Finite Group Databasebeta

← #81684

Subgroups — #81684

Order 384.

444 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 111 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#5725C211yesmaximal, characteristic, complemented
2D3×#124C211yesmaximal, characteristic, complemented
3C3×#489C211yesmaximal, characteristic, complemented
4#5089C211yesmaximal, characteristic, complemented
5#5058C211yesmaximal, characteristic, complemented
6#4819C212yesmaximal, complemented
7#4806C212yesmaximal, complemented
8#4787C212yesmaximal, complemented
9#4773C212yesmaximal, complemented
10#4712C211yesmaximal, characteristic, complemented
11#4705C211yesmaximal, characteristic, complemented
12#3213—31nomaximal, Sylow
13C2×#239—21no—
14C2×D3×Q8C2211yescharacteristic
15C2×D3×D4C2211yescharacteristic
16C3×C4⋊Q8C2211yescharacteristic, complemented
17C3×#124C2211yescharacteristic
18#947—21no—
19#947C2211yescharacteristic
20C2×#215C2212yescomplemented
21#936C2211yescharacteristic
22C2×#213C2212yes—
23C2×D24C2212yes—
24C2×#204C2212yescomplemented
25#895—21no—
26#895C2212yes—
27#893—21no—
28#893C2212yescomplemented
29#885—21no—
30#885—21no—
31D3×C22⋊C4—21no—
32D3×C22⋊C4—21no—
33#875C2211yescharacteristic
34#874C2211yescharacteristic, complemented
35C42×D3C2211yescharacteristic
36C3×#103C2212yescomplemented
37C3×#102C2212yes—
38C3×#97C2211yescharacteristic, complemented
39(C2×D3)⋊C8C2212yes—
40Dic3⋊C8C2212yescomplemented
41#810C2212yescomplemented
42#809C2212yes—
43#803C2211yescharacteristic, complemented
44#532—31no—
45C2×#124—31no—
46#496—31no—
47#496—31no—
48#489—31no—
49#489D311yescharacteristic, complemented
50#463—32no—
51#463—32no—
52#448—32no—
53#447—32no—
54#432—31no—
55C23×D3—21no—
56C2×C3×Q8—21no—
57C2×C3×Q8C2311yescharacteristic
58C2×C3×D4C2311yescharacteristic
59C2×#67—21no—
60#239—44no—
61D3×Q8—42no—
62D3×D4—42no—
63C2×D12—22no—
64C2×D12—21no—
65C2×D12—21no—
66C2×D12C2311yescharacteristic
67C2×D12C2312yes—
68C2×C4×D3—21no—
69C2×C4×D3—22no—
70C2×C4×D3D411yescharacteristic
71C2×C4×D3D411yescharacteristic
72C2×C4×D3C2311yescharacteristic
73C2×Dic6—21no—
74C2×Dic6C2311yescharacteristic
75C2×C3×C8C2312yes—
76C3×C4⋊C4—21no—
77C3×C4⋊C4C2312yes—
78C3×C22⋊C4—21no—
79C3×C22⋊C4—21no—
80C3×C42C2311yescharacteristic
81#217—21no—
82#217—21no—
83#215—42no—
84#213—42no—
85(C2×D3)⋊C4—21no—
86(C2×D3)⋊C4—21no—
87(C2×D3)⋊C4—21no—
88(C2×D3)⋊C4—21no—
89(C2×D3)⋊C4—22no—
90(C2×D3)⋊C4—22no—
91C4×Dic3D411yescharacteristic
92C4×Dic3D411yescharacteristic
93C4×Dic3C2311yescharacteristic
94C2×C3⋊C8C2312yes—
95D24—42no—
96#204—42no—
97C2×#41—61no—
98C22×Q8—31no—
99C22×D4—31no—
100C2×SD16—32no—
101C2×SD16—32no—
102C2×D8—32no—
103C2×D8—32no—
104C4⋊Q8C2×D311yescharacteristic, complemented
105#127—31no—
106#124—31no—
107#124—31no—
108#124—31no—
109#124—61no—
110#124—61no—
111#124—61no—
112#124C2×D311yescharacteristic
113#121—32no—
114#121—61no—
115C4×D4—32no—
116C4×D4—61no—
117C2×C22⋊C4—61no—
118C2×C22⋊C4—61no—
119C2×C42—31no—
120C4⋊C8—32no—
121#103—32no—
122#103C2×D312yescomplemented
123#102—32no—
124#102C2×D312yes—
125C22⋊C8—32no—
126#97—31no—
127#97C2×D311yescharacteristic, complemented
128C23×C3—21no—
129C22×D3—21no—
130C22×D3—22no—
131C22×D3—42no—
132C22×D3—42no—
133C22×D3—42no—
134C22×D3C2×D411yescharacteristic
135C3×Q8—22no—
136C3×Q8—42no—
137C3×D4—22no—
138C2×C3×C4—21no—
139C2×C3×C4—22no—
140C2×C3×C4C2×D411yescharacteristic
141C2×C3×C4C2×D411yescharacteristic
142C2×C3×C4C2411yescharacteristic
143#67—42no—
144C2×Dic3—21no—
145C2×Dic3C2×D411yescharacteristic
146C2×Dic3C2×D411yescharacteristic
147C2×Dic3C2×D411yescharacteristic
148D12—22no—
149D12—22no—
150D12—22no—
151D12—42no—
152D12—42no—
153D12—44no—
154C4×D3—22no—
155C4×D3—22no—
156C4×D3—41no—
157C4×D3—44no—
158C4×D3—42no—
159Dic6—22no—
160Dic6—42no—
161C3×C8—22no—
162C3⋊C8—22no—
163C24—61no—
164#41—124no—
165C2×Q8—21no—
166C2×Q8—31no—
167C2×Q8—61no—
168C2×Q8—122no—
169C2×Q8C22×D311yescharacteristic
170C2×D4—31no—
171C2×D4—32no—
172C2×D4—62no—
173C2×D4—61no—
174C2×D4—61no—
175C2×D4—61no—
176C2×D4—122no—
177C2×D4C22×D311yescharacteristic
178C22×C4—31no—
179C22×C4—31no—
180C22×C4—31no—
181C22×C4—62no—
182C22×C4—61no—
183SD16—122no—
184SD16—122no—
185D8—122no—
186D8—122no—
187C2×C8—32no—
188C2×C8C22×D312yes—
189C4⋊C4—21no—
190C4⋊C4C22×D312yes—
191C22⋊C4—21no—
192C22⋊C4—21no—
193C22⋊C4—61no—
194C22⋊C4—61no—
195C22⋊C4—62no—
196C22⋊C4—62no—
197C22⋊C4—61no—
198C22⋊C4—61no—
199C22⋊C4—61no—
200C22⋊C4—61no—
201C42—31no—
202C42—31no—
203C42—31no—
204C42C22×D311yescharacteristic
205C22×C3—41no—
206C22×C3—41no—
207C22×C3—41no—
208C22×C3C22×D411yescharacteristic
209C2×D3—21no—
210C2×D3—21no—
211C2×D3—41no—
212C2×D3—42no—
213C2×D3—42no—
214C2×D3—41no—
215C2×D3—42no—
216C2×D3—41no—
217C2×D3—84no—
218C2×D3AGL(1,ℤ/8)12yes—
219C3×C4—21no—
220C3×C4—21no—
221C3×C4—41no—
222C3×C4—42no—
223C3×C4H(5,𝔽2)12yes—
224Dic3—21no—
225Dic3—21no—
226Dic3—21no—
227Dic3—41no—
228C23—21no—
229C23—31no—
230C23—61no—
231C23—62no—
232C23—122no—
233C23—122no—
234C23—122no—
235Q8—22no—
236Q8—42no—
237Q8—62no—
238Q8—122no—
239D4—22no—
240D4—62no—
241D4—62no—
242D4—62no—
243D4—122no—
244D4—122no—
245D4—124no—
246D4—122no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—31no—
250C2×C4—31no—
251C2×C4—31no—
252C2×C4—62no—
253C2×C4—62no—
254C2×C4—61no—
255C2×C4—121no—
256C2×C4—124no—
257C2×C4—122no—
258C2×C4D3×D411yescharacteristic
259C2×C4D3×D411yescharacteristic
260C2×C4C23×D311yescharacteristic
261C8—22no—
262C8—62no—
263C2×C3—81no—
264C2×C3#53511yescharacteristic
265C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
266C2×C3#57511yescharacteristic
267D3—22no—
268D3—82no—
269D3—81no—
270C22—32no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—61no—
275C22—61no—
276C22—121no—
277C22—122no—
278C22—122no—
279C22—121no—
280C22—122no—
281C22—121no—
282C22—244no—
283C22C2×D3×D411yescharacteristic
284C4—21no—
285C4—21no—
286C4—41no—
287C4—42no—
288C4—61no—
289C4—61no—
290C4—61no—
291C4—121no—
292C4#100912yes—
293C3#321311yescharacteristic, complemented, Sylow
294C2—62no—
295C2—81no—
296C2—241no—
297C2—242no—
298C2#567411yescharacteristic
299C2D3×AGL(1,ℤ/8)11yescharacteristic
300C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#489C2
socleC22×C3C22×D4
hypercenter#489D3
hypocenterC3#3213
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #5725C2semidirect
2. D3×#124C2semidirect
3. C3×#489C2semidirect
4. #5089C2semidirect
5. #5058C2semidirect
6. #4819C2semidirect
7. #4806C2semidirect
8. #4787C2semidirect
9. #4773C2semidirect
10. #4712C2semidirect
11. #4705C2semidirect
16. C3×C4⋊Q8C22semidirect
20. C2×#215C22semidirect
24. C2×#204C22semidirect
28. #893C22semidirect
34. #874C22semidirect
36. C3×#103C22semidirect
38. C3×#97C22semidirect
40. Dic3⋊C8C22semidirect
41. #810C22semidirect
43. #803C22semidirect
49. #489D3semidirect
104. C4⋊Q8C2×D3semidirect
122. #103C2×D3semidirect
127. #97C2×D3semidirect
293. C3#3213semidirect

Maximal subgroups

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

subgroupindex
#57252
D3×#1242
C3×#4892
#50892
#50582
#48192
#48062
#47872
#47732
#47122
#47052
#32133