Finite Group Databasebeta

← #78613

Subgroups — #78613

Order 384.

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

normal subgroups only

This group has 472 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
1#5791C211yesmaximal, characteristic, complemented
2C2×#930C211yesmaximal, characteristic, complemented
3C2×C3×C22⋊C8C211yesmaximal, characteristic, complemented
4#5115C212yesmaximal, complemented
5#5114C212yesmaximal, complemented
6#4737C212yesmaximal, complemented
7#4735C211yesmaximal, characteristic, complemented
8#4734C212yesmaximal, complemented
9#4732C211yesmaximal, characteristic, complemented
10#4731C211yesmaximal, characteristic, complemented
11#4722C211yesmaximal, characteristic, complemented
12#3188—31nomaximal, Sylow
13C23×C3×C4C2211yescharacteristic
14C22×#67—21no—
15C2×#235C2211yescharacteristic, complemented
16C22×D12C2211yescharacteristic
17C22×C3×C8C2212yescomplemented
18#953—21no—
19#930—21no—
20#930—21no—
21#930C2211yescharacteristic
22#930C2212yes—
23#930C2212yes—
24#930C2211yescharacteristic, complemented
25#929—21no—
26C4×#67—21no—
27C4×#67—21no—
28C2×(C2×D3)⋊C4—21no—
29#926C2211yescharacteristic, complemented
30C2×C4⋊Dic3C2211yescharacteristic
31#924C2211yescharacteristic
32#924C2211yescharacteristic, complemented
33C2×D24C2212yescomplemented
34C2×#204C2212yescomplemented
35C3×C22⋊C8C2211yescharacteristic
36C3×C22⋊C8C2211yescharacteristic
37C3×C22⋊C8C2212yescomplemented
38#821C2212yescomplemented
39#821C2212yescomplemented
40#821C2212yescomplemented
41#818C2212yescomplemented
42#817C2212yescomplemented
43#816C2212yescomplemented
44#525—31no—
45C2×#121—31no—
46#484—31no—
47#482—32no—
48#481—31no—
49#466—32no—
50#465—32no—
51#450—31no—
52#449—32no—
53#447—31no—
54C2×C22⋊C8D311yescharacteristic, complemented
55C24×C3D411yescharacteristic
56C23×D3—21no—
57C22×C3×C4—42no—
58C22×C3×C4D411yescharacteristic
59C22×C3×C4D411yescharacteristic
60C22×C3×C4D411yescharacteristic
61C22×C3×C4C2311yescharacteristic
62C22×C3×C4C2311yescharacteristic
63C22×C3×C4C2311yescharacteristic
64C2×#67—21no—
65C2×#67—21no—
66C2×#67—21no—
67C2×#67—22no—
68C2×#67—22no—
69C2×#67—44no—
70C22×Dic3—21no—
71#235—42no—
72C2×D12—21no—
73C2×D12—42no—
74C2×D12C2312yes—
75C2×D12C2311yescharacteristic
76C2×C4×D3—21no—
77C2×Dic6C2311yescharacteristic
78C2×C3×C8—42no—
79C2×C3×C8C2312yes—
80C2×C3×C8C2312yes—
81#217—21no—
82#217—21no—
83#217—21no—
84(C2×D3)⋊C4—22no—
85(C2×D3)⋊C4—22no—
86(C2×D3)⋊C4—21no—
87(C2×D3)⋊C4—21no—
88C4⋊Dic3—21no—
89C4⋊Dic3C2312yes—
90C4⋊Dic3C2311yescharacteristic
91C4⋊Dic3C2311yescharacteristic
92Dic3⋊C4—21no—
93Dic3⋊C4—21no—
94C4×Dic3—21no—
95D24—42no—
96#204—42no—
97C2×#41—31no—
98C22×D4—31no—
99C22×D4—61no—
100C23×C4C2×D311yescharacteristic
101C2×SD16—32no—
102C2×D8—32no—
103C22×C8C2×D312yescomplemented
104#123—61no—
105C22⋊Q8—31no—
106C22⋊Q8—31no—
107#121—32no—
108#121—32no—
109#121—31no—
110#121—31no—
111#121—61no—
112#121—61no—
113#120—61no—
114C4×D4—61no—
115C4×D4—61no—
116#117—31no—
117C2×C4⋊C4—31no—
118C2×C22⋊C4—61no—
119#107—32no—
120#106—32no—
121#103—32no—
122#102—32no—
123#102—32no—
124#102—32no—
125C22⋊C8C2×D311yescharacteristic
126C22⋊C8C2×D311yescharacteristic
127C22⋊C8C2×D312yescomplemented
128C23×C3—22no—
129C23×C3—22no—
130C23×C3—41no—
131C23×C3C2×D411yescharacteristic
132C23×C3C2×D411yescharacteristic
133C23×C3C2×D411yescharacteristic
134C22×D3—21no—
135C22×D3—22no—
136C22×D3—42no—
137C22×D3—42no—
138C22×D3—42no—
139C2×C3×C4—22no—
140C2×C3×C4—22no—
141C2×C3×C4—22no—
142C2×C3×C4—41no—
143C2×C3×C4—41no—
144C2×C3×C4—41no—
145C2×C3×C4C2×D411yescharacteristic
146C2×C3×C4C2×D411yescharacteristic
147C2×C3×C4C2×D411yescharacteristic
148C2×C3×C4C2411yescharacteristic
149#67—42no—
150#67—44no—
151#67—42no—
152#67—42no—
153#67—44no—
154C2×Dic3—21no—
155C2×Dic3—21no—
156C2×Dic3—21no—
157C2×Dic3—22no—
158C2×Dic3—42no—
159D12—22no—
160D12—22no—
161D12—22no—
162D12—42no—
163C4×D3—42no—
164Dic6—22no—
165C3×C8—22no—
166C3×C8—22no—
167C24—61no—
168C24D1211yescharacteristic
169#41—122no—
170C2×Q8—31no—
171C2×D4—31no—
172C2×D4—32no—
173C2×D4—61no—
174C2×D4—61no—
175C2×D4—61no—
176C2×D4—62no—
177C2×D4—62no—
178C2×D4—61no—
179C2×D4—122no—
180C2×D4—124no—
181C22×C4—42no—
182C22×C4—61no—
183C22×C4—61no—
184C22×C4D1211yescharacteristic
185C22×C4D1211yescharacteristic
186C22×C4D1211yescharacteristic
187C22×C4C22×D311yescharacteristic
188C22×C4C22×D311yescharacteristic
189C22×C4C22×D311yescharacteristic
190SD16—122no—
191D8—122no—
192C2×C8—42no—
193C2×C8C22×D312yes—
194C2×C8C22×D312yes—
195C4⋊C4—31no—
196C4⋊C4—31no—
197C4⋊C4—32no—
198C4⋊C4—61no—
199C4⋊C4—61no—
200C4⋊C4—61no—
201C22⋊C4—61no—
202C22⋊C4—61no—
203C22⋊C4—61no—
204C22⋊C4—62no—
205C22⋊C4—62no—
206C22⋊C4—61no—
207C22⋊C4—61no—
208C42—61no—
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—21no—
212C22×C3—21no—
213C22×C3—21no—
214C22×C3—21no—
215C22×C3—21no—
216C22×C3—44no—
217C22×C3#13512yes—
218C22×C3AGL(1,ℤ/8)12yes—
219C22×C3C22×D411yescharacteristic
220C2×D3—41no—
221C2×D3—42no—
222C2×D3—41no—
223C2×D3—42no—
224C2×D3—42no—
225C2×D3—41no—
226C2×D3—84no—
227C3×C4—21no—
228C3×C4—21no—
229C3×C4—21no—
230C3×C4H(5,𝔽2)12yes—
231Dic3—41no—
232Dic3—41no—
233Dic3—42no—
234Dic3—41no—
235C23—22no—
236C23—22no—
237C23—41no—
238C23—61no—
239C23—62no—
240C23—122no—
241C23—122no—
242C23—122no—
243C23C2×D1211yescharacteristic
244C23C2×D1211yescharacteristic
245C23C2×D1211yescharacteristic
246Q8—62no—
247D4—62no—
248D4—62no—
249D4—62no—
250D4—122no—
251D4—124no—
252D4—122no—
253D4—122no—
254D4—124no—
255D4—122no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—61no—
263C2×C4—61no—
264C2×C4—62no—
265C2×C4—61no—
266C2×C4—122no—
267C2×C4—122no—
268C2×C4C2×D1211yescharacteristic
269C2×C4C2×D1211yescharacteristic
270C2×C4C2×D1211yescharacteristic
271C2×C4C23×D311yescharacteristic
272C8—22no—
273C8—22no—
274C2×C3—22no—
275C2×C3—22no—
276C2×C3—41no—
277C2×C3#53411yescharacteristic
278C2×C3C2×#13511yescharacteristic
279C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
280D3—81no—
281D3—82no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—44no—
290C22—121no—
291C22—122no—
292C22—122no—
293C22—121no—
294C22—121no—
295C22—122no—
296C22—244no—
297C22#90412yes—
298C22#90812yes—
299C22C22×D1211yescharacteristic
300C4—21no—
301C4—21no—
302C4—21no—
303C4—121no—
304C4—122no—
305C4—121no—
306C4—121no—
307C4#100412yes—
308C3#318811yescharacteristic, complemented, Sylow
309C2—22no—
310C2—22no—
311C2—41no—
312C2—242no—
313C2—241no—
314C2#549311yescharacteristic
315C2C2×#90411yescharacteristic
316C2C2×#90811yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×C22⋊C8C2
socleC22×C3C22×D4
hypercenterC2×C22⋊C8D3
hypocenterC3#3188
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
1. #5791C2semidirect
2. C2×#930C2semidirect
3. C2×C3×C22⋊C8C2semidirect
4. #5115C2semidirect
5. #5114C2semidirect
6. #4737C2semidirect
7. #4735C2semidirect
8. #4734C2semidirect
9. #4732C2semidirect
10. #4731C2semidirect
11. #4722C2semidirect
15. C2×#235C22semidirect
17. C22×C3×C8C22semidirect
24. #930C22semidirect
29. #926C22semidirect
32. #924C22semidirect
33. C2×D24C22semidirect
34. C2×#204C22semidirect
37. C3×C22⋊C8C22semidirect
38. #821C22semidirect
39. #821C22semidirect
40. #821C22semidirect
41. #818C22semidirect
42. #817C22semidirect
43. #816C22semidirect
54. C2×C22⋊C8D3semidirect
103. C22×C8C2×D3semidirect
127. C22⋊C8C2×D3semidirect
308. C3#3188semidirect

Maximal subgroups

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

subgroupindex
#57912
C2×#9302
C2×C3×C22⋊C82
#51152
#51142
#47372
#47352
#47342
#47322
#47312
#47222
#31883