Finite Group Databasebeta

← #80314

Subgroups — #80314

Order 384.

488 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 121 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#5716C211yesmaximal, characteristic, complemented
2C4×D3×D4C211yesmaximal, characteristic, complemented
3C3×#459C211yesmaximal, characteristic, complemented
4#5156C212yesmaximal, complemented
5#5152C212yesmaximal, complemented
6#5072C211yesmaximal, characteristic, complemented
7#5008C211yesmaximal, characteristic, complemented
8#4842C211yesmaximal, characteristic, complemented
9#4835C211yesmaximal, characteristic, complemented
10#4785C212yesmaximal, complemented
11#4777C212yesmaximal, complemented
12#3109—31nomaximal, Sylow
13C2×#237—21no—
14C2×D3×D4C2211yescharacteristic
15C22×C4×D3—21no—
16C2×C3×D8C2212yes—
17C3×#127C2211yescharacteristic, complemented
18C3×C4×D4C2211yescharacteristic
19#938—21no—
20#938C2212yes—
21#936—21no—
22D4×Dic3—21no—
23D4×Dic3C2211yescharacteristic
24D4×Dic3C2212yescomplemented
25C2×#213C2212yes—
26C4×#67—21no—
27#893C2211yescharacteristic
28D3×C4⋊C4C2211yescharacteristic
29#67⋊C4—21no—
30D3×C22⋊C4—21no—
31C4×D12C2211yescharacteristic
32C42×D3C2211yescharacteristic
33#869C2211yescharacteristic, complemented
34C3×C4⋊C8C2211yescharacteristic, complemented
35C3×#102C2212yescomplemented
36#832C2212yes—
37#832C2212yescomplemented
38#821C2212yes—
39(C2×D3)⋊C8C2212yes—
40#817C2212yescomplemented
41#815C2212yescomplemented
42#808C2212yescomplemented
43C4⋊(C3⋊C8)C2211yescharacteristic, complemented
44#532—31no—
45C2×C4×D4—31no—
46#481—32no—
47#476—31no—
48#476—31no—
49#459—31no—
50#459D311yescharacteristic, complemented
51#447—32no—
52#442—32no—
53#442—32no—
54#432—31no—
55C23×D3—21no—
56C2×C3×D4—21no—
57C2×C3×D4—21no—
58C2×C3×D4C2312yes—
59C2×C3×D4C2311yescharacteristic
60C22×C3×C4—21no—
61C2×#67—22no—
62C2×#67—21no—
63C22×Dic3—21no—
64C22×Dic3—22no—
65#237—44no—
66D3×D4—22no—
67D3×D4—22no—
68C2×D12C2311yescharacteristic
69C2×C4×D3—21no—
70C2×C4×D3—21no—
71C2×C4×D3—21no—
72C2×C4×D3—44no—
73C2×C4×D3D411yescharacteristic
74C2×C4×D3D411yescharacteristic
75C2×C4×D3C2311yescharacteristic
76C2×Dic6—21no—
77C3×D8—24no—
78C2×C3×C8C2312yes—
79C3×C4⋊C4C2311yescharacteristic
80C3×C22⋊C4—21no—
81C3×C42C2311yescharacteristic
82#217—21no—
83#217—22no—
84#217—22no—
85#213—24no—
86(C2×D3)⋊C4—21no—
87(C2×D3)⋊C4—21no—
88C4⋊Dic3—21no—
89C4⋊Dic3C2312yes—
90C4⋊Dic3C2311yescharacteristic
91Dic3⋊C4—21no—
92C4×Dic3D411yescharacteristic
93C4×Dic3D411yescharacteristic
94C4×Dic3C2311yescharacteristic
95C2×C3⋊C8C2312yes—
96C2×#41—61no—
97C22×D4—31no—
98C23×C4—61no—
99C2×D8—32no—
100C2×D8C2×D312yes—
101C4⋊Q8—31no—
102#127C2×D311yescharacteristic, complemented
103#124—61no—
104#121—32no—
105#121—61no—
106C4×D4—31no—
107C4×D4—32no—
108C4×D4—31no—
109C4×D4—31no—
110C4×D4—61no—
111C4×D4—61no—
112C4×D4—61no—
113C4×D4C2×D311yescharacteristic
114C2×C4⋊C4—31no—
115C2×C22⋊C4—61no—
116C2×C42—31no—
117#106—32no—
118#106—32no—
119C4⋊C8—31no—
120C4⋊C8C2×D311yescharacteristic, complemented
121#102—32no—
122#102—32no—
123#102—32no—
124#102C2×D312yescomplemented
125C22⋊C8—32no—
126#97—32no—
127C23×C3—21no—
128C23×C3—22no—
129C22×D3—24no—
130C22×D3—21no—
131C22×D3—42no—
132C22×D3—42no—
133C22×D3C2×D411yescharacteristic
134C3×D4—21no—
135C3×D4—22no—
136C3×D4—22no—
137C3×D4—42no—
138C3×D4—42no—
139C3×D4#4112yescomplemented
140C2×C3×C4—21no—
141C2×C3×C4—42no—
142C2×C3×C4C2×D411yescharacteristic
143C2×C3×C4C2×D411yescharacteristic
144C2×C3×C4C2411yescharacteristic
145#67—24no—
146#67—44no—
147C2×Dic3—21no—
148C2×Dic3—22no—
149C2×Dic3—42no—
150C2×Dic3—44no—
151C2×Dic3C2×D411yescharacteristic
152C2×Dic3C2×D411yescharacteristic
153C2×Dic3C2×D411yescharacteristic
154D12—21no—
155D12#4112yescomplemented
156C4×D3—22no—
157C4×D3—22no—
158C4×D3—21no—
159C4×D3—21no—
160C4×D3—42no—
161C4×D3—42no—
162C4×D3—42no—
163Dic6—42no—
164C3×C8—22no—
165C3⋊C8—22no—
166C24—61no—
167#41—124no—
168C2×Q8—61no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—31no—
172C2×D4—61no—
173C2×D4—62no—
174C2×D4—62no—
175C2×D4—62no—
176C2×D4C22×D312yes—
177C2×D4C22×D311yescharacteristic
178C22×C4—21no—
179C22×C4—31no—
180C22×C4—31no—
181C22×C4—31no—
182C22×C4—61no—
183C22×C4—61no—
184C22×C4—61no—
185C22×C4—62no—
186C22×C4—61no—
187C22×C4—124no—
188D8—24no—
189D8—64no—
190C2×C8—32no—
191C2×C8C22×D312yes—
192C4⋊C4—31no—
193C4⋊C4—32no—
194C4⋊C4—61no—
195C4⋊C4—61no—
196C4⋊C4C22×D311yescharacteristic
197C22⋊C4—21no—
198C22⋊C4—61no—
199C22⋊C4—61no—
200C22⋊C4—62no—
201C22⋊C4—62no—
202C22⋊C4—61no—
203C42—31no—
204C42—31no—
205C42—31no—
206C42C22×D311yescharacteristic
207C22×C3—22no—
208C22×C3—41no—
209C22×C3—41no—
210C22×C3—42no—
211C22×C3—42no—
212C22×C3—42no—
213C22×C3C22×D411yescharacteristic
214C2×D3—21no—
215C2×D3—21no—
216C2×D3—22no—
217C2×D3—48no—
218C2×D3—41no—
219C2×D3—41no—
220C2×D3AGL(1,ℤ/8)12yes—
221C3×C4—21no—
222C3×C4—41no—
223C3×C4AGL(1,ℤ/8)12yes—
224C3×C4C2×#4111yescharacteristic
225C3×C4C2×#4111yescharacteristic
226Dic3—21no—
227Dic3—21no—
228Dic3—21no—
229Dic3—42no—
230Dic3—41no—
231C23—21no—
232C23—22no—
233C23—31no—
234C23—64no—
235C23—61no—
236C23—122no—
237C23—122no—
238Q8—122no—
239D4—22no—
240D4—21no—
241D4—22no—
242D4—32no—
243D4—42no—
244D4—42no—
245D4—64no—
246D4—61no—
247D4—124no—
248D4#23712yescomplemented
249C2×C4—21no—
250C2×C4—31no—
251C2×C4—31no—
252C2×C4—31no—
253C2×C4—42no—
254C2×C4—62no—
255C2×C4—62no—
256C2×C4—61no—
257C2×C4—61no—
258C2×C4—61no—
259C2×C4—62no—
260C2×C4—122no—
261C2×C4—122no—
262C2×C4—122no—
263C2×C4—124no—
264C2×C4—122no—
265C2×C4D3×D411yescharacteristic
266C2×C4D3×D411yescharacteristic
267C2×C4C23×D311yescharacteristic
268C8—22no—
269C8—62no—
270C2×C3—42no—
271C2×C3—82no—
272C2×C3#52511yescharacteristic
273C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
274C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
275D3—22no—
276D3—42no—
277C22—22no—
278C22—32no—
279C22—41no—
280C22—41no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—61no—
285C22—61no—
286C22—62no—
287C22—128no—
288C22—121no—
289C22—121no—
290C22C2×D3×D411yescharacteristic
291C4—21no—
292C4—41no—
293C4—61no—
294C4—61no—
295C4—61no—
296C4—121no—
297C4—122no—
298C4#91112yes—
299C4C2×#23711yescharacteristic
300C4D3×#4111yescharacteristic
301C3#310911yescharacteristic, complemented, Sylow
302C2—42no—
303C2—62no—
304C2—82no—
305C2—122no—
306C2#560611yescharacteristic
307C2C2×#91111yescharacteristic
308C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3109
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #5716C2semidirect
2. C4×D3×D4C2semidirect
3. C3×#459C2semidirect
4. #5156C2semidirect
5. #5152C2semidirect
6. #5072C2semidirect
7. #5008C2semidirect
8. #4842C2semidirect
9. #4835C2semidirect
10. #4785C2semidirect
11. #4777C2semidirect
17. C3×#127C22semidirect
24. D4×Dic3C22semidirect
33. #869C22semidirect
34. C3×C4⋊C8C22semidirect
35. C3×#102C22semidirect
37. #832C22semidirect
40. #817C22semidirect
41. #815C22semidirect
42. #808C22semidirect
43. C4⋊(C3⋊C8)C22semidirect
50. #459D3semidirect
102. #127C2×D3semidirect
120. C4⋊C8C2×D3semidirect
124. #102C2×D3semidirect
139. C3×D4#41semidirect
155. D12#41semidirect
248. D4#237semidirect
301. C3#3109semidirect

Maximal subgroups

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

subgroupindex
#57162
C4×D3×D42
C3×#4592
#51562
#51522
#50722
#50082
#48422
#48352
#47852
#47772
#31093