Finite Group Databasebeta

← #83654

Subgroups — #83654

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
1C3×#555C211yesmaximal, characteristic, complemented
2D4×D12C211yesmaximal, characteristic, complemented
3#5088C211yesmaximal, characteristic, complemented
4#5084C211yesmaximal, characteristic, complemented
5#5039C212yesmaximal, complemented
6#5036C212yesmaximal, complemented
7#5015C211yesmaximal, characteristic, complemented
8C4×#213C211yesmaximal, characteristic, complemented
9#5012C211yesmaximal, characteristic, complemented
10#5003C212yesmaximal, complemented
11#5001C212yesmaximal, complemented
12#3325—31nomaximal, Sylow
13C2×D3×D4—21no—
14C22×D12C2212yes—
15C3×C4⋊Q8C2211yescharacteristic, complemented
16C3×#124—22no—
17C3×#121—21no—
18C3×#121C2212yescomplemented
19C3×C4×D4C2211yescharacteristic
20C3×C4×D4C2211yescharacteristic, complemented
21C3×#117—21no—
22C3×#117C2212yescomplemented
23C2×#213C2211yescharacteristic
24C2×#213C2212yescomplemented
25#930C2212yes—
26C2×#208C2212yescomplemented
27#895—21no—
28#882—22no—
29#874C2211yescharacteristic
30C4×D12C2211yescharacteristic
31#832C2211yescharacteristic, complemented
32#830C2212yescomplemented
33#809C2211yescharacteristic
34#809C2212yes—
35#809C2212yescomplemented
36#809C2212yescomplemented
37#809C2212yescomplemented
38#808C2212yescomplemented
39#807C2211yescharacteristic, complemented
40C4⋊(C3⋊C8)C2211yescharacteristic, complemented
41C4×C3⋊C8C2211yescharacteristic, complemented
42#555D311yescharacteristic, complemented
43D42—31no—
44#486—31no—
45#482—32no—
46#476—31no—
47#469—32no—
48#459—31no—
49#447—32no—
50C4×D8—31no—
51#436—31no—
52#418—32no—
53C23×D3—22no—
54C2×C3×Q8—21no—
55C2×C3×D4—22no—
56C2×C3×D4D411yescharacteristic
57C2×C3×D4C2311yescharacteristic
58C22×C3×C4—21no—
59C22×C3×C4C2312yes—
60C2×#67—22no—
61D3×D4—44no—
62C2×D12—24no—
63C2×D12—21no—
64C2×D12—22no—
65C2×D12—21no—
66C2×D12—42no—
67C2×D12C2312yes—
68C2×D12C2311yescharacteristic
69C2×C4×D3—21no—
70C3×C4⋊C4—21no—
71C3×C4⋊C4D411yescharacteristic
72C3×C4⋊C4C2311yescharacteristic
73C3×C4⋊C4C2312yes—
74C3×C22⋊C4—22no—
75C3×C22⋊C4—22no—
76C3×C22⋊C4—21no—
77C3×C22⋊C4D412yes—
78C3×C42—22no—
79C3×C42C2311yescharacteristic
80#213—22no—
81#213—42no—
82(C2×D3)⋊C4—21no—
83(C2×D3)⋊C4—22no—
84C4⋊Dic3C2311yescharacteristic
85#208—42no—
86C2×C3⋊C8C2311yescharacteristic
87C2×C3⋊C8C2311yescharacteristic
88C2×C3⋊C8C2312yes—
89C22×D4—32no—
90C22×D4—61no—
91C2×D8—31no—
92C2×D8—32no—
93C2×M16—32no—
94C4⋊Q8C2×D311yescharacteristic, complemented
95#127—31no—
96#124—22no—
97#121—21no—
98#121—32no—
99#121—61no—
100#121C2×D312yescomplemented
101#120—62no—
102C4×D4—31no—
103C4×D4C2×D311yescharacteristic
104C4×D4C2×D311yescharacteristic, complemented
105#117—21no—
106#117C2×D312yescomplemented
107#107—31no—
108#106—32no—
109C4⋊C8—31no—
110#102—31no—
111#102—31no—
112#102—32no—
113#102—32no—
114#102—32no—
115#102—32no—
116C22⋊C8—32no—
117C4×C8—31no—
118C23×C3—21no—
119C23×C3C2×D412yes—
120C22×D3—21no—
121C22×D3—22no—
122C22×D3—44no—
123C22×D3—44no—
124C22×D3—82no—
125C3×Q8—81no—
126C3×D4—22no—
127C3×D4—21no—
128C3×D4—21no—
129C3×D4—82no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—21no—
133C2×C3×C4—22no—
134C2×C3×C4—42no—
135C2×C3×C4C2×D412yes—
136C2×C3×C4C2×D411yescharacteristic
137C2×C3×C4C2×D411yescharacteristic
138C2×C3×C4C2411yescharacteristic
139#67—44no—
140C2×Dic3—21no—
141D12—22no—
142D12—21no—
143D12—22no—
144D12—44no—
145D12—42no—
146D12—81no—
147D12#4112yes—
148C4×D3—42no—
149C3⋊C8—21no—
150C3⋊C8—21no—
151C3⋊C8—22no—
152C24—62no—
153C2×Q8—21no—
154C2×D4—22no—
155C2×D4—31no—
156C2×D4—32no—
157C2×D4—64no—
158C2×D4—62no—
159C2×D4—62no—
160C2×D4—61no—
161C2×D4—61no—
162C2×D4—122no—
163C2×D4—124no—
164C2×D4#6711yescharacteristic
165C2×D4C22×D311yescharacteristic
166C22×C4—21no—
167C22×C4—61no—
168C22×C4C22×D312yes—
169D8—62no—
170D8—122no—
171M16—122no—
172C2×C8—31no—
173C2×C8—31no—
174C2×C8—32no—
175C4⋊C4—21no—
176C4⋊C4—31no—
177C4⋊C4#6711yescharacteristic
178C4⋊C4C22×D312yes—
179C4⋊C4C22×D311yescharacteristic
180C22⋊C4—22no—
181C22⋊C4—22no—
182C22⋊C4—21no—
183C22⋊C4—61no—
184C22⋊C4—62no—
185C22⋊C4#6712yes—
186C42—22no—
187C42C22×D311yescharacteristic
188C22×C3—22no—
189C22×C3—22no—
190C22×C3—22no—
191C22×C3—41no—
192C22×C3—41no—
193C22×C3—41no—
194C22×C3C22×D411yescharacteristic
195C2×D3—22no—
196C2×D3—41no—
197C2×D3—41no—
198C2×D3—42no—
199C2×D3—42no—
200C2×D3—42no—
201C2×D3—88no—
202C3×C4—22no—
203C3×C4—21no—
204C3×C4—41no—
205C3×C4—42no—
206C3×C4AGL(1,ℤ/8)12yes—
207C3×C4C2×#4111yescharacteristic
208C3×C4H(5,𝔽2)11yescharacteristic
209Dic3—41no—
210C23—21no—
211C23—61no—
212C23—62no—
213C23—124no—
214C23—124no—
215C23—242no—
216C23C2×#6712yes—
217Q8—81no—
218D4—22no—
219D4—21no—
220D4—21no—
221D4—32no—
222D4—62no—
223D4—61no—
224D4—62no—
225D4—82no—
226D4—124no—
227D4—124no—
228D4—122no—
229D4—241no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—61no—
236C2×C4—122no—
237C2×C4C2×#6712yes—
238C2×C4C2×#6711yescharacteristic
239C2×C4C2×#6711yescharacteristic
240C2×C4C23×D311yescharacteristic
241C8—61no—
242C8—61no—
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—22no—
253C22—22no—
254C22—41no—
255C22—41no—
256C22—41no—
257C22—62no—
258C22—121no—
259C22—121no—
260C22—122no—
261C22—122no—
262C22—122no—
263C22—248no—
264C22C22×#6711yescharacteristic
265C4—22no—
266C4—21no—
267C4—41no—
268C4—42no—
269C4—121no—
270C4#94912yes—
271C4D3×#4111yescharacteristic
272C4#100911yescharacteristic
273C3#332511yescharacteristic, complemented, Sylow
274C2—42no—
275C2—81no—
276C2—122no—
277C2—242no—
278C2C2×#94911yescharacteristic
279C2#582911yescharacteristic
280C2#583511yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#555C2
socleC22×C3C22×D4
hypercenter#555D3
hypocenterC3#3325
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #555

Complements

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

NH ≅ G/Ntype
1. C3×#555C2semidirect
2. D4×D12C2semidirect
3. #5088C2semidirect
4. #5084C2semidirect
5. #5039C2semidirect
6. #5036C2semidirect
7. #5015C2semidirect
8. C4×#213C2semidirect
9. #5012C2semidirect
10. #5003C2semidirect
11. #5001C2semidirect
15. C3×C4⋊Q8C22semidirect
18. C3×#121C22semidirect
20. C3×C4×D4C22semidirect
22. C3×#117C22semidirect
24. C2×#213C22semidirect
26. C2×#208C22semidirect
31. #832C22semidirect
32. #830C22semidirect
35. #809C22semidirect
36. #809C22semidirect
37. #809C22semidirect
38. #808C22semidirect
39. #807C22semidirect
40. C4⋊(C3⋊C8)C22semidirect
41. C4×C3⋊C8C22semidirect
42. #555D3semidirect
94. C4⋊Q8C2×D3semidirect
100. #121C2×D3semidirect
104. C4×D4C2×D3semidirect
106. #117C2×D3semidirect
273. C3#3325semidirect

Maximal subgroups

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

subgroupindex
C3×#5552
D4×D122
#50882
#50842
#50392
#50362
#50152
C4×#2132
#50122
#50032
#50012
#33253