Finite Group Databasebeta

← #83435

Subgroups — #83435

Order 384.

418 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C3×#540C211yesmaximal, characteristic, complemented
2#5521C211yesmaximal, characteristic, complemented
3#5072C212yesmaximal, complemented
4#5066C212yesmaximal, complemented
5#5036C212yesmaximal, complemented
6#5035C212yesmaximal, complemented
7#5014C212yesmaximal, complemented
8C4×#213C212yesmaximal, complemented
9#5000C211yesmaximal, characteristic, complemented
10#3122—31nomaximal, Sylow
11C22×D12C2211yescharacteristic
12C3×#127C2211yescharacteristic, complemented
13C3×#125C2211yescharacteristic, complemented
14C3×#123—22no—
15C3×#121—22no—
16C3×#121C2212yes—
17C3×C4×D4C2212yescomplemented
18C3×#117C2211yescharacteristic
19C2×#213C2212yes—
20C2×#213C2212yes—
21#926C2211yescharacteristic
22#893—22no—
23D3×C22⋊C4—22no—
24C4×D12C2212yes—
25C4×D12C2212yes—
26#832C2212yescomplemented
27#832C2212yescomplemented
28#830C2212yes—
29#809C2212yes—
30#809C2212yes—
31#808C2212yescomplemented
32#807C2212yescomplemented
33C4⋊(C3⋊C8)C2212yescomplemented
34#803C2211yescharacteristic, complemented
35C4×C3⋊C8C2211yescharacteristic, complemented
36#540D311yescharacteristic, complemented
37#518—31no—
38#482—32no—
39#478—32no—
40#459—32no—
41#447—32no—
42#442—32no—
43C4×D8—32no—
44#433—31no—
45C23×D3—21no—
46C2×C3×D4—22no—
47C2×C3×D4—22no—
48C2×C3×D4C2312yes—
49C22×C3×C4—22no—
50C22×C3×C4C2311yescharacteristic
51C2×D12—21no—
52C2×D12—24no—
53C2×D12C2312yes—
54C2×C4×D3—22no—
55C2×C4×D3—22no—
56C3×C4⋊C4—22no—
57C3×C4⋊C4D412yes—
58C3×C4⋊C4C2312yes—
59C3×C22⋊C4—22no—
60C3×C22⋊C4—22no—
61C3×C22⋊C4D412yes—
62C3×C42C2311yescharacteristic
63C3×C42C2311yescharacteristic
64#217—21no—
65#213—24no—
66#213—24no—
67(C2×D3)⋊C4—22no—
68(C2×D3)⋊C4—22no—
69C4⋊Dic3C2312yes—
70C4×Dic3—21no—
71C2×C3⋊C8C2312yes—
72C2×C3⋊C8C2312yes—
73C22×D4—31no—
74C2×D8—32no—
75C2×D8—32no—
76#127C2×D311yescharacteristic, complemented
77#125C2×D311yescharacteristic, complemented
78#123—22no—
79#121—22no—
80#121C2×D312yes—
81C4×D4—32no—
82C4×D4—32no—
83C4×D4—62no—
84C4×D4C2×D312yescomplemented
85#117—31no—
86#117C2×D311yescharacteristic
87C2×C22⋊C4—62no—
88#107—32no—
89#106—32no—
90C4⋊C8—32no—
91#102—32no—
92#102—32no—
93#102—32no—
94#102—32no—
95C22⋊C8—32no—
96#97—31no—
97C4×C8—31no—
98C23×C3—22no—
99C23×C3C2×D411yescharacteristic
100C22×D3—22no—
101C22×D3—24no—
102C22×D3—82no—
103C3×D4—22no—
104C3×D4—22no—
105C3×D4—82no—
106C3×D4—82no—
107C2×C3×C4—22no—
108C2×C3×C4—22no—
109C2×C3×C4—82no—
110C2×C3×C4C2×D411yescharacteristic
111C2×C3×C4C2×D412yes—
112C2×C3×C4C2×D412yes—
113C2×C3×C4C2411yescharacteristic
114C2×Dic3—22no—
115D12—24no—
116D12—22no—
117D12#4114yes—
118C4×D3—82no—
119C4×D3—82no—
120C3⋊C8—22no—
121C3⋊C8—22no—
122C24—61no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—32no—
126C2×D4—64no—
127C2×D4—61no—
128C2×D4C22×D312yes—
129C22×C4—22no—
130C22×C4—62no—
131C22×C4—62no—
132C22×C4C22×D311yescharacteristic
133D8—64no—
134D8—64no—
135C2×C8—32no—
136C2×C8—32no—
137C4⋊C4—22no—
138C4⋊C4—32no—
139C4⋊C4#6712yes—
140C4⋊C4C22×D312yes—
141C22⋊C4—22no—
142C22⋊C4—22no—
143C22⋊C4—61no—
144C22⋊C4—62no—
145C22⋊C4—62no—
146C22⋊C4#6712yes—
147C42—61no—
148C42C22×D311yescharacteristic
149C42C22×D311yescharacteristic
150C22×C3—21no—
151C22×C3—22no—
152C22×C3—42no—
153C22×C3—42no—
154C22×C3—42no—
155C22×C3C22×D411yescharacteristic
156C2×D3—24no—
157C2×D3—42no—
158C2×D3—42no—
159C2×D3—84no—
160C3×C4—21no—
161C3×C4—22no—
162C3×C4—22no—
163C3×C4—42no—
164C3×C4C2×#4112yes—
165Dic3—42no—
166C23—22no—
167C23—64no—
168C23—62no—
169C23—242no—
170C23C2×#6711yescharacteristic
171D4—22no—
172D4—22no—
173D4—34no—
174D4—64no—
175D4—62no—
176D4—82no—
177D4—82no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—62no—
181C2×C4—82no—
182C2×C4—242no—
183C2×C4—242no—
184C2×C4C2×#6711yescharacteristic
185C2×C4C2×#6712yes—
186C2×C4C2×#6712yes—
187C2×C4C23×D311yescharacteristic
188C8—62no—
189C8—62no—
190C2×C3—41no—
191C2×C3—82no—
192C2×C3#52511yescharacteristic
193C2×C3#57612yes—
194D3—44no—
195C22—21no—
196C22—22no—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—64no—
201C22—122no—
202C22—122no—
203C22—244no—
204C22C22×#6711yescharacteristic
205C4—21no—
206C4—22no—
207C4—22no—
208C4—42no—
209C4—122no—
210C4D3×#4112yes—
211C3#312211yescharacteristic, complemented, Sylow
212C2—41no—
213C2—82no—
214C2—124no—
215C2#582811yescharacteristic
216C2#583512yes—
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#540C2
socleC22×C3C22×D4
hypercenter#540D3
hypocenterC3#3122
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. C3×#540C2semidirect
2. #5521C2semidirect
3. #5072C2semidirect
4. #5066C2semidirect
5. #5036C2semidirect
6. #5035C2semidirect
7. #5014C2semidirect
8. C4×#213C2semidirect
9. #5000C2semidirect
12. C3×#127C22semidirect
13. C3×#125C22semidirect
17. C3×C4×D4C22semidirect
26. #832C22semidirect
27. #832C22semidirect
31. #808C22semidirect
32. #807C22semidirect
33. C4⋊(C3⋊C8)C22semidirect
34. #803C22semidirect
35. C4×C3⋊C8C22semidirect
36. #540D3semidirect
76. #127C2×D3semidirect
77. #125C2×D3semidirect
84. C4×D4C2×D3semidirect
211. C3#3122semidirect

Maximal subgroups

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

subgroupindex
C3×#5402
#55212
#50722
#50662
#50362
#50352
#50142
C4×#2132
#50002
#31223