Finite Group Databasebeta

← #79670

Subgroups — #79670

Order 384.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#914C212yesmaximal, complemented
2#5577C211yesmaximal, characteristic, complemented
3D4×D12C211yesmaximal, characteristic, complemented
4C3×C4×SD16C211yesmaximal, characteristic, complemented
5#5027C211yesmaximal, characteristic, complemented
6#5015C211yesmaximal, characteristic, complemented
7#4865C212yesmaximal, complemented
8#4810C212yesmaximal, complemented
9#4773C212yesmaximal, complemented
10#4695C211yesmaximal, characteristic, complemented
11#4688C211yesmaximal, characteristic, complemented
12#3274—31nomaximal, Sylow
13C2×#239C2212yescomplemented
14C2×D3×D4C2212yes—
15C22×D12—21no—
16C2×C3×SD16C2211yescharacteristic
17C3×C4×Q8C2211yescharacteristic, complemented
18C3×C4×D4C2211yescharacteristic
19C2×#215C2212yescomplemented
20C2×#213C2212yes—
21#930—21no—
22#914—28no—
23C2×D24C2212yescomplemented
24C2×D24C2212yescomplemented
25C2×ΓL(1,𝔽25)C2212yescomplemented
26#895—22no—
27#895C2212yes—
28#895C2212yescomplemented
29#882—22no—
30#874—21no—
31#874C2211yescharacteristic
32#874C2211yescharacteristic, complemented
33C4×D12—21no—
34C4×D12C2211yescharacteristic
35C3×#106C2211yescharacteristic, complemented
36C3×#103C2211yescharacteristic, complemented
37C3×#102C2211yescharacteristic
38C3×C4×C8C2211yescharacteristic, complemented
39(C2×D3)⋊C8C2212yescomplemented
40#809C2212yescomplemented
41#809C2212yescomplemented
42C4⋊(C3⋊C8)C2211yescharacteristic, complemented
43C2×AGL(1,ℤ/8)—32no—
44#550—31no—
45D42—31no—
46#493—31no—
47#469—32no—
48#460—31no—
49#459—31no—
50#449—32no—
51#447—32no—
52C4×SD16D311yescharacteristic, complemented
53#436—31no—
54C23×D3—22no—
55C2×C3×Q8C2311yescharacteristic
56C2×C3×D4C2311yescharacteristic
57C22×C3×C4—21no—
58C2×#67—22no—
59#239—24no—
60#239—42no—
61D3×D4—24no—
62D3×D4—42no—
63C2×D12—22no—
64C2×D12—22no—
65C2×D12—21no—
66C2×D12—21no—
67C2×D12—22no—
68C2×D12—21no—
69C2×D12—21no—
70C2×D12—44no—
71C2×D12D411yescharacteristic
72C2×D12C2312yes—
73C2×D12C2312yes—
74C2×C4×D3—22no—
75C2×C4×D3C2312yes—
76C3×SD16D414yescomplemented
77C2×C3×C8C2311yescharacteristic
78C2×C3×C8C2311yescharacteristic
79C3×C4⋊C4—21no—
80C3×C4⋊C4C2311yescharacteristic
81C3×C4⋊C4C2311yescharacteristic
82C3×C22⋊C4—21no—
83C3×C42—21no—
84C3×C42C2311yescharacteristic
85#215—24no—
86#213—24no—
87(C2×D3)⋊C4—22no—
88(C2×D3)⋊C4—22no—
89(C2×D3)⋊C4D412yes—
90C4⋊Dic3D411yescharacteristic
91C2×C3⋊C8C2312yes—
92D24—24no—
93D24—42no—
94ΓL(1,𝔽25)—24no—
95C2×#41—32no—
96C22×D4—32no—
97C22×D4—61no—
98AGL(1,ℤ/8)—68no—
99C2×SD16—32no—
100C2×SD16C2×D311yescharacteristic
101C2×D8—32no—
102C2×D8—32no—
103C2×D8—32no—
104C2×M16—32no—
105#127—31no—
106#127—31no—
107#127—61no—
108#121—32no—
109#121—32no—
110#121—62no—
111#121—61no—
112#120—62no—
113C4×Q8C2×D311yescharacteristic, complemented
114C4×D4—31no—
115C4×D4—61no—
116C4×D4C2×D311yescharacteristic
117#106C2×D311yescharacteristic, complemented
118C4⋊C8—31no—
119#103C2×D311yescharacteristic, complemented
120#102—32no—
121#102—32no—
122#102C2×D311yescharacteristic
123C22⋊C8—32no—
124C4×C8C2×D311yescharacteristic, complemented
125C23×C3—21no—
126C22×D3—22no—
127C22×D3—22no—
128C22×D3—44no—
129C22×D3—44no—
130C22×D3—82no—
131C22×D3C2×D412yes—
132C3×Q8—21no—
133C3×Q8C2×D412yescomplemented
134C3×D4—21no—
135C3×D4C2×D412yes—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4—42no—
140C2×C3×C4C2×D411yescharacteristic
141C2×C3×C4C2×D411yescharacteristic
142C2×C3×C4C2411yescharacteristic
143#67—44no—
144C2×Dic3C2×D412yes—
145D12—22no—
146D12—22no—
147D12—22no—
148D12—22no—
149D12—22no—
150D12—44no—
151D12—44no—
152D12—42no—
153D12—42no—
154D12—81no—
155D12—81no—
156D12—82no—
157C4×D3—22no—
158C4×D3—22no—
159C4×D3—44no—
160C3×C8—21no—
161C3×C8C2×D412yescomplemented
162C3⋊C8—22no—
163C24—62no—
164#41—64no—
165#41—122no—
166C2×Q8C22×D311yescharacteristic
167C2×D4—31no—
168C2×D4—32no—
169C2×D4—32no—
170C2×D4—62no—
171C2×D4—62no—
172C2×D4—62no—
173C2×D4—62no—
174C2×D4—64no—
175C2×D4—61no—
176C2×D4—61no—
177C2×D4—61no—
178C2×D4—61no—
179C2×D4—124no—
180C2×D4—122no—
181C2×D4C22×D311yescharacteristic
182C22×C4—21no—
183C22×C4—32no—
184C22×C4—62no—
185SD16—64no—
186SD16D1214yescomplemented
187D8—64no—
188D8—64no—
189D8—122no—
190M16—64no—
191C2×C8—32no—
192C2×C8C22×D311yescharacteristic
193C2×C8C22×D311yescharacteristic
194C4⋊C4—21no—
195C4⋊C4—31no—
196C4⋊C4C22×D311yescharacteristic
197C4⋊C4C22×D311yescharacteristic
198C22⋊C4—21no—
199C22⋊C4—32no—
200C22⋊C4—62no—
201C22⋊C4—62no—
202C42—21no—
203C42C22×D311yescharacteristic
204C22×C3—22no—
205C22×C3—41no—
206C22×C3—41no—
207C22×C3C22×D411yescharacteristic
208C2×D3—22no—
209C2×D3—22no—
210C2×D3—22no—
211C2×D3—42no—
212C2×D3—42no—
213C2×D3—42no—
214C2×D3—42no—
215C2×D3—42no—
216C2×D3—42no—
217C2×D3—88no—
218C3×C4—22no—
219C3×C4—21no—
220C3×C4—41no—
221C3×C4—41no—
222C3×C4AGL(1,ℤ/8)12yescomplemented
223C3×C4C22×D411yescharacteristic
224C3×C4H(5,𝔽2)11yescharacteristic
225Dic3—22no—
226C23—21no—
227C23—32no—
228C23—62no—
229C23—62no—
230C23—124no—
231C23—124no—
232C23—242no—
233Q8—21no—
234Q8C2×D1212yescomplemented
235D4—21no—
236D4—62no—
237D4—62no—
238D4—62no—
239D4—62no—
240D4—62no—
241D4—124no—
242D4—124no—
243D4—124no—
244D4—122no—
245D4—122no—
246D4—241no—
247D4—241no—
248D4—242no—
249D4C2×D1212yes—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—32no—
254C2×C4—42no—
255C2×C4—62no—
256C2×C4—62no—
257C2×C4—124no—
258C2×C4D3×D411yescharacteristic
259C2×C4D3×D411yescharacteristic
260C2×C4C23×D311yescharacteristic
261C8—21no—
262C8—62no—
263C8C2×D1212yescomplemented
264C2×C3—42no—
265C2×C3D4211yescharacteristic
266C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
267C2×C3#57611yescharacteristic
268D3—42no—
269D3—82no—
270D3—82no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—62no—
275C22—62no—
276C22—62no—
277C22—122no—
278C22—122no—
279C22—122no—
280C22—122no—
281C22—122no—
282C22—122no—
283C22—248no—
284C22C2×D3×D411yescharacteristic
285C4—21no—
286C4—22no—
287C4—41no—
288C4—41no—
289C4—62no—
290C4#91412yescomplemented
291C4C22×D1211yescharacteristic
292C4#100911yescharacteristic
293C3#327411yescharacteristic, complemented, Sylow
294C2—42no—
295C2—122no—
296C2—242no—
297C2—242no—
298C2D4×D1211yescharacteristic
299C2C2×#91411yescharacteristic
300C2#576911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×C4×SD16C2
socleC22×C3C22×D4
hypercenterC4×SD16D3
hypocenterC3#3274
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#914C2semidirect
2. #5577C2semidirect
3. D4×D12C2semidirect
4. C3×C4×SD16C2semidirect
5. #5027C2semidirect
6. #5015C2semidirect
7. #4865C2semidirect
8. #4810C2semidirect
9. #4773C2semidirect
10. #4695C2semidirect
11. #4688C2semidirect
13. C2×#239C22semidirect
17. C3×C4×Q8C22semidirect
19. C2×#215C22semidirect
23. C2×D24C22semidirect
24. C2×D24C22semidirect
25. C2×ΓL(1,𝔽25)C22semidirect
28. #895C22semidirect
32. #874C22semidirect
35. C3×#106C22semidirect
36. C3×#103C22semidirect
38. C3×C4×C8C22semidirect
39. (C2×D3)⋊C8C22semidirect
40. #809C22semidirect
41. #809C22semidirect
42. C4⋊(C3⋊C8)C22semidirect
52. C4×SD16D3semidirect
76. C3×SD16D4semidirect
113. C4×Q8C2×D3semidirect
117. #106C2×D3semidirect
119. #103C2×D3semidirect
124. C4×C8C2×D3semidirect
133. C3×Q8C2×D4semidirect
161. C3×C8C2×D4semidirect
186. SD16D12semidirect
222. C3×C4AGL(1,ℤ/8)semidirect
234. Q8C2×D12semidirect
263. C8C2×D12semidirect
290. C4#914semidirect
293. C3#3274semidirect

Maximal subgroups

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

subgroupindex
C2×#9142
#55772
D4×D122
C3×C4×SD162
#50272
#50152
#48652
#48102
#47732
#46952
#46882
#32743