Finite Group Databasebeta

← #11724596

Subgroups — #11724596

Order 864.

1136 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1136 conjugacy classes of subgroups — too many to draw. Only its 202 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
1Q8×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C3×#3832C211yesmaximal, characteristic, complemented
3C32×#239C211yesmaximal, characteristic, complemented
4#89026C213yesmaximal, complemented
5C4×D3×C3⋊D3C213yesmaximal, complemented
6#89021C213yesmaximal, complemented
7#89014C213yesmaximal, complemented
8#41×C3⋊D3—31nomaximal
9#64319—34nomaximal
10C2×D3×C3⋊D3C2213yes—
11C33×Q8C2211yescharacteristic, complemented
12C4×C3⋊(C3⋊D3)C2213yes—
13#6308C2213yescomplemented
14C3×C4⋊(C3⋊D3)C2213yescomplemented
15C3×C4×C3⋊D3C2213yescomplemented
16C32×D12C2213yescomplemented
17C32×C4×D3C2213yescomplemented
18#6290C2216yescomplemented
19#6289C2211yescharacteristic
20Dic3×C3⋊D3C2213yescomplemented
21D3×C3⋊Dic3C2213yescomplemented
22C32×#41—31no—
23#3832D311yescharacteristic, complemented
24Q8×C3⋊D3—31no—
25Q8×C3⋊D3—34no—
26Q8×C3⋊D3—64no—
27#3830—33no—
28D4×C3⋊D3—33no—
29#3828—33no—
30C2×C4×C3⋊D3—33no—
31C3×#239—34no—
32C3×#239D314yescomplemented
33#3802—312no—
34C4×D32—312no—
35#3795—312no—
36#3795—312no—
37C2×C3⋊(C3⋊D3)C2311yescharacteristic
38C2×C3×C3⋊D3C2313yes—
39C2×C32×D3C2313yes—
40D3×C3⋊D3—26no—
41C33×C4C2313yes—
42C3⋊(C3⋊Dic3)C2313yes—
43C3×C3⋊Dic3C2311yescharacteristic
44C32×Dic3C2311yescharacteristic
45D3×#41—94no—
46D3×#41—91no—
47C22×C3⋊D3—33no—
48C2×D32—312no—
49C32×Q8—24no—
50C32×Q8C2×D314yescomplemented
51C32×Q8C2×D311yescharacteristic, complemented
52C32×D4—33no—
53C2×C32×C4—33no—
54#638—36no—
55C2×C3⋊Dic3—33no—
56C4⋊(C3⋊D3)C2×D313yescomplemented
57C4×C3⋊D3—33no—
58C4×C3⋊D3—33no—
59C4×C3⋊D3—31no—
60C4×C3⋊D3—312no—
61C4×C3⋊D3—612no—
62C4×C3⋊D3C2×D313yescomplemented
63#634—33no—
64#634—312no—
65#634—612no—
66C3×D12—312no—
67C3×D12C2×D3112yescomplemented
68C3×C4×D3—312no—
69C3×C4×D3C2×D3112yescomplemented
70#625—324no—
71#624—34no—
72D3×Dic3—312no—
73D3×Dic3—312no—
74C2×C33C2411yescharacteristic
75C3⋊(C3⋊D3)#4112yescomplemented
76C3×C3⋊D3—23no—
77C32×D3—23no—
78C3×#41—34no—
79C3×#41—91no—
80#239—34no—
81#239C3⋊D311yescharacteristic, complemented
82D3×Q8—94no—
83D3×Q8—91no—
84D3×Q8—184no—
85#237—912no—
86#237—93no—
87D3×D4—912no—
88D3×D4—93no—
89#235—912no—
90#235—93no—
91C2×C4×D3—912no—
92C2×C4×D3—93no—
93C22×C32—33no—
94C2×C3⋊D3—31no—
95C2×C3⋊D3—34no—
96C2×C3⋊D3—64no—
97C2×C3⋊D3—66no—
98C2×C3⋊D3C22×D313yes—
99C2×C3×D3—312no—
100C2×C3×D3C22×D3112yes—
101D32—624no—
102C32×C4—212no—
103C32×C4—31no—
104C32×C4C22×D3112yes—
105C32×C4C22×D313yes—
106C3⋊Dic3—33no—
107C3⋊Dic3—312no—
108C3⋊Dic3—612no—
109C3⋊Dic3C22×D311yescharacteristic
110C3×Dic3—34no—
111C3×Dic3C22×D314yes—
112C2×#41—271noSylow
113C33C2×#4111yescharacteristic, complemented, Sylow
114C22×D3—912no—
115C22×D3—93no—
116C3×Q8—24no—
117C3×Q8D3214yescomplemented
118C3×Q8C2×C3⋊D311yescharacteristic, complemented
119C3×D4—312no—
120C3×D4—93no—
121C2×C3×C4—312no—
122C2×C3×C4—93no—
123#67—924no—
124#67—96no—
125C2×Dic3—912no—
126C2×Dic3—93no—
127D12—312no—
128D12C2×C3⋊D313yescomplemented
129C4×D3—312no—
130C4×D3—94no—
131C4×D3—912no—
132C4×D3—93no—
133C4×D3—912no—
134C4×D3—93no—
135C4×D3—91no—
136C4×D3—1812no—
137C4×D3C2×C3⋊D313yescomplemented
138Dic6—912no—
139Dic6—93no—
140Dic6—1812no—
141C2×C32—24no—
142C2×C32—63no—
143C2×C32C23×D314yes—
144C2×C32C23×D311yescharacteristic
145C3⋊D3—23no—
146C3⋊D3—32no—
147C3⋊D3—38no—
148C3⋊D3—68no—
149C3×D3—212no—
150C3×D3—612no—
151#41—31no—
152#41—91no—
153#41—273no—
154#41—273no—
155C2×Q8—271no—
156C2×D4—273no—
157C22×C4—273no—
158C22×C3—312no—
159C22×C3—93no—
160C2×D3—312no—
161C2×D3—94no—
162C2×D3—91no—
163C2×D3—184no—
164C2×D3—186no—
165C2×D3—1824no—
166C2×D3C22×C3⋊D313yes—
167C3×C4—212no—
168C3×C4—34no—
169C3×C4—91no—
170C3×C4C2×D32112yes—
171C3×C4C22×C3⋊D313yes—
172Dic3—34no—
173Dic3—912no—
174Dic3—93no—
175Dic3—1812no—
176Dic3C22×C3⋊D311yescharacteristic
177C32—24no—
178C32D3×#4114yescomplemented
179C32D3×#4111yescharacteristic, complemented
180C23—273no—
181Q8—273no—
182Q8D3×C3⋊D311yescharacteristic, complemented
183D4—33no—
184D4—93no—
185D4—276no—
186C2×C4—33no—
187C2×C4—93no—
188C2×C4—271no—
189C2×C4—273no—
190C2×C4—273no—
191C2×C4—273no—
192C2×C3—24no—
193C2×C3—612no—
194C2×C3—183no—
195C2×C3C22×D3214yes—
196C2×C3C23×C3⋊D311yescharacteristic
197D3—23no—
198D3—612no—
199D3—98no—
200D3—92no—
201D3—188no—
202C22—33no—
203C22—93no—
204C22—271no—
205C22—546no—
206C4—31no—
207C4—91no—
208C4—273no—
209C4C2×D3×C3⋊D313yes—
210C3—24no—
211C3#6431914yescomplemented
212C3#41×C3⋊D311yescharacteristic, complemented
213C2—63no—
214C2—183no—
215C2—272no—
216C2C22×D3×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C22×D3×C3⋊D3
commutator, socle, cosocleC2×C33C24
FittingC33×Q8C22
hypercenterQ8D3×C3⋊D3
hypocenterC33C2×#41
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. Q8×C3⋊(C3⋊D3)C2semidirect
2. C3×#3832C2semidirect
3. C32×#239C2semidirect
4. #89026C2semidirect
5. C4×D3×C3⋊D3C2semidirect
6. #89021C2semidirect
7. #89014C2semidirect
11. C33×Q8C22semidirect
13. #6308C22semidirect
14. C3×C4⋊(C3⋊D3)C22semidirect
15. C3×C4×C3⋊D3C22semidirect
16. C32×D12C22semidirect
17. C32×C4×D3C22semidirect
18. #6290C22semidirect
20. Dic3×C3⋊D3C22semidirect
21. D3×C3⋊Dic3C22semidirect
23. #3832D3semidirect
32. C3×#239D3semidirect
50. C32×Q8C2×D3semidirect
51. C32×Q8C2×D3semidirect
56. C4⋊(C3⋊D3)C2×D3semidirect
62. C4×C3⋊D3C2×D3semidirect
67. C3×D12C2×D3semidirect
69. C3×C4×D3C2×D3semidirect
75. C3⋊(C3⋊D3)#41semidirect
81. #239C3⋊D3semidirect
113. C33C2×#41semidirect
117. C3×Q8D32semidirect
118. C3×Q8C2×C3⋊D3semidirect
128. D12C2×C3⋊D3semidirect
137. C4×D3C2×C3⋊D3semidirect
178. C32D3×#41semidirect
179. C32D3×#41semidirect
182. Q8D3×C3⋊D3semidirect
211. C3#64319semidirect
212. C3#41×C3⋊D3semidirect

Maximal subgroups

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

subgroupindex
Q8×C3⋊(C3⋊D3)2
C3×#38322
C32×#2392
#890262
C4×D3×C3⋊D32
#890212
#890142
#41×C3⋊D33
#643193