Finite Group Databasebeta

← #11721201

Subgroups — #11721201

Order 864.

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

normal subgroups only

This group has 237 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#3830⋊C3C211yesmaximal, characteristic, complemented
2#88648C211yesmaximal, characteristic, complemented
3#88546C211yesmaximal, characteristic, complemented
4#3753⋊C3C211yesmaximal, characteristic, complemented
5#88439C211yesmaximal, characteristic, complemented
6#88431C211yesmaximal, characteristic, complemented
7((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic, complemented
8#63936—31nomaximal
9#63935—31nomaximal
10D4×H(3,𝔽3)C2211yescharacteristic
11#6229C2211yescharacteristic
12#638⋊C3—21no—
13C2×(C3⋊Dic3)⋊C3—21no—
14C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C4×(C3⋊D3)⋊C3C2211yescharacteristic
16#634⋊C3C2211yescharacteristic
17#6198—21no—
18((C3⋊D3)⋊C3)⋊C4—21no—
19#6180C2211yescharacteristic
20(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#3830D311yescharacteristic, complemented
22C3×#237—31no—
23#3799—31no—
24#3796—31no—
25#3753D311yescharacteristic, complemented
26C3×#214—31no—
27C3×#214—31no—
28C3×#214—61no—
29C3×#213—31no—
30#3719—31no—
31#3718—31no—
32#3717—31no—
33#3716—31no—
34#3710—31no—
35D3×C3⋊C8—31no—
36C22×H(3,𝔽3)—21no—
37C2×(C3⋊D3)⋊C3—21no—
38C2×(C3⋊D3)⋊C3D411yescharacteristic
39C4×H(3,𝔽3)C2311yescharacteristic
40#1084—21no—
41(C3⋊Dic3)⋊C3—21no—
42(C3⋊Dic3)⋊C3D411yescharacteristic
43#950—91no—
44#916—91no—
45#912—91no—
46C32×D4—21no—
47C32×D4C2×D311yescharacteristic
48C32×D4C2×D311yescharacteristic
49#638—21no—
50C2×C3⋊Dic3—21no—
51C4⋊(C3⋊D3)C2×D311yescharacteristic
52C4×C3⋊D3C2×D311yescharacteristic
53#634C2×D311yescharacteristic
54C3×#67—61no—
55C2×C3×Dic3—61no—
56C3×D12—31no—
57C3×C4×D3—31no—
58C3×Dic6—31no—
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)—41no—
72C2×H(3,𝔽3)C2×D411yescharacteristic
73(C3⋊D3)⋊C3—21no—
74(C3⋊D3)⋊C3—41no—
75C3×#41—91no—
76#239—91no—
77#237—31no—
78#237—91no—
79#237—91no—
80#235—91no—
81C3×SD16—91no—
82C3×D8—91no—
83#216—91no—
84#216—91no—
85#215—91no—
86#214—31no—
87#214—31no—
88#214—61no—
89#214—91no—
90#213—31no—
91#213—91no—
92C2×C3⋊C8—91no—
93Dic12—91no—
94#204—91no—
95D3×C8—91no—
96D3×C8—91no—
97C22×C32—21no—
98C22×C32—21no—
99C22×C32—41no—
100C2×C3⋊D3—21no—
101C2×C3⋊D3#6711yescharacteristic
102C2×C3×D3—31no—
103C2×C3×D3—61no—
104C32×C4—21no—
105C32×C4C22×D311yescharacteristic
106C32×C4C22×D311yescharacteristic
107C3⋊Dic3—21no—
108C3⋊Dic3#6711yescharacteristic
109C3×Dic3—31no—
110C3×Dic3—61no—
111C3×Dic3—61no—
112C3×Dic3—61no—
113C3×Dic3—121no—
114#135—271noSylow
115H(3,𝔽3)#13511yescharacteristic, complemented, Sylow
116C3×Q8—91no—
117C3×Q8—91no—
118C3×D4—31no—
119C3×D4—31no—
120C3×D4—61no—
121C3×D4—91no—
122C3×D4—181no—
123C3×D4D3211yescharacteristic
124C2×C3×C4—91no—
125C2×C3×C4—181no—
126#67—61no—
127#67—181no—
128#67—181no—
129#67—181no—
130C2×Dic3—61no—
131C2×Dic3—181no—
132C2×Dic3—181no—
133D12—31no—
134D12—91no—
135D12—181no—
136C4×D3—31no—
137C4×D3—91no—
138C4×D3—181no—
139C4×D3—181no—
140Dic6—31no—
141Dic6—31no—
142Dic6—31no—
143Dic6—61no—
144Dic6—91no—
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×C32—41no—
154C2×C32—41no—
155C2×C32—81no—
156C2×C32D3×D411yescharacteristic
157C2×C32C2×#6711yescharacteristic
158C3⋊D3—21no—
159C3⋊D3—41no—
160C3×D3—61no—
161C3×D3—121no—
162#41—91no—
163#41—271no—
164Q16—271no—
165SD16—91no—
166SD16—271no—
167D8—91no—
168C2×C8—271no—
169C22×C3—21no—
170C22×C3—61no—
171C22×C3—61no—
172C22×C3—91no—
173C22×C3—121no—
174C22×C3—181no—
175C2×D3—31no—
176C2×D3—61no—
177C2×D3—91no—
178C2×D3—181no—
179C3×C4—31no—
180C3×C4—31no—
181C3×C4—61no—
182C3×C4—91no—
183C3×C4—181no—
184C3×C4—181no—
185C3×C4C2×D3211yescharacteristic
186Dic3—31no—
187Dic3—61no—
188Dic3—61no—
189Dic3—61no—
190Dic3—91no—
191Dic3—121no—
192Dic3—181no—
193C32—21no—
194C32#91211yescharacteristic, complemented
195C32#95011yescharacteristic, complemented
196Q8—91no—
197Q8—91no—
198D4—91no—
199D4—181no—
200D4—541no—
201D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
202C2×C4—91no—
203C2×C4—181no—
204C2×C4—541no—
205C8—91no—
206C8—91no—
207C2×C3—31no—
208C2×C3—31no—
209C2×C3—41no—
210C2×C3—61no—
211C2×C3—121no—
212C2×C3—121no—
213C2×C3—181no—
214C2×C3—241no—
215C2×C3—361no—
216C2×C3D3×#6711yescharacteristic
217D3—61no—
218D3—121no—
219D3—181no—
220D3—361no—
221C22—21no—
222C22—91no—
223C22—181no—
224C4—91no—
225C4—181no—
226C4—181no—
227C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
228C3—31no—
229C3—31no—
230C3—61no—
231C3#6393311yescharacteristic
232C2—41no—
233C2—181no—
234C2—361no—
235C2#8867611yescharacteristic
subgroupG/H
centerC2#88676
commutator, cosocleC4×H(3,𝔽3)C23
FrattiniC3×C4C2×D32
FittingD4×H(3,𝔽3)C22
socleC2×C3D3×#67
hypercenterD4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#135
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
1. #3830⋊C3C2semidirect
2. #88648C2semidirect
3. #88546C2semidirect
4. #3753⋊C3C2semidirect
5. #88439C2semidirect
6. #88431C2semidirect
7. ((C3⋊D3)⋊C3)⋊C8C2semidirect
21. #3830D3semidirect
25. #3753D3semidirect
115. H(3,𝔽3)#135semidirect
194. C32#912semidirect
195. C32#950semidirect

Maximal subgroups

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

subgroupindex
#3830⋊C32
#886482
#885462
#3753⋊C32
#884392
#884312
((C3⋊D3)⋊C3)⋊C82
#639363
#639353