Finite Group Databasebeta

← #80152

Subgroups — #80152

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×#938C211yesmaximal, characteristic
2D3×C22⋊Q8C211yesmaximal, characteristic, complemented
3C3×#450C211yesmaximal, characteristic, complemented
4#5218C211yesmaximal, characteristic, complemented
5#5174C212yesmaximal, complemented
6#5170C212yesmaximal, complemented
7#5044C211yesmaximal, characteristic, complemented
8#4780C212yesmaximal, complemented
9#4774C212yesmaximal, complemented
10#4726C211yesmaximal, characteristic, complemented
11#4722C211yesmaximal, characteristic, complemented
12#3193—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×#67—21no—
15C2×D3×Q8C2211yescharacteristic, complemented
16C22×C4×D3C2211yescharacteristic
17C2×C3×SD16C2212yescomplemented
18C3×C22⋊Q8C2211yescharacteristic
19C2×#217—21no—
20#946C2211yescharacteristic, complemented
21#938—21no—
22#938—21no—
23#938C2212yes—
24#938C2212yes—
25C2×#214C2212yescomplemented
26C2×C4⋊Dic3C2211yescharacteristic
27#924C2211yescharacteristic
28C2×ΓL(1,𝔽25)C2212yescomplemented
29#897C2211yescharacteristic, complemented
30#896—21no—
31D3×C4⋊C4—21no—
32D3×C4⋊C4C2211yescharacteristic, complemented
33D3×C22⋊C4—21no—
34#878—21no—
35C3×#102C2212yescomplemented
36C3×C22⋊C8C2211yescharacteristic
37#835C2212yescomplemented
38#832C2212yescomplemented
39#830C2211yescharacteristic
40(C2×D3)⋊C8C2212yescomplemented
41#818C2212yescomplemented
42#816C2212yescomplemented
43#807C2212yescomplemented
44C2×C22⋊Q8—31no—
45C2×#121—31no—
46#484—31no—
47#484—31no—
48#480—32no—
49#468—32no—
50#468—32no—
51#450—31no—
52#450D311yescharacteristic, complemented
53#448—32no—
54#407—31no—
55C24×C3—21no—
56C23×D3D411yescharacteristic
57C2×C3×Q8C2311yescharacteristic
58C2×C3×D4—21no—
59C2×C3×D4—42no—
60C2×C3×D4C2312yes—
61C22×C3×C4C2311yescharacteristic
62C2×#67—22no—
63C2×#67—22no—
64C2×#67—44no—
65C22×Dic3—21no—
66C22×Dic3D411yescharacteristic
67D3×Q8—42no—
68C2×C4×D3—21no—
69C2×C4×D3—21no—
70C2×C4×D3—42no—
71C2×C4×D3D411yescharacteristic
72C2×C4×D3D411yescharacteristic
73C2×C4×D3C2311yescharacteristic
74C2×C4×D3C2311yescharacteristic
75C2×Dic6—21no—
76C2×Dic6C2311yescharacteristic
77C3×SD16—42no—
78C2×C3×C8C2312yes—
79C3×C4⋊C4—21no—
80C3×C4⋊C4C2311yescharacteristic
81C3×C22⋊C4—21no—
82#217—21no—
83#217—22no—
84#217—22no—
85#214—42no—
86(C2×D3)⋊C4—21no—
87(C2×D3)⋊C4—21no—
88C4⋊Dic3—21no—
89C4⋊Dic3—21no—
90C4⋊Dic3C2312yes—
91C4⋊Dic3C2311yescharacteristic
92Dic3⋊C4—21no—
93Dic3⋊C4—21no—
94Dic3⋊C4—21no—
95C2×C3⋊C8C2312yes—
96ΓL(1,𝔽25)—42no—
97C22×Q8—31no—
98C22×D4—61no—
99C22×D4C2×D311yescharacteristic
100C23×C4—31no—
101C2×SD16—32no—
102C2×SD16C2×D312yescomplemented
103C2×M16—32no—
104C22⋊Q8—31no—
105C22⋊Q8—31no—
106C22⋊Q8—31no—
107C22⋊Q8—61no—
108C22⋊Q8—61no—
109C22⋊Q8C2×D311yescharacteristic
110#121—32no—
111#121—32no—
112#121—61no—
113#121—61no—
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#103—32no—
123#102—32no—
124#102C2×D312yescomplemented
125C22⋊C8—31no—
126C22⋊C8—32no—
127C22⋊C8C2×D311yescharacteristic
128C23×C3—22no—
129C23×C3—42no—
130C23×C3—42no—
131C23×C3—42no—
132C23×C3C2×D411yescharacteristic
133C22×D3—22no—
134C22×D3—22no—
135C22×D3—41no—
136C22×D3C2×D411yescharacteristic
137C22×D3C2×D411yescharacteristic
138C3×Q8—22no—
139C3×D4—22no—
140C3×D4—22no—
141C3×D4—42no—
142C2×C3×C4—22no—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4C2×D411yescharacteristic
146C2×C3×C4C2411yescharacteristic
147#67—44no—
148#67—44no—
149C2×Dic3—21no—
150C2×Dic3—21no—
151C2×Dic3—22no—
152C2×Dic3—41no—
153C2×Dic3—42no—
154C2×Dic3C2×D411yescharacteristic
155C2×Dic3C2×D411yescharacteristic
156C4×D3—22no—
157C4×D3—22no—
158C4×D3—41no—
159C4×D3—41no—
160C4×D3—42no—
161C4×D3—42no—
162Dic6—22no—
163Dic6—42no—
164C3×C8—22no—
165C3⋊C8—22no—
166C24—21no—
167C24—31no—
168C2×Q8—31no—
169C2×Q8—61no—
170C2×Q8—122no—
171C2×Q8C22×D311yescharacteristic
172C2×D4—21no—
173C2×D4—42no—
174C2×D4—62no—
175C2×D4—62no—
176C2×D4—124no—
177C2×D4C22×D312yes—
178C22×C4—31no—
179C22×C4—31no—
180C22×C4—31no—
181C22×C4—31no—
182C22×C4—31no—
183C22×C4—61no—
184C22×C4—61no—
185C22×C4—61no—
186C22×C4—122no—
187C22×C4C22×D311yescharacteristic
188SD16—42no—
189SD16—122no—
190M16—122no—
191C2×C8—32no—
192C2×C8C22×D312yes—
193C4⋊C4—21no—
194C4⋊C4—31no—
195C4⋊C4—32no—
196C4⋊C4—61no—
197C4⋊C4—61no—
198C4⋊C4—61no—
199C4⋊C4—61no—
200C4⋊C4—61no—
201C4⋊C4C22×D311yescharacteristic
202C22⋊C4—21no—
203C22⋊C4—61no—
204C22⋊C4—61no—
205C22⋊C4—62no—
206C22⋊C4—62no—
207C22⋊C4—61no—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3—42no—
211C22×C3—42no—
212C22×C3—42no—
213C22×C3—84no—
214C22×C3#13712yes—
215C22×C3C22×D411yescharacteristic
216C2×D3—21no—
217C2×D3—21no—
218C2×D3—21no—
219C2×D3—21no—
220C2×D3—21no—
221C2×D3—44no—
222C2×D3AGL(1,ℤ/8)12yes—
223C3×C4—21no—
224C3×C4—41no—
225C3×C4—41no—
226C3×C4H(5,𝔽2)12yes—
227Dic3—21no—
228Dic3—21no—
229Dic3—41no—
230Dic3—41no—
231Dic3—42no—
232C23—22no—
233C23—31no—
234C23—31no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—62no—
239C23—62no—
240C23—121no—
241C23D3×D411yescharacteristic
242Q8—22no—
243Q8—62no—
244Q8—122no—
245D4—22no—
246D4—22no—
247D4—42no—
248D4—124no—
249D4—124no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—31no—
254C2×C4—31no—
255C2×C4—62no—
256C2×C4—62no—
257C2×C4—61no—
258C2×C4—62no—
259C2×C4—61no—
260C2×C4—121no—
261C2×C4—121no—
262C2×C4—122no—
263C2×C4—122no—
264C2×C4—121no—
265C2×C4—122no—
266C2×C4D3×D411yescharacteristic
267C2×C4C23×D311yescharacteristic
268C8—22no—
269C8—62no—
270C2×C3—22no—
271C2×C3—82no—
272C2×C3#53411yescharacteristic
273C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
274C2×C3C2×#13711yescharacteristic
275D3—22no—
276D3—41no—
277C22—21no—
278C22—21no—
279C22—32no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—61no—
284C22—61no—
285C22—61no—
286C22—61no—
287C22—61no—
288C22—84no—
289C22—124no—
290C22#91512yes—
291C22C2×D3×D411yescharacteristic
292C4—21no—
293C4—41no—
294C4—41no—
295C4—61no—
296C4—61no—
297C4—122no—
298C4—121no—
299C4—121no—
300C4#100412yes—
301C3#319311yescharacteristic, complemented, Sylow
302C2—22no—
303C2—62no—
304C2—82no—
305C2—121no—
306C2#558911yescharacteristic
307C2C2×#91511yescharacteristic
308C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#450C2
socleC22×C3C22×D4
prime radicalC2×#938C2
hypercenter#450D3
hypocenterC3#3193
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. D3×C22⋊Q8C2semidirect
3. C3×#450C2semidirect
4. #5218C2semidirect
5. #5174C2semidirect
6. #5170C2semidirect
7. #5044C2semidirect
8. #4780C2semidirect
9. #4774C2semidirect
10. #4726C2semidirect
11. #4722C2semidirect
15. C2×D3×Q8C22semidirect
17. C2×C3×SD16C22semidirect
20. #946C22semidirect
25. C2×#214C22semidirect
28. C2×ΓL(1,𝔽25)C22semidirect
29. #897C22semidirect
32. D3×C4⋊C4C22semidirect
35. C3×#102C22semidirect
37. #835C22semidirect
38. #832C22semidirect
40. (C2×D3)⋊C8C22semidirect
41. #818C22semidirect
42. #816C22semidirect
43. #807C22semidirect
52. #450D3semidirect
102. C2×SD16C2×D3semidirect
124. #102C2×D3semidirect
301. C3#3193semidirect

Maximal subgroups

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

subgroupindex
C2×#9382
D3×C22⋊Q82
C3×#4502
#52182
#51742
#51702
#50442
#47802
#47742
#47262
#47222
#31933