Finite Group Databasebeta

← #10696477

Subgroups — #10696477

Order 768.

552 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 95 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×#5071C212yesmaximal
2#74217C211yesmaximal, characteristic, complemented
3C3×#1961C211yesmaximal, characteristic
4C2×#4465C212yesmaximal, complemented
5#67630C211yesmaximal, characteristic, complemented
6#10080—31nomaximal, Sylow
7C2×C3×#127C2211yescharacteristic
8C2×#870C2212yes—
9C2×C3×#102—22no—
10C2×C3×#97C2211yescharacteristic
11C3×#382C2211yescharacteristic
12C2×#832—22no—
13C2×#832—22no—
14#5071—24no—
15#5071—24no—
16#4981—22no—
17C2×#803C2212yes—
18#4661—22no—
19#4659—22no—
20#4465—24no—
21C2×#490—32no—
22#2822—31no—
23#1961D311yescharacteristic
24C2×#330—32no—
25#1295—31no—
26C22×C3×D4—22no—
27C22×C3×D4—21no—
28C22×C3×C8—21no—
29C3×#127—22no—
30C3×#127—21no—
31C3×#127D412yes—
32C2×C3×C4⋊C4D412yes—
33C2×C3×C42C2311yescharacteristic
34C2×C4⋊Dic3—22no—
35C2×C4⋊Dic3D412yes—
36C2×C4⋊Dic3D412yes—
37C2×Dic3⋊C4—22no—
38C2×Dic3⋊C4—22no—
39C22×C3⋊C8—22no—
40#870—22no—
41#870—22no—
42#870—22no—
43#870—22no—
44C3×#102—44no—
45C3×#97—22no—
46C3×#95—22no—
47#832—28no—
48#832—28no—
49C3⋊#95—22no—
50C3⋊#95—24no—
51#803—22no—
52#803D414yes—
53C2×#127C2×D311yescharacteristic
54C2×#125—32no—
55#490—64no—
56#490—64no—
57C2×#102—22no—
58C2×#102—62no—
59C2×#102—62no—
60C2×#97—32no—
61C2×#97C2×D311yescharacteristic
62#400—62no—
63#398—62no—
64#398—62no—
65#382C2×D311yescharacteristic
66#330—64no—
67C24×C3—41no—
68C2×C3×D4—24no—
69C2×C3×D4—22no—
70C2×C3×D4—42no—
71C2×C3×D4—48no—
72C2×C3×D4—42no—
73C2×C3×D4—42no—
74C2×C3×D4—41no—
75C22×C3×C4—22no—
76C22×C3×C4C2×D412yes—
77C22×C3×C4C2×D411yescharacteristic
78C22×Dic3—22no—
79C22×Dic3—22no—
80C2×C3×C8—22no—
81C2×C3×C8—41no—
82C2×C3×C8—41no—
83C3×C4⋊C4—42no—
84C3×C42C2×D411yescharacteristic
85C3×C42C2×D412yes—
86C3×C42#4111yescharacteristic
87C4⋊Dic3—22no—
88C4⋊Dic3—22no—
89C4⋊Dic3—22no—
90C4⋊Dic3—22no—
91C4⋊Dic3—82no—
92Dic3⋊C4—42no—
93Dic3⋊C4—42no—
94Dic3⋊C4—42no—
95Dic3⋊C4—42no—
96C2×C3⋊C8—24no—
97C2×C3⋊C8—44no—
98C22×D4—22no—
99C22×D4—21no—
100C22×C8—21no—
101C22×C8—62no—
102#127—21no—
103#127—22no—
104#127D1212yes—
105#125—62no—
106#125—62no—
107#125—62no—
108#125—62no—
109C2×C4⋊C4—32no—
110C2×C4⋊C4—32no—
111C2×C4⋊C4—62no—
112C2×C4⋊C4—62no—
113C2×C4⋊C4—62no—
114C2×C4⋊C4#6712yes—
115C2×C42C22×D311yescharacteristic
116#102—44no—
117#102—68no—
118#102—68no—
119#97—22no—
120#97—34no—
121#97—62no—
122#95—22no—
123#95—64no—
124#95—62no—
125C23×C3—42no—
126C23×C3—81no—
127C23×C3—84no—
128C23×C3—81no—
129C23×C3#12011yescharacteristic
130C3×D4—48no—
131C3×D4—44no—
132C3×D4—84no—
133C3×D4—82no—
134C2×C3×C4—21no—
135C2×C3×C4—24no—
136C2×C3×C4—21no—
137C2×C3×C4—42no—
138C2×C3×C4—44no—
139C2×C3×C4#12012yes—
140C2×C3×C4#12112yes—
141C2×C3×C4#12111yescharacteristic
142C2×C3×C4#12711yescharacteristic
143C2×Dic3—42no—
144C2×Dic3—42no—
145C2×Dic3—42no—
146C2×Dic3—44no—
147C2×Dic3—42no—
148C3×C8—42no—
149C3⋊C8—44no—
150C24—41no—
151C2×D4—24no—
152C2×D4—22no—
153C2×D4—42no—
154C2×D4—48no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—41no—
158C22×C4—22no—
159C22×C4—62no—
160C22×C4—62no—
161C22×C4D3×D412yes—
162C22×C4C2×#6711yescharacteristic
163C2×C8—22no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—64no—
167C2×C8—124no—
168C4⋊C4—42no—
169C4⋊C4—62no—
170C4⋊C4—62no—
171C4⋊C4—62no—
172C4⋊C4—62no—
173C4⋊C4—122no—
174C4⋊C4—122no—
175C4⋊C4—122no—
176C4⋊C4—122no—
177C4⋊C4—242no—
178C42C2×D1211yescharacteristic
179C42#23511yescharacteristic
180C42D3×D412yes—
181C22×C3—84no—
182C22×C3—82no—
183C22×C3—162no—
184C22×C3—162no—
185C22×C3#39211yescharacteristic
186C22×C3#45311yescharacteristic
187C22×C3#45414yes—
188C22×C3#45411yescharacteristic
189C3×C4—22no—
190C3×C4—22no—
191C3×C4—21no—
192C3×C4—21no—
193C3×C4—82no—
194Dic3—82no—
195Dic3—82no—
196C23—42no—
197C23—84no—
198C23—81no—
199C23—81no—
200C23#93711yescharacteristic
201D4—44no—
202D4—48no—
203D4—82no—
204D4—84no—
205C2×C4—24no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—44no—
209C2×C4—42no—
210C2×C4—124no—
211C2×C4—122no—
212C2×C4—122no—
213C2×C4—122no—
214C2×C4—122no—
215C2×C4#88212yes—
216C2×C4#88412yes—
217C2×C4#93011yescharacteristic
218C2×C4#94011yescharacteristic
219C8—42no—
220C8—124no—
221C2×C3—162no—
222C2×C3#165712yes—
223C2×C3#165812yes—
224C2×C3#200312yes—
225C2×C3#200611yescharacteristic
226C22—82no—
227C22—84no—
228C22—162no—
229C22—162no—
230C22#496011yescharacteristic
231C22#506211yescharacteristic
232C22#507711yescharacteristic
233C22#520114yes—
234C4—21no—
235C4—21no—
236C4—22no—
237C4—22no—
238C4—82no—
239C4—242no—
240C4—242no—
241C3#1008011yescharacteristic, complemented, Sylow
242C2—162no—
243C2#6966411yescharacteristic
244C2#7032612yes—
245C2#7125012yes—
246C2#7132812yes—
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1961C2
socleC23×C3#120
prime radicalC2×C3×#127C22
hypercenter#1961D3
hypocenterC3#10080
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1961

Complements

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

NH ≅ G/Ntype
2. #74217C2semidirect
4. C2×#4465C2semidirect
5. #67630C2semidirect
241. C3#10080semidirect

Maximal subgroups

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

subgroupindex
C2×#50712
#742172
C3×#19612
C2×#44652
#676302
#100803