Finite Group Databasebeta

← #83448

Subgroups — #83448

Order 384.

396 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 117 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×#541C211yesmaximal, characteristic, complemented
2#5525C211yesmaximal, characteristic
3#5086C211yesmaximal, characteristic, complemented
4#5085C211yesmaximal, characteristic, complemented
5#5068C211yesmaximal, characteristic, complemented
6#5067C211yesmaximal, characteristic, complemented
7#5047C212yesmaximal, complemented
8#5046C212yesmaximal, complemented
9#5028C212yesmaximal, complemented
10#5027C212yesmaximal, complemented
11#5000C211yesmaximal, characteristic, complemented
12#3222—31nomaximal, Sylow
13C2×#235C2211yescharacteristic
14C22×D12C2211yescharacteristic
15C3×C4⋊Q8C2211yescharacteristic, complemented
16C3×#126—22no—
17C3×#125—22no—
18C3×#125C2211yescharacteristic, complemented
19C3×C22⋊Q8C2212yes—
20C3×C4×Q8C2212yescomplemented
21C3×#117C2211yescharacteristic
22C2×#216C2212yescomplemented
23C2×#215C2212yes—
24C2×#215C2212yes—
25C2×#215C2212yescomplemented
26#895—22no—
27#882—22no—
28#875C2211yescharacteristic
29#875C2211yescharacteristic
30#874C2211yescharacteristic
31#874C2211yescharacteristic
32#830C2212yes—
33#810C2212yescomplemented
34#809C2212yes—
35#809C2212yes—
36#809C2212yescomplemented
37C4⋊(C3⋊C8)C2212yescomplemented
38#803C2211yescharacteristic, complemented
39C4×C3⋊C8C2211yescharacteristic, complemented
40#541D311yescharacteristic, complemented
41#535—31no—
42#497—31no—
43#492—31no—
44#490—31no—
45#488—31no—
46#464—32no—
47#460—32no—
48#452—32no—
49#450—32no—
50#433—31no—
51C23×D3—21no—
52C2×C3×Q8C2312yes—
53C22×C3×C4C2311yescharacteristic
54C2×#67—21no—
55#235—42no—
56C2×D12—21no—
57C2×D12—22no—
58C2×D12—22no—
59C2×D12—42no—
60C2×D12C2311yescharacteristic
61C2×D12C2311yescharacteristic
62C2×D12C2311yescharacteristic
63C2×C4×D3—21no—
64C2×Dic6C2311yescharacteristic
65C3×C4⋊C4—22no—
66C3×C4⋊C4—22no—
67C3×C4⋊C4—22no—
68C3×C4⋊C4—22no—
69C3×C4⋊C4D412yes—
70C3×C4⋊C4C2312yes—
71C3×C22⋊C4—22no—
72C3×C22⋊C4D412yes—
73C3×C42—22no—
74C3×C42C2311yescharacteristic
75C3×C42C2311yescharacteristic
76#216—42no—
77#215—42no—
78#215—42no—
79#215—42no—
80(C2×D3)⋊C4—22no—
81(C2×D3)⋊C4—22no—
82C2×C3⋊C8C2312yes—
83C2×C3⋊C8C2312yes—
84C2×#41—31no—
85C22×D4—31no—
86C2×Q16—32no—
87C2×SD16—32no—
88C2×SD16—32no—
89C2×SD16—32no—
90C4⋊Q8C2×D311yescharacteristic, complemented
91#127—31no—
92#127—31no—
93#126—22no—
94#125—22no—
95#125C2×D311yescharacteristic, complemented
96#124—31no—
97#124—31no—
98C22⋊Q8C2×D312yes—
99#121—62no—
100#120—62no—
101C4×Q8C2×D312yescomplemented
102#117C2×D311yescharacteristic
103C4⋊C8—32no—
104#103—32no—
105#102—32no—
106#102—32no—
107#102—32no—
108C22⋊C8—32no—
109#97—31no—
110C4×C8—31no—
111C23×C3C2×D411yescharacteristic
112C22×D3—21no—
113C22×D3—21no—
114C22×D3—21no—
115C22×D3—42no—
116C22×D3—82no—
117C3×Q8—22no—
118C3×Q8—22no—
119C2×C3×C4—22no—
120C2×C3×C4—22no—
121C2×C3×C4—22no—
122C2×C3×C4C2×D412yes—
123C2×C3×C4C2×D412yes—
124C2×C3×C4C2×D411yescharacteristic
125C2×C3×C4C2411yescharacteristic
126#67—42no—
127C2×Dic3—21no—
128D12—22no—
129D12—22no—
130D12—22no—
131D12—42no—
132D12—82no—
133D12—82no—
134C4×D3—42no—
135Dic6—22no—
136C3⋊C8—22no—
137C3⋊C8—22no—
138C24—61no—
139#41—122no—
140C2×Q8—31no—
141C2×Q8C22×D312yes—
142C2×D4—31no—
143C2×D4—31no—
144C2×D4—31no—
145C2×D4—61no—
146C2×D4—61no—
147C2×D4—62no—
148C2×D4—62no—
149C2×D4—122no—
150C22×C4—61no—
151C22×C4C22×D311yescharacteristic
152Q16—122no—
153SD16—122no—
154SD16—122no—
155SD16—122no—
156C2×C8—32no—
157C2×C8—32no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4#6712yes—
163C4⋊C4C22×D312yes—
164C22⋊C4—22no—
165C22⋊C4—62no—
166C22⋊C4—62no—
167C22⋊C4#6712yes—
168C42—22no—
169C42C22×D311yescharacteristic
170C42C22×D311yescharacteristic
171C22×C3—21no—
172C22×C3—22no—
173C22×C3C22×D411yescharacteristic
174C2×D3—41no—
175C2×D3—41no—
176C2×D3—41no—
177C2×D3—42no—
178C2×D3—42no—
179C2×D3—42no—
180C2×D3—84no—
181C3×C4—21no—
182C3×C4—22no—
183C3×C4—22no—
184C3×C4—42no—
185C3×C4—42no—
186C3×C4H(5,𝔽2)12yes—
187Dic3—41no—
188C23—61no—
189C23—61no—
190C23—61no—
191C23—122no—
192C23—242no—
193C23C2×#6711yescharacteristic
194Q8—22no—
195Q8—22no—
196Q8—62no—
197D4—62no—
198D4—62no—
199D4—62no—
200D4—122no—
201D4—122no—
202D4—242no—
203D4—242no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—61no—
208C2×C4—122no—
209C2×C4C2×#6712yes—
210C2×C4C2×#6712yes—
211C2×C4C2×#6711yescharacteristic
212C2×C4C23×D311yescharacteristic
213C8—62no—
214C8—62no—
215C2×C3—41no—
216C2×C3#53511yescharacteristic
217C2×C3#57712yes—
218D3—81no—
219D3—81no—
220D3—81no—
221C22—21no—
222C22—22no—
223C22—121no—
224C22—121no—
225C22—121no—
226C22—122no—
227C22—122no—
228C22—122no—
229C22—244no—
230C22C22×#6711yescharacteristic
231C4—21no—
232C4—22no—
233C4—22no—
234C4—42no—
235C4—42no—
236C4—121no—
237C4#100912yes—
238C3#322211yescharacteristic, complemented, Sylow
239C2—41no—
240C2—241no—
241C2—241no—
242C2—241no—
243C2#583011yescharacteristic
244C2#583712yes—
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#541C2
socleC22×C3C22×D4
prime radical#5525C2
hypercenter#541D3
hypocenterC3#3222
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
1. C3×#541C2semidirect
3. #5086C2semidirect
4. #5085C2semidirect
5. #5068C2semidirect
6. #5067C2semidirect
7. #5047C2semidirect
8. #5046C2semidirect
9. #5028C2semidirect
10. #5027C2semidirect
11. #5000C2semidirect
15. C3×C4⋊Q8C22semidirect
18. C3×#125C22semidirect
20. C3×C4×Q8C22semidirect
22. C2×#216C22semidirect
25. C2×#215C22semidirect
33. #810C22semidirect
36. #809C22semidirect
37. C4⋊(C3⋊C8)C22semidirect
38. #803C22semidirect
39. C4×C3⋊C8C22semidirect
40. #541D3semidirect
90. C4⋊Q8C2×D3semidirect
95. #125C2×D3semidirect
101. C4×Q8C2×D3semidirect
238. C3#3222semidirect

Maximal subgroups

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

subgroupindex
C3×#5412
#55252
#50862
#50852
#50682
#50672
#50472
#50462
#50282
#50272
#50002
#32223