Finite Group Databasebeta

← #80387

Subgroups — #80387

Order 384.

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

normal subgroups only

This group has 488 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
1D3×#127C211yesmaximal, characteristic
2#5578C211yesmaximal, characteristic, complemented
3C3×#460C211yesmaximal, characteristic, complemented
4#5175C212yesmaximal, complemented
5#5169C212yesmaximal, complemented
6#5075C211yesmaximal, characteristic, complemented
7#5027C211yesmaximal, characteristic, complemented
8#4848C211yesmaximal, characteristic, complemented
9#4833C211yesmaximal, characteristic, complemented
10#4776C212yesmaximal, complemented
11#4760C212yesmaximal, complemented
12#3208—31nomaximal, Sylow
13C2×D3×D4—21no—
14C2×D3×D4—21no—
15C2×D3×D4C2212yes—
16C2×C3×SD16C2212yescomplemented
17C3×#127C2211yescharacteristic
18C3×C4×Q8C2211yescharacteristic, complemented
19#946C2211yescharacteristic
20#944C2211yescharacteristic, complemented
21C2×#215C2212yescomplemented
22#940—21no—
23#940—21no—
24#940C2212yes—
25C2×#214C2212yescomplemented
26C2×#204C2212yescomplemented
27#897C2211yescharacteristic
28#896—21no—
29#889—21no—
30#888C2211yescharacteristic, complemented
31#874C2211yescharacteristic
32#872—21no—
33C42×D3C2211yescharacteristic
34C4×Dic6—21no—
35C4×Dic6C2211yescharacteristic, complemented
36C3×C4⋊C8C2211yescharacteristic, complemented
37C3×#102C2212yescomplemented
38#832C2212yescomplemented
39#821C2212yescomplemented
40(C2×D3)⋊C8C2212yes—
41C8×Dic3C2212yescomplemented
42#809C2212yescomplemented
43C4⋊(C3⋊C8)C2211yescharacteristic, complemented
44#533—31no—
45C2×#127—31no—
46#492—32no—
47#486—32no—
48#460—31no—
49#460—31no—
50#460—31no—
51#460D311yescharacteristic, complemented
52#450—32no—
53#450—32no—
54#431—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×D12C2312yes—
70C2×C4×D3D411yescharacteristic
71C2×C4×D3D411yescharacteristic
72C2×C4×D3C2311yescharacteristic
73C2×Dic6—21no—
74C2×Dic6C2311yescharacteristic
75C3×SD16—42no—
76C2×C3×C8C2312yes—
77C3×C4⋊C4—21no—
78C3×C4⋊C4C2311yescharacteristic
79C3×C42—21no—
80C3×C42C2311yescharacteristic
81#215—42no—
82#214—42no—
83(C2×D3)⋊C4—21no—
84(C2×D3)⋊C4—21no—
85C4⋊Dic3—21no—
86C4⋊Dic3C2311yescharacteristic
87Dic3⋊C4—21no—
88Dic3⋊C4—21no—
89Dic3⋊C4—21no—
90Dic3⋊C4—21no—
91Dic3⋊C4—21no—
92C4×Dic3—21no—
93C4×Dic3D411yescharacteristic
94C4×Dic3D411yescharacteristic
95C4×Dic3C2311yescharacteristic
96C2×C3⋊C8C2312yes—
97#204—42no—
98C22×D4—32no—
99C22×D4—61no—
100C22×D4—61no—
101C2×SD16—32no—
102C2×SD16—32no—
103C2×SD16—32no—
104C2×SD16C2×D312yescomplemented
105C4⋊Q8—31no—
106C4⋊Q8—31no—
107#127—32no—
108#127—31no—
109#127—61no—
110#127—61no—
111#127C2×D311yescharacteristic
112#125—61no—
113C22⋊Q8—31no—
114C22⋊Q8—31no—
115C22⋊Q8—61no—
116C4×Q8—31no—
117C4×Q8—61no—
118C4×Q8C2×D311yescharacteristic, complemented
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—
128C4×C8—32no—
129C23×C3—22no—
130C22×D3—22no—
131C22×D3—44no—
132C22×D3—44no—
133C22×D3—44no—
134C22×D3C2×D411yescharacteristic
135C3×Q8—21no—
136C3×Q8—21no—
137C3×D4—22no—
138C3×D4—22no—
139C3×D4—42no—
140C3×D4—42no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4C2×D411yescharacteristic
144C2×C3×C4C2×D411yescharacteristic
145C2×C3×C4C2411yescharacteristic
146#67—44no—
147#67—44no—
148#67—44no—
149C2×Dic3—21no—
150C2×Dic3—21no—
151C2×Dic3C2×D411yescharacteristic
152C2×Dic3C2×D411yescharacteristic
153C2×Dic3C2×D411yescharacteristic
154D12—22no—
155D12—22no—
156D12—42no—
157D12—42no—
158C4×D3—21no—
159C4×D3—21no—
160C4×D3—22no—
161C4×D3SD1614yescomplemented
162Dic6—21no—
163Dic6—21no—
164Dic6—42no—
165C3×C8—22no—
166C3⋊C8—22no—
167C24—62no—
168C2×Q8—31no—
169C2×Q8—61no—
170C2×Q8C22×D311yescharacteristic
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—32no—
174C2×D4—62no—
175C2×D4—62no—
176C2×D4—62no—
177C2×D4—61no—
178C2×D4—61no—
179C2×D4—124no—
180C2×D4—122no—
181C2×D4—122no—
182C2×D4—124no—
183C2×D4C22×D312yes—
184C22×C4—31no—
185C22×C4—31no—
186C22×C4—31no—
187SD16—42no—
188SD16—122no—
189SD16—122no—
190SD16—122no—
191C2×C8—32no—
192C2×C8C22×D312yes—
193C4⋊C4—21no—
194C4⋊C4—31no—
195C4⋊C4—61no—
196C4⋊C4—61no—
197C4⋊C4—61no—
198C4⋊C4—61no—
199C4⋊C4—61no—
200C4⋊C4—61no—
201C4⋊C4C22×D311yescharacteristic
202C22⋊C4—61no—
203C22⋊C4—61no—
204C42—21no—
205C42—31no—
206C42—31no—
207C42—31no—
208C42—61no—
209C42C22×D311yescharacteristic
210C22×C3—42no—
211C22×C3—42no—
212C22×C3—42no—
213C22×C3C22×D411yescharacteristic
214C2×D3—21no—
215C2×D3—21no—
216C2×D3—42no—
217C2×D3—42no—
218C2×D3—42no—
219C2×D3—88no—
220C2×D3C2×SD1612yes—
221C3×C4—21no—
222C3×C4—41no—
223C3×C4—41no—
224C3×C4C2×SD1612yes—
225C3×C4H(5,𝔽2)11yescharacteristic
226C3×C4H(5,𝔽2)11yescharacteristic
227Dic3—21no—
228Dic3—41no—
229Dic3—41no—
230Dic3C2×SD1612yes—
231Dic3AGL(1,ℤ/8)12yes—
232C23—22no—
233C23—31no—
234C23—62no—
235C23—124no—
236C23—124no—
237C23—124no—
238Q8—21no—
239Q8—21no—
240Q8—61no—
241Q8—61no—
242Q8—122no—
243D4—22no—
244D4—22no—
245D4—42no—
246D4—42no—
247D4—62no—
248D4—62no—
249D4—122no—
250D4—124no—
251D4—124no—
252D4—124no—
253D4—122no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—31no—
257C2×C4—31no—
258C2×C4—34no—
259C2×C4—31no—
260C2×C4—62no—
261C2×C4—61no—
262C2×C4—61no—
263C2×C4—61no—
264C2×C4—61no—
265C2×C4D3×D411yescharacteristic
266C2×C4D3×D411yescharacteristic
267C2×C4C23×D311yescharacteristic
268C8—22no—
269C8—62no—
270C2×C3—82no—
271C2×C3#53511yescharacteristic
272C2×C3C22×SD1611yescharacteristic
273C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
274D3—22no—
275D3—82no—
276C22—32no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—61no—
281C22—61no—
282C22—122no—
283C22—122no—
284C22—122no—
285C22—248no—
286C22C2×D3×D411yescharacteristic
287C4—21no—
288C4—32no—
289C4—32no—
290C4—41no—
291C4—41no—
292C4—61no—
293C4—121no—
294C4—121no—
295C4D3×SD1612yes—
296C4#100411yescharacteristic
297C4#100911yescharacteristic
298C3#320811yescharacteristic, complemented, Sylow
299C2—62no—
300C2—82no—
301C2—242no—
302C2#560711yescharacteristic
303C2C2×D3×SD1611yescharacteristic
304C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#460C2
socleC22×C3C22×D4
prime radicalD3×#127C2
hypercenter#460D3
hypocenterC3#3208
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
2. #5578C2semidirect
3. C3×#460C2semidirect
4. #5175C2semidirect
5. #5169C2semidirect
6. #5075C2semidirect
7. #5027C2semidirect
8. #4848C2semidirect
9. #4833C2semidirect
10. #4776C2semidirect
11. #4760C2semidirect
16. C2×C3×SD16C22semidirect
18. C3×C4×Q8C22semidirect
20. #944C22semidirect
21. C2×#215C22semidirect
25. C2×#214C22semidirect
26. C2×#204C22semidirect
30. #888C22semidirect
35. C4×Dic6C22semidirect
36. C3×C4⋊C8C22semidirect
37. C3×#102C22semidirect
38. #832C22semidirect
39. #821C22semidirect
41. C8×Dic3C22semidirect
42. #809C22semidirect
43. C4⋊(C3⋊C8)C22semidirect
51. #460D3semidirect
104. C2×SD16C2×D3semidirect
118. C4×Q8C2×D3semidirect
122. C4⋊C8C2×D3semidirect
126. #102C2×D3semidirect
161. C4×D3SD16semidirect
298. C3#3208semidirect

Maximal subgroups

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

subgroupindex
D3×#1272
#55782
C3×#4602
#51752
#51692
#50752
#50272
#48482
#48332
#47762
#47602
#32083