Finite Group Databasebeta

← #83388

Subgroups — #83388

Order 384.

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

normal subgroups only

This group has 420 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×#535C211yesmaximal, characteristic, complemented
2#5535C211yesmaximal, characteristic, complemented
3#5238C212yesmaximal, complemented
4#5217C212yesmaximal, complemented
5#5073C211yesmaximal, characteristic, complemented
6#5072C212yesmaximal, complemented
7#5071C211yesmaximal, characteristic, complemented
8#5058C211yesmaximal, characteristic, complemented
9#5057C212yesmaximal, complemented
10#5056C211yesmaximal, characteristic, complemented
11#5000C211yesmaximal, characteristic, complemented
12#3220—31nomaximal, Sylow
13C2×C3×#41C2211yescharacteristic
14C22×C3×D4C2211yescharacteristic
15C3×#127C2211yescharacteristic
16C3×#127C2211yescharacteristic
17C3×#124C2211yescharacteristic
18C3×#124C2211yescharacteristic
19C3×#121—22no—
20C3×#120—22no—
21C3×#117C2211yescharacteristic
22C2×#215C2212yescomplemented
23C2×#213C2212yes—
24C2×#213C2212yes—
25C2×#213C2212yescomplemented
26#930C2212yes—
27#894—22no—
28#884—22no—
29#874C2211yescharacteristic, complemented
30C4×D12C2212yescomplemented
31#870C2211yescharacteristic, complemented
32#835C2212yescomplemented
33#832C2212yes—
34#832C2212yes—
35#832C2212yescomplemented
36#830C2212yes—
37C4⋊(C3⋊C8)C2212yescomplemented
38#803C2211yescharacteristic, complemented
39C4×C3⋊C8C2211yescharacteristic, complemented
40#540—31no—
41#535D311yescharacteristic, complemented
42#496—31no—
43#493—31no—
44#490—31no—
45#488—31no—
46#463—32no—
47#459—32no—
48#449—32no—
49#447—32no—
50#433—31no—
51C24×C3—21no—
52C3×#41—42no—
53C2×C3×Q8C2311yescharacteristic
54C2×C3×D4—21no—
55C2×C3×D4—21no—
56C2×C3×D4—22no—
57C2×C3×D4—22no—
58C2×C3×D4—42no—
59C2×C3×D4C2311yescharacteristic
60C2×C3×D4C2311yescharacteristic
61C2×C3×D4C2311yescharacteristic
62C22×C3×C4—21no—
63C22×C3×C4C2311yescharacteristic
64C2×#67—22no—
65C2×D12—22no—
66C2×D12C2312yes—
67C2×C4×D3—22no—
68C3×C4⋊C4D412yes—
69C3×C22⋊C4—22no—
70C3×C22⋊C4—22no—
71C3×C22⋊C4D412yes—
72C3×C42C2311yescharacteristic
73C3×C42C2311yescharacteristic
74#215—42no—
75#213—42no—
76#213—42no—
77#213—42no—
78(C2×D3)⋊C4—22no—
79(C2×D3)⋊C4—22no—
80C4⋊Dic3C2312yes—
81Dic3⋊C4—22no—
82C2×C3⋊C8C2312yes—
83C2×C3⋊C8C2312yes—
84C2×#41C2×D311yescharacteristic
85C22×D4C2×D311yescharacteristic
86C2×SD16—32no—
87C2×D8—32no—
88C2×D8—32no—
89C2×D8—32no—
90#127—31no—
91#127C2×D311yescharacteristic
92#127C2×D311yescharacteristic
93#125—31no—
94#124C2×D311yescharacteristic
95#124C2×D311yescharacteristic
96#123—62no—
97#121—22no—
98#121—32no—
99#121—62no—
100#120—22no—
101C4×D4—32no—
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×C3—21no—
112C23×C3—21no—
113C23×C3—21no—
114C23×C3—42no—
115C23×C3—82no—
116C23×C3C2×D411yescharacteristic
117C22×D3—22no—
118C3×Q8—22no—
119C3×D4—22no—
120C3×D4—22no—
121C3×D4—22no—
122C3×D4—42no—
123C3×D4—42no—
124C3×D4—82no—
125C3×D4—82no—
126C2×C3×C4—22no—
127C2×C3×C4—21no—
128C2×C3×C4—42no—
129C2×C3×C4C2×D412yes—
130C2×C3×C4C2×D411yescharacteristic
131C2×C3×C4C2×D412yes—
132C2×C3×C4C2411yescharacteristic
133#67—82no—
134C2×Dic3—22no—
135D12—22no—
136D12—22no—
137D12—82no—
138C4×D3—82no—
139C3⋊C8—22no—
140C3⋊C8—22no—
141C24—21no—
142#41—42no—
143C2×Q8C22×D311yescharacteristic
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—22no—
147C2×D4—22no—
148C2×D4—32no—
149C2×D4—42no—
150C2×D4—62no—
151C2×D4—62no—
152C2×D4C22×D311yescharacteristic
153C2×D4C22×D311yescharacteristic
154C2×D4C22×D311yescharacteristic
155C22×C4—21no—
156C22×C4—62no—
157C22×C4C22×D311yescharacteristic
158SD16—122no—
159D8—122no—
160D8—122no—
161D8—122no—
162C2×C8—32no—
163C2×C8—32no—
164C4⋊C4—32no—
165C4⋊C4—62no—
166C4⋊C4#6712yes—
167C22⋊C4—22no—
168C22⋊C4—22no—
169C22⋊C4—62no—
170C22⋊C4—62no—
171C22⋊C4#6712yes—
172C42C22×D311yescharacteristic
173C42C22×D311yescharacteristic
174C22×C3—21no—
175C22×C3—22no—
176C22×C3—41no—
177C22×C3—41no—
178C22×C3—41no—
179C22×C3—42no—
180C22×C3—42no—
181C22×C3—42no—
182C22×C3—84no—
183C22×C3C22×D411yescharacteristic
184C2×D3—42no—
185C2×D3—42no—
186C2×D3—42no—
187C3×C4—21no—
188C3×C4—22no—
189C3×C4—22no—
190C3×C4—41no—
191C3×C4H(5,𝔽2)12yes—
192Dic3—42no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23—42no—
197C23—62no—
198C23—82no—
199C23C2×#6711yescharacteristic
200Q8—22no—
201D4—22no—
202D4—22no—
203D4—22no—
204D4—42no—
205D4—42no—
206D4—62no—
207D4—62no—
208D4—82no—
209D4—82no—
210D4—242no—
211D4—242no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—42no—
215C2×C4—62no—
216C2×C4—242no—
217C2×C4C2×#6712yes—
218C2×C4C2×#6712yes—
219C2×C4C2×#6711yescharacteristic
220C2×C4C23×D311yescharacteristic
221C8—62no—
222C8—62no—
223C2×C3—41no—
224C2×C3—81no—
225C2×C3—81no—
226C2×C3—81no—
227C2×C3#53511yescharacteristic
228C2×C3#57612yes—
229D3—82no—
230C22—21no—
231C22—22no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—42no—
236C22—42no—
237C22—42no—
238C22—84no—
239C22—122no—
240C22—122no—
241C22—122no—
242C22C22×#6711yescharacteristic
243C4—21no—
244C4—22no—
245C4—22no—
246C4—41no—
247C4—122no—
248C4#100412yes—
249C3#322011yescharacteristic, complemented, Sylow
250C2—41no—
251C2—81no—
252C2—81no—
253C2—81no—
254C2—242no—
255C2#580511yescharacteristic
256C2#583512yes—
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#535C2
socleC22×C3C22×D4
hypercenter#535D3
hypocenterC3#3220
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C3×#535C2semidirect
2. #5535C2semidirect
3. #5238C2semidirect
4. #5217C2semidirect
5. #5073C2semidirect
6. #5072C2semidirect
7. #5071C2semidirect
8. #5058C2semidirect
9. #5057C2semidirect
10. #5056C2semidirect
11. #5000C2semidirect
22. C2×#215C22semidirect
25. C2×#213C22semidirect
29. #874C22semidirect
30. C4×D12C22semidirect
31. #870C22semidirect
32. #835C22semidirect
35. #832C22semidirect
37. C4⋊(C3⋊C8)C22semidirect
38. #803C22semidirect
39. C4×C3⋊C8C22semidirect
41. #535D3semidirect
249. C3#3220semidirect

Maximal subgroups

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

subgroupindex
C3×#5352
#55352
#52382
#52172
#50732
#50722
#50712
#50582
#50572
#50562
#50002
#32203