Finite Group Databasebeta

← #11721193

Subgroups — #11721193

Order 864.

292 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 42 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
1D4×#272C211yesmaximal, characteristic, complemented
2#88649C211yesmaximal, characteristic, complemented
3#3753⋊C3C211yesmaximal, characteristic, complemented
4#3754⋊C3C211yesmaximal, characteristic, complemented
5#88437C211yesmaximal, characteristic, complemented
6#88436C211yesmaximal, characteristic, complemented
7#88435C211yesmaximal, characteristic, complemented
8#63934—31nomaximal
9#63932—31nomaximal
10C22×#272—21no—
11D4×H(3,𝔽3)C2211yescharacteristic
12#6235—21no—
13C4⋊#272C2211yescharacteristic
14C4×#272C2211yescharacteristic
15C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
16#634⋊C3C2211yescharacteristic, complemented
17#6200—21no—
18((C3⋊D3)⋊C3)⋊C4—21no—
19(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
20(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
21C3×D3×D4—31no—
22C3×D3×D4—31no—
23C3×D3×D4—61no—
24#3799—31no—
25#3795—31no—
26#3754D311yescharacteristic, complemented
27#3753D311yescharacteristic, complemented
28C3×#214—31no—
29C3×#213—31no—
30#3716—31no—
31#3715—31no—
32#3714—31no—
33#3713—31no—
34#3711—31no—
35#3711—31no—
36C22×H(3,𝔽3)—21no—
37C2×#272—21no—
38C2×#272—42no—
39C2×#272D411yescharacteristic
40C2×(C3⋊D3)⋊C3—21no—
41C4×H(3,𝔽3)C2311yescharacteristic
42#1084D411yescharacteristic
43(C3⋊Dic3)⋊C3—21no—
44#932—91no—
45#914—91no—
46#911—91no—
47C22×C3×D3—61no—
48C22×C3×D3—61no—
49C22×C3×D3—62no—
50C32×D4—21no—
51C32×D4C2×D311yescharacteristic
52C32×D4C2×D311yescharacteristic
53C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
54#634C2×D311yescharacteristic, complemented
55C3×#67—61no—
56C3×#67—61no—
57C3×#67—62no—
58C3×D12—31no—
59C3×D12—31no—
60C3×D12—31no—
61C3×D12—61no—
62C3×C4×D3—31no—
63C3×C4×D3—31no—
64C3×C4×D3—61no—
65C3×Dic6—31no—
66#626—61no—
67#625—61no—
68#624—61no—
69D3×Dic3—61no—
70C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
71C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
72C3×C3⋊C8—31no—
73C3×C3⋊C8—31no—
74C2×H(3,𝔽3)—41no—
75C2×H(3,𝔽3)C2×D411yescharacteristic
76#272—21no—
77#272—41no—
78(C3⋊D3)⋊C3—41no—
79C2×C3×D4—91no—
80#239—91no—
81#237—91no—
82D3×D4—31no—
83D3×D4—31no—
84D3×D4—61no—
85#235—91no—
86C3×SD16—91no—
87C3×D8—91no—
88#215—91no—
89#214—31no—
90#214—91no—
91#214—91no—
92#213—31no—
93#213—91no—
94#213—91no—
95#213—91no—
96#208—91no—
97D24—91no—
98#204—91no—
99ΓL(1,𝔽25)—91no—
100ΓL(1,𝔽25)—91no—
101C22×C32—22no—
102C22×C32—21no—
103C22×C32—21no—
104C2×C3⋊D3—21no—
105C2×C3×D3—31no—
106C2×C3×D3—31no—
107C2×C3×D3—61no—
108C2×C3×D3—62no—
109C2×C3×D3—61no—
110C2×C3×D3—61no—
111C2×C3×D3—61no—
112C2×C3×D3—124no—
113C2×C3×D3—122no—
114C2×C3×D3—122no—
115C32×C4—21no—
116C32×C4C22×D311yescharacteristic
117C32×C4C22×D311yescharacteristic
118C3⋊Dic3—21no—
119C3×Dic3—31no—
120C3×Dic3—31no—
121C3×Dic3—61no—
122C3×Dic3—61no—
123AGL(1,ℤ/8)—271noSylow
124H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
125C23×C3—181no—
126C22×D3—61no—
127C22×D3—61no—
128C22×D3—62no—
129C3×Q8—91no—
130C3×D4—31no—
131C3×D4—31no—
132C3×D4—61no—
133C3×D4—91no—
134C3×D4—91no—
135C3×D4—181no—
136C3×D4D3211yescharacteristic
137C2×C3×C4—91no—
138#67—61no—
139#67—61no—
140#67—62no—
141#67—181no—
142#67—181no—
143C2×Dic3—181no—
144D12—31no—
145D12—31no—
146D12—31no—
147D12—61no—
148D12—91no—
149D12—181no—
150C4×D3—31no—
151C4×D3—31no—
152C4×D3—61no—
153C4×D3—181no—
154C4×D3—181no—
155Dic6—31no—
156Dic6—91no—
157C3×C8—91no—
158C3×C8—91no—
159C3⋊C8—31no—
160C3⋊C8—31no—
161C3⋊C8—91no—
162C3⋊C8—91no—
163C2×C32—21no—
164C2×C32—42no—
165C2×C32—41no—
166C2×C32—41no—
167C2×C32D3×D411yescharacteristic
168C2×C32D3×D411yescharacteristic
169C3⋊D3—41no—
170C3×D3—62no—
171C3×D3—61no—
172C3×D3—61no—
173C3×D3—122no—
174C3×D3—121no—
175C3×D3—121no—
176C3×D3—121no—
177#41—271no—
178C2×D4—91no—
179SD16—91no—
180SD16—271no—
181D8—91no—
182D8—271no—
183M16—271no—
184C22×C3—21no—
185C22×C3—61no—
186C22×C3—61no—
187C22×C3—62no—
188C22×C3—91no—
189C22×C3—181no—
190C22×C3—181no—
191C22×C3—362no—
192C2×D3—31no—
193C2×D3—31no—
194C2×D3—61no—
195C2×D3—61no—
196C2×D3—61no—
197C2×D3—62no—
198C2×D3—61no—
199C2×D3—122no—
200C2×D3—122no—
201C2×D3—124no—
202C2×D3—181no—
203C3×C4—31no—
204C3×C4—31no—
205C3×C4—61no—
206C3×C4—91no—
207C3×C4—181no—
208C3×C4C2×D3211yescharacteristic
209Dic3—31no—
210Dic3—31no—
211Dic3—61no—
212Dic3—61no—
213Dic3—181no—
214C32—21no—
215C32#91111yescharacteristic, complemented
216C32#91411yescharacteristic, complemented
217C23—181no—
218Q8—91no—
219D4—91no—
220D4—91no—
221D4—181no—
222D4—541no—
223D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
224C2×C4—91no—
225C2×C4—541no—
226C8—91no—
227C8—91no—
228C2×C3—31no—
229C2×C3—31no—
230C2×C3—41no—
231C2×C3—61no—
232C2×C3—121no—
233C2×C3—121no—
234C2×C3—122no—
235C2×C3—181no—
236C2×C3—361no—
237C2×C3—361no—
238C2×C3#381111yescharacteristic
239D3—61no—
240D3—61no—
241D3—62no—
242D3—121no—
243D3—121no—
244D3—122no—
245D3—121no—
246D3—361no—
247C22—21no—
248C22—91no—
249C22—181no—
250C22—181no—
251C22—362no—
252C4—91no—
253C4—181no—
254C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
255C3—31no—
256C3—31no—
257C3—61no—
258C3#6393711yescharacteristic
259C2—41no—
260C2—181no—
261C2—361no—
262C2—361no—
263C2#8867711yescharacteristic
subgroupG/H
centerC2#88677
commutator, cosocleC4×H(3,𝔽3)C23
FrattiniC3×C4C2×D32
FittingD4×H(3,𝔽3)C22
socleC2×C3#3811
hypercenterD4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)AGL(1,ℤ/8)
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. D4×#272C2semidirect
2. #88649C2semidirect
3. #3753⋊C3C2semidirect
4. #3754⋊C3C2semidirect
5. #88437C2semidirect
6. #88436C2semidirect
7. #88435C2semidirect
15. C4⋊((C3⋊D3)⋊C3)C22semidirect
16. #634⋊C3C22semidirect
19. (C3⋊(C3⋊C8))⋊C3C22semidirect
20. (C3⋊(C3⋊C8))⋊C3C22semidirect
26. #3754D3semidirect
27. #3753D3semidirect
53. C4⋊(C3⋊D3)C2×D3semidirect
54. #634C2×D3semidirect
70. C3⋊(C3⋊C8)C2×D3semidirect
71. C3⋊(C3⋊C8)C2×D3semidirect
124. H(3,𝔽3)AGL(1,ℤ/8)semidirect
215. C32#911semidirect
216. C32#914semidirect

Maximal subgroups

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

subgroupindex
D4×#2722
#886492
#3753⋊C32
#3754⋊C32
#884372
#884362
#884352
#639343
#639323