Finite Group Databasebeta

← #10698217

Subgroups — #10698217

Order 768.

416 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 123 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#74342C211yesmaximal, characteristic
2C3×#1956C211yesmaximal, characteristic
3#71779C212yesmaximal
4#67655C212yesmaximal
5#67634C211yesmaximal, characteristic
6#10352—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8C2×C3×#103—22no—
9C2×C3×C4×C8C2211yescharacteristic
10C3×#384C2211yescharacteristic
11#4986—22no—
12#4984—22no—
13#4977—22no—
14C4×C4⋊Dic3C2212yes—
15C2×C4⋊(C3⋊C8)C2212yes—
16#4529—22no—
17#4529—22no—
18#2571—31no—
19#1987—32no—
20#1956D311yescharacteristic
21#1296—31no—
22#1279—32no—
23C22×C3×Q8—21no—
24C22×C3×C8—21no—
25C3×C4⋊Q8—22no—
26C3×C4⋊Q8—21no—
27C3×C4⋊Q8Q812yes—
28C2×C3×C4⋊C4—22no—
29C2×C3×C4⋊C4—22no—
30C2×C3×C4⋊C4D412yes—
31C2×C3×C42C2311yescharacteristic
32C2×C4⋊Dic3—22no—
33C2×C4⋊Dic3D412yes—
34C2×C4⋊Dic3D412yes—
35C2×C4×Dic3—22no—
36C22×C3⋊C8—22no—
37C3×#103—44no—
38C3×C4×C8—22no—
39C3⋊#95—22no—
40C3⋊#95—24no—
41C3⋊#95—22no—
42C4⋊(C3⋊C8)—22no—
43C4⋊(C3⋊C8)—22no—
44C2×C4⋊Q8C2×D311yescharacteristic
45C2×C4⋊C8—32no—
46C2×#103—22no—
47C2×C4×C8C2×D311yescharacteristic
48#400—62no—
49#389—62no—
50#384C2×D311yescharacteristic
51#382—62no—
52C4×C4⋊C4—32no—
53#340—62no—
54#340—62no—
55C2×C3×Q8—22no—
56C2×C3×Q8—42no—
57C2×C3×Q8—42no—
58C2×C3×Q8—41no—
59C22×C3×C4—22no—
60C22×C3×C4—41no—
61C22×C3×C4C2×D412yes—
62C22×C3×C4C2×D411yescharacteristic
63C22×Dic3—22no—
64C22×Dic3—22no—
65C2×C3×C8—22no—
66C2×C3×C8—41no—
67C2×C3×C8—41no—
68C3×C4⋊C4—24no—
69C3×C4⋊C4—42no—
70C3×C4⋊C4—42no—
71C3×C4⋊C4—42no—
72C3×C4⋊C4—42no—
73C3×C42C2×Q811yescharacteristic
74C3×C42#4111yescharacteristic
75C3×C42#4112yes—
76C4⋊Dic3—22no—
77C4⋊Dic3—22no—
78C4⋊Dic3—22no—
79C4⋊Dic3—22no—
80C4⋊Dic3—82no—
81C4×Dic3—42no—
82C4×Dic3—42no—
83C2×C3⋊C8—42no—
84C2×C3⋊C8—42no—
85C2×C3⋊C8—42no—
86C22×Q8—21no—
87C22×C8—21no—
88C22×C8—62no—
89C4⋊Q8—21no—
90C4⋊Q8—22no—
91C4⋊Q8Dic612yes—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—32no—
95C2×C4⋊C4—32no—
96C2×C4⋊C4—62no—
97C2×C4⋊C4#6712yes—
98C2×C42—62no—
99C2×C42C22×D311yescharacteristic
100C4⋊C8—62no—
101C4⋊C8—62no—
102#103—44no—
103C4×C8—22no—
104#95—62no—
105#95—64no—
106#95—62no—
107C23×C3#12011yescharacteristic
108C3×Q8—44no—
109C3×Q8—82no—
110C2×C3×C4—21no—
111C2×C3×C4—21no—
112C2×C3×C4—42no—
113C2×C3×C4—44no—
114C2×C3×C4—42no—
115C2×C3×C4—81no—
116C2×C3×C4—81no—
117C2×C3×C4C22⋊Q812yes—
118C2×C3×C4C22⋊Q811yescharacteristic
119C2×C3×C4#12412yes—
120C2×C3×C4#12411yescharacteristic
121C2×C3×C4#13512yes—
122C2×C3×C4#13512yes—
123C2×C3×C4#13712yes—
124C2×C3×C4#13712yes—
125C2×Dic3—42no—
126C2×Dic3—42no—
127C2×Dic3—42no—
128C2×Dic3—42no—
129C2×Dic3—42no—
130C2×Dic3—42no—
131C3×C8—42no—
132C3⋊C8—82no—
133C2×Q8—22no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—41no—
137C22×C4—22no—
138C22×C4—41no—
139C22×C4—62no—
140C22×C4—62no—
141C22×C4D3×D412yes—
142C22×C4C2×#6711yescharacteristic
143C2×C8—22no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—122no—
147C2×C8—122no—
148C2×C8—122no—
149C4⋊C4—24no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—62no—
155C4⋊C4—62no—
156C4⋊C4—62no—
157C4⋊C4—62no—
158C4⋊C4—242no—
159C42—122no—
160C42—122no—
161C42C2×Dic611yescharacteristic
162C42#23511yescharacteristic
163C42#23712yes—
164C22×C3#39311yescharacteristic
165C22×C3#45212yes—
166C22×C3#45212yes—
167C22×C3#45311yescharacteristic
168C22×C3#45411yescharacteristic
169C3×C4—21no—
170C3×C4—21no—
171C3×C4—82no—
172C3×C4—82no—
173C3×C4#47912yes—
174C3×C4#47912yes—
175C3×C4#48812yes—
176C3×C4#49112yes—
177Dic3—82no—
178Dic3—82no—
179C23#93711yescharacteristic
180Q8—44no—
181Q8—82no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—44no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4—122no—
190C2×C4—122no—
191C2×C4—122no—
192C2×C4—122no—
193C2×C4—122no—
194C2×C4—122no—
195C2×C4#87812yes—
196C2×C4#88512yes—
197C2×C4#91512yes—
198C2×C4#91612yes—
199C2×C4#91812yes—
200C2×C4#91912yes—
201C2×C4#92411yescharacteristic
202C2×C4#93611yescharacteristic
203C8—42no—
204C8—242no—
205C2×C3#163011yescharacteristic
206C2×C3#163411yescharacteristic
207C2×C3#163812yes—
208C2×C3#202612yes—
209C2×C3#202711yescharacteristic
210C22#494711yescharacteristic
211C22#506211yescharacteristic
212C22#507711yescharacteristic
213C22#517312yes—
214C22#518712yes—
215C4—21no—
216C4—21no—
217C4—82no—
218C4—82no—
219C4—242no—
220C4—242no—
221C4#479612yes—
222C4#479712yes—
223C4#479912yes—
224C4#480012yes—
225C3#1035211yescharacteristic, complemented, Sylow
226C2#6964311yescharacteristic
227C2#7023212yes—
228C2#7122912yes—
229C2#7127011yescharacteristic
230C2#7130411yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1956C2
socle, prime radicalC23×C3#120
hypercenter#1956D3
hypocenterC3#10352
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
225. C3#10352semidirect

Maximal subgroups

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

subgroupindex
#743422
C3×#19562
#717792
#676552
#676342
#103523