Finite Group Databasebeta

← #81835

Subgroups — #81835

Order 384.

454 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 117 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#5766C211yesmaximal, characteristic, complemented
2#5762C212yesmaximal, complemented
3#5757C211yesmaximal, characteristic, complemented
4C3×#495C211yesmaximal, characteristic, complemented
5#5192C211yesmaximal, characteristic, complemented
6#5177C212yesmaximal, complemented
7#5160C211yesmaximal, characteristic, complemented
8#5062C212yesmaximal, complemented
9#5061C212yesmaximal, complemented
10#4700C211yesmaximal, characteristic, complemented
11#4700C211yesmaximal, characteristic, complemented
12#3290—31nomaximal, Sylow
13#1007C2212yescomplemented
14#1004C2212yes—
15C2×C3×Q16C2211yescharacteristic, complemented
16C2×C3×SD16C2212yescomplemented
17C2×C3×D8C2211yescharacteristic
18C3×#124C2212yescomplemented
19#942C2212yes—
20#942C2212yescomplemented
21#932C2212yes—
22#932C2212yescomplemented
23#919—21no—
24#918—22no—
25D3×Q16—21no—
26#916—22no—
27#915—22no—
28#914—22no—
29D3×SD16—22no—
30#912—21no—
31#911—22no—
32D3×D8—21no—
33#904C2211yescharacteristic
34#904C2211yescharacteristic
35#901C2211yescharacteristic
36#901C2211yescharacteristic, complemented
37#901C2212yescomplemented
38C3×C4×C8C2211yescharacteristic, complemented
39#836C2212yescomplemented
40#833C2212yescomplemented
41#819C2211yescharacteristic, complemented
42#819C2211yescharacteristic, complemented
43#805C2211yescharacteristic
44#805C2212yescomplemented
45#805C2211yescharacteristic, complemented
46#578—31no—
47#577—32no—
48#576—31no—
49#495D311yescharacteristic, complemented
50#473—31no—
51#472—31no—
52#471—32no—
53#455—32no—
54#454—32no—
55#443—31no—
56#443—31no—
57C2×C3×Q8C2312yes—
58C2×C3×D4C2312yes—
59C2×#67—42no—
60#239—22no—
61#239—22no—
62D3×Q8—22no—
63D3×Q8—22no—
64#237—22no—
65#237—22no—
66D3×D4—22no—
67D3×D4—22no—
68#235—42no—
69#235C2311yescharacteristic
70#235C2312yes—
71#235C2311yescharacteristic
72C3×Q16—21no—
73C3×Q16—21no—
74C3×SD16—22no—
75C3×SD16—22no—
76C3×D8—21no—
77C3×D8—21no—
78C2×C3×C8C2311yescharacteristic
79C2×C3×C8C2311yescharacteristic
80C3×C22⋊C4—42no—
81C3×C42C2311yescharacteristic
82#216—22no—
83#216—22no—
84#215—22no—
85#215—22no—
86#214—22no—
87#214—22no—
88#213—22no—
89#213—22no—
90#208C2311yescharacteristic
91#208C2312yes—
92#208C2311yescharacteristic
93Dic12D411yescharacteristic
94Dic12D411yescharacteristic
95D24D411yescharacteristic
96D24D411yescharacteristic
97#204D412yes—
98#204D411yescharacteristic
99#204D411yescharacteristic
100ΓL(1,𝔽25)—21no—
101ΓL(1,𝔽25)—22no—
102ΓL(1,𝔽25)—21no—
103D3×C8—21no—
104D3×C8—22no—
105D3×C8—21no—
1062−1+4—32no—
107H(5,𝔽2)—32no—
108#137—32no—
109#137—32no—
110#137—62no—
111#137—62no—
112AGL(1,ℤ/8)—32no—
113AGL(1,ℤ/8)—32no—
114AGL(1,ℤ/8)—62no—
115AGL(1,ℤ/8)—62no—
116#135—31no—
117#135—31no—
118#135—61no—
119#135—61no—
120#135—62no—
121C2×Q16—61no—
122C2×Q16C2×D311yescharacteristic, complemented
123C2×SD16—62no—
124C2×SD16C2×D312yescomplemented
125C2×D8—61no—
126C2×D8C2×D311yescharacteristic
127#131—31no—
128#131—32no—
129#131—31no—
130#124C2×D312yescomplemented
131#108—31no—
132#108—31no—
133#104—31no—
134#104—31no—
135#104—32no—
136#101—32no—
137#100—32no—
138C4×C8C2×D311yescharacteristic, complemented
139C23×C3—22no—
140C22×D3—42no—
141C3×Q8—22no—
142C3×Q8—22no—
143C3×D4—22no—
144C3×D4—22no—
145C2×C3×C4—21no—
146C2×C3×C4—22no—
147C2×C3×C4C2411yescharacteristic
148#67—21no—
149#67—22no—
150#67—21no—
151#67—42no—
152#67—42no—
153C2×Dic3—42no—
154D12—42no—
155D12C2×D411yescharacteristic
156D12C2×D412yes—
157D12C2×D411yescharacteristic
158C4×D3—21no—
159C4×D3—22no—
160C4×D3—21no—
161C4×D3—42no—
162C4×D3—42no—
163Dic6—42no—
164Dic6C2×D411yescharacteristic
165Dic6C2×D412yes—
166Dic6C2×D411yescharacteristic
167C3×C8C2×D411yescharacteristic
168C3×C8C2×D411yescharacteristic
169C3×C8C2×D411yescharacteristic
170C3×C8C2×D411yescharacteristic
171C3⋊C8—21no—
172C3⋊C8—22no—
173C3⋊C8—21no—
174#41—31no—
175#41—32no—
176#41—31no—
177#41—62no—
178#41—62no—
179#41—62no—
180#41—62no—
181#41—122no—
182C2×Q8—62no—
183C2×Q8—62no—
184C2×Q8C22×D312yes—
185C2×D4—62no—
186C2×D4—62no—
187C2×D4—122no—
188C2×D4C22×D312yes—
189Q16—21no—
190Q16—21no—
191Q16—31no—
192Q16—31no—
193Q16—62no—
194Q16—62no—
195SD16—22no—
196SD16—22no—
197SD16—31no—
198SD16—31no—
199SD16—32no—
200SD16—62no—
201SD16—62no—
202SD16—62no—
203SD16—62no—
204D8—21no—
205D8—21no—
206D8—31no—
207D8—31no—
208D8—62no—
209D8—62no—
210M16—31no—
211M16—32no—
212M16—31no—
213M16—61no—
214M16—62no—
215M16—61no—
216C2×C8—61no—
217C2×C8—62no—
218C2×C8—61no—
219C2×C8C22×D311yescharacteristic
220C2×C8C22×D311yescharacteristic
221C22⋊C4—42no—
222C42C22×D311yescharacteristic
223C22×C3—42no—
224C22×C3—82no—
225C22×C3H(5,𝔽2)11yescharacteristic
226C2×D3—21no—
227C2×D3—22no—
228C2×D3—21no—
229C2×D3—162no—
230C3×C4—41no—
231C3×C4—42no—
232C3×C4C22×D411yescharacteristic
233C3×C4C22×D411yescharacteristic
234Dic3—21no—
235Dic3—22no—
236Dic3—21no—
237C23—22no—
238C23—122no—
239Q8—22no—
240Q8—22no—
241Q8—31no—
242Q8—32no—
243Q8—31no—
244Q8—122no—
245D4—22no—
246D4—22no—
247D4—31no—
248D4—32no—
249D4—31no—
250D4—61no—
251D4—62no—
252D4—61no—
253D4—122no—
254D4—122no—
255D4—122no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—61no—
259C2×C4—62no—
260C2×C4—61no—
261C2×C4—122no—
262C2×C4—122no—
263C2×C4—122no—
264C2×C4C23×D311yescharacteristic
265C8—61no—
266C8—62no—
267C8—61no—
268C8D3×D411yescharacteristic
269C8D3×D411yescharacteristic
270C8D3×D411yescharacteristic
271C8D3×D411yescharacteristic
272C2×C3—21no—
273C2×C3—82no—
274C2×C3D4211yescharacteristic
275D3—41no—
276D3—42no—
277D3—41no—
278C22—42no—
279C22—61no—
280C22—62no—
281C22—61no—
282C22—82no—
283C22—482no—
284C22#100411yescharacteristic
285C4—41no—
286C4—42no—
287C4—61no—
288C4—62no—
289C4—61no—
290C4C2×D3×D411yescharacteristic
291C4C2×D3×D411yescharacteristic
292C3#329011yescharacteristic, complemented, Sylow
293C2—21no—
294C2—82no—
295C2—121no—
296C2—122no—
297C2—121no—
298C2#571711yescharacteristic
subgroupG/H
centerC2#5717
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#495C2
socleC2×C3D42
hypercenter#495D3
hypocenterC3#3290
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #5766C2semidirect
2. #5762C2semidirect
3. #5757C2semidirect
4. C3×#495C2semidirect
5. #5192C2semidirect
6. #5177C2semidirect
7. #5160C2semidirect
8. #5062C2semidirect
9. #5061C2semidirect
10. #4700C2semidirect
11. #4700C2semidirect
13. #1007C22semidirect
15. C2×C3×Q16C22semidirect
16. C2×C3×SD16C22semidirect
18. C3×#124C22semidirect
20. #942C22semidirect
22. #932C22semidirect
36. #901C22semidirect
37. #901C22semidirect
38. C3×C4×C8C22semidirect
39. #836C22semidirect
40. #833C22semidirect
41. #819C22semidirect
42. #819C22semidirect
44. #805C22semidirect
45. #805C22semidirect
49. #495D3semidirect
122. C2×Q16C2×D3semidirect
124. C2×SD16C2×D3semidirect
130. #124C2×D3semidirect
138. C4×C8C2×D3semidirect
292. C3#3290semidirect

Maximal subgroups

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

subgroupindex
#57662
#57622
#57572
C3×#4952
#51922
#51772
#51602
#50622
#50612
#47002
#32903