Finite Group Databasebeta

← #10698440

Subgroups — #10698440

Order 768.

542 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 75 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#73844C211yesmaximal, characteristic
2C3×#1819C211yesmaximal, characteristic
3#71542C212yesmaximal
4C3⋊#1304C211yesmaximal, characteristic
5C3⋊#1304C212yesmaximal
6#10391—31nomaximal, Sylow
7C22×C3×C22⋊C4C2211yescharacteristic
8C3×#379C2211yescharacteristic
9C3×#379C2211yescharacteristic
10#5247C2212yes—
11#5247C2212yes—
12#4947—22no—
13#4947—24no—
14C3×#342—22no—
15#4526—24no—
16#2840—31no—
17#1819—32no—
18#1819D311yescharacteristic
19#1304—32no—
20#1304—31no—
21C25×C3C2311yescharacteristic
22C23×C3×C4Q812yescomplemented
23C22×Dic6—22no—
24C2×C3×C22⋊C4—22no—
25C2×C3×C22⋊C4—21no—
26C2×C3×C22⋊C4—21no—
27C2×C3×C22⋊C4—22no—
28C2×C3×C22⋊C4—21no—
29C2×C3×C22⋊C4—22no—
30C2×C3×C22⋊C4—22no—
31C2×C3×C22⋊C4D412yes—
32C2×#217—22no—
33C2×#217—22no—
34C2×#217—22no—
35C2×#217—22no—
36C2×#217—22no—
37C2×#217D414yes—
38C3⋊#95—26no—
39C22×C22⋊C4C2×D311yescharacteristic
40#393—64no—
41#393—62no—
42#379—32no—
43#379—32no—
44#379C2×D311yescharacteristic
45#379C2×D311yescharacteristic
46#342—22no—
47#342—64no—
48C24×C3—42no—
49C24×C3—82no—
50C24×C3C2×D412yes—
51C24×C3C2×D411yescharacteristic
52C24×C3C2×Q811yescharacteristic
53C24×C3#4111yescharacteristic
54C24×C3#4112yes—
55C22×C3×C4—22no—
56C22×C3×C4—22no—
57C22×C3×C4—22no—
58C22×C3×C4—22no—
59C22×C3×C4—41no—
60C22×C3×C4—82no—
61C22×Dic3—24no—
62C22×Dic3—42no—
63C2×Dic6—84no—
64C2×Dic6—82no—
65C3×C22⋊C4—41no—
66C3×C22⋊C4—42no—
67C3×C22⋊C4—41no—
68C3×C22⋊C4—42no—
69C3×C22⋊C4—42no—
70C3×C22⋊C4—82no—
71C3×C22⋊C4—82no—
72#217—44no—
73#217—44no—
74#217—82no—
75#217—82no—
76#217—82no—
77#217—82no—
78C25C22×D311yescharacteristic
79C22×Q8—62no—
80C23×C4Dic612yescomplemented
81C2×C22⋊C4—21no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—34no—
89C2×C22⋊C4—62no—
90C2×C22⋊C4—62no—
91C2×C22⋊C4—62no—
92C2×C22⋊C4—62no—
93C2×C22⋊C4—62no—
94C2×C22⋊C4#6712yes—
95#95—66no—
96C23×C3—24no—
97C23×C3—22no—
98C23×C3—42no—
99C23×C3—41no—
100C23×C3—42no—
101C23×C3—42no—
102C23×C3—42no—
103C23×C3—42no—
104C23×C3—42no—
105C23×C3—42no—
106C23×C3—41no—
107C23×C3—42no—
108C23×C3—88no—
109C23×C3#12011yescharacteristic
110C23×C3C22⋊Q812yes—
111C23×C3C22⋊Q811yescharacteristic
112C23×C3#12412yes—
113C23×C3#12411yescharacteristic
114C2×C3×C4—44no—
115C2×C3×C4—42no—
116C2×C3×C4—42no—
117C2×C3×C4—82no—
118C2×C3×C4—82no—
119C2×C3×C4—82no—
120C2×C3×C4—81no—
121C2×C3×C4—82no—
122C2×Dic3—44no—
123C2×Dic3—44no—
124C2×Dic3—44no—
125C2×Dic3—82no—
126C2×Dic3—82no—
127C2×Dic3—82no—
128Dic6—88no—
129C24—42no—
130C24—82no—
131C24C2×Dic611yescharacteristic
132C24#23511yescharacteristic
133C24D3×D412yes—
134C24#23712yes—
135C24C2×#6711yescharacteristic
136C2×Q8—244no—
137C2×Q8—242no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—41no—
143C22×C4—64no—
144C22×C4—82no—
145C22×C4—122no—
146C22⋊C4—42no—
147C22⋊C4—41no—
148C22⋊C4—41no—
149C22⋊C4—42no—
150C22⋊C4—42no—
151C22⋊C4—82no—
152C22⋊C4—82no—
153C22⋊C4—124no—
154C22⋊C4—124no—
155C22⋊C4—242no—
156C22⋊C4—242no—
157C22⋊C4—242no—
158C22⋊C4—242no—
159C22×C3—22no—
160C22×C3—22no—
161C22×C3—21no—
162C22×C3—22no—
163C22×C3—21no—
164C22×C3—22no—
165C22×C3—22no—
166C22×C3—22no—
167C22×C3—22no—
168C22×C3—22no—
169C22×C3—22no—
170C22×C3—44no—
171C22×C3—84no—
172C22×C3—82no—
173C22×C3—82no—
174C22×C3—84no—
175C22×C3#39311yescharacteristic
176C22×C3#45812yes—
177C3×C4—82no—
178C3×C4—82no—
179C3×C4—161no—
180Dic3—84no—
181Dic3—162no—
182C23—22no—
183C23—24no—
184C23—42no—
185C23—41no—
186C23—42no—
187C23—42no—
188C23—42no—
189C23—42no—
190C23—42no—
191C23—42no—
192C23—41no—
193C23—42no—
194C23—88no—
195C23#87812yes—
196C23#88512yes—
197C23#92411yescharacteristic
198C23#93611yescharacteristic
199C23#93711yescharacteristic
200Q8—248no—
201C2×C4—44no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—81no—
208C2×C4—82no—
209C2×C4—124no—
210C2×C4—124no—
211C2×C4—124no—
212C2×C4—242no—
213C2×C4—242no—
214C2×C4—242no—
215C2×C3—22no—
216C2×C3—42no—
217C2×C3—41no—
218C2×C3—42no—
219C2×C3—41no—
220C2×C3#203011yescharacteristic
221C2×C3#203212yes—
222C22—22no—
223C22—22no—
224C22—22no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—22no—
229C22—21no—
230C22—22no—
231C22—22no—
232C22—21no—
233C22—44no—
234C22—82no—
235C22—84no—
236C22—84no—
237C22—82no—
238C22#494711yescharacteristic
239C22#505312yes—
240C4—82no—
241C4—82no—
242C4—161no—
243C4—244no—
244C4—482no—
245C3#1039111yescharacteristic, complemented, Sylow
246C2—22no—
247C2—41no—
248C2—42no—
249C2—42no—
250C2—41no—
251C2#6953611yescharacteristic
252C2#7002512yes—
subgroupG/H
centerC22#4947
commutator, prime radical, cosocleC25×C3C23
FrattiniC25C22×D3
FittingC3×#1819C2
socleC22×C3#393
hypercenter#1819D3
hypocenterC3#10391
LCSG ▷ C25×C3 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
22. C23×C3×C4Q8semidirect
80. C23×C4Dic6semidirect
245. C3#10391semidirect

Maximal subgroups

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

subgroupindex
#738442
C3×#18192
#715422
C3⋊#13042
#103913