Finite Group Databasebeta

← #78638

Subgroups — #78638

Order 384.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#235C211yesmaximal, characteristic, complemented
2#5791C211yesmaximal, characteristic, complemented
3C2×#909C211yesmaximal, characteristic, complemented
4C2×#908C211yesmaximal, characteristic, complemented
5C3×#407C211yesmaximal, characteristic, complemented
6#5131C211yesmaximal, characteristic, complemented
7#5120C211yesmaximal, characteristic, complemented
8#4734C214yesmaximal, complemented
9#4733C214yesmaximal, complemented
10#3005—31nomaximal, Sylow
11C23×C3×C4C2211yescharacteristic
12C22×#67—21no—
13C2×#235—22no—
14C2×#235—24no—
15C2×#235—24no—
16C2×#235C2214yes—
17C2×#235C2211yescharacteristic
18C22×D12C2211yescharacteristic
19C22×C4×D3—21no—
20C22×Dic6C2211yescharacteristic
21C2×C3×M16C2211yescharacteristic
22C2×C3×M16C2211yescharacteristic
23#953—21no—
24#930C2212yescomplemented
25#929—21no—
26C4×#67—22no—
27#926C2211yescharacteristic
28#924C2212yescomplemented
29#909—24no—
30#908—24no—
31C2×Dic12C2212yescomplemented
32C2×D24C2212yescomplemented
33C2×#204C2212yescomplemented
34C2×#204C2212yescomplemented
35C3×C22⋊C8C2214yescomplemented
36#821C2214yescomplemented
37#816C2214yescomplemented
38C22×#41—31no—
39C2×#137—31no—
40C2×AGL(1,ℤ/8)—31no—
41#525—31no—
42#452—34no—
43#449—34no—
44#419—31no—
45#418—31no—
46#407D311yescharacteristic, complemented
47C24×C3D411yescharacteristic
48C23×D3—21no—
49C22×C3×C4—22no—
50C22×C3×C4—22no—
51C22×C3×C4D412yes—
52C22×C3×C4D411yescharacteristic
53C22×C3×C4C2312yes—
54C22×C3×C4C2311yescharacteristic
55C2×#67—28no—
56C2×#67—24no—
57C2×#67—22no—
58C2×#67—21no—
59C22×Dic3—21no—
60#235—216no—
61#235—28no—
62#235—22no—
63#235—22no—
64#235—28no—
65C2×D12—21no—
66C2×D12—22no—
67C2×D12D414yescomplemented
68C2×D12C2311yescharacteristic
69C2×D12C2312yes—
70C2×C4×D3—28no—
71C2×C4×D3—24no—
72C2×C4×D3—21no—
73C2×Dic6—21no—
74C2×Dic6—22no—
75C2×Dic6D414yescomplemented
76C2×Dic6C2311yescharacteristic
77C2×Dic6C2312yes—
78C3×M16—22no—
79C3×M16—22no—
80C2×C3×C8C2312yes—
81C2×C3×C8C2312yes—
82#217—22no—
83#217—21no—
84(C2×D3)⋊C4—22no—
85C4⋊Dic3C2312yes—
86Dic3⋊C4—22no—
87C4×Dic3—21no—
88Dic12—24no—
89D24—24no—
90#204—24no—
91#204—24no—
92C2×#41—31no—
93C2×#41—34no—
94C2×#41—64no—
95C2×#41—64no—
96C2×#41—62no—
97C22×Q8—31no—
98C22×D4—31no—
99C22×D4—61no—
100C23×C4—61no—
101C23×C4C2×D311yescharacteristic
102#137—64no—
103AGL(1,ℤ/8)—64no—
104C2×Q16—32no—
105C2×SD16—32no—
106C2×SD16—32no—
107C2×D8—32no—
108C2×M16C2×D311yescharacteristic
109C2×M16C2×D311yescharacteristic
110#123—61no—
111C22⋊Q8—32no—
112#121—32no—
113#120—61no—
114C4×D4—62no—
115#117—31no—
116#103—34no—
117#102—34no—
118C22⋊C8C2×D314yescomplemented
119C23×C3—22no—
120C23×C3—42no—
121C23×C3C2×D412yes—
122C23×C3C2×D411yescharacteristic
123C22×D3—24no—
124C22×D3—22no—
125C22×D3—21no—
126C22×D3—44no—
127C2×C3×C4—22no—
128C2×C3×C4—24no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4C2×D412yes—
132C2×C3×C4C2×D411yescharacteristic
133C2×C3×C4C2×D412yes—
134C2×C3×C4C2×D412yes—
135C2×C3×C4C2411yescharacteristic
136#67—216no—
137#67—44no—
138#67—42no—
139C2×Dic3—24no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—21no—
143D12—24no—
144D12—22no—
145D12—21no—
146D12—21no—
147D12C2×D414yes—
148C4×D3—216no—
149C4×D3—42no—
150Dic6—24no—
151Dic6—22no—
152Dic6—21no—
153Dic6—21no—
154Dic6C2×D414yes—
155C3×C8—22no—
156C3×C8—22no—
157C24—61no—
158C24D1211yescharacteristic
159#41—68no—
160#41—68no—
161#41—616no—
162#41—62no—
163#41—62no—
164C2×Q8—34no—
165C2×Q8—32no—
166C2×Q8—31no—
167C2×Q8—61no—
168C2×Q8—62no—
169C2×D4—34no—
170C2×D4—32no—
171C2×D4—31no—
172C2×D4—68no—
173C2×D4—64no—
174C2×D4—62no—
175C2×D4—61no—
176C2×D4—61no—
177C2×D4—62no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—68no—
181C22×C4—61no—
182C22×C4—64no—
183C22×C4—61no—
184C22×C4D1212yes—
185C22×C4D1211yescharacteristic
186C22×C4C22×D312yes—
187C22×C4C22×D311yescharacteristic
188Q16—64no—
189SD16—64no—
190SD16—64no—
191D8—64no—
192M16—22no—
193M16—22no—
194C2×C8C22×D312yes—
195C2×C8C22×D312yes—
196C4⋊C4—32no—
197C4⋊C4—62no—
198C22⋊C4—61no—
199C22⋊C4—62no—
200C22⋊C4—62no—
201C42—61no—
202C22×C3—22no—
203C22×C3—22no—
204C22×C3—22no—
205C22×C3—22no—
206C22×C3—44no—
207C22×C3#12012yes—
208C22×C3C22×D411yescharacteristic
209C2×D3—24no—
210C2×D3—48no—
211C2×D3—44no—
212C2×D3—41no—
213C2×D3—42no—
214C3×C4—22no—
215C3×C4#12012yes—
216C3×C4C22×D411yescharacteristic
217C3×C4C22×D411yescharacteristic
218Dic3—24no—
219Dic3—41no—
220Dic3—42no—
221C23—22no—
222C23—42no—
223C23—64no—
224C23—62no—
225C23—61no—
226C23—124no—
227C23C2×D1212yes—
228C23C2×D1211yescharacteristic
229Q8—34no—
230Q8—64no—
231Q8—62no—
232Q8—61no—
233Q8—61no—
234D4—34no—
235D4—616no—
236D4—64no—
237D4—62no—
238D4—61no—
239D4—61no—
240D4—124no—
241D4—122no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—616no—
247C2×C4—64no—
248C2×C4—62no—
249C2×C4—62no—
250C2×C4—61no—
251C2×C4—122no—
252C2×C4C2×D1212yes—
253C2×C4C2×D1211yescharacteristic
254C2×C4D3×D412yes—
255C2×C4D3×D412yes—
256C2×C4C23×D311yescharacteristic
257C8—22no—
258C8—22no—
259C2×C3—22no—
260C2×C3—42no—
261C2×C3C2×#12011yescharacteristic
262C2×C3#57512yes—
263D3—44no—
264D3—81no—
265C22—22no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—44no—
270C22—64no—
271C22—128no—
272C22—124no—
273C22—121no—
274C22—122no—
275C22#88212yes—
276C22C22×D1211yescharacteristic
277C4—22no—
278C4—64no—
279C4—121no—
280C4—122no—
281C4#88212yes—
282C4C2×D3×D411yescharacteristic
283C4C2×D3×D411yescharacteristic
284C3#300511yescharacteristic, complemented, Sylow
285C2—22no—
286C2—42no—
287C2—124no—
288C2—241no—
289C2C2×#88211yescharacteristic
290C2#574812yes—
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#407C2
socleC22×C3C22×D4
hypercenter#407D3
hypocenterC3#3005
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C22×#235C2semidirect
2. #5791C2semidirect
3. C2×#909C2semidirect
4. C2×#908C2semidirect
5. C3×#407C2semidirect
6. #5131C2semidirect
7. #5120C2semidirect
8. #4734C2semidirect
9. #4733C2semidirect
24. #930C22semidirect
28. #924C22semidirect
31. C2×Dic12C22semidirect
32. C2×D24C22semidirect
33. C2×#204C22semidirect
34. C2×#204C22semidirect
35. C3×C22⋊C8C22semidirect
36. #821C22semidirect
37. #816C22semidirect
46. #407D3semidirect
67. C2×D12D4semidirect
75. C2×Dic6D4semidirect
118. C22⋊C8C2×D3semidirect
284. C3#3005semidirect

Maximal subgroups

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

subgroupindex
C22×#2352
#57912
C2×#9092
C2×#9082
C3×#4072
#51312
#51202
#47342
#47332
#30053