Finite Group Databasebeta

← #79556

Subgroups — #79556

Order 384.

460 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 123 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×#436C211yesmaximal, characteristic, complemented
4#5135C212yesmaximal, complemented
5#5131C212yesmaximal, complemented
6#4843C211yesmaximal, characteristic, complemented
7#4842C211yesmaximal, characteristic, complemented
8#4737C212yesmaximal, complemented
9#4732C212yesmaximal, complemented
10#4694C211yesmaximal, characteristic, complemented
11C4×D24C211yesmaximal, characteristic, complemented
12#3325—31nomaximal, Sylow
13C2×D3×D4—21no—
14C22×D12C2212yes—
15C2×C3×M16C2212yescomplemented
16C3×C4×D4C2211yescharacteristic
17#938—21no—
18#930C2212yes—
19#930C2212yescomplemented
20#926C2212yescomplemented
21C2×D24C2211yescharacteristic
22C2×D24C2212yescomplemented
23#895—21no—
24#892—21no—
25#885—22no—
26#882—22no—
27#874C2211yescharacteristic
28C4×D12C2211yescharacteristic
29C4×D12C2211yescharacteristic, complemented
30#869C2211yescharacteristic, complemented
31C3×C4⋊C8C2211yescharacteristic, complemented
32C3×C22⋊C8C2212yescomplemented
33C3×C4×C8C2211yescharacteristic, complemented
34#821C2211yescharacteristic
35#821C2212yes—
36#821C2212yescomplemented
37#821C2212yescomplemented
38#821C2211yescharacteristic, complemented
39#821C2212yescomplemented
40#818C2211yescharacteristic, complemented
41#817C2212yescomplemented
42#555—31no—
43D42—31no—
44#486—31no—
45#482—32no—
46#476—31no—
47#469—32no—
48#459—31no—
49#447—32no—
50C4×D8—31no—
51#436D311yescharacteristic, complemented
52#418—32no—
53C23×D3—22no—
54C2×C3×D4D411yescharacteristic
55C22×C3×C4C2312yes—
56C2×#67—22no—
57C2×#67—22no—
58D3×D4—44no—
59C2×D12—24no—
60C2×D12—21no—
61C2×D12—22no—
62C2×D12—21no—
63C2×D12—42no—
64C2×D12C2312yes—
65C2×D12C2311yescharacteristic
66C2×D12C2311yescharacteristic
67C2×C4×D3—21no—
68C2×C4×D3—21no—
69C2×Dic6—21no—
70C3×M16—42no—
71C2×C3×C8C2312yes—
72C2×C3×C8C2311yescharacteristic
73C2×C3×C8C2311yescharacteristic
74C3×C4⋊C4D411yescharacteristic
75C3×C22⋊C4D412yes—
76C3×C42C2311yescharacteristic
77#217—22no—
78(C2×D3)⋊C4—21no—
79(C2×D3)⋊C4—21no—
80(C2×D3)⋊C4—22no—
81(C2×D3)⋊C4—22no—
82C4⋊Dic3—21no—
83C4⋊Dic3C2311yescharacteristic
84C4⋊Dic3C2312yes—
85C4⋊Dic3C2311yescharacteristic
86C4×Dic3—22no—
87D24—22no—
88D24—42no—
89C22×D4—32no—
90C22×D4—61no—
91C2×D8—31no—
92C2×D8—32no—
93C2×M16C2×D312yescomplemented
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×D4—31no—
104C4×D4C2×D311yescharacteristic
105#117—32no—
106#117—61no—
107#107—31no—
108#106—32no—
109C4⋊C8C2×D311yescharacteristic, complemented
110#102—31no—
111#102—32no—
112#102—32no—
113#102—32no—
114#102—31no—
115#102—32no—
116C22⋊C8C2×D312yescomplemented
117C4×C8C2×D311yescharacteristic, complemented
118C23×C3C2×D412yes—
119C22×D3—21no—
120C22×D3—21no—
121C22×D3—22no—
122C22×D3—44no—
123C22×D3—44no—
124C22×D3—82no—
125C3×D4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4C2×D412yes—
129C2×C3×C4C2×D411yescharacteristic
130C2×C3×C4C2×D411yescharacteristic
131C2×C3×C4C2411yescharacteristic
132#67—44no—
133#67—82no—
134C2×Dic3—21no—
135C2×Dic3—22no—
136C2×Dic3—21no—
137D12—21no—
138D12—22no—
139D12—21no—
140D12—21no—
141D12—22no—
142D12—44no—
143D12—42no—
144D12—81no—
145D12#4112yes—
146C4×D3—42no—
147C4×D3—42no—
148Dic6—81no—
149C3×C8—21no—
150C3×C8—22no—
151C3×C8—21no—
152C24—62no—
153C2×Q8—61no—
154C2×D4—31no—
155C2×D4—32no—
156C2×D4—31no—
157C2×D4—64no—
158C2×D4—62no—
159C2×D4—62no—
160C2×D4—62no—
161C2×D4—61no—
162C2×D4—61no—
163C2×D4—122no—
164C2×D4—124no—
165C2×D4D1211yescharacteristic
166C22×C4—61no—
167C22×C4—61no—
168C22×C4C22×D312yes—
169D8—62no—
170D8—122no—
171M16—42no—
172C2×C8C22×D311yescharacteristic
173C2×C8C22×D311yescharacteristic
174C2×C8C22×D312yes—
175C4⋊C4—32no—
176C4⋊C4—31no—
177C4⋊C4—31no—
178C4⋊C4—61no—
179C4⋊C4D1211yescharacteristic
180C22⋊C4—62no—
181C22⋊C4—62no—
182C22⋊C4—61no—
183C22⋊C4—61no—
184C22⋊C4—62no—
185C22⋊C4D1212yes—
186C42—62no—
187C42C22×D311yescharacteristic
188C22×C3—22no—
189C22×C3—22no—
190C22×C3—22no—
191C22×C3C22×D411yescharacteristic
192C2×D3—22no—
193C2×D3—41no—
194C2×D3—41no—
195C2×D3—42no—
196C2×D3—41no—
197C2×D3—41no—
198C2×D3—42no—
199C2×D3—42no—
200C2×D3—41no—
201C2×D3—88no—
202C3×C4—22no—
203C3×C4—21no—
204C3×C4AGL(1,ℤ/8)12yes—
205C3×C4C2×#4111yescharacteristic
206C3×C4H(5,𝔽2)11yescharacteristic
207Dic3—41no—
208Dic3—41no—
209Dic3—42no—
210C23—61no—
211C23—61no—
212C23—62no—
213C23—124no—
214C23—124no—
215C23—242no—
216C23C2×D1212yes—
217Q8—241no—
218D4—22no—
219D4—32no—
220D4—61no—
221D4—61no—
222D4—62no—
223D4—61no—
224D4—62no—
225D4—124no—
226D4—124no—
227D4—122no—
228D4—241no—
229D4—242no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—61no—
233C2×C4—62no—
234C2×C4—61no—
235C2×C4—122no—
236C2×C4—122no—
237C2×C4C2×D1212yes—
238C2×C4C2×D1211yescharacteristic
239C2×C4C2×D1211yescharacteristic
240C2×C4C23×D311yescharacteristic
241C8—21no—
242C8—22no—
243C8—21no—
244C2×C3—42no—
245C2×C3#54611yescharacteristic
246C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
247C2×C3#57611yescharacteristic
248D3—42no—
249D3—81no—
250D3—82no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—62no—
255C22—121no—
256C22—121no—
257C22—121no—
258C22—122no—
259C22—121no—
260C22—122no—
261C22—122no—
262C22—121no—
263C22—248no—
264C22C22×D1211yescharacteristic
265C4—21no—
266C4—22no—
267C4—121no—
268C4—121no—
269C4—122no—
270C4#90812yes—
271C4#100411yescharacteristic
272C4D3×#4111yescharacteristic
273C3#332511yescharacteristic, complemented, Sylow
274C2—42no—
275C2—122no—
276C2—242no—
277C2—241no—
278C2#555411yescharacteristic
279C2C2×#90811yescharacteristic
280C2#575211yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#436C2
socleC22×C3C22×D4
hypercenter#436D3
hypocenterC3#3325
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. #5563C2semidirect
2. D4×D12C2semidirect
3. C3×#436C2semidirect
4. #5135C2semidirect
5. #5131C2semidirect
6. #4843C2semidirect
7. #4842C2semidirect
8. #4737C2semidirect
9. #4732C2semidirect
10. #4694C2semidirect
11. C4×D24C2semidirect
15. C2×C3×M16C22semidirect
19. #930C22semidirect
20. #926C22semidirect
22. C2×D24C22semidirect
29. C4×D12C22semidirect
30. #869C22semidirect
31. C3×C4⋊C8C22semidirect
32. C3×C22⋊C8C22semidirect
33. C3×C4×C8C22semidirect
36. #821C22semidirect
37. #821C22semidirect
38. #821C22semidirect
39. #821C22semidirect
40. #818C22semidirect
41. #817C22semidirect
51. #436D3semidirect
93. C2×M16C2×D3semidirect
109. C4⋊C8C2×D3semidirect
116. C22⋊C8C2×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×#4362
#51352
#51312
#48432
#48422
#47372
#47322
#46942
C4×D242
#33253