Finite Group Databasebeta

← #80390

Subgroups — #80390

Order 384.

490 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 115 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
1D3×#127C211yesmaximal, characteristic, complemented
2#5580C211yesmaximal, characteristic, complemented
3C3×#460C211yesmaximal, characteristic, complemented
4#5176C212yesmaximal, complemented
5#5169C212yesmaximal, complemented
6#5072C211yesmaximal, characteristic, complemented
7#5027C211yesmaximal, characteristic, complemented
8#4843C211yesmaximal, characteristic, complemented
9#4835C211yesmaximal, characteristic, complemented
10#4773C212yesmaximal, complemented
11#4764C212yesmaximal, complemented
12#3217—31nomaximal, Sylow
13C2×D3×D4—21no—
14C2×D3×D4—21no—
15C2×D3×D4C2212yes—
16C2×C3×SD16C2212yescomplemented
17C3×#127C2211yescharacteristic
18C3×C4×Q8C2211yescharacteristic, complemented
19#947C2211yescharacteristic, complemented
20#946C2211yescharacteristic
21C2×#215C2212yescomplemented
22#940—21no—
23#940—21no—
24#940C2212yes—
25C2×#213C2212yescomplemented
26C2×D24C2212yescomplemented
27#898—21no—
28#895C2211yescharacteristic
29#894—21no—
30#890C2211yescharacteristic, complemented
31#874C2211yescharacteristic
32C4×D12—21no—
33C4×D12C2211yescharacteristic, complemented
34#872—21no—
35C42×D3C2211yescharacteristic
36C3×C4⋊C8C2211yescharacteristic, complemented
37C3×#102C2212yescomplemented
38#832C2212yescomplemented
39#821C2212yescomplemented
40(C2×D3)⋊C8C2212yes—
41#815C2212yescomplemented
42#809C2212yescomplemented
43C4⋊(C3⋊C8)C2211yescharacteristic, complemented
44C2×#127—31no—
45#529—31no—
46#496—32no—
47#490—32no—
48#460—31no—
49#460D311yescharacteristic, complemented
50#459—31no—
51#459—31no—
52#450—32no—
53#447—32no—
54#432—31no—
55C23×D3—22no—
56C2×C3×Q8C2311yescharacteristic
57C2×C3×D4—21no—
58C2×C3×D4—21no—
59C2×C3×D4C2312yes—
60C2×#67—22no—
61C2×#67—22no—
62C2×#67—22no—
63D3×D4—42no—
64D3×D4—42no—
65D3×D4—44no—
66D3×D4—44no—
67C2×D12—21no—
68C2×D12—21no—
69C2×D12—21no—
70C2×D12C2312yes—
71C2×D12C2311yescharacteristic
72C2×C4×D3—21no—
73C2×C4×D3D411yescharacteristic
74C2×C4×D3D411yescharacteristic
75C2×C4×D3C2311yescharacteristic
76C3×SD16—42no—
77C2×C3×C8C2312yes—
78C3×C4⋊C4—21no—
79C3×C4⋊C4C2311yescharacteristic
80C3×C42—21no—
81C3×C42C2311yescharacteristic
82#215—42no—
83#213—42no—
84(C2×D3)⋊C4—21no—
85(C2×D3)⋊C4—21no—
86(C2×D3)⋊C4—21no—
87(C2×D3)⋊C4—21no—
88(C2×D3)⋊C4—21no—
89C4⋊Dic3—21no—
90C4⋊Dic3C2311yescharacteristic
91Dic3⋊C4—21no—
92Dic3⋊C4—21no—
93C4×Dic3D411yescharacteristic
94C4×Dic3D411yescharacteristic
95C4×Dic3C2311yescharacteristic
96C2×C3⋊C8C2312yes—
97D24—42no—
98C22×D4—32no—
99C22×D4—61no—
100C22×D4—61no—
101C2×SD16—32no—
102C2×SD16C2×D312yescomplemented
103C2×D8—32no—
104C2×D8—32no—
105#127—32no—
106#127—31no—
107#127—61no—
108#127—61no—
109#127C2×D311yescharacteristic
110#126—61no—
111#125—31no—
112#124—31no—
113#123—61no—
114C22⋊Q8—31no—
115#121—31no—
116C4×Q8C2×D311yescharacteristic, complemented
117C4×D4—31no—
118C4×D4—61no—
119#117—61no—
120C2×C42—31no—
121C4⋊C8—31no—
122C4⋊C8C2×D311yescharacteristic, complemented
123#102—32no—
124#102—32no—
125#102—32no—
126#102C2×D312yescomplemented
127C22⋊C8—32no—
128#97—32no—
129C23×C3—22no—
130C22×D3—21no—
131C22×D3—22no—
132C22×D3—44no—
133C22×D3—44no—
134C22×D3—44no—
135C22×D3C2×D411yescharacteristic
136C3×Q8—21no—
137C3×Q8—21no—
138C3×D4—22no—
139C3×D4—22no—
140C3×D4—42no—
141C3×D4—42no—
142C2×C3×C4—21no—
143C2×C3×C4—21no—
144C2×C3×C4C2×D411yescharacteristic
145C2×C3×C4C2×D411yescharacteristic
146C2×C3×C4C2411yescharacteristic
147#67—44no—
148#67—44no—
149#67—44no—
150C2×Dic3—21no—
151C2×Dic3C2×D411yescharacteristic
152C2×Dic3C2×D411yescharacteristic
153C2×Dic3C2×D411yescharacteristic
154D12—21no—
155D12—22no—
156D12—21no—
157D12—22no—
158D12—42no—
159D12—42no—
160D12—42no—
161C4×D3—22no—
162C4×D3—22no—
163C4×D3—21no—
164C4×D3—21no—
165C4×D3—42no—
166C3×C8—22no—
167C3⋊C8—22no—
168C24—62no—
169C2×Q8C22×D311yescharacteristic
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—32no—
173C2×D4—31no—
174C2×D4—61no—
175C2×D4—62no—
176C2×D4—62no—
177C2×D4—62no—
178C2×D4—61no—
179C2×D4—61no—
180C2×D4—122no—
181C2×D4—122no—
182C2×D4—124no—
183C2×D4—124no—
184C2×D4C22×D312yes—
185C22×C4—31no—
186C22×C4—31no—
187C22×C4—31no—
188C22×C4—61no—
189SD16—42no—
190SD16—122no—
191D8—122no—
192D8—122no—
193C2×C8—32no—
194C2×C8C22×D312yes—
195C4⋊C4—21no—
196C4⋊C4—31no—
197C4⋊C4—61no—
198C4⋊C4—61no—
199C4⋊C4—61no—
200C4⋊C4C22×D311yescharacteristic
201C22⋊C4—61no—
202C22⋊C4—61no—
203C22⋊C4—61no—
204C22⋊C4—61no—
205C22⋊C4—61no—
206C42—21no—
207C42—31no—
208C42—31no—
209C42—31no—
210C42C22×D311yescharacteristic
211C22×C3—42no—
212C22×C3—42no—
213C22×C3—42no—
214C22×C3C22×D411yescharacteristic
215C2×D3—21no—
216C2×D3—21no—
217C2×D3—42no—
218C2×D3—41no—
219C2×D3—41no—
220C2×D3—42no—
221C2×D3—42no—
222C2×D3—41no—
223C2×D3—88no—
224C2×D3AGL(1,ℤ/8)12yes—
225C3×C4—21no—
226C3×C4—41no—
227C3×C4—41no—
228C3×C4AGL(1,ℤ/8)12yes—
229C3×C4H(5,𝔽2)11yescharacteristic
230C3×C4H(5,𝔽2)11yescharacteristic
231Dic3—21no—
232Dic3—21no—
233Dic3—21no—
234Dic3—41no—
235C23—22no—
236C23—31no—
237C23—61no—
238C23—62no—
239C23—124no—
240C23—124no—
241C23—124no—
242Q8—21no—
243Q8—21no—
244D4—22no—
245D4—22no—
246D4—42no—
247D4—42no—
248D4—61no—
249D4—62no—
250D4—61no—
251D4—62no—
252D4—122no—
253D4—122no—
254D4—124no—
255D4—124no—
256D4—124no—
257D4—122no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—31no—
261C2×C4—31no—
262C2×C4—31no—
263C2×C4—62no—
264C2×C4—62no—
265C2×C4—61no—
266C2×C4—61no—
267C2×C4—61no—
268C2×C4—122no—
269C2×C4D3×D411yescharacteristic
270C2×C4D3×D411yescharacteristic
271C2×C4C23×D311yescharacteristic
272C8—22no—
273C8—62no—
274C2×C3—82no—
275C2×C3#53511yescharacteristic
276C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
277C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
278D3—22no—
279D3—81no—
280D3—82no—
281C22—32no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—61no—
286C22—61no—
287C22—121no—
288C22—122no—
289C22—122no—
290C22—121no—
291C22—122no—
292C22—121no—
293C22—248no—
294C22C2×D3×D411yescharacteristic
295C4—21no—
296C4—41no—
297C4—41no—
298C4—61no—
299C4—61no—
300C4—61no—
301C4—121no—
302C4#91412yes—
303C4#100411yescharacteristic
304C4#100911yescharacteristic
305C3#321711yescharacteristic, complemented, Sylow
306C2—62no—
307C2—82no—
308C2—242no—
309C2—241no—
310C2#560711yescharacteristic
311C2C2×#91411yescharacteristic
312C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#460C2
socleC22×C3C22×D4
hypercenter#460D3
hypocenterC3#3217
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. D3×#127C2semidirect
2. #5580C2semidirect
3. C3×#460C2semidirect
4. #5176C2semidirect
5. #5169C2semidirect
6. #5072C2semidirect
7. #5027C2semidirect
8. #4843C2semidirect
9. #4835C2semidirect
10. #4773C2semidirect
11. #4764C2semidirect
16. C2×C3×SD16C22semidirect
18. C3×C4×Q8C22semidirect
19. #947C22semidirect
21. C2×#215C22semidirect
25. C2×#213C22semidirect
26. C2×D24C22semidirect
30. #890C22semidirect
33. C4×D12C22semidirect
36. C3×C4⋊C8C22semidirect
37. C3×#102C22semidirect
38. #832C22semidirect
39. #821C22semidirect
41. #815C22semidirect
42. #809C22semidirect
43. C4⋊(C3⋊C8)C22semidirect
49. #460D3semidirect
102. C2×SD16C2×D3semidirect
116. C4×Q8C2×D3semidirect
122. C4⋊C8C2×D3semidirect
126. #102C2×D3semidirect
305. C3#3217semidirect

Maximal subgroups

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

subgroupindex
D3×#1272
#55802
C3×#4602
#51762
#51692
#50722
#50272
#48432
#48352
#47732
#47642
#32173