Finite Group Databasebeta

← #79390

Subgroups — #79390

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
1#5535C211yesmaximal, characteristic, complemented
2#5525C211yesmaximal, characteristic, complemented
3C3×#433C211yesmaximal, characteristic, complemented
4#4844C212yesmaximal, complemented
5#4843C212yesmaximal, complemented
6#4734C212yesmaximal, complemented
7#4732C212yesmaximal, complemented
8#4713C211yesmaximal, characteristic, complemented
9#4712C211yesmaximal, characteristic, complemented
10#4697C211yesmaximal, characteristic, complemented
11#4695C211yesmaximal, characteristic, complemented
12#3220—31nomaximal, Sylow
13C2×#235C2211yescharacteristic
14C22×D12C2211yescharacteristic
15C3×#117C2211yescharacteristic
16#930C2212yes—
17C2×D24C2212yes—
18C2×D24C2212yes—
19C2×D24C2212yescomplemented
20C2×#204C2212yescomplemented
21#895—22no—
22#894—22no—
23#884—22no—
24#882—22no—
25#875C2211yescharacteristic
26#875C2211yescharacteristic
27#874C2211yescharacteristic
28#874C2211yescharacteristic
29#874C2211yescharacteristic, complemented
30C4×D12C2212yescomplemented
31#870C2211yescharacteristic, complemented
32C3×C4⋊C8C2212yescomplemented
33C3×C22⋊C8C2212yes—
34C3×#97C2211yescharacteristic, complemented
35C3×C4×C8C2211yescharacteristic, complemented
36#821C2212yes—
37#821C2212yes—
38#821C2212yescomplemented
39#816C2212yescomplemented
40#540—31no—
41#535—31no—
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#433D311yescharacteristic, complemented
51C23×D3—21no—
52C22×C3×C4C2311yescharacteristic
53C2×#67—21no—
54C2×#67—22no—
55#235—42no—
56C2×D12—21no—
57C2×D12—22no—
58C2×D12—22no—
59C2×D12—22no—
60C2×D12—42no—
61C2×D12C2311yescharacteristic
62C2×D12C2311yescharacteristic
63C2×D12C2311yescharacteristic
64C2×D12C2312yes—
65C2×C4×D3—21no—
66C2×C4×D3—22no—
67C2×Dic6C2311yescharacteristic
68C2×C3×C8C2312yes—
69C2×C3×C8C2312yes—
70C3×C4⋊C4D412yes—
71C3×C22⋊C4D412yes—
72C3×C42C2311yescharacteristic
73C3×C42C2311yescharacteristic
74(C2×D3)⋊C4—22no—
75(C2×D3)⋊C4—22no—
76(C2×D3)⋊C4—22no—
77(C2×D3)⋊C4—22no—
78C4⋊Dic3C2312yes—
79Dic3⋊C4—22no—
80D24—42no—
81D24—42no—
82D24—42no—
83#204—42no—
84C2×#41—31no—
85C22×D4—31no—
86C2×SD16—32no—
87C2×D8—32no—
88C2×D8—32no—
89C2×D8—32no—
90#127—31no—
91#127—31no—
92#127—31no—
93#125—31no—
94#124—31no—
95#124—31no—
96#123—62no—
97#121—32no—
98#121—62no—
99#121—62no—
100#120—62no—
101C4×D4—32no—
102#117C2×D311yescharacteristic
103C4⋊C8C2×D312yescomplemented
104#103—32no—
105#102—32no—
106#102—32no—
107#102—32no—
108C22⋊C8C2×D312yes—
109#97C2×D311yescharacteristic, complemented
110C4×C8C2×D311yescharacteristic, complemented
111C23×C3C2×D411yescharacteristic
112C22×D3—22no—
113C22×D3—21no—
114C22×D3—21no—
115C22×D3—21no—
116C22×D3—42no—
117C22×D3—82no—
118C2×C3×C4—22no—
119C2×C3×C4C2×D412yes—
120C2×C3×C4C2×D412yes—
121C2×C3×C4C2×D411yescharacteristic
122C2×C3×C4C2411yescharacteristic
123#67—42no—
124#67—82no—
125C2×Dic3—22no—
126C2×Dic3—21no—
127D12—22no—
128D12—22no—
129D12—22no—
130D12—22no—
131D12—22no—
132D12—42no—
133D12—82no—
134D12—82no—
135D12—82no—
136C4×D3—42no—
137C4×D3—82no—
138Dic6—22no—
139C3×C8—22no—
140C3×C8—22no—
141C24—61no—
142#41—122no—
143C2×Q8—31no—
144C2×D4—31no—
145C2×D4—31no—
146C2×D4—31no—
147C2×D4—32no—
148C2×D4—61no—
149C2×D4—62no—
150C2×D4—61no—
151C2×D4—62no—
152C2×D4—62no—
153C2×D4—62no—
154C2×D4—122no—
155C22×C4—61no—
156C22×C4—62no—
157C22×C4C22×D311yescharacteristic
158SD16—122no—
159D8—122no—
160D8—122no—
161D8—122no—
162C2×C8C22×D312yes—
163C2×C8C22×D312yes—
164C4⋊C4—32no—
165C4⋊C4—62no—
166C4⋊C4D1212yes—
167C22⋊C4—62no—
168C22⋊C4—62no—
169C22⋊C4—62no—
170C22⋊C4—62no—
171C22⋊C4D1212yes—
172C42C22×D311yescharacteristic
173C42C22×D311yescharacteristic
174C22×C3—21no—
175C22×C3—22no—
176C22×C3C22×D411yescharacteristic
177C2×D3—41no—
178C2×D3—41no—
179C2×D3—41no—
180C2×D3—42no—
181C2×D3—42no—
182C2×D3—42no—
183C2×D3—42no—
184C2×D3—42no—
185C2×D3—42no—
186C2×D3—84no—
187C3×C4—21no—
188C3×C4—22no—
189C3×C4—22no—
190C3×C4H(5,𝔽2)12yes—
191Dic3—41no—
192Dic3—42no—
193C23—62no—
194C23—61no—
195C23—61no—
196C23—61no—
197C23—122no—
198C23—242no—
199C23C2×D1211yescharacteristic
200Q8—62no—
201D4—62no—
202D4—62no—
203D4—62no—
204D4—62no—
205D4—62no—
206D4—122no—
207D4—122no—
208D4—242no—
209D4—242no—
210D4—242no—
211D4—242no—
212C2×C4—22no—
213C2×C4—61no—
214C2×C4—62no—
215C2×C4—122no—
216C2×C4—242no—
217C2×C4C2×D1212yes—
218C2×C4C2×D1212yes—
219C2×C4C2×D1211yescharacteristic
220C2×C4C23×D311yescharacteristic
221C8—22no—
222C8—22no—
223C2×C3—41no—
224C2×C3#53511yescharacteristic
225C2×C3#57612yes—
226D3—81no—
227D3—82no—
228D3—81no—
229D3—81no—
230C22—22no—
231C22—21no—
232C22—122no—
233C22—122no—
234C22—122no—
235C22—122no—
236C22—122no—
237C22—121no—
238C22—121no—
239C22—121no—
240C22—122no—
241C22—244no—
242C22C22×D1211yescharacteristic
243C4—22no—
244C4—22no—
245C4—21no—
246C4—122no—
247C4—121no—
248C4#100912yes—
249C3#322011yescharacteristic, complemented, Sylow
250C2—41no—
251C2—241no—
252C2—241no—
253C2—242no—
254C2—241no—
255C2#552511yescharacteristic
256C2#575212yes—
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#433C2
socleC22×C3C22×D4
hypercenter#433D3
hypocenterC3#3220
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
1. #5535C2semidirect
2. #5525C2semidirect
3. C3×#433C2semidirect
4. #4844C2semidirect
5. #4843C2semidirect
6. #4734C2semidirect
7. #4732C2semidirect
8. #4713C2semidirect
9. #4712C2semidirect
10. #4697C2semidirect
11. #4695C2semidirect
19. C2×D24C22semidirect
20. C2×#204C22semidirect
29. #874C22semidirect
30. C4×D12C22semidirect
31. #870C22semidirect
32. C3×C4⋊C8C22semidirect
34. C3×#97C22semidirect
35. C3×C4×C8C22semidirect
38. #821C22semidirect
39. #816C22semidirect
50. #433D3semidirect
103. C4⋊C8C2×D3semidirect
109. #97C2×D3semidirect
110. C4×C8C2×D3semidirect
249. C3#3220semidirect

Maximal subgroups

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

subgroupindex
#55352
#55252
C3×#4332
#48442
#48432
#47342
#47322
#47132
#47122
#46972
#46952
#32203