Finite Group Databasebeta

← #78591

Subgroups — #78591

Order 384.

420 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 137 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#5791C211yesmaximal, characteristic
2C2×#926C211yesmaximal, characteristic, complemented
3C2×C3×C22⋊C8C211yesmaximal, characteristic, complemented
4#5113C212yesmaximal, complemented
5#5106C212yesmaximal, complemented
6#4737C212yesmaximal, complemented
7#4736C211yesmaximal, characteristic, complemented
8#4735C211yesmaximal, characteristic, complemented
9#4723C212yesmaximal, complemented
10#4722C211yesmaximal, characteristic, complemented
11#4721C211yesmaximal, characteristic, complemented
12#3091—31nomaximal, Sylow
13C23×C3×C4C2211yescharacteristic
14C2×#235C2211yescharacteristic
15C22×C3×C8C2212yes—
16#953—21no—
17C2×#217—21no—
18#930C2211yescharacteristic
19#930C2211yescharacteristic
20#929—21no—
21C4×#67—21no—
22C4×#67—21no—
23#926—21no—
24#926—21no—
25#926C2212yes—
26#926C2212yes—
27#926C2211yescharacteristic
28C2×C4⋊Dic3C2211yescharacteristic
29C2×C4⋊Dic3C2211yescharacteristic
30#924C2211yescharacteristic
31#924C2211yescharacteristic
32C2×C4×Dic3—21no—
33C3×C22⋊C8C2211yescharacteristic
34C3×C22⋊C8C2211yescharacteristic
35C3×C22⋊C8C2212yes—
36#821C2212yescomplemented
37#821C2212yescomplemented
38#818C2212yescomplemented
39#818C2212yescomplemented
40#817C2212yescomplemented
41#817C2212yescomplemented
42#816C2212yescomplemented
43#816C2212yescomplemented
44#525—31no—
45C2×#117—31no—
46#485—32no—
47#484—31no—
48#483—31no—
49#482—32no—
50#481—31no—
51#480—31no—
52#427—32no—
53#416—32no—
54C2×C22⋊C8D311yescharacteristic, complemented
55C24×C3D411yescharacteristic
56C22×C3×C4—24no—
57C22×C3×C4D411yescharacteristic
58C22×C3×C4D411yescharacteristic
59C22×C3×C4D411yescharacteristic
60C22×C3×C4C2311yescharacteristic
61C22×C3×C4C2311yescharacteristic
62C22×C3×C4C2311yescharacteristic
63C2×#67—21no—
64C2×#67—21no—
65C2×#67—21no—
66C22×Dic3—21no—
67C22×Dic3—21no—
68#235—24no—
69C2×D12C2311yescharacteristic
70C2×C4×D3—21no—
71C2×Dic6C2311yescharacteristic
72C2×C3×C8—24no—
73C2×C3×C8C2312yes—
74C2×C3×C8C2312yes—
75#217—21no—
76#217—21no—
77#217—22no—
78#217—22no—
79#217—21no—
80(C2×D3)⋊C4—21no—
81(C2×D3)⋊C4—21no—
82C4⋊Dic3—21no—
83C4⋊Dic3—21no—
84C4⋊Dic3C2312yes—
85C4⋊Dic3C2312yes—
86C4⋊Dic3C2311yescharacteristic
87C4⋊Dic3C2311yescharacteristic
88Dic3⋊C4—21no—
89Dic3⋊C4—21no—
90C4×Dic3—22no—
91C4×Dic3—22no—
92C4×Dic3—21no—
93C2×#41—31no—
94C23×C4C2×D311yescharacteristic
95C22×C8C2×D312yes—
96#123—61no—
97C22⋊Q8—31no—
98C22⋊Q8—31no—
99#121—31no—
100#121—31no—
101#120—61no—
102C4×D4—61no—
103C4×D4—61no—
104#117—32no—
105#117—32no—
106#117—31no—
107#117—61no—
108#117—61no—
109C2×C4⋊C4—31no—
110C2×C4⋊C4—31no—
111C2×C22⋊C4—61no—
112C2×C42—61no—
113#107—32no—
114#107—32no—
115#106—32no—
116#106—32no—
117#103—32no—
118#103—32no—
119#102—32no—
120#102—32no—
121C22⋊C8C2×D311yescharacteristic
122C22⋊C8C2×D311yescharacteristic
123C22⋊C8C2×D312yes—
124C23×C3—22no—
125C23×C3—22no—
126C23×C3—41no—
127C23×C3C2×D411yescharacteristic
128C23×C3C2×D411yescharacteristic
129C23×C3C2×D411yescharacteristic
130C22×D3—21no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4C2×D411yescharacteristic
137C2×C3×C4C2×D411yescharacteristic
138C2×C3×C4C2×D411yescharacteristic
139C2×C3×C4#4114yes—
140C2×C3×C4C2411yescharacteristic
141#67—42no—
142#67—42no—
143#67—42no—
144C2×Dic3—22no—
145C2×Dic3—22no—
146C2×Dic3—21no—
147C2×Dic3—21no—
148C2×Dic3—21no—
149C2×Dic3—42no—
150C2×Dic3—42no—
151D12—22no—
152C4×D3—42no—
153Dic6—22no—
154C3×C8—22no—
155C3×C8—22no—
156C24D1211yescharacteristic
157#41—64no—
158C2×Q8—31no—
159C2×D4—31no—
160C2×D4—61no—
161C2×D4—61no—
162C2×D4—61no—
163C22×C4—24no—
164C22×C4—61no—
165C22×C4—61no—
166C22×C4—61no—
167C22×C4D1211yescharacteristic
168C22×C4D1211yescharacteristic
169C22×C4D1211yescharacteristic
170C22×C4C22×D311yescharacteristic
171C22×C4C22×D311yescharacteristic
172C22×C4C22×D311yescharacteristic
173C2×C8—24no—
174C2×C8C22×D312yes—
175C2×C8C22×D312yes—
176C4⋊C4—32no—
177C4⋊C4—32no—
178C4⋊C4—31no—
179C4⋊C4—31no—
180C4⋊C4—61no—
181C4⋊C4—61no—
182C4⋊C4—61no—
183C4⋊C4—61no—
184C22⋊C4—62no—
185C22⋊C4—62no—
186C22⋊C4—61no—
187C22⋊C4—61no—
188C22⋊C4—61no—
189C22⋊C4—61no—
190C22⋊C4—61no—
191C42—62no—
192C42—62no—
193C42—61no—
194C22×C3—21no—
195C22×C3—21no—
196C22×C3—21no—
197C22×C3—21no—
198C22×C3—21no—
199C22×C3—21no—
200C22×C3—21no—
201C22×C3—44no—
202C22×C3#12312yes—
203C22×C3#13512yes—
204C22×C3C22×D411yescharacteristic
205C2×D3—41no—
206C2×D3—41no—
207C2×D3—41no—
208C3×C4—21no—
209C3×C4—21no—
210C3×C4#12312yes—
211C3×C4C2×#4112yes—
212Dic3—41no—
213Dic3—42no—
214Dic3—41no—
215Dic3—42no—
216Dic3—41no—
217C23—22no—
218C23—22no—
219C23—41no—
220C23—61no—
221C23C2×D1211yescharacteristic
222C23C2×D1211yescharacteristic
223C23C2×D1211yescharacteristic
224Q8—62no—
225D4—62no—
226D4—122no—
227D4—122no—
228D4—122no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—62no—
235C2×C4—62no—
236C2×C4—61no—
237C2×C4—61no—
238C2×C4—61no—
239C2×C4—122no—
240C2×C4—122no—
241C2×C4—122no—
242C2×C4C2×D1211yescharacteristic
243C2×C4C2×D1211yescharacteristic
244C2×C4C2×D1211yescharacteristic
245C2×C4#23714yes—
246C2×C4C23×D311yescharacteristic
247C8—22no—
248C8—22no—
249C2×C3—22no—
250C2×C3—22no—
251C2×C3—41no—
252C2×C3C2×#12311yescharacteristic
253C2×C3C2×#13511yescharacteristic
254C2×C3#57511yescharacteristic
255D3—81no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—44no—
264C22—121no—
265C22—121no—
266C22—121no—
267C22#88612yes—
268C22#90412yes—
269C22C22×D1211yescharacteristic
270C4—21no—
271C4—21no—
272C4—122no—
273C4—121no—
274C4—121no—
275C4—122no—
276C4—121no—
277C4#88612yes—
278C4C2×#23712yes—
279C3#309111yescharacteristic, complemented, Sylow
280C2—22no—
281C2—22no—
282C2—41no—
283C2—241no—
284C2C2×#88611yescharacteristic
285C2C2×#90411yescharacteristic
286C2#574811yescharacteristic
subgroupG/H
centerC2×C4C2×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×C22⋊C8C2
socleC22×C3C22×D4
prime radical#5791C2
hypercenterC2×C22⋊C8D3
hypocenterC3#3091
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#926C2semidirect
3. C2×C3×C22⋊C8C2semidirect
4. #5113C2semidirect
5. #5106C2semidirect
6. #4737C2semidirect
7. #4736C2semidirect
8. #4735C2semidirect
9. #4723C2semidirect
10. #4722C2semidirect
11. #4721C2semidirect
36. #821C22semidirect
37. #821C22semidirect
38. #818C22semidirect
39. #818C22semidirect
40. #817C22semidirect
41. #817C22semidirect
42. #816C22semidirect
43. #816C22semidirect
54. C2×C22⋊C8D3semidirect
279. C3#3091semidirect

Maximal subgroups

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

subgroupindex
#57912
C2×#9262
C2×C3×C22⋊C82
#51132
#51062
#47372
#47362
#47352
#47232
#47222
#47212
#30913