Finite Group Databasebeta

← #79612

Subgroups — #79612

Order 384.

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

normal subgroups only

This group has 460 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#5563C211yesmaximal, characteristic, complemented
2D4×D12C211yesmaximal, characteristic, complemented
3C3×C4×D8C211yesmaximal, characteristic, complemented
4#5156C212yesmaximal, complemented
5#5015C211yesmaximal, characteristic, complemented
6#5008C211yesmaximal, characteristic, complemented
7#4885C212yesmaximal, complemented
8#4782C212yesmaximal, complemented
9#4770C212yesmaximal, complemented
10#4694C211yesmaximal, characteristic, complemented
11#4688C211yesmaximal, characteristic, complemented
12#3325—31nomaximal, Sylow
13C2×D3×D4C2212yes—
14C22×D12—21no—
15C2×C3×D8C2211yescharacteristic
16C3×C4×D4C2211yescharacteristic
17C3×C4×D4C2211yescharacteristic, complemented
18#938C2212yescomplemented
19C2×#213C2212yescomplemented
20#930—21no—
21#930—21no—
22#926—21no—
23C2×ΓL(1,𝔽25)C2212yescomplemented
24#895C2212yes—
25#892C2212yescomplemented
26#885—22no—
27#882—22no—
28#874C2211yescharacteristic
29C4×D12C2211yescharacteristic
30#869C2211yescharacteristic, complemented
31C3×#107C2211yescharacteristic, complemented
32C3×#102C2211yescharacteristic
33C3×#102C2211yescharacteristic, complemented
34C3×C4×C8C2211yescharacteristic, complemented
35#832C2212yes—
36#821C2212yescomplemented
37#821C2212yescomplemented
38(C2×D3)⋊C8C2212yescomplemented
39#809C2212yescomplemented
40#808C2212yescomplemented
41C4⋊(C3⋊C8)C2211yescharacteristic, complemented
42#555—31no—
43D42—31no—
44#486—31no—
45#482—32no—
46#476—31no—
47#469—32no—
48#459—31no—
49#447—32no—
50C4×D8D311yescharacteristic, complemented
51#436—31no—
52#418—32no—
53C23×D3—22no—
54C2×C3×D4C2311yescharacteristic
55C2×C3×D4C2311yescharacteristic
56C22×C3×C4—21no—
57C22×C3×C4—21no—
58C2×#67—22no—
59C2×#67—22no—
60D3×D4—24no—
61D3×D4—42no—
62C2×D12—22no—
63C2×D12—21no—
64C2×D12—21no—
65C2×D12—44no—
66C2×D12D411yescharacteristic
67C2×D12C2312yes—
68C2×C4×D3C2312yes—
69C2×Dic6—21no—
70C3×D8—22no—
71C2×C3×C8C2311yescharacteristic
72C2×C3×C8C2311yescharacteristic
73C3×C4⋊C4C2311yescharacteristic
74C3×C4⋊C4C2311yescharacteristic
75C3×C22⋊C4—21no—
76C3×C22⋊C4—21no—
77C3×C42C2311yescharacteristic
78#217—22no—
79#213—42no—
80(C2×D3)⋊C4—22no—
81(C2×D3)⋊C4—22no—
82(C2×D3)⋊C4D412yes—
83C4⋊Dic3—21no—
84C4⋊Dic3D411yescharacteristic
85C4⋊Dic3C2312yes—
86C4×Dic3—22no—
87C2×C3⋊C8C2312yes—
88ΓL(1,𝔽25)—42no—
89C22×D4—32no—
90C22×D4—61no—
91C2×D8—32no—
92C2×D8C2×D311yescharacteristic
93C2×M16—32no—
94C4⋊Q8—31no—
95#127—31no—
96#124—62no—
97#121—32no—
98#121—32no—
99#121—61no—
100#121—61no—
101#120—62no—
102C4×D4—31no—
103C4×D4C2×D311yescharacteristic
104C4×D4C2×D311yescharacteristic, complemented
105#117—32no—
106#117—61no—
107#107C2×D311yescharacteristic, complemented
108#106—32no—
109C4⋊C8—31no—
110#102—32no—
111#102—32no—
112#102—32no—
113#102—32no—
114#102C2×D311yescharacteristic
115#102C2×D311yescharacteristic, complemented
116C22⋊C8—32no—
117C4×C8C2×D311yescharacteristic, complemented
118C23×C3—21no—
119C23×C3—21no—
120C22×D3—22no—
121C22×D3—44no—
122C22×D3—44no—
123C22×D3—82no—
124C22×D3C2×D412yes—
125C3×D4—21no—
126C3×D4—21no—
127C3×D4—21no—
128C3×D4#4112yes—
129C2×C3×C4—21no—
130C2×C3×C4—21no—
131C2×C3×C4—42no—
132C2×C3×C4—42no—
133C2×C3×C4C2×D411yescharacteristic
134C2×C3×C4C2×D411yescharacteristic
135C2×C3×C4C2411yescharacteristic
136#67—44no—
137#67—82no—
138C2×Dic3—22no—
139C2×Dic3C2×D412yes—
140D12—22no—
141D12—22no—
142D12—22no—
143D12—44no—
144D12—42no—
145D12—81no—
146C4×D3—22no—
147C4×D3—22no—
148Dic6—81no—
149C3×C8—21no—
150C3×C8—21no—
151C3⋊C8—22no—
152C24—62no—
153C2×Q8—61no—
154C2×D4—31no—
155C2×D4—32no—
156C2×D4—62no—
157C2×D4—62no—
158C2×D4—64no—
159C2×D4—62no—
160C2×D4—61no—
161C2×D4—61no—
162C2×D4—124no—
163C2×D4—122no—
164C2×D4C22×D311yescharacteristic
165C2×D4C22×D311yescharacteristic
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—32no—
169D8—22no—
170D8—122no—
171M16—122no—
172C2×C8—32no—
173C2×C8C22×D311yescharacteristic
174C2×C8C22×D311yescharacteristic
175C4⋊C4—31no—
176C4⋊C4—32no—
177C4⋊C4—61no—
178C4⋊C4C22×D311yescharacteristic
179C4⋊C4C22×D311yescharacteristic
180C22⋊C4—21no—
181C22⋊C4—21no—
182C22⋊C4—32no—
183C22⋊C4—62no—
184C22⋊C4—62no—
185C22⋊C4—62no—
186C42—62no—
187C42C22×D311yescharacteristic
188C22×C3—22no—
189C22×C3—41no—
190C22×C3—41no—
191C22×C3—41no—
192C22×C3—41no—
193C22×C3—41no—
194C22×C3C22×D411yescharacteristic
195C2×D3—22no—
196C2×D3—22no—
197C2×D3—22no—
198C2×D3—42no—
199C2×D3—42no—
200C2×D3—42no—
201C2×D3—88no—
202C3×C4—21no—
203C3×C4—41no—
204C3×C4—41no—
205C3×C4AGL(1,ℤ/8)12yes—
206C3×C4C2×#4111yescharacteristic
207C3×C4H(5,𝔽2)11yescharacteristic
208Dic3—22no—
209Dic3—42no—
210C23—21no—
211C23—21no—
212C23—32no—
213C23—62no—
214C23—124no—
215C23—124no—
216C23—242no—
217Q8—241no—
218D4—21no—
219D4—21no—
220D4—21no—
221D4—62no—
222D4—62no—
223D4—62no—
224D4—124no—
225D4—124no—
226D4—122no—
227D4—241no—
228D4—242no—
229D4#23512yes—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—32no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—62no—
236C2×C4—62no—
237C2×C4—62no—
238C2×C4D3×D411yescharacteristic
239C2×C4D3×D411yescharacteristic
240C2×C4C23×D311yescharacteristic
241C8—21no—
242C8—21no—
243C8—62no—
244C2×C3—42no—
245C2×C3—81no—
246C2×C3#54611yescharacteristic
247C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
248C2×C3#57611yescharacteristic
249D3—42no—
250D3—82no—
251C22—22no—
252C22—41no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—41no—
257C22—62no—
258C22—62no—
259C22—62no—
260C22—122no—
261C22—122no—
262C22—122no—
263C22—248no—
264C22C2×D3×D411yescharacteristic
265C4—21no—
266C4—41no—
267C4—41no—
268C4—62no—
269C4—122no—
270C4#91112yes—
271C4C2×#23511yescharacteristic
272C4#100911yescharacteristic
273C3#332511yescharacteristic, complemented, Sylow
274C2—42no—
275C2—81no—
276C2—122no—
277C2—242no—
278C2#555011yescharacteristic
279C2C2×#91111yescharacteristic
280C2#576911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×C4×D8C2
socleC22×C3C22×D4
hypercenterC4×D8D3
hypocenterC3#3325
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #5563C2semidirect
2. D4×D12C2semidirect
3. C3×C4×D8C2semidirect
4. #5156C2semidirect
5. #5015C2semidirect
6. #5008C2semidirect
7. #4885C2semidirect
8. #4782C2semidirect
9. #4770C2semidirect
10. #4694C2semidirect
11. #4688C2semidirect
17. C3×C4×D4C22semidirect
18. #938C22semidirect
19. C2×#213C22semidirect
23. C2×ΓL(1,𝔽25)C22semidirect
25. #892C22semidirect
30. #869C22semidirect
31. C3×#107C22semidirect
33. C3×#102C22semidirect
34. C3×C4×C8C22semidirect
36. #821C22semidirect
37. #821C22semidirect
38. (C2×D3)⋊C8C22semidirect
39. #809C22semidirect
40. #808C22semidirect
41. C4⋊(C3⋊C8)C22semidirect
50. C4×D8D3semidirect
104. C4×D4C2×D3semidirect
107. #107C2×D3semidirect
115. #102C2×D3semidirect
117. C4×C8C2×D3semidirect
273. C3#3325semidirect

Maximal subgroups

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

subgroupindex
#55632
D4×D122
C3×C4×D82
#51562
#50152
#50082
#48852
#47822
#47702
#46942
#46882
#33253