Finite Group Databasebeta

← #11721060

Subgroups — #11721060

Order 864.

239 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 239 conjugacy classes of subgroups — too many to draw. Only its 50 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#3828⋊C3C211yesmaximal, characteristic, complemented
2#3828⋊C3C211yesmaximal, characteristic, complemented
3C2×#6180C211yesmaximal, characteristic, complemented
4#88433C211yesmaximal, characteristic, complemented
5#88432C211yesmaximal, characteristic, complemented
6#88431C211yesmaximal, characteristic, complemented
7#88431C211yesmaximal, characteristic, complemented
8#63829—31nomaximal
9#63829—31nomaximal
10C2×C4×H(3,𝔽3)C2211yescharacteristic
11#638⋊C3—21no—
12#638⋊C3—21no—
13C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C4×(C3⋊D3)⋊C3—21no—
16C4×(C3⋊D3)⋊C3—21no—
17#634⋊C3C2211yescharacteristic
18#634⋊C3C2211yescharacteristic
19#6180C2211yescharacteristic
20#6180C2211yescharacteristic
21#3828D311yescharacteristic, complemented
22#3828D311yescharacteristic, complemented
23C3×#235—31no—
24C3×#235—31no—
25C2×C3×C3⋊C8—31no—
26C2×C3×C3⋊C8—31no—
27C2×C3×C3⋊C8—61no—
28#3719—31no—
29#3719—31no—
30#3717—31no—
31#3717—31no—
32#3716—31no—
33#3716—31no—
34#3714—31no—
35#3714—31no—
36C22×H(3,𝔽3)D411yescharacteristic
37C2×(C3⋊D3)⋊C3—21no—
38C2×(C3⋊D3)⋊C3—21no—
39C4×H(3,𝔽3)D411yescharacteristic
40C4×H(3,𝔽3)C2311yescharacteristic
41(C3⋊Dic3)⋊C3—21no—
42(C3⋊Dic3)⋊C3—21no—
43#950—91no—
44#950—91no—
45#904—91no—
46C2×C32×C4—21no—
47C2×C32×C4C2×D311yescharacteristic
48C2×C32×C4C2×D311yescharacteristic
49#638—21no—
50#638—21no—
51C4⋊(C3⋊D3)C2×D311yescharacteristic
52C4⋊(C3⋊D3)C2×D311yescharacteristic
53C4×C3⋊D3—21no—
54C4×C3⋊D3—21no—
55#634C2×D311yescharacteristic
56#634C2×D311yescharacteristic
57C3×#67—61no—
58C3×#67—61no—
59C3×D12—31no—
60C3×D12—31no—
61C3×C4×D3—61no—
62C3×C4×D3—61no—
63C3×Dic6—31no—
64C3×Dic6—31no—
65C3×C3⋊C8—31no—
66C3×C3⋊C8—31no—
67C3×C3⋊C8—31no—
68C3×C3⋊C8—31no—
69C3×C3⋊C8—61no—
70C3×C3⋊C8—61no—
71C2×H(3,𝔽3)—21no—
72C2×H(3,𝔽3)C2×D411yescharacteristic
73(C3⋊D3)⋊C3—41no—
74(C3⋊D3)⋊C3—41no—
75C3×#41—91no—
76C3×#41—91no—
77#235—31no—
78#235—31no—
79#235—91no—
80#235—181no—
81C2×C3×C8—91no—
82#216—91no—
83#216—91no—
84#215—91no—
85#215—91no—
86#214—91no—
87#214—91no—
88#213—91no—
89#213—91no—
90C2×C3⋊C8—31no—
91C2×C3⋊C8—31no—
92C2×C3⋊C8—61no—
93Dic12—91no—
94D24—91no—
95#204—91no—
96#204—91no—
97C22×C32—21no—
98C22×C32#6711yescharacteristic
99C22×C32#6711yescharacteristic
100C2×C3⋊D3—21no—
101C2×C3⋊D3—21no—
102C2×C3×D3—61no—
103C2×C3×D3—61no—
104C32×C4—21no—
105C32×C4—21no—
106C32×C4#6711yescharacteristic
107C32×C4#6711yescharacteristic
108C32×C4C22×D311yescharacteristic
109C32×C4C22×D311yescharacteristic
110C3⋊Dic3—21no—
111C3⋊Dic3—21no—
112C3×Dic3—61no—
113C3×Dic3—61no—
114#135—271noSylow
115H(3,𝔽3)#13511yescharacteristic, complemented, Sylow
116C3×Q8—91no—
117C3×Q8—91no—
118C3×D4—91no—
119C3×D4—91no—
120C3×D4—181no—
121C3×D4—181no—
122C2×C3×C4—31no—
123C2×C3×C4—61no—
124C2×C3×C4—121no—
125C2×C3×C4—181no—
126C2×C3×C4—181no—
127C2×C3×C4D3211yescharacteristic
128#67—61no—
129#67—61no—
130#67—181no—
131#67—181no—
132D12—31no—
133D12—31no—
134D12—91no—
135D12—181no—
136C4×D3—61no—
137C4×D3—61no—
138C4×D3—181no—
139C4×D3—181no—
140Dic6—31no—
141Dic6—31no—
142Dic6—91no—
143Dic6—181no—
144C3×C8—91no—
145C3×C8—91no—
146C3⋊C8—31no—
147C3⋊C8—31no—
148C3⋊C8—31no—
149C3⋊C8—31no—
150C3⋊C8—61no—
151C3⋊C8—61no—
152C2×C32—21no—
153C2×C32—22no—
154C2×C32—21no—
155C2×C32—21no—
156C2×C32C2×#6711yescharacteristic
157C2×C32C2×#6711yescharacteristic
158C3⋊D3—41no—
159C3⋊D3—41no—
160C3×D3—121no—
161C3×D3—121no—
162#41—91no—
163#41—91no—
164Q16—271no—
165SD16—271no—
166SD16—271no—
167D8—271no—
168C2×C8—91no—
169C22×C3—31no—
170C22×C3—61no—
171C22×C3—121no—
172C22×C3—181no—
173C22×C3—181no—
174C22×C3#62511yescharacteristic
175C2×D3—61no—
176C2×D3—61no—
177C2×D3—181no—
178C2×D3—181no—
179C3×C4—31no—
180C3×C4—31no—
181C3×C4—61no—
182C3×C4—61no—
183C3×C4—121no—
184C3×C4—121no—
185C3×C4—181no—
186C3×C4—181no—
187C3×C4#62511yescharacteristic
188C3×C4C2×D3211yescharacteristic
189Dic3—61no—
190Dic3—61no—
191Dic3—181no—
192Dic3—181no—
193C32—21no—
194C32#95011yescharacteristic, complemented
195C32#95011yescharacteristic, complemented
196Q8—91no—
197Q8—91no—
198D4—91no—
199D4—91no—
200D4—181no—
201D4—181no—
202C2×C4—181no—
203C2×C4—181no—
204C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
205C8—91no—
206C8—91no—
207C2×C3—21no—
208C2×C3—31no—
209C2×C3—61no—
210C2×C3—61no—
211C2×C3—121no—
212C2×C3—122no—
213C2×C3—121no—
214C2×C3—361no—
215C2×C3—361no—
216C2×C3C2×#62511yescharacteristic
217D3—121no—
218D3—121no—
219D3—361no—
220D3—361no—
221C22—181no—
222C22—181no—
223C22#619811yescharacteristic
224C4—181no—
225C4—181no—
226C4#619811yescharacteristic
227C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
228C3—31no—
229C3—61no—
230C3—121no—
231C3#6382211yescharacteristic
232C2—21no—
233C2—361no—
234C2—361no—
235C2C2×#619811yescharacteristic
subgroupG/H
centerC4#6198
commutator, cosocleC4×H(3,𝔽3)C23
FrattiniC3×C4C2×D32
FittingC2×C4×H(3,𝔽3)C22
socleC2×C3C2×#625
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#135
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #3828⋊C3C2semidirect
2. #3828⋊C3C2semidirect
3. C2×#6180C2semidirect
4. #88433C2semidirect
5. #88432C2semidirect
6. #88431C2semidirect
7. #88431C2semidirect
21. #3828D3semidirect
22. #3828D3semidirect
115. H(3,𝔽3)#135semidirect
194. C32#950semidirect
195. C32#950semidirect

Maximal subgroups

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

subgroupindex
#3828⋊C32
C2×#61802
#884332
#884322
#884312
#638293