127#67—181no—
128C2×Dic3—181no—
129D12—31no—
130D12—31no—
131D12—61no—
132D12—91no—
133D12—91no—
134D12—181no—
135D12—181no—
136C4×D3—31no—
137C4×D3—61no—
138C4×D3—91no—
139C4×D3—181no—
140C4×D3—181no—
141C4×D3—181no—
142Dic6—31no—
143Dic6—31no—
144Dic6—61no—
145C3×C8—91no—
146C3×C8—91no—
147C3⋊C8—31no—
148C3⋊C8—31no—
149C3⋊C8—31no—
150C3⋊C8—61no—
151C3⋊C8—91no—
152C2×C32—21no—
153C2×C32D3×D411yescharacteristic
154C2×C32C2×#6711yescharacteristic
155C3⋊D3—21no—
156C3⋊D3—41no—
157C3⋊D3—41no—
158C3×D3—61no—
159C3×D3—121no—
160C3×D3—121no—
161#41—91no—
162#41—271no—
163Q16—91no—
164SD16—91no—
165SD16—271no—
166D8—271no—
167C2×C8—271no—
168C22×C3—91no—
169C22×C3—181no—
170C22×C3—181no—
171C2×D3—31no—
172C2×D3—61no—
173C2×D3—61no—
174C2×D3—91no—
175C2×D3—181no—
176C2×D3—181no—
177C3×C4—21no—
178C3×C4—31no—
179C3×C4—31no—
180C3×C4—61no—
181C3×C4—61no—
182C3×C4—61no—
183C3×C4—91no—
184C3×C4—121no—
185C3×C4—181no—
186C3×C4C2×D3211yescharacteristic
187Dic3—31no—
188Dic3—61no—
189Dic3—61no—
190Dic3—91no—
191Dic3—121no—
192C32—21no—
193C32#91611yescharacteristic, complemented
194C32#95011yescharacteristic, complemented
195Q8—91no—
196Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
197D4—91no—
198D4—91no—
199D4—181no—
200D4—541no—
201C2×C4—91no—
202C2×C4—181no—
203C2×C4—541no—
204C8—91no—
205C8—91no—
206C2×C3—31no—
207C2×C3—31no—
208C2×C3—61no—
209C2×C3—181no—
210C2×C3—361no—
211C2×C3—361no—
212C2×C3D3×#6711yescharacteristic
213D3—61no—
214D3—121no—
215D3—121no—
216D3—181no—
217D3—361no—
218D3—361no—
219C22—91no—
220C22—181no—
221C22—181no—
222C4—21no—
223C4—91no—
224C4—181no—
225C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
226C3—31no—
227C3—31no—
228C3—61no—
229C3#6394711yescharacteristic
230C2—181no—
231C2—361no—
232C2—361no—
233C2#8867611yescharacteristic
subgroupG/H
centerC2#88676
commutator, cosocleC4×H(3,𝔽3)C23
FrattiniC3×C4C2×D32
FittingQ8×H(3,𝔽3)C22
socleC2×C3D3×#67
hypercenterQ8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#135
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
1. #88722C2semidirect
2. #88648C2semidirect
3. #88550C2semidirect
4. #3755⋊C3C2semidirect
5. #88438C2semidirect
6. #88432C2semidirect
7. ((C3⋊D3)⋊C3)⋊C8C2semidirect
21. #3832D3semidirect
25. #3755D3semidirect
115. H(3,𝔽3)#135semidirect
193. C32#916semidirect
194. C32#950semidirect

Maximal subgroups

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

subgroupindex
#887222
#886482
#885502
#3755⋊C32
#884382
#884322
((C3⋊D3)⋊C3)⋊C82
#639503
#639493