Finite Group Databasebeta

← #11713707

Subgroups — #11713707

Order 768.

474 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 53 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
1A4×#125C211yesmaximal, characteristic, complemented
2#85290C211yesmaximal, characteristic
3#73053C212yesmaximal
4#73052C212yesmaximal, complemented
5C4×A4⋊C8C211yesmaximal, characteristic, complemented
6#25907—31nomaximal, Sylow
7C2×C4⋊S4C2211yescharacteristic
8C2×A4⋊Q8C2211yescharacteristic
9A4×C4⋊C4—22no—
10A4×C4⋊C4C2212yes—
11C42×A4C2211yescharacteristic
12(C2×S4)⋊C4—22no—
13C2×A4⋊C8C2212yes—
14#5068—41nomaximal
15C22×#125D311yescharacteristic, complemented
16C2×#488—31no—
17C2×#488—31no—
18#2850—31no—
19#2026—32no—
20#2025—32no—
21#1991—31no—
22#1991—31no—
23#1874—32no—
24#1872—32no—
25C4×C22⋊C8—31no—
26C22×S4—21no—
27C2×C4×A4—22no—
28C2×C4×A4D412yes—
29C2×C4×A4C2311yescharacteristic
30C2×SL(2,ℤ/4)—21no—
31C4⋊S4—22no—
32A4⋊Q8—22no—
33C3×#125—41no—
34#875—41no—
35A4⋊C8—22no—
36#810—42no—
37#809—42no—
38C4×C3⋊C8—41no—
39C2×#125—31no—
40C2×#125—31no—
41C2×#125—32no—
42C2×#125—62no—
43C2×#124—31no—
44C2×C22⋊Q8—31no—
45C2×#121—31no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4C2×D312yes—
48C22×C42C2×D311yescharacteristic
49#488—64no—
50#488—62no—
51#488—61no—
52#488—61no—
53C2×#103—32no—
54C2×#103—32no—
55C2×#102—32no—
56C2×#102—32no—
57C2×C22⋊C8—32no—
58C2×C4×C8—31no—
59C2×C4×C8—31no—
60#391—31no—
61#390—31no—
62#388—62no—
63#386—62no—
64#382—31no—
65#340—32no—
66#340—32no—
67#340—32no—
68#340—32no—
69#336—32no—
70C22×A4C2×D411yescharacteristic
71C2×S4—41no—
72C2×S4—42no—
73C2×D12—41no—
74C2×Dic6—41no—
75C4×A4—22no—
76C4×A4—42no—
77C4×A4#4112yes—
78SL(2,ℤ/4)—41no—
79C3×C4⋊C4—42no—
80C3×C4⋊C4—82no—
81C3×C42—41no—
82(C2×D3)⋊C4—82no—
83C2×C3⋊C8—42no—
84C22×Q8—31no—
85C22×D4—31no—
86C22×D4—61no—
87C23×C4—22no—
88C23×C4#6712yes—
89C23×C4C22×D311yescharacteristic
90C22×C8—32no—
91C22×C8—32no—
92#125—34no—
93#125—32no—
94#125—31no—
95#125—62no—
96#125—61no—
97#125—64no—
98#125S411yescharacteristic, complemented
99#124—62no—
100#124—61no—
101#124—61no—
102C22⋊Q8—62no—
103C22⋊Q8—62no—
104#121—62no—
105#121—62no—
106C2×C4⋊C4—32no—
107C2×C4⋊C4—32no—
108C2×C4⋊C4—31no—
109C2×C4⋊C4—31no—
110C2×C4⋊C4—62no—
111C2×C4⋊C4—62no—
112C2×C4⋊C4—64no—
113C2×C4⋊C4—62no—
114C2×C4⋊C4—61no—
115C2×C22⋊C4—61no—
116C2×C22⋊C4—62no—
117C2×C22⋊C4—62no—
118C2×C22⋊C4—61no—
119C2×C42—31no—
120C2×C42—31no—
121C2×C42—32no—
122#103—64no—
123#103—62no—
124#103—62no—
125#102—64no—
126#102—62no—
127#102—62no—
128C22⋊C8—62no—
129C22⋊C8—62no—
130C4×C8—62no—
131C4×C8—61no—
132C4×C8—61no—
133#95—62no—
134#95—62no—
135C22×D3—81no—
136C2×A4#12411yescharacteristic
137C2×A4#13512yes—
138S4—81no—
139C2×C3×C4—42no—
140C2×C3×C4—41no—
141C2×C3×C4—82no—
142C2×Dic3—81no—
143D12—82no—
144Dic6—82no—
145C3⋊C8—82no—
146C24—61no—
147C24C2×#6711yescharacteristic
148C2×Q8—61no—
149C2×Q8—62no—
150C2×Q8—62no—
151C2×Q8—61no—
152C2×D4—61no—
153C2×D4—62no—
154C2×D4—62no—
155C2×D4—61no—
156C2×D4—121no—
157C2×D4—122no—
158C2×D4—121no—
159C2×D4—122no—
160C22×C4—22no—
161C22×C4—31no—
162C22×C4—32no—
163C22×C4—31no—
164C22×C4—32no—
165C22×C4—32no—
166C22×C4—42no—
167C22×C4—62no—
168C22×C4—62no—
169C22×C4—62no—
170C22×C4—61no—
171C22×C4—61no—
172C22×C4—61no—
173C22×C4—122no—
174C22×C4#23912yes—
175C2×C8—62no—
176C2×C8—62no—
177C2×C8—62no—
178C2×C8—62no—
179C2×C8—62no—
180C2×C8—62no—
181C4⋊C4—22no—
182C4⋊C4—34no—
183C4⋊C4—32no—
184C4⋊C4—62no—
185C4⋊C4—64no—
186C4⋊C4—64no—
187C4⋊C4—62no—
188C4⋊C4—62no—
189C4⋊C4—62no—
190C4⋊C4—122no—
191C4⋊C4C2×S412yes—
192C22⋊C4—122no—
193C22⋊C4—122no—
194C22⋊C4—124no—
195C22⋊C4—121no—
196C22⋊C4—121no—
197C22⋊C4—122no—
198C42—32no—
199C42—31no—
200C42—61no—
201C42C2×S411yescharacteristic
202C22×C3—41no—
203C2×D3—161no—
204C2×D3—162no—
205A4#48811yescharacteristic, complemented
206C3×C4—42no—
207C3×C4—82no—
208C3×C4—162no—
209Dic3—161no—
210C23—31no—
211C23—31no—
212C23—32no—
213C23—121no—
214C23—122no—
215C23—121no—
216C23—122no—
217C23—121no—
218C23#94711yescharacteristic
219C23#95012yes—
220Q8—122no—
221Q8—122no—
222D4—122no—
223D4—122no—
224D4—241no—
225D4—242no—
226D4—241no—
227C2×C4—22no—
228C2×C4—31no—
229C2×C4—32no—
230C2×C4—32no—
231C2×C4—32no—
232C2×C4—32no—
233C2×C4—62no—
234C2×C4—62no—
235C2×C4—61no—
236C2×C4—62no—
237C2×C4—62no—
238C2×C4—62no—
239C2×C4—122no—
240C2×C4—122no—
241C2×C4—122no—
242C2×C4—122no—
243C2×C4—121no—
244C2×C4—121no—
245C2×C4—121no—
246C2×C4—122no—
247C2×C4—122no—
248C2×C4—121no—
249C2×C4—121no—
250C2×C4GL(2,ℤ/4)12yes—
251C2×C4C22×S411yescharacteristic
252C8—122no—
253C8—122no—
254C2×C3—41no—
255C2×C3—42no—
256D3—321no—
257C22—31no—
258C22—32no—
259C22—31no—
260C22—31no—
261C22—32no—
262C22—32no—
263C22—61no—
264C22—241no—
265C22—241no—
266C22—242no—
267C22—241no—
268C22—242no—
269C22#506811yescharacteristic, complemented
270C22C2×GL(2,ℤ/4)11yescharacteristic
271C4—22no—
272C4—32no—
273C4—42no—
274C4—62no—
275C4—122no—
276C4—241no—
277C4—241no—
278C4—241no—
279C4#591912yes—
280C3—41noSylow
281C2—31no—
282C2—31no—
283C2—32no—
284C2—481no—
285C2#8550811yescharacteristic
286C2#8551412yes—
subgroupG/H
centerC22C2×GL(2,ℤ/4)
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC22×#125D3
socleC24C2×#67
prime radicalC2×C4⋊S4C22
hypercenter#125S4
hypocenterA4#488
LCSG ▷ C2×C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
1. A4×#125C2semidirect
4. #73052C2semidirect
5. C4×A4⋊C8C2semidirect
15. C22×#125D3semidirect
98. #125S4semidirect
205. A4#488semidirect
269. C22#5068semidirect

Maximal subgroups

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

subgroupindex
A4×#1252
#852902
#730532
#730522
C4×A4⋊C82
#259073
#50684