Finite Group Databasebeta

← #11721073

Subgroups — #11721073

Order 864.

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

normal subgroups only

This group has 257 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#88740C211yesmaximal, characteristic, complemented
2#3828⋊C3C211yesmaximal, characteristic, complemented
3C2×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
4#88439C211yesmaximal, characteristic, complemented
5#88438C211yesmaximal, characteristic, complemented
6#88437C211yesmaximal, characteristic, complemented
7#88436C211yesmaximal, characteristic, complemented
8#63829—31nomaximal
9#63822—31nomaximal
10C2×C4×H(3,𝔽3)C2211yescharacteristic
11#6235—21no—
12C4⋊#272C2211yescharacteristic
13C4×#272—21no—
14#6229C2211yescharacteristic
15#638⋊C3—21no—
16C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
17C4×(C3⋊D3)⋊C3—21no—
18#634⋊C3C2211yescharacteristic
19(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#3828D311yescharacteristic, complemented
22C3×#235—31no—
23C3×#235—31no—
24C3×#235—31no—
25C3×#235—61no—
26C2×C3⋊(C3⋊C8)D311yescharacteristic, complemented
27C2×C3×C3⋊C8—31no—
28#3719—31no—
29#3718—31no—
30#3717—31no—
31#3716—31no—
32#3715—31no—
33#3715—31no—
34#3714—31no—
35#3713—31no—
36C22×H(3,𝔽3)D411yescharacteristic
37C2×#272—21no—
38C2×(C3⋊D3)⋊C3—21no—
39C4×H(3,𝔽3)D411yescharacteristic
40C4×H(3,𝔽3)C2311yescharacteristic
41#1084—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—
50C4⋊(C3⋊D3)C2×D311yescharacteristic
51C4×C3⋊D3—21no—
52#634C2×D311yescharacteristic
53C3×#67—62no—
54C3×#67—61no—
55C3×#67—61no—
56C3×#67—61no—
57C3×D12—31no—
58C3×D12—31no—
59C3×D12—31no—
60C3×D12—61no—
61C3×C4×D3—62no—
62C3×C4×D3—61no—
63C3×C4×D3—61no—
64C3×C4×D3—61no—
65C3×Dic6—31no—
66C3×Dic6—31no—
67C3×Dic6—31no—
68C3×Dic6—61no—
69C3⋊(C3⋊C8)C2×D311yescharacteristic
70C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×C3⋊C8—31no—
72C3×C3⋊C8—31no—
73C2×H(3,𝔽3)—21no—
74C2×H(3,𝔽3)C2×D411yescharacteristic
75#272—41no—
76(C3⋊D3)⋊C3—41no—
77C3×#41—91no—
78C3×#41—91no—
79#235—31no—
80#235—31no—
81#235—31no—
82#235—61no—
83#235—181no—
84C2×C3×C8—91no—
85#216—91no—
86#216—91no—
87#215—91no—
88#215—91no—
89#214—91no—
90#214—91no—
91#213—91no—
92#213—91no—
93C2×C3⋊C8—31no—
94C2×C3⋊C8—91no—
95Dic12—91no—
96D24—91no—
97#204—91no—
98#204—91no—
99C22×C32—21no—
100C22×C32D1211yescharacteristic
101C22×C32#6711yescharacteristic
102C2×C3⋊D3—21no—
103C2×C3×D3—62no—
104C2×C3×D3—61no—
105C2×C3×D3—61no—
106C2×C3×D3—61no—
107C32×C4—21no—
108C32×C4—21no—
109C32×C4D1211yescharacteristic
110C32×C4#6711yescharacteristic
111C32×C4C22×D311yescharacteristic
112C32×C4C22×D311yescharacteristic
113C3⋊Dic3—21no—
114C3×Dic3—62no—
115C3×Dic3—61no—
116C3×Dic3—61no—
117C3×Dic3—61no—
118#135—271noSylow
119H(3,𝔽3)#13511yescharacteristic, complemented, Sylow
120C3×Q8—91no—
121C3×Q8—91no—
122C3×D4—91no—
123C3×D4—91no—
124C3×D4—181no—
125C3×D4—181no—
126C2×C3×C4—31no—
127C2×C3×C4—61no—
128C2×C3×C4—121no—
129C2×C3×C4—181no—
130C2×C3×C4—181no—
131C2×C3×C4D3211yescharacteristic
132#67—61no—
133#67—61no—
134#67—62no—
135#67—61no—
136#67—181no—
137D12—31no—
138D12—31no—
139D12—31no—
140D12—61no—
141D12—181no—
142C4×D3—61no—
143C4×D3—61no—
144C4×D3—62no—
145C4×D3—61no—
146C4×D3—181no—
147Dic6—31no—
148Dic6—31no—
149Dic6—31no—
150Dic6—61no—
151Dic6—181no—
152C3×C8—91no—
153C3×C8—91no—
154C3⋊C8—31no—
155C3⋊C8—31no—
156C3⋊C8—91no—
157C3⋊C8—91no—
158C2×C32—21no—
159C2×C32—21no—
160C2×C32—21no—
161C2×C32—41no—
162C2×C32C2×D1211yescharacteristic
163C2×C32C2×#6711yescharacteristic
164C3⋊D3—41no—
165C3×D3—122no—
166C3×D3—121no—
167C3×D3—121no—
168C3×D3—121no—
169#41—91no—
170#41—91no—
171Q16—271no—
172SD16—271no—
173SD16—271no—
174D8—271no—
175C2×C8—91no—
176C22×C3—31no—
177C22×C3—61no—
178C22×C3—121no—
179C22×C3—181no—
180C22×C3—181no—
181C22×C3#62611yescharacteristic
182C2×D3—61no—
183C2×D3—61no—
184C2×D3—62no—
185C2×D3—61no—
186C2×D3—181no—
187C3×C4—31no—
188C3×C4—31no—
189C3×C4—61no—
190C3×C4—61no—
191C3×C4—121no—
192C3×C4—121no—
193C3×C4—181no—
194C3×C4—181no—
195C3×C4#62611yescharacteristic
196C3×C4C2×D3211yescharacteristic
197Dic3—61no—
198Dic3—61no—
199Dic3—62no—
200Dic3—61no—
201Dic3—181no—
202C32—21no—
203C32#90411yescharacteristic, complemented
204C32#95011yescharacteristic, complemented
205Q8—91no—
206Q8—91no—
207D4—91no—
208D4—91no—
209D4—181no—
210D4—181no—
211C2×C4—181no—
212C2×C4—181no—
213C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
214C8—91no—
215C8—91no—
216C2×C3—21no—
217C2×C3—31no—
218C2×C3—61no—
219C2×C3—61no—
220C2×C3—121no—
221C2×C3—121no—
222C2×C3—121no—
223C2×C3—361no—
224C2×C3—361no—
225C2×C3C2×#62611yescharacteristic
226D3—121no—
227D3—121no—
228D3—122no—
229D3—121no—
230D3—361no—
231C22—181no—
232C22—181no—
233C22#620011yescharacteristic
234C4—181no—
235C4—181no—
236C4#620011yescharacteristic
237C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
238C3—31no—
239C3—61no—
240C3—121no—
241C3#6382911yescharacteristic
242C2—21no—
243C2—361no—
244C2—361no—
245C2C2×#620011yescharacteristic
subgroupG/H
centerC4#6200
commutator, cosocleC4×H(3,𝔽3)C23
FrattiniC3×C4C2×D32
FittingC2×C4×H(3,𝔽3)C22
socleC2×C3C2×#626
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. #88740C2semidirect
2. #3828⋊C3C2semidirect
3. C2×(C3⋊(C3⋊C8))⋊C3C2semidirect
4. #88439C2semidirect
5. #88438C2semidirect
6. #88437C2semidirect
7. #88436C2semidirect
21. #3828D3semidirect
26. C2×C3⋊(C3⋊C8)D3semidirect
119. H(3,𝔽3)#135semidirect
203. C32#904semidirect
204. C32#950semidirect

Maximal subgroups

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

subgroupindex
#887402
#3828⋊C32
C2×(C3⋊(C3⋊C8))⋊C32
#884392
#884382
#884372
#884362
#638293
#638223