Finite Group Databasebeta

← #11714096

Subgroups — #11714096

Order 768.

671 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 671 conjugacy classes of subgroups — too many to draw. Only its 77 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×#5401C214yesmaximal, complemented
2#85289C211yesmaximal, characteristic
3#85283C211yesmaximal, characteristic, complemented
4C4×C8×A4C211yesmaximal, characteristic, complemented
5#33170—31nomaximal, Sylow
6C2×C4⋊S4—22no—
7C2×C4⋊S4C2212yes—
8C2×A4⋊Q8C2212yes—
9C2×C8×A4C2212yes—
10C42×A4C2211yescharacteristic
11C4⋊SL(2,ℤ/4)—22no—
12#5401—28no—
13#4693—41nomaximal
14C2×#492—31no—
15C2×#465—34no—
16C22×C4×C8D311yescharacteristic, complemented
17#2857—31no—
18#2856—31no—
19#1968—34no—
20#1957—31no—
21#1956—31no—
22#1844—31no—
23C22×S4—22no—
24C2×C4×A4D412yes—
25C2×C4×A4C2311yescharacteristic
26C2×SL(2,ℤ/4)—22no—
27C4⋊S4—24no—
28C4⋊S4—44no—
29A4⋊Q8—24no—
30C2×#204—44no—
31#874—41no—
32#869—41no—
33C8×A4D414yescomplemented
34C3×C4×C8—41no—
35C22×SD16—34no—
36C23×C8C2×D312yes—
37C2×C4⋊Q8—31no—
38C2×#127—31no—
39C2×C22⋊Q8—32no—
40C2×#121—32no—
41C2×#121—62no—
42C22×C42C2×D311yescharacteristic
43#492—62no—
44#492—61no—
45#492—61no—
46#465—68no—
47#465—68no—
48C2×#106—34no—
49C2×#103—34no—
50C2×#102—34no—
51C2×C4×C8—31no—
52C2×C4×C8—31no—
53C2×C4×C8—32no—
54#391—32no—
55#390—32no—
56#390—62no—
57#389—62no—
58#384—31no—
59#384—31no—
60#380—62no—
61C22×A4C2×D411yescharacteristic
62C2×S4—42no—
63C2×S4—44no—
64C2×D12—42no—
65C2×D12—82no—
66C2×Dic6—42no—
67C4×A4SD1614yescomplemented
68C4×A4C2×D412yes—
69SL(2,ℤ/4)—42no—
70C2×C3×C8—42no—
71C3×C42—41no—
72C4⋊Dic3—82no—
73#204—88no—
74C22×Q8—32no—
75C22×D4—32no—
76C22×D4—62no—
77C22×D4—62no—
78C23×C4D1212yes—
79C23×C4C22×D311yescharacteristic
80C2×SD16—68no—
81C2×SD16—68no—
82C2×SD16—64no—
83C2×SD16—64no—
84C22×C8—32no—
85C22×C8—32no—
86C22×C8—34no—
87C22×C8D1214yescomplemented
88C4⋊Q8—62no—
89C4⋊Q8—61no—
90C4⋊Q8—61no—
91#127—62no—
92#127—61no—
93#127—61no—
94C22⋊Q8—64no—
95C22⋊Q8—64no—
96#121—64no—
97#121—64no—
98#121—124no—
99#121—124no—
100C2×C4⋊C4—32no—
101C2×C4⋊C4—32no—
102C2×C4⋊C4—62no—
103C2×C4⋊C4—62no—
104C2×C4⋊C4—62no—
105C2×C4⋊C4—62no—
106C2×C22⋊C4—62no—
107C2×C22⋊C4—64no—
108C2×C22⋊C4—62no—
109C2×C42—31no—
110C2×C42—32no—
111C2×C42—31no—
112#106—68no—
113#103—68no—
114#102—68no—
115C4×C8—32no—
116C4×C8—31no—
117C4×C8—61no—
118C4×C8S411yescharacteristic, complemented
119#95—64no—
120C22×D3—82no—
121C2×A4#12711yescharacteristic
122C2×A4C2×SD1612yes—
123S4—82no—
124C2×C3×C4—42no—
125C2×C3×C4—41no—
126C2×Dic3—82no—
127D12—84no—
128D12—164no—
129Dic6—84no—
130C3×C8—44no—
131C24—62no—
132C24C2×D1211yescharacteristic
133C2×Q8—64no—
134C2×Q8—62no—
135C2×Q8—64no—
136C2×Q8—62no—
137C2×D4—64no—
138C2×D4—62no—
139C2×D4—64no—
140C2×D4—62no—
141C2×D4—124no—
142C2×D4—122no—
143C2×D4—124no—
144C2×D4—122no—
145C2×D4—124no—
146C2×D4—122no—
147C2×D4—124no—
148C2×D4—122no—
149C22×C4—32no—
150C22×C4—31no—
151C22×C4—32no—
152C22×C4—31no—
153C22×C4—32no—
154C22×C4—34no—
155C22×C4—62no—
156C22×C4—62no—
157C22×C4—62no—
158C22×C4#20414yescomplemented
159C22×C4C2×D1212yes—
160SD16—128no—
161SD16—128no—
162C2×C8—32no—
163C2×C8—34no—
164C2×C8—34no—
165C2×C8—34no—
166C2×C8—62no—
167C2×C8C2×S412yes—
168C4⋊C4—64no—
169C4⋊C4—64no—
170C4⋊C4—124no—
171C4⋊C4—124no—
172C4⋊C4—124no—
173C4⋊C4—122no—
174C4⋊C4—122no—
175C22⋊C4—124no—
176C22⋊C4—128no—
177C22⋊C4—122no—
178C22⋊C4—122no—
179C42—32no—
180C42—31no—
181C42—61no—
182C42C2×S411yescharacteristic
183C22×C3—41no—
184C2×D3—162no—
185C2×D3—164no—
186A4#49211yescharacteristic, complemented
187C3×C4—42no—
188C3×C4—44no—
189Dic3—162no—
190C23—31no—
191C23—31no—
192C23—32no—
193C23—122no—
194C23—124no—
195C23—122no—
196C23—124no—
197C23—122no—
198C23#87411yescharacteristic
199C23C2×#20412yes—
200Q8—124no—
201Q8—124no—
202D4—124no—
203D4—124no—
204D4—244no—
205D4—242no—
206D4—244no—
207D4—242no—
208D4—244no—
209C2×C4—32no—
210C2×C4—31no—
211C2×C4—32no—
212C2×C4—32no—
213C2×C4—34no—
214C2×C4—34no—
215C2×C4—32no—
216C2×C4—34no—
217C2×C4—61no—
218C2×C4—62no—
219C2×C4—122no—
220C2×C4—122no—
221C2×C4—124no—
222C2×C4—122no—
223C2×C4—124no—
224C2×C4—122no—
225C2×C4—122no—
226C2×C4C4⋊S412yes—
227C2×C4C22×S411yescharacteristic
228C8—34no—
229C8C4⋊S414yescomplemented
230C2×C3—41no—
231C2×C3—42no—
232D3—322no—
233C22—31no—
234C22—32no—
235C22—31no—
236C22—31no—
237C22—32no—
238C22—32no—
239C22—61no—
240C22—242no—
241C22—242no—
242C22—244no—
243C22—242no—
244C22—244no—
245C22#469311yescharacteristic, complemented
246C22C2×C4⋊S411yescharacteristic
247C4—32no—
248C4—34no—
249C4—242no—
250C4—242no—
251C4—242no—
252C4#540114yescomplemented
253C4C2×C4⋊S412yes—
254C3—41noSylow
255C2—31no—
256C2—31no—
257C2—32no—
258C2—482no—
259C2#8528911yescharacteristic
260C2C2×#540112yes—
subgroupG/H
centerC22C2×C4⋊S4
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC22×C4×C8D3
socleC24C2×D12
prime radical#85289C2
hypercenterC4×C8S4
hypocenterA4#492
LCSG ▷ C2×C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8

Complements

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

NH ≅ G/Ntype
1. C2×#5401C2semidirect
3. #85283C2semidirect
4. C4×C8×A4C2semidirect
16. C22×C4×C8D3semidirect
33. C8×A4D4semidirect
67. C4×A4SD16semidirect
87. C22×C8D12semidirect
118. C4×C8S4semidirect
158. C22×C4#204semidirect
186. A4#492semidirect
229. C8C4⋊S4semidirect
245. C22#4693semidirect
252. C4#5401semidirect

Maximal subgroups

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

subgroupindex
C2×#54012
#852892
#852832
C4×C8×A42
#331703
#46934