Finite Group Databasebeta

← #84020

Subgroups — #84020

Order 384.

454 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 125 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×#578C211yesmaximal, characteristic, complemented
2#5837C212yesmaximal, complemented
3#5769C211yesmaximal, characteristic, complemented
4#5245C212yesmaximal, complemented
5#5203C212yesmaximal, complemented
6#5202C212yesmaximal, complemented
7#5196C211yesmaximal, characteristic, complemented
8#5192C211yesmaximal, characteristic, complemented
9#5191C211yesmaximal, characteristic, complemented
10#5144C211yesmaximal, characteristic, complemented
11#5142C211yesmaximal, characteristic, complemented
12#3290—31nomaximal, Sylow
13C3×2−1+4C2212yescomplemented
14#1009C2212yes—
15C3×#137—22no—
16C3×#137C2212yescomplemented
17C3×#135—21no—
18C3×#135C2211yescharacteristic
19C2×C3×Q16—21no—
20C2×C3×Q16C2211yescharacteristic, complemented
21C3×#131C2211yescharacteristic, complemented
22#950—22no—
23#950C2211yescharacteristic
24#949—22no—
25#949C2212yes—
26#949C2212yescomplemented
27#948C2212yescomplemented
28#947C2212yescomplemented
29#942—22no—
30#942C2212yes—
31C2×#215—22no—
32C2×#215C2212yescomplemented
33#919—21no—
34#914—22no—
35D3×D8—21no—
36C2×D24C2211yescharacteristic
37#901C2211yescharacteristic
38#837C2211yescharacteristic, complemented
39#837C2212yescomplemented
40#836C2212yescomplemented
41#825C2211yescharacteristic
42#823C2212yescomplemented
43#822C2211yescharacteristic, complemented
44#819C2211yescharacteristic, complemented
45C8×Dic3C2211yescharacteristic, complemented
46#578D311yescharacteristic, complemented
47#577—32no—
48#576—31no—
49#495—31no—
50#473—31no—
51#472—31no—
52#471—32no—
53#455—32no—
54#454—32no—
55#443—31no—
56#443—31no—
57C3×#41—22no—
58C3×#41—22no—
59C3×#41—42no—
60C3×#41C2312yes—
61C3×#41C2311yescharacteristic
62C2×C3×Q8—22no—
63C2×C3×Q8—22no—
64C2×C3×Q8C2312yes—
65#239—22no—
66D3×D4—22no—
67D3×D4—42no—
68#235—22no—
69#235C2311yescharacteristic
70C2×D12—22no—
71C2×D12C2312yes—
72C3×Q16—22no—
73C3×Q16—21no—
74C3×Q16—21no—
75C3×Q16D411yescharacteristic
76C3×SD16—22no—
77C3×SD16D412yes—
78C3×D8D411yescharacteristic
79C3×M16—21no—
80C3×M16C2311yescharacteristic
81C2×C3×C8—21no—
82C2×C3×C8C2311yescharacteristic
83#216—22no—
84#216D411yescharacteristic
85#215—22no—
86#215—22no—
87#215—22no—
88#215—22no—
89#215—22no—
90#215D411yescharacteristic
91#214D411yescharacteristic
92#213—22no—
93#213—22no—
94#213D411yescharacteristic
95(C2×D3)⋊C4—42no—
96C4×Dic3C2311yescharacteristic
97#208—22no—
98#208C2312yes—
99#208C2311yescharacteristic
100C2×C3⋊C8—22no—
101C2×C3⋊C8C2311yescharacteristic
102D24—21no—
103D24—21no—
104ΓL(1,𝔽25)—21no—
105D3×C8—21no—
1062−1+4C2×D312yescomplemented
107H(5,𝔽2)—32no—
108#137—22no—
109#137—32no—
110#137—62no—
111#137C2×D312yescomplemented
112AGL(1,ℤ/8)—32no—
113AGL(1,ℤ/8)—32no—
114AGL(1,ℤ/8)—62no—
115AGL(1,ℤ/8)—62no—
116#135—21no—
117#135—31no—
118#135—62no—
119#135—61no—
120#135C2×D311yescharacteristic
121C2×Q16—21no—
122C2×Q16C2×D311yescharacteristic, complemented
123C2×SD16—32no—
124C2×SD16—62no—
125C2×D8—31no—
126C2×D8—61no—
127#131—32no—
128#131—31no—
129#131C2×D311yescharacteristic, complemented
130#124—32no—
131#108—31no—
132#108—31no—
133#104—31no—
134#104—32no—
135#104—31no—
136#101—32no—
137#100—32no—
138C4×C8—31no—
139C22×D3—22no—
140C22×D3—42no—
141C3×Q8—22no—
142C3×Q8—22no—
143C3×Q8—42no—
144C3×Q8C2×D411yescharacteristic
145C3×Q8C2×D412yes—
146C3×D4—22no—
147C3×D4—21no—
148C3×D4—42no—
149C3×D4C2×D411yescharacteristic
150C3×D4C2×D412yes—
151C2×C3×C4—22no—
152C2×C3×C4—21no—
153C2×C3×C4—22no—
154C2×C3×C4—42no—
155C2×C3×C4—42no—
156C2×C3×C4C2411yescharacteristic
157#67—21no—
158#67—42no—
159C2×Dic3—21no—
160D12—22no—
161D12—22no—
162D12—42no—
163D12C2×D411yescharacteristic
164C4×D3—21no—
165C4×D3—42no—
166Dic6C2×D411yescharacteristic
167C3×C8—21no—
168C3×C8C2×D411yescharacteristic
169C3×C8C2×D411yescharacteristic
170C3⋊C8—21no—
171C3⋊C8—22no—
172C3⋊C8C2×D411yescharacteristic
173C3⋊C8C2×D411yescharacteristic
174#41—22no—
175#41—22no—
176#41—31no—
177#41—42no—
178#41—62no—
179#41—62no—
180#41C22×D311yescharacteristic
181#41C22×D312yes—
182C2×Q8—22no—
183C2×Q8—22no—
184C2×Q8C22×D312yes—
185C2×D4—32no—
186C2×D4—62no—
187C2×D4—62no—
188C2×D4—122no—
189Q16—22no—
190Q16—21no—
191Q16—21no—
192Q16—31no—
193Q16—62no—
194Q16#6711yescharacteristic
195SD16—22no—
196SD16—31no—
197SD16—31no—
198SD16—62no—
199SD16—62no—
200SD16—62no—
201SD16—62no—
202SD16—62no—
203SD16#6712yes—
204D8—31no—
205D8—62no—
206D8—62no—
207D8—61no—
208D8—61no—
209D8#6711yescharacteristic
210M16—21no—
211M16—32no—
212M16—31no—
213M16—62no—
214M16—61no—
215M16C22×D311yescharacteristic
216C2×C8—21no—
217C2×C8—31no—
218C2×C8—62no—
219C2×C8—61no—
220C2×C8C22×D311yescharacteristic
221C22⋊C4—122no—
222C42—31no—
223C22×C3—22no—
224C22×C3—21no—
225C22×C3H(5,𝔽2)11yescharacteristic
226C2×D3—21no—
227C2×D3—42no—
228C2×D3—82no—
229C2×D3—162no—
230C3×C4—22no—
231C3×C4—21no—
232C3×C4—42no—
233C3×C4C22×D411yescharacteristic
234C3×C4C22×D411yescharacteristic
235Dic3—21no—
236Dic3—41no—
237C23—62no—
238C23—122no—
239Q8—22no—
240Q8—22no—
241Q8—31no—
242Q8—42no—
243Q8D3×D411yescharacteristic
244Q8C2×#6712yes—
245D4—22no—
246D4—21no—
247D4—31no—
248D4—42no—
249D4—61no—
250D4—62no—
251D4—62no—
252D4—122no—
253D4—122no—
254D4D3×D411yescharacteristic
255D4C2×#6712yes—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—61no—
262C2×C4—61no—
263C2×C4—122no—
264C2×C4C23×D311yescharacteristic
265C8—21no—
266C8—31no—
267C8—31no—
268C8—62no—
269C8—61no—
270C8C2×#6711yescharacteristic
271C8C2×#6711yescharacteristic
272C2×C3—21no—
273C2×C3—42no—
274C2×C3—41no—
275C2×C3D4211yescharacteristic
276D3—41no—
277D3—82no—
278C22—21no—
279C22—22no—
280C22—61no—
281C22—122no—
282C22—242no—
283C22—482no—
284C22#100411yescharacteristic
285C4—21no—
286C4—22no—
287C4—42no—
288C4—61no—
289C4—121no—
290C4C2×D3×D411yescharacteristic
291C4C22×#6711yescharacteristic
292C3#329011yescharacteristic, complemented, Sylow
293C2—21no—
294C2—42no—
295C2—41no—
296C2—121no—
297C2—242no—
298C2D4×#6711yescharacteristic
subgroupG/H
centerC2D4×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#578C2
socleC2×C3D42
hypercenter#578D3
hypocenterC3#3290
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #578

Complements

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

NH ≅ G/Ntype
1. C3×#578C2semidirect
2. #5837C2semidirect
3. #5769C2semidirect
4. #5245C2semidirect
5. #5203C2semidirect
6. #5202C2semidirect
7. #5196C2semidirect
8. #5192C2semidirect
9. #5191C2semidirect
10. #5144C2semidirect
11. #5142C2semidirect
13. C3×2−1+4C22semidirect
16. C3×#137C22semidirect
20. C2×C3×Q16C22semidirect
21. C3×#131C22semidirect
26. #949C22semidirect
27. #948C22semidirect
28. #947C22semidirect
32. C2×#215C22semidirect
38. #837C22semidirect
39. #837C22semidirect
40. #836C22semidirect
42. #823C22semidirect
43. #822C22semidirect
44. #819C22semidirect
45. C8×Dic3C22semidirect
46. #578D3semidirect
106. 2−1+4C2×D3semidirect
111. #137C2×D3semidirect
122. C2×Q16C2×D3semidirect
129. #131C2×D3semidirect
292. C3#3290semidirect

Maximal subgroups

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

subgroupindex
C3×#5782
#58372
#57692
#52452
#52032
#52022
#51962
#51922
#51912
#51442
#51422
#32903