Finite Group Databasebeta

← #80135

Subgroups — #80135

Order 384.

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

normal subgroups only

This group has 450 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
1C2×D4×Dic3C211yesmaximal, characteristic, complemented
2#5624C211yesmaximal, characteristic, complemented
3C3×#450C211yesmaximal, characteristic, complemented
4#5217C211yesmaximal, characteristic
5#5165C212yesmaximal, complemented
6#5164C212yesmaximal, complemented
7#5043C211yesmaximal, characteristic, complemented
8#4785C212yesmaximal, complemented
9#4766C212yesmaximal, complemented
10#4736C211yesmaximal, characteristic, complemented
11#4719C211yesmaximal, characteristic, complemented
12#3114—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×Dic3—21no—
15C2×C3×SD16C2212yes—
16C3×C22⋊Q8C2211yescharacteristic
17C2×#217—21no—
18#947C2211yescharacteristic, complemented
19Q8×Dic3C2211yescharacteristic, complemented
20D4×Dic3—21no—
21D4×Dic3—21no—
22D4×Dic3C2212yes—
23D4×Dic3C2212yes—
24C2×#213C2212yes—
25#930C2211yescharacteristic
26C2×C4⋊Dic3C2211yescharacteristic
27C2×C4×Dic3C2211yescharacteristic
28#898—21no—
29#893C2211yescharacteristic, complemented
30#892—21no—
31#890C2211yescharacteristic, complemented
32#886—21no—
33#67⋊C4—21no—
34C3×#102C2212yes—
35C3×C22⋊C8C2211yescharacteristic
36#835C2212yescomplemented
37#832C2212yes—
38#830C2211yescharacteristic
39#821C2212yescomplemented
40#818C2212yescomplemented
41#815C2212yescomplemented
42Dic3⋊C8C2212yescomplemented
43#808C2212yescomplemented
44#529—31no—
45C2×C4×D4—31no—
46#483—31no—
47#480—31no—
48#478—32no—
49#463—32no—
50#450D311yescharacteristic, complemented
51#447—31no—
52#442—32no—
53#440—32no—
54#432—31no—
55C24×C3—21no—
56C2×C3×Q8C2311yescharacteristic
57C2×C3×D4—24no—
58C2×C3×D4—21no—
59C2×C3×D4C2312yes—
60C22×C3×C4C2311yescharacteristic
61C2×#67—21no—
62C22×Dic3—21no—
63C22×Dic3—22no—
64C22×Dic3—22no—
65C22×Dic3—44no—
66C22×Dic3D411yescharacteristic
67C22×Dic3D411yescharacteristic
68C2×D12C2311yescharacteristic
69C2×C4×D3—21no—
70C3×SD16—24no—
71C2×C3×C8C2312yes—
72C3×C4⋊C4—21no—
73C3×C4⋊C4C2311yescharacteristic
74C3×C22⋊C4—21no—
75#217—22no—
76#217—22no—
77#213—24no—
78(C2×D3)⋊C4—21no—
79(C2×D3)⋊C4—21no—
80(C2×D3)⋊C4—21no—
81(C2×D3)⋊C4—21no—
82C4⋊Dic3—21no—
83C4⋊Dic3—21no—
84C4⋊Dic3—21no—
85C4⋊Dic3C2312yes—
86C4⋊Dic3C2311yescharacteristic
87Dic3⋊C4—21no—
88C4×Dic3—21no—
89C4×Dic3D411yescharacteristic
90C4×Dic3D411yescharacteristic
91C4×Dic3C2311yescharacteristic
92C4×Dic3C2311yescharacteristic
93C2×C3⋊C8C2312yes—
94C22×D4C2×D311yescharacteristic
95C23×C4—61no—
96C2×SD16C2×D312yes—
97C2×D8—32no—
98#126—61no—
99#125—31no—
100#124—31no—
101#123—61no—
102C22⋊Q8C2×D311yescharacteristic
103#121—31no—
104C4×Q8—31no—
105C4×D4—32no—
106C4×D4—32no—
107C4×D4—31no—
108C4×D4—61no—
109C4×D4—61no—
110C4×D4—61no—
111#117—61no—
112C2×C4⋊C4—31no—
113C2×C22⋊C4—61no—
114C2×C42—31no—
115#107—32no—
116#106—32no—
117C4⋊C8—32no—
118#103—32no—
119#102—32no—
120#102—32no—
121#102C2×D312yes—
122C22⋊C8—31no—
123C22⋊C8C2×D311yescharacteristic
124#97—32no—
125C23×C3—24no—
126C23×C3—22no—
127C23×C3—42no—
128C23×C3—42no—
129C23×C3C2×D411yescharacteristic
130C22×D3—21no—
131C3×Q8—22no—
132C3×D4—24no—
133C3×D4—22no—
134C3×D4#4114yes—
135C2×C3×C4—22no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4C2×D411yescharacteristic
139C2×C3×C4C2411yescharacteristic
140#67—42no—
141C2×Dic3—22no—
142C2×Dic3—21no—
143C2×Dic3—22no—
144C2×Dic3—41no—
145C2×Dic3—44no—
146C2×Dic3—44no—
147C2×Dic3—42no—
148C2×Dic3C2×D411yescharacteristic
149C2×Dic3C2×D411yescharacteristic
150C2×Dic3C2×D411yescharacteristic
151C2×Dic3C2×D411yescharacteristic
152D12—22no—
153C4×D3—42no—
154C3×C8—22no—
155C3⋊C8—22no—
156C24—21no—
157C2×Q8C22×D311yescharacteristic
158C2×D4—24no—
159C2×D4—21no—
160C2×D4—31no—
161C2×D4—61no—
162C2×D4C22×D312yes—
163C22×C4—31no—
164C22×C4—31no—
165C22×C4—61no—
166C22×C4—62no—
167C22×C4—62no—
168C22×C4—61no—
169C22×C4—124no—
170C22×C4C22×D311yescharacteristic
171SD16—24no—
172D8—64no—
173C2×C8—32no—
174C2×C8C22×D312yes—
175C4⋊C4—21no—
176C4⋊C4—31no—
177C4⋊C4—32no—
178C4⋊C4—61no—
179C4⋊C4—61no—
180C4⋊C4—61no—
181C4⋊C4—61no—
182C4⋊C4C22×D311yescharacteristic
183C22⋊C4—21no—
184C22⋊C4—61no—
185C22⋊C4—61no—
186C22⋊C4—61no—
187C22⋊C4—62no—
188C22⋊C4—62no—
189C22⋊C4—61no—
190C42—31no—
191C42—31no—
192C42—31no—
193C42—31no—
194C42—61no—
195C22×C3—21no—
196C22×C3—21no—
197C22×C3—24no—
198C22×C3—48no—
199C22×C3—42no—
200C22×C3—42no—
201C22×C3AGL(1,ℤ/8)12yes—
202C22×C3C22×D411yescharacteristic
203C2×D3—41no—
204C2×D3—41no—
205C2×D3—41no—
206C3×C4—21no—
207C3×C4—41no—
208C3×C4—41no—
209C3×C4C2×#4112yes—
210Dic3—21no—
211Dic3—21no—
212Dic3—21no—
213Dic3—21no—
214Dic3—42no—
215Dic3—41no—
216C23—24no—
217C23—22no—
218C23—42no—
219C23—42no—
220C23—61no—
221C23D3×D411yescharacteristic
222Q8—22no—
223D4—24no—
224D4—22no—
225D4—62no—
226D4—122no—
227D4#23714yes—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—31no—
232C2×C4—31no—
233C2×C4—31no—
234C2×C4—31no—
235C2×C4—62no—
236C2×C4—61no—
237C2×C4—62no—
238C2×C4—122no—
239C2×C4—121no—
240C2×C4—124no—
241C2×C4—124no—
242C2×C4—122no—
243C2×C4D3×D411yescharacteristic
244C2×C4C23×D311yescharacteristic
245C8—22no—
246C8—62no—
247C2×C3—22no—
248C2×C3—44no—
249C2×C3#52511yescharacteristic
250C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
251C2×C3#57511yescharacteristic
252D3—81no—
253C22—21no—
254C22—21no—
255C22—24no—
256C22—48no—
257C22—42no—
258C22—42no—
259C22—121no—
260C22—121no—
261C22—121no—
262C22#91412yes—
263C22C2×D3×D411yescharacteristic
264C4—21no—
265C4—41no—
266C4—41no—
267C4—61no—
268C4—61no—
269C4—61no—
270C4—61no—
271C4—122no—
272C4—121no—
273C4C2×#23712yes—
274C3#311411yescharacteristic, complemented, Sylow
275C2—22no—
276C2—44no—
277C2—241no—
278C2#558611yescharacteristic
279C2C2×#91411yescharacteristic
280C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#450C2
socleC22×C3C22×D4
prime radical#5217C2
hypercenter#450D3
hypocenterC3#3114
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
1. C2×D4×Dic3C2semidirect
2. #5624C2semidirect
3. C3×#450C2semidirect
5. #5165C2semidirect
6. #5164C2semidirect
7. #5043C2semidirect
8. #4785C2semidirect
9. #4766C2semidirect
10. #4736C2semidirect
11. #4719C2semidirect
18. #947C22semidirect
19. Q8×Dic3C22semidirect
29. #893C22semidirect
31. #890C22semidirect
36. #835C22semidirect
39. #821C22semidirect
40. #818C22semidirect
41. #815C22semidirect
42. Dic3⋊C8C22semidirect
43. #808C22semidirect
50. #450D3semidirect
274. C3#3114semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic32
#56242
C3×#4502
#52172
#51652
#51642
#50432
#47852
#47662
#47362
#47192
#31143