Finite Group Databasebeta

← #80045

Subgroups — #80045

Order 384.

462 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 115 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×#946C211yesmaximal, characteristic, complemented
2D3×#121C211yesmaximal, characteristic
3C3×#448C211yesmaximal, characteristic, complemented
4#5227C211yesmaximal, characteristic, complemented
5#5174C212yesmaximal, complemented
6#5169C212yesmaximal, complemented
7#5033C211yesmaximal, characteristic, complemented
8#4812C212yesmaximal, complemented
9#4811C212yesmaximal, complemented
10#4736C211yesmaximal, characteristic, complemented
11#4726C211yesmaximal, characteristic, complemented
12#3193—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic, complemented
14C2×D3×D4—21no—
15C2×D3×D4C2211yescharacteristic
16C22×C4×D3C2211yescharacteristic
17C2×C3×SD16C2212yescomplemented
18C3×#121C2211yescharacteristic
19#946—21no—
20#946—21no—
21#946C2212yes—
22#946C2212yescomplemented
23C2×#215C2212yescomplemented
24#939—21no—
25#938C2211yescharacteristic
26#930C2211yescharacteristic
27C2×(C2×D3)⋊C4—21no—
28C2×C4⋊Dic3C2211yescharacteristic, complemented
29C2×Dic3⋊C4—21no—
30C2×ΓL(1,𝔽25)C2212yescomplemented
31#895C2211yescharacteristic
32D3×C4⋊C4C2211yescharacteristic
33#884—21no—
34D3×C22⋊C4—21no—
35C3×#103C2212yescomplemented
36C3×C22⋊C8C2211yescharacteristic, complemented
37#835C2212yescomplemented
38#832C2212yescomplemented
39#830C2211yescharacteristic, complemented
40#821C2212yescomplemented
41(C2×D3)⋊C8C2212yes—
42#818C2212yescomplemented
43#807C2212yescomplemented
44C2×C22⋊Q8—31no—
45C2×#121—31no—
46#484—32no—
47#480—31no—
48#480—31no—
49#468—32no—
50#468—32no—
51#450—32no—
52#448—31no—
53#448D311yescharacteristic, complemented
54#407—31no—
55C23×D3—21no—
56C23×D3D411yescharacteristic
57C2×C3×Q8—21no—
58C2×C3×Q8—42no—
59C2×C3×Q8C2312yes—
60C2×C3×D4—21no—
61C2×C3×D4C2311yescharacteristic
62C22×C3×C4—21no—
63C22×C3×C4C2311yescharacteristic
64C2×#67—21no—
65C2×#67—21no—
66C2×#67—21no—
67C22×Dic3—21no—
68C22×Dic3D411yescharacteristic
69D3×D4—42no—
70D3×D4—44no—
71C2×D12—21no—
72C2×D12C2311yescharacteristic
73C2×C4×D3—21no—
74C2×C4×D3—42no—
75C2×C4×D3D411yescharacteristic
76C2×C4×D3D411yescharacteristic
77C2×C4×D3C2311yescharacteristic
78C2×C4×D3C2311yescharacteristic
79C3×SD16—42no—
80C2×C3×C8C2312yes—
81C3×C4⋊C4C2311yescharacteristic
82C3×C22⋊C4—21no—
83#217—21no—
84#215—42no—
85(C2×D3)⋊C4—22no—
86(C2×D3)⋊C4—22no—
87(C2×D3)⋊C4—21no—
88(C2×D3)⋊C4—21no—
89C4⋊Dic3—21no—
90C4⋊Dic3C2312yes—
91C4⋊Dic3C2311yescharacteristic
92Dic3⋊C4—21no—
93Dic3⋊C4—22no—
94Dic3⋊C4—22no—
95C2×C3⋊C8C2312yes—
96ΓL(1,𝔽25)—42no—
97C22×Q8C2×D311yescharacteristic, complemented
98C22×D4—31no—
99C22×D4—61no—
100C23×C4—31no—
101C2×SD16—32no—
102C2×SD16C2×D312yescomplemented
103C2×M16—32no—
104C22⋊Q8—32no—
105C22⋊Q8—32no—
106C22⋊Q8—61no—
107C22⋊Q8—61no—
108#121—31no—
109#121—31no—
110#121—31no—
111#121—61no—
112#121—61no—
113#121C2×D311yescharacteristic
114C2×C4⋊C4—31no—
115C2×C4⋊C4—31no—
116C2×C4⋊C4—61no—
117C2×C22⋊C4—61no—
118C2×C22⋊C4—61no—
119#107—32no—
120#107—32no—
121#103—32no—
122#103C2×D312yescomplemented
123#102—32no—
124#102—32no—
125C22⋊C8—31no—
126C22⋊C8—32no—
127C22⋊C8C2×D311yescharacteristic, complemented
128C23×C3—21no—
129C23×C3C2×D411yescharacteristic
130C22×D3—22no—
131C22×D3—22no—
132C22×D3—21no—
133C22×D3—41no—
134C22×D3—42no—
135C22×D3—42no—
136C22×D3—42no—
137C22×D3C2×D411yescharacteristic
138C22×D3C2×D411yescharacteristic
139C3×Q8—22no—
140C3×Q8—22no—
141C3×Q8—42no—
142C3×D4—22no—
143C3×D4—42no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—21no—
147C2×C3×C4—42no—
148C2×C3×C4C2×D411yescharacteristic
149C2×C3×C4C2411yescharacteristic
150#67—42no—
151#67—42no—
152#67—42no—
153C2×Dic3—22no—
154C2×Dic3—21no—
155C2×Dic3—41no—
156C2×Dic3—42no—
157C2×Dic3C2×D411yescharacteristic
158C2×Dic3C2×D411yescharacteristic
159D12—22no—
160D12—42no—
161C4×D3—22no—
162C4×D3—22no—
163C4×D3—41no—
164C4×D3—41no—
165C4×D3—42no—
166C3×C8—22no—
167C3⋊C8—22no—
168C24—31no—
169C24—61no—
170C2×Q8—21no—
171C2×Q8—42no—
172C2×Q8C22×D312yes—
173C2×D4—21no—
174C2×D4—31no—
175C2×D4—61no—
176C2×D4—61no—
177C2×D4—61no—
178C2×D4—61no—
179C2×D4—122no—
180C2×D4—124no—
181C2×D4C22×D311yescharacteristic
182C22×C4—21no—
183C22×C4—31no—
184C22×C4—31no—
185C22×C4—31no—
186C22×C4—31no—
187C22×C4—31no—
188C22×C4—61no—
189C22×C4—61no—
190C22×C4—122no—
191C22×C4C22×D311yescharacteristic
192SD16—42no—
193SD16—122no—
194M16—122no—
195C2×C8—32no—
196C2×C8C22×D312yes—
197C4⋊C4—32no—
198C4⋊C4—31no—
199C4⋊C4—61no—
200C4⋊C4—62no—
201C4⋊C4—62no—
202C4⋊C4—61no—
203C4⋊C4C22×D311yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—61no—
206C22⋊C4—62no—
207C22⋊C4—62no—
208C22⋊C4—61no—
209C22⋊C4—61no—
210C22×C3—21no—
211C22×C3—21no—
212C22×C3—41no—
213C22×C3—41no—
214C22×C3—41no—
215C22×C3AGL(1,ℤ/8)12yes—
216C22×C3C22×D411yescharacteristic
217C2×D3—21no—
218C2×D3—21no—
219C2×D3—21no—
220C2×D3—21no—
221C2×D3—21no—
222C2×D3—44no—
223C2×D3—41no—
224C2×D3—41no—
225C2×D3—41no—
226C2×D3—84no—
227C2×D3#13712yes—
228C3×C4—21no—
229C3×C4—42no—
230C3×C4—41no—
231C3×C4H(5,𝔽2)12yes—
232Dic3—21no—
233Dic3—21no—
234Dic3—41no—
235Dic3—42no—
236C23—21no—
237C23—31no—
238C23—31no—
239C23—62no—
240C23—62no—
241C23—61no—
242C23—121no—
243C23—122no—
244C23—122no—
245C23—122no—
246C23D3×D411yescharacteristic
247Q8—22no—
248Q8—22no—
249Q8—42no—
250D4—22no—
251D4—42no—
252D4—62no—
253D4—122no—
254D4—122no—
255D4—122no—
256D4—122no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—31no—
261C2×C4—31no—
262C2×C4—42no—
263C2×C4—62no—
264C2×C4—62no—
265C2×C4—62no—
266C2×C4—61no—
267C2×C4—121no—
268C2×C4—121no—
269C2×C4—122no—
270C2×C4—121no—
271C2×C4—122no—
272C2×C4D3×D411yescharacteristic
273C2×C4C23×D311yescharacteristic
274C8—22no—
275C8—62no—
276C2×C3—22no—
277C2×C3—81no—
278C2×C3#53411yescharacteristic
279C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
280C2×C3C2×#13711yescharacteristic
281D3—22no—
282D3—41no—
283D3—81no—
284C22—21no—
285C22—21no—
286C22—32no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—61no—
291C22—61no—
292C22—61no—
293C22—61no—
294C22—61no—
295C22—124no—
296C22—121no—
297C22—121no—
298C22—121no—
299C22—244no—
300C22#91412yes—
301C22C2×D3×D411yescharacteristic
302C4—21no—
303C4—41no—
304C4—42no—
305C4—61no—
306C4—61no—
307C4—121no—
308C4—122no—
309C4#100412yes—
310C3#319311yescharacteristic, complemented, Sylow
311C2—22no—
312C2—62no—
313C2—81no—
314C2—121no—
315C2—241no—
316C2#558911yescharacteristic
317C2C2×#91411yescharacteristic
318C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#448C2
socleC22×C3C22×D4
prime radicalD3×#121C2
hypercenter#448D3
hypocenterC3#3193
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#946C2semidirect
3. C3×#448C2semidirect
4. #5227C2semidirect
5. #5174C2semidirect
6. #5169C2semidirect
7. #5033C2semidirect
8. #4812C2semidirect
9. #4811C2semidirect
10. #4736C2semidirect
11. #4726C2semidirect
13. C22×C3×Q8C22semidirect
17. C2×C3×SD16C22semidirect
22. #946C22semidirect
23. C2×#215C22semidirect
28. C2×C4⋊Dic3C22semidirect
30. C2×ΓL(1,𝔽25)C22semidirect
35. C3×#103C22semidirect
36. C3×C22⋊C8C22semidirect
37. #835C22semidirect
38. #832C22semidirect
39. #830C22semidirect
40. #821C22semidirect
42. #818C22semidirect
43. #807C22semidirect
53. #448D3semidirect
97. C22×Q8C2×D3semidirect
102. C2×SD16C2×D3semidirect
122. #103C2×D3semidirect
127. C22⋊C8C2×D3semidirect
310. C3#3193semidirect

Maximal subgroups

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

subgroupindex
C2×#9462
D3×#1212
C3×#4482
#52272
#51742
#51692
#50332
#48122
#48112
#47362
#47262
#31933