Finite Group Databasebeta

← #79550

Subgroups — #79550

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×#908C212yesmaximal, complemented
2#5553C211yesmaximal, characteristic, complemented
3D4×D12C211yesmaximal, characteristic, complemented
4C3×#436C211yesmaximal, characteristic, complemented
5#5135C212yesmaximal, complemented
6#4848C211yesmaximal, characteristic, complemented
7#4843C211yesmaximal, characteristic, complemented
8#4734C212yesmaximal, complemented
9#4732C212yesmaximal, complemented
10#4695C211yesmaximal, characteristic, complemented
11C4×#204C211yesmaximal, characteristic, complemented
12#3274—31nomaximal, Sylow
13C2×D3×D4—21no—
14C2×#235C2212yescomplemented
15C22×D12C2212yes—
16C2×C3×M16C2212yescomplemented
17C3×C4×D4C2211yescharacteristic
18#940—21no—
19#930C2212yes—
20#930C2212yescomplemented
21#908—28no—
22C2×D24C2212yes—
23C2×D24C2212yescomplemented
24C2×D24C2212yescomplemented
25C2×#204C2211yescharacteristic
26C2×#204C2212yescomplemented
27#895—21no—
28#893—21no—
29#884—22no—
30#882—22no—
31#874C2211yescharacteristic
32#874C2211yescharacteristic, complemented
33C4×D12C2211yescharacteristic
34C4×Dic6C2211yescharacteristic, complemented
35C3×C4⋊C8C2211yescharacteristic, complemented
36C3×C22⋊C8C2212yescomplemented
37C3×C4×C8C2211yescharacteristic, complemented
38#821C2211yescharacteristic
39#821C2212yescomplemented
40#821C2212yescomplemented
41#817C2211yescharacteristic, complemented
42#816C2211yescharacteristic, 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×SD16—31no—
53#436D311yescharacteristic, complemented
54C23×D3—22no—
55C2×C3×D4D411yescharacteristic
56C22×C3×C4C2312yes—
57C2×#67—22no—
58C2×#67—22no—
59C2×#67—22no—
60D3×D4—44no—
61#235—24no—
62#235—42no—
63C2×D12—24no—
64C2×D12—21no—
65C2×D12—21no—
66C2×D12—22no—
67C2×D12—21no—
68C2×D12—21no—
69C2×D12—42no—
70C2×D12C2312yes—
71C2×D12C2312yes—
72C2×D12C2311yescharacteristic
73C2×C4×D3—21no—
74C2×C4×D3—22no—
75C2×Dic6C2311yescharacteristic
76C3×M16—24no—
77C2×C3×C8C2311yescharacteristic
78C2×C3×C8C2312yes—
79C2×C3×C8C2311yescharacteristic
80C3×C4⋊C4D411yescharacteristic
81C3×C22⋊C4D412yes—
82C3×C42C2311yescharacteristic
83(C2×D3)⋊C4—21no—
84(C2×D3)⋊C4—22no—
85(C2×D3)⋊C4—22no—
86C4⋊Dic3C2311yescharacteristic
87C4⋊Dic3C2311yescharacteristic
88Dic3⋊C4—21no—
89C4×Dic3—21no—
90D24—24no—
91D24—24no—
92D24—42no—
93#204—24no—
94#204D414yescomplemented
95C2×#41—32no—
96C22×D4—32no—
97C22×D4—61no—
98AGL(1,ℤ/8)—68no—
99C2×SD16—31no—
100C2×SD16—32no—
101C2×D8—32no—
102C2×D8—32no—
103C2×D8—32no—
104C2×M16C2×D312yescomplemented
105#127—31no—
106#127—31no—
107#127—61no—
108#121—32no—
109#121—32no—
110#121—61no—
111#121—62no—
112#120—62no—
113C4×Q8—31no—
114C4×D4—31no—
115C4×D4—61no—
116C4×D4C2×D311yescharacteristic
117#106—31no—
118C4⋊C8C2×D311yescharacteristic, complemented
119#103—31no—
120#102—31no—
121#102—32no—
122#102—32no—
123C22⋊C8C2×D312yescomplemented
124C4×C8C2×D311yescharacteristic, complemented
125C23×C3C2×D412yes—
126C22×D3—21no—
127C22×D3—22no—
128C22×D3—22no—
129C22×D3—44no—
130C22×D3—44no—
131C22×D3—82no—
132C3×D4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4C2×D411yescharacteristic
136C2×C3×C4C2×D412yes—
137C2×C3×C4C2×D411yescharacteristic
138C2×C3×C4C2411yescharacteristic
139#67—44no—
140#67—44no—
141#67—82no—
142C2×Dic3—21no—
143C2×Dic3—21no—
144C2×Dic3—21no—
145D12—22no—
146D12—22no—
147D12—21no—
148D12—22no—
149D12—22no—
150D12—44no—
151D12—42no—
152D12—42no—
153D12—81no—
154D12—81no—
155D12C2×D412yes—
156C4×D3—44no—
157C4×D3—42no—
158Dic6—21no—
159Dic6C2×D412yescomplemented
160C3×C8—22no—
161C3×C8—21no—
162C3×C8C2×D412yescomplemented
163C24—62no—
164#41—64no—
165#41—122no—
166C2×Q8—31no—
167C2×D4—31no—
168C2×D4—32no—
169C2×D4—32no—
170C2×D4—64no—
171C2×D4—62no—
172C2×D4—62no—
173C2×D4—62no—
174C2×D4—61no—
175C2×D4—62no—
176C2×D4—61no—
177C2×D4—61no—
178C2×D4—61no—
179C2×D4—122no—
180C2×D4—124no—
181C2×D4D1211yescharacteristic
182C22×C4—61no—
183C22×C4—62no—
184C22×C4C22×D312yes—
185SD16—34no—
186SD16—64no—
187D8—64no—
188D8—64no—
189D8—122no—
190M16—24no—
191C2×C8C22×D311yescharacteristic
192C2×C8C22×D312yes—
193C2×C8C22×D311yescharacteristic
194C4⋊C4—31no—
195C4⋊C4—31no—
196C4⋊C4—61no—
197C4⋊C4D1211yescharacteristic
198C22⋊C4—61no—
199C22⋊C4—62no—
200C22⋊C4—62no—
201C22⋊C4D1212yes—
202C42—61no—
203C42C22×D311yescharacteristic
204C22×C3—22no—
205C22×C3—22no—
206C22×C3—22no—
207C22×C3C22×D411yescharacteristic
208C2×D3—22no—
209C2×D3—41no—
210C2×D3—41no—
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×C4AGL(1,ℤ/8)12yescomplemented
221C3×C4C22×D411yescharacteristic
222C3×C4H(5,𝔽2)11yescharacteristic
223Dic3—22no—
224Dic3—41no—
225Dic3—41no—
226C23—61no—
227C23—62no—
228C23—62no—
229C23—124no—
230C23—124no—
231C23—242no—
232C23C2×D1212yes—
233Q8—32no—
234Q8—61no—
235D4—22no—
236D4—32no—
237D4—62no—
238D4—62no—
239D4—61no—
240D4—62no—
241D4—62no—
242D4—124no—
243D4—124no—
244D4—124no—
245D4—122no—
246D4—122no—
247D4—241no—
248D4—241no—
249D4—242no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—61no—
253C2×C4—61no—
254C2×C4—61no—
255C2×C4—124no—
256C2×C4—122no—
257C2×C4C2×D1211yescharacteristic
258C2×C4C2×D1212yes—
259C2×C4C2×D1211yescharacteristic
260C2×C4C23×D311yescharacteristic
261C8—22no—
262C8—21no—
263C8D3×D412yescomplemented
264C2×C3—42no—
265C2×C3D4211yescharacteristic
266C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
267C2×C3#57611yescharacteristic
268D3—42no—
269D3—82no—
270D3—82no—
271C22—22no—
272C22—22no—
273C22—22no—
274C22—62no—
275C22—121no—
276C22—121no—
277C22—122no—
278C22—122no—
279C22—122no—
280C22—122no—
281C22—122no—
282C22—122no—
283C22—248no—
284C22C22×D1211yescharacteristic
285C4—21no—
286C4—22no—
287C4—62no—
288C4—121no—
289C4—121no—
290C4#90812yescomplemented
291C4C2×D3×D411yescharacteristic
292C4#100911yescharacteristic
293C3#327411yescharacteristic, complemented, Sylow
294C2—42no—
295C2—122no—
296C2—242no—
297C2—242no—
298C2D4×D1211yescharacteristic
299C2C2×#90811yescharacteristic
300C2#575211yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#436C2
socleC22×C3C22×D4
hypercenter#436D3
hypocenterC3#3274
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#908C2semidirect
2. #5553C2semidirect
3. D4×D12C2semidirect
4. C3×#436C2semidirect
5. #5135C2semidirect
6. #4848C2semidirect
7. #4843C2semidirect
8. #4734C2semidirect
9. #4732C2semidirect
10. #4695C2semidirect
11. C4×#204C2semidirect
14. C2×#235C22semidirect
16. C2×C3×M16C22semidirect
20. #930C22semidirect
23. C2×D24C22semidirect
24. C2×D24C22semidirect
26. C2×#204C22semidirect
32. #874C22semidirect
34. C4×Dic6C22semidirect
35. C3×C4⋊C8C22semidirect
36. C3×C22⋊C8C22semidirect
37. C3×C4×C8C22semidirect
39. #821C22semidirect
40. #821C22semidirect
41. #817C22semidirect
42. #816C22semidirect
53. #436D3semidirect
94. #204D4semidirect
104. C2×M16C2×D3semidirect
118. C4⋊C8C2×D3semidirect
123. C22⋊C8C2×D3semidirect
124. C4×C8C2×D3semidirect
159. Dic6C2×D4semidirect
162. C3×C8C2×D4semidirect
220. C3×C4AGL(1,ℤ/8)semidirect
263. C8D3×D4semidirect
290. C4#908semidirect
293. C3#3274semidirect

Maximal subgroups

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

subgroupindex
C2×#9082
#55532
D4×D122
C3×#4362
#51352
#48482
#48432
#47342
#47322
#46952
C4×#2042
#32743