Finite Group Databasebeta

← #11722545

Subgroups — #11722545

Order 864.

635 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 635 conjugacy classes of subgroups — too many to draw. Only its 114 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
1C2×#6235C211yesmaximal, characteristic, complemented
2C2×#638⋊C3C212yesmaximal, complemented
3#88677C212yesmaximal, complemented
4#88676C214yesmaximal, complemented
5#88672C212yesmaximal, complemented
6#88671C214yesmaximal, complemented
7#64330—32nomaximal
8C23×H(3,𝔽3)C2211yescharacteristic, complemented
9C22×#272C2211yescharacteristic, complemented
10C22×(C3⋊D3)⋊C3C2212yescomplemented
11C2×((C3⋊D3)⋊C3)⋊C2C2212yescomplemented
12#6235C2214yescomplemented
13C2×#1084C2211yescharacteristic, complemented
14#638⋊C3C2214yescomplemented
15#638⋊C3C2214yescomplemented
16C2×(C3⋊Dic3)⋊C3C2212yescomplemented
17#6200C2214yescomplemented
18#6199C2212yescomplemented
19#6198C2212yescomplemented
20((C3⋊D3)⋊C3)⋊C4C2214yescomplemented
21#6196C2212yescomplemented
22C2×#638D312yescomplemented
23C2×C3×#67—32no—
24C2×C3×#67—32no—
25C2×C3×#67—61no—
26#3811—34no—
27D3×#67—34no—
28D3×#67—34no—
29#3805—34no—
30#3804—34no—
31#3804—34no—
32C22×H(3,𝔽3)—41no—
33C22×H(3,𝔽3)C2312yes—
34C22×H(3,𝔽3)C2311yescharacteristic
35C2×#272—41no—
36C2×#272C2312yes—
37C2×(C3⋊D3)⋊C3—42no—
38C2×(C3⋊D3)⋊C3C2314yes—
39((C3⋊D3)⋊C3)⋊C2—42no—
40#1084C2312yes—
41(C3⋊Dic3)⋊C3C2314yes—
42#1004—91no—
43#1004—92no—
44C23×C32—21no—
45C23×C32C2×D312yescomplemented
46C22×C3⋊D3C2×D312yescomplemented
47C22×C3×D3—32no—
48C22×C3×D3—32no—
49C22×C3×D3—61no—
50C2×D32—34no—
51#638C2×D314yescomplemented
52#638C2×D314yescomplemented
53C2×C3⋊Dic3C2×D312yescomplemented
54C3×#67—34no—
55C3×#67—34no—
56C3×#67—34no—
57C3×#67—34no—
58C3×#67—64no—
59C2×C3×Dic3—32no—
60C2×C3×Dic3—32no—
61C2×C3×Dic3—61no—
62#627—34no—
63#626—34no—
64#626—34no—
65#625—34no—
66#624—34no—
67D3×Dic3—34no—
68D3×Dic3—34no—
69C2×H(3,𝔽3)—21no—
70C2×H(3,𝔽3)—22no—
71C2×H(3,𝔽3)C2411yescharacteristic
72#272—22no—
73(C3⋊D3)⋊C3—24no—
74C2×C3×D4—92no—
75C2×C3×D4—91no—
76C2×#67—32no—
77C2×#67—32no—
78C2×#67—61no—
79C2×#67—92no—
80C2×#67—92no—
81C2×#67—94no—
82#237—94no—
83#237—94no—
84#237—94no—
85D3×D4—94no—
86D3×D4—94no—
87D3×D4—94no—
88#235—92no—
89#235—94no—
90C22×C32—21no—
91C22×C32—22no—
92C22×C32—42no—
93C22×C32—42no—
94C22×C32C22×D312yes—
95C22×C32C22×D312yes—
96C22×C32C22×D312yes—
97C2×C3⋊D3—42no—
98C2×C3⋊D3C22×D314yes—
99C2×C3×D3—34no—
100C2×C3×D3—34no—
101C2×C3×D3—62no—
102C2×C3×D3—122no—
103C2×C3×D3—122no—
104C2×C3×D3—122no—
105D32—124no—
106C3⋊Dic3C22×D314yes—
107C3×Dic3—34no—
108C3×Dic3—34no—
109C3×Dic3—62no—
110H(5,𝔽2)—271noSylow
111H(3,𝔽3)H(5,𝔽2)11yescharacteristic, complemented, Sylow
112C23×C3—32no—
113C23×C3—61no—
114C23×C3—92no—
115C23×C3—91no—
116C23×C3D3211yescharacteristic
117C22×D3—32no—
118C22×D3—32no—
119C22×D3—61no—
120C22×D3—92no—
121C22×D3—92no—
122C22×D3—94no—
123C3×D4—94no—
124C3×D4—94no—
125C3×D4—94no—
126C2×C3×C4—92no—
127C2×C3×C4—91no—
128#67—34no—
129#67—34no—
130#67—34no—
131#67—34no—
132#67—64no—
133#67—94no—
134#67—94no—
135#67—94no—
136#67—94no—
137#67—94no—
138C2×Dic3—32no—
139C2×Dic3—32no—
140C2×Dic3—61no—
141C2×Dic3—92no—
142C2×Dic3—92no—
143C2×Dic3—94no—
144D12—92no—
145D12—94no—
146C4×D3—94no—
147C4×D3—94no—
148C4×D3—94no—
149Dic6—92no—
150Dic6—94no—
151C2×C32—22no—
152C2×C32—22no—
153C2×C32—22no—
154C2×C32—22no—
155C2×C32—21no—
156C2×C32—42no—
157C2×C32C23×D312yes—
158C3⋊D3—24no—
159C3×D3—64no—
160C3×D3—64no—
161C3×D3—122no—
162#41—272no—
163#41—274no—
164C2×D4—91no—
165C2×D4—92no—
166C2×D4—274no—
167C2×D4—272no—
168C22×C3—32no—
169C22×C3—32no—
170C22×C3—32no—
171C22×C3—41no—
172C22×C3—61no—
173C22×C3—62no—
174C22×C3—94no—
175C22×C3—92no—
176C22×C3—122no—
177C22×C3—122no—
178C22×C3—361no—
179C22×C3—362no—
180C22×C3C2×D3212yes—
181C22×C3C2×D3211yescharacteristic
182C2×D3—34no—
183C2×D3—34no—
184C2×D3—62no—
185C2×D3—94no—
186C2×D3—122no—
187C2×D3—122no—
188C2×D3—122no—
189C2×D3—362no—
190C2×D3—362no—
191C2×D3—364no—
192C3×C4—94no—
193C3×C4—92no—
194Dic3—34no—
195Dic3—34no—
196Dic3—62no—
197Dic3—94no—
198C32—21no—
199C32#100412yescomplemented
200C23—91no—
201C23—92no—
202C23—272no—
203C23((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
204Q8—272no—
205D4—94no—
206D4—94no—
207D4—94no—
208D4—272no—
209D4—274no—
210C2×C4—91no—
211C2×C4—92no—
212C2×C4—274no—
213C2×C4—272no—
214C2×C3—22no—
215C2×C3—21no—
216C2×C3—32no—
217C2×C3—62no—
218C2×C3—62no—
219C2×C3—62no—
220C2×C3—62no—
221C2×C3—61no—
222C2×C3—122no—
223C2×C3—184no—
224C2×C3—182no—
225C2×C3C22×D3211yescharacteristic
226D3—64no—
227D3—64no—
228D3—122no—
229D3—184no—
230C22—41no—
231C22—92no—
232C22—94no—
233C22—361no—
234C22—362no—
235C22—1082no—
236C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
237C22C2×((C3⋊D3)⋊C3)⋊C212yes—
238C4—92no—
239C4—94no—
240C3—32no—
241C3—61no—
242C3#6433011yescharacteristic
243C2—21no—
244C2—22no—
245C2—182no—
246C2—184no—
247C2C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC2C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC2×H(3,𝔽3)C24
Frattini, socleC2×C3C22×D32
FittingC23×H(3,𝔽3)C22
hypercenterC23((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)H(5,𝔽2)
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C2×#6235C2semidirect
2. C2×#638⋊C3C2semidirect
3. #88677C2semidirect
4. #88676C2semidirect
5. #88672C2semidirect
6. #88671C2semidirect
8. C23×H(3,𝔽3)C22semidirect
9. C22×#272C22semidirect
10. C22×(C3⋊D3)⋊C3C22semidirect
11. C2×((C3⋊D3)⋊C3)⋊C2C22semidirect
12. #6235C22semidirect
13. C2×#1084C22semidirect
14. #638⋊C3C22semidirect
15. #638⋊C3C22semidirect
16. C2×(C3⋊Dic3)⋊C3C22semidirect
17. #6200C22semidirect
18. #6199C22semidirect
19. #6198C22semidirect
20. ((C3⋊D3)⋊C3)⋊C4C22semidirect
21. #6196C22semidirect
22. C2×#638D3semidirect
45. C23×C32C2×D3semidirect
46. C22×C3⋊D3C2×D3semidirect
51. #638C2×D3semidirect
52. #638C2×D3semidirect
53. C2×C3⋊Dic3C2×D3semidirect
111. H(3,𝔽3)H(5,𝔽2)semidirect
199. C32#1004semidirect
203. C23((C3⋊D3)⋊C3)⋊C2semidirect

Maximal subgroups

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

subgroupindex
C2×#62352
C2×#638⋊C32
#886772
#886762
#886722
#886712
#643303