Finite Group Databasebeta

← #82309

Subgroups — #82309

Order 384.

454 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 133 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×C3×C4×D4C211yesmaximal, characteristic, complemented
2#5479C211yesmaximal, characteristic, complemented
3#5218C211yesmaximal, characteristic, complemented
4#5217C211yesmaximal, characteristic
5C4×#214C212yesmaximal, complemented
6#5016C212yesmaximal, complemented
7C4×#213C212yesmaximal, complemented
8#5009C212yesmaximal, complemented
9#4970C211yesmaximal, characteristic, complemented
10#4968C211yesmaximal, characteristic, complemented
11#4925C211yesmaximal, characteristic, complemented
12#3105—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×C3×C4—21no—
15C3×C4×D4—21no—
16C3×C4×D4—21no—
17C3×C4×D4C2212yes—
18C3×C4×D4C2212yes—
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4—21no—
21C2×C3×C42C2211yescharacteristic
22C2×#214C2212yes—
23C2×#213C2212yes—
24#930C2211yescharacteristic
25#929—21no—
26C4×#67—21no—
27#924C2211yescharacteristic
28#876—21no—
29#875C2211yescharacteristic, complemented
30C4×D12C2211yescharacteristic, complemented
31#872—21no—
32#870C2211yescharacteristic, complemented
33C4×Dic6C2211yescharacteristic, complemented
34#832C2212yes—
35#832C2212yes—
36#830C2211yescharacteristic
37#830C2211yescharacteristic
38#810C2212yescomplemented
39#809C2212yescomplemented
40#808C2212yescomplemented
41#807C2212yescomplemented
42C4⋊(C3⋊C8)C2212yescomplemented
43C4×C3⋊C8C2212yescomplemented
44#529—31no—
45C2×C4×D4D311yescharacteristic, complemented
46#484—31no—
47#481—31no—
48#478—32no—
49#463—32no—
50#450—31no—
51#447—31no—
52C4×SD16—32no—
53C4×D8—32no—
54#431—31no—
55C24×C3—21no—
56C2×C3×D4—24no—
57C2×C3×D4—21no—
58C2×C3×D4C2312yes—
59C22×C3×C4—21no—
60C22×C3×C4—22no—
61C22×C3×C4—22no—
62C22×C3×C4—44no—
63C22×C3×C4D411yescharacteristic
64C22×C3×C4D411yescharacteristic
65C22×C3×C4C2311yescharacteristic
66C2×#67—21no—
67C2×D12C2311yescharacteristic
68C2×C4×D3—21no—
69C2×Dic6C2311yescharacteristic
70C3×C4⋊C4—21no—
71C3×C4⋊C4C2312yes—
72C3×C22⋊C4—22no—
73C3×C22⋊C4—22no—
74C3×C42D411yescharacteristic
75C3×C42D411yescharacteristic
76C3×C42C2311yescharacteristic
77C3×C42C2311yescharacteristic
78#217—21no—
79#214—24no—
80#213—24no—
81(C2×D3)⋊C4—21no—
82(C2×D3)⋊C4—21no—
83(C2×D3)⋊C4—21no—
84(C2×D3)⋊C4—21no—
85C4⋊Dic3C2311yescharacteristic
86C4⋊Dic3C2311yescharacteristic
87Dic3⋊C4—21no—
88Dic3⋊C4—21no—
89Dic3⋊C4—21no—
90Dic3⋊C4—21no—
91C4×Dic3—21no—
92C2×C3⋊C8C2312yes—
93C2×C3⋊C8C2312yes—
94C22×D4C2×D311yescharacteristic
95C23×C4—21no—
96C2×SD16—32no—
97C2×D8—32no—
98#126—61no—
99#125—31no—
100#124—31no—
101#123—61no—
102C22⋊Q8—31no—
103#121—31no—
104C4×Q8—31no—
105C4×D4—21no—
106C4×D4—21no—
107C4×D4—31no—
108C4×D4—61no—
109C4×D4C2×D312yes—
110C4×D4C2×D312yes—
111#117—61no—
112C2×C4⋊C4C2×D311yescharacteristic
113C2×C22⋊C4—21no—
114C2×C42C2×D311yescharacteristic
115#107—32no—
116#106—32no—
117C4⋊C8—32no—
118#103—32no—
119#102—32no—
120#102—32no—
121#102—32no—
122C22⋊C8—31no—
123C22⋊C8—31no—
124C4×C8—32no—
125C23×C3—24no—
126C23×C3—22no—
127C23×C3—42no—
128C23×C3—42no—
129C23×C3C2×D411yescharacteristic
130C22×D3—21no—
131C3×D4—24no—
132C3×D4—22no—
133C3×D4#4114yes—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—44no—
139C2×C3×C4—42no—
140C2×C3×C4—44no—
141C2×C3×C4C2×D411yescharacteristic
142C2×C3×C4C2×D411yescharacteristic
143C2×C3×C4C2×D411yescharacteristic
144C2×C3×C4C2×D411yescharacteristic
145C2×C3×C4C2×D411yescharacteristic
146C2×C3×C4C2411yescharacteristic
147#67—42no—
148C2×Dic3—21no—
149C2×Dic3—21no—
150C2×Dic3—21no—
151D12—22no—
152C4×D3—42no—
153Dic6—22no—
154C3⋊C8—22no—
155C3⋊C8—22no—
156C24—21no—
157C2×Q8—31no—
158C2×D4—24no—
159C2×D4—21no—
160C2×D4—31no—
161C2×D4—61no—
162C2×D4C22×D312yes—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—44no—
167C22×C4—61no—
168C22×C4#6711yescharacteristic
169C22×C4#6711yescharacteristic
170C22×C4C22×D311yescharacteristic
171SD16—64no—
172D8—64no—
173C2×C8—32no—
174C2×C8—32no—
175C4⋊C4—21no—
176C4⋊C4—31no—
177C4⋊C4—31no—
178C4⋊C4—61no—
179C4⋊C4—61no—
180C4⋊C4—61no—
181C4⋊C4—61no—
182C4⋊C4C22×D312yes—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—61no—
186C22⋊C4—61no—
187C22⋊C4—61no—
188C22⋊C4—61no—
189C22⋊C4—61no—
190C42—61no—
191C42#6711yescharacteristic
192C42#6711yescharacteristic
193C42C22×D311yescharacteristic
194C42C22×D311yescharacteristic
195C22×C3—21no—
196C22×C3—21no—
197C22×C3—24no—
198C22×C3—48no—
199C22×C3—42no—
200C22×C3—42no—
201C22×C3#13512yes—
202C22×C3C22×D411yescharacteristic
203C2×D3—41no—
204C2×D3—41no—
205C2×D3—41no—
206C3×C4—21no—
207C3×C4—21no—
208C3×C4—21no—
209C3×C4—21no—
210C3×C4—42no—
211C3×C4AGL(1,ℤ/8)12yes—
212C3×C4C2×#4112yes—
213Dic3—41no—
214Dic3—41no—
215Dic3—41no—
216C23—24no—
217C23—22no—
218C23—42no—
219C23—42no—
220C23—61no—
221C23C2×#6711yescharacteristic
222Q8—62no—
223D4—24no—
224D4—22no—
225D4—62no—
226D4—122no—
227D4#23514yes—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—44no—
233C2×C4—42no—
234C2×C4—44no—
235C2×C4—61no—
236C2×C4—61no—
237C2×C4—61no—
238C2×C4—122no—
239C2×C4C2×#6711yescharacteristic
240C2×C4C2×#6711yescharacteristic
241C2×C4C2×#6711yescharacteristic
242C2×C4C2×#6711yescharacteristic
243C2×C4C2×#6711yescharacteristic
244C2×C4C23×D311yescharacteristic
245C8—62no—
246C8—62no—
247C2×C3—22no—
248C2×C3—44no—
249C2×C3#52511yescharacteristic
250C2×C3C2×#13511yescharacteristic
251C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
252D3—81no—
253C22—21no—
254C22—21no—
255C22—24no—
256C22—48no—
257C22—42no—
258C22—42no—
259C22—121no—
260C22—121no—
261C22—121no—
262C22#95012yes—
263C22C22×#6711yescharacteristic
264C4—21no—
265C4—21no—
266C4—21no—
267C4—21no—
268C4—42no—
269C4—121no—
270C4—121no—
271C4—121no—
272C4#93212yes—
273C4C2×#23512yes—
274C3#310511yescharacteristic, complemented, Sylow
275C2—22no—
276C2—44no—
277C2—241no—
278C2#579111yescharacteristic
279C2C2×#93211yescharacteristic
280C2C2×#95011yescharacteristic
subgroupG/H
centerC2×C4C2×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×C4×D4C2
socleC22×C3C22×D4
prime radical#5217C2
hypercenterC2×C4×D4D3
hypocenterC3#3105
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×D4C2semidirect
2. #5479C2semidirect
3. #5218C2semidirect
5. C4×#214C2semidirect
6. #5016C2semidirect
7. C4×#213C2semidirect
8. #5009C2semidirect
9. #4970C2semidirect
10. #4968C2semidirect
11. #4925C2semidirect
29. #875C22semidirect
30. C4×D12C22semidirect
32. #870C22semidirect
33. C4×Dic6C22semidirect
38. #810C22semidirect
39. #809C22semidirect
40. #808C22semidirect
41. #807C22semidirect
42. C4⋊(C3⋊C8)C22semidirect
43. C4×C3⋊C8C22semidirect
45. C2×C4×D4D3semidirect
274. C3#3105semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D42
#54792
#52182
#52172
C4×#2142
#50162
C4×#2132
#50092
#49702
#49682
#49252
#31053