Finite Group Databasebeta

← #80723

Subgroups — #80723

Order 384.

442 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 109 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#5607C212yesmaximal, complemented
2C3×#466C211yesmaximal, characteristic, complemented
3#5156C212yesmaximal, complemented
4#5115C211yesmaximal, characteristic, complemented
5#5110C211yesmaximal, characteristic, complemented
6#5039C212yesmaximal, complemented
7#4885C212yesmaximal, complemented
8#4784C212yesmaximal, complemented
9#4773C212yesmaximal, complemented
10#3199—31nomaximal, Sylow
11C2×D3×D4C2212yescomplemented
12C2×#235C2211yescharacteristic
13C2×C3×D8C2211yescharacteristic
14C22×C3×C8C2211yescharacteristic, complemented
15C3×#121C2212yescomplemented
16#940—22no—
17#938C2212yes—
18#937—22no—
19C2×#213C2212yes—
20#930C2212yescomplemented
21C2×#208C2211yescharacteristic, complemented
22C2×D24C2211yescharacteristic
23#895C2212yes—
24#892C2212yescomplemented
25#885—22no—
26#882—22no—
27C3×#107C2211yescharacteristic, complemented
28C3×#102C2212yescomplemented
29#832C2212yescomplemented
30#821C2212yescomplemented
31(C2×D3)⋊C8C2212yescomplemented
32(C2×D3)⋊C8C2212yescomplemented
33#818C2211yescharacteristic, complemented
34#809C2212yescomplemented
35#808C2212yescomplemented
36#535—32no—
37#482—32no—
38#482—32no—
39#469—32no—
40#466—31no—
41#466D311yescharacteristic, complemented
42#447—32no—
43#447—32no—
44#408—31no—
45C23×D3—22no—
46C2×C3×D4—22no—
47C2×C3×D4C2312yes—
48C22×C3×C4C2311yescharacteristic
49C2×#67—22no—
50C2×#67—22no—
51C2×#67—22no—
52C2×#67D412yes—
53D3×D4—42no—
54D3×D4—42no—
55#235—41no—
56#235—41no—
57C2×D12—22no—
58C2×D12D411yescharacteristic
59C2×D12C2312yes—
60C2×C4×D3C2312yes—
61C2×Dic6D411yescharacteristic
62C3×D8—41no—
63C2×C3×C8—41no—
64C2×C3×C8C2311yescharacteristic
65C2×C3×C8C2311yescharacteristic
66C3×C4⋊C4C2312yes—
67C3×C22⋊C4—22no—
68#217—22no—
69#213—42no—
70(C2×D3)⋊C4—22no—
71(C2×D3)⋊C4—22no—
72(C2×D3)⋊C4—22no—
73C4⋊Dic3C2312yes—
74C4×Dic3—22no—
75#208—41no—
76C2×C3⋊C8C2312yes—
77D24—41no—
78C2×#41—31no—
79C22×D4—32no—
80C2×D8—32no—
81C2×D8—31no—
82C2×D8C2×D311yescharacteristic
83C2×M16—31no—
84C22×C8C2×D311yescharacteristic, complemented
85#127—62no—
86#124—62no—
87#121—32no—
88#121—32no—
89#121—32no—
90#121C2×D312yescomplemented
91#120—62no—
92#120—62no—
93#117—32no—
94#107—31no—
95#107C2×D311yescharacteristic, complemented
96#106—32no—
97#102—32no—
98#102—32no—
99#102—32no—
100#102C2×D312yescomplemented
101C22⋊C8—32no—
102C22⋊C8—32no—
103C23×C3—22no—
104C23×C3C2×D411yescharacteristic
105C22×D3—22no—
106C22×D3—44no—
107C22×D3—82no—
108C22×D3—82no—
109C22×D3C2×D412yes—
110C3×D4—22no—
111C3×D4—22no—
112C3×D4—82no—
113C2×C3×C4—21no—
114C2×C3×C4—21no—
115C2×C3×C4—22no—
116C2×C3×C4C2×D411yescharacteristic
117C2×C3×C4C2411yescharacteristic
118#67—42no—
119#67—44no—
120#67—82no—
121#67—82no—
122C2×Dic3—22no—
123C2×Dic3C2×D412yes—
124D12—22no—
125D12—22no—
126D12—41no—
127D12—82no—
128C4×D3—22no—
129C4×D3—22no—
130Dic6—41no—
131C3×C8—21no—
132C3×C8—21no—
133C3⋊C8—22no—
134C24—62no—
135#41—121no—
136#41—121no—
137C2×Q8—31no—
138C2×D4—22no—
139C2×D4—32no—
140C2×D4—31no—
141C2×D4—32no—
142C2×D4—62no—
143C2×D4—62no—
144C2×D4—62no—
145C2×D4—62no—
146C2×D4—122no—
147C2×D4—122no—
148C2×D4C22×D312yes—
149C22×C4—32no—
150C22×C4C22×D311yescharacteristic
151D8—41no—
152D8—121no—
153D8—122no—
154M16—121no—
155C2×C8—32no—
156C2×C8—41no—
157C2×C8C22×D311yescharacteristic
158C2×C8C22×D311yescharacteristic
159C4⋊C4—32no—
160C4⋊C4C22×D312yes—
161C22⋊C4—22no—
162C22⋊C4—62no—
163C22⋊C4—62no—
164C22⋊C4—62no—
165C22⋊C4—62no—
166C42—62no—
167C22×C3—21no—
168C22×C3—21no—
169C22×C3—21no—
170C22×C3—42no—
171C22×C3—42no—
172C22×C3—42no—
173C22×C3C22×D411yescharacteristic
174C2×D3—22no—
175C2×D3—22no—
176C2×D3—22no—
177C2×D3—42no—
178C2×D3—42no—
179C2×D3—42no—
180C2×D3—88no—
181C3×C4—21no—
182C3×C4—42no—
183C3×C4H(5,𝔽2)11yescharacteristic
184C3×C4H(5,𝔽2)11yescharacteristic
185Dic3—22no—
186Dic3—42no—
187C23—22no—
188C23—32no—
189C23—62no—
190C23—124no—
191C23—242no—
192C23—242no—
193C23D3×D411yescharacteristic
194Q8—121no—
195D4—22no—
196D4—22no—
197D4—62no—
198D4—62no—
199D4—82no—
200D4—121no—
201D4—122no—
202D4—124no—
203D4—242no—
204D4—242no—
205D4—242no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—32no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4D3×D411yescharacteristic
214C2×C4C23×D311yescharacteristic
215C8—21no—
216C8—21no—
217C8—62no—
218C2×C3—41no—
219C2×C3—82no—
220C2×C3#53411yescharacteristic
221C2×C3#57611yescharacteristic
222C2×C3#57611yescharacteristic
223D3—42no—
224D3—82no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—62no—
232C22—62no—
233C22—62no—
234C22—122no—
235C22—122no—
236C22—122no—
237C22—248no—
238C22C2×D3×D411yescharacteristic
239C4—21no—
240C4—42no—
241C4—62no—
242C4—122no—
243C4#100411yescharacteristic
244C4#100911yescharacteristic
245C3#319911yescharacteristic, complemented, Sylow
246C2—41no—
247C2—82no—
248C2—122no—
249C2—242no—
250C2#560511yescharacteristic
251C2#575711yescharacteristic
252C2#576911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#466C2
socleC22×C3C22×D4
hypercenter#466D3
hypocenterC3#3199
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #5607C2semidirect
2. C3×#466C2semidirect
3. #5156C2semidirect
4. #5115C2semidirect
5. #5110C2semidirect
6. #5039C2semidirect
7. #4885C2semidirect
8. #4784C2semidirect
9. #4773C2semidirect
11. C2×D3×D4C22semidirect
14. C22×C3×C8C22semidirect
15. C3×#121C22semidirect
20. #930C22semidirect
21. C2×#208C22semidirect
24. #892C22semidirect
27. C3×#107C22semidirect
28. C3×#102C22semidirect
29. #832C22semidirect
30. #821C22semidirect
31. (C2×D3)⋊C8C22semidirect
32. (C2×D3)⋊C8C22semidirect
33. #818C22semidirect
34. #809C22semidirect
35. #808C22semidirect
41. #466D3semidirect
84. C22×C8C2×D3semidirect
90. #121C2×D3semidirect
95. #107C2×D3semidirect
100. #102C2×D3semidirect
245. C3#3199semidirect

Maximal subgroups

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

subgroupindex
#56072
C3×#4662
#51562
#51152
#51102
#50392
#48852
#47842
#47732
#31993