Finite Group Databasebeta

← #10698216

Subgroups — #10698216

Order 768.

416 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C3×#2571C211yesmaximal, characteristic
2#69579C211yesmaximal, characteristic
3#69489C212yesmaximal
4#67634C211yesmaximal, characteristic
5#67619C212yesmaximal
6#10352—31nomaximal, Sylow
7C2×#869C2211yescharacteristic
8C3×#400—22no—
9C3×#389—22no—
10C3×#384C2211yescharacteristic
11C3×#382—22no—
12C3×C4×C4⋊C4C2212yes—
13C2×#810—22no—
14C2×C4⋊(C3⋊C8)C2212yes—
15C2×C4×C3⋊C8C2211yescharacteristic
16#4529—22no—
17#4529—22no—
18#2571D311yescharacteristic
19#1987—32no—
20#1956—31no—
21#1296—31no—
22#1279—32no—
23C22×Dic6—21no—
24C2×C3×C4⋊C4—22no—
25C2×C3×C4⋊C4—22no—
26C2×C3×C4⋊C4D412yes—
27C2×C3×C4⋊C4D412yes—
28C2×C3×C4⋊C4D412yes—
29C2×C3×C42—22no—
30C2×C3×C42C2311yescharacteristic
31C2×C4⋊Dic3—22no—
32C22×C3⋊C8—21no—
33C22×C3⋊C8—22no—
34#869—22no—
35#869—21no—
36#869Q812yes—
37C3×#95—22no—
38C3×#95—24no—
39C3×#95—22no—
40#810—44no—
41C4⋊(C3⋊C8)—22no—
42C4⋊(C3⋊C8)—22no—
43C4×C3⋊C8—22no—
44C2×C4⋊Q8—31no—
45C2×C4⋊C8—32no—
46C2×#103—62no—
47C2×C4×C8—31no—
48#400—22no—
49#389—22no—
50#384C2×D311yescharacteristic
51#382—22no—
52C4×C4⋊C4C2×D312yes—
53#340—62no—
54#340—62no—
55C22×C3×C4—22no—
56C22×C3×C4—22no—
57C22×C3×C4—22no—
58C22×C3×C4C2×D411yescharacteristic
59C22×C3×C4C2×D412yes—
60C22×Dic3—41no—
61C2×Dic6—22no—
62C2×Dic6—42no—
63C2×Dic6—42no—
64C2×Dic6—41no—
65C3×C4⋊C4—22no—
66C3×C4⋊C4—22no—
67C3×C4⋊C4—22no—
68C3×C4⋊C4—22no—
69C3×C4⋊C4—42no—
70C3×C4⋊C4—42no—
71C3×C4⋊C4—42no—
72C3×C4⋊C4—82no—
73C3×C42—42no—
74C3×C42—42no—
75C3×C42C2×Q811yescharacteristic
76C3×C42#4112yes—
77C3×C42#4111yescharacteristic
78C4⋊Dic3—24no—
79C4⋊Dic3—42no—
80C2×C3⋊C8—22no—
81C2×C3⋊C8—41no—
82C2×C3⋊C8—41no—
83C2×C3⋊C8—42no—
84C2×C3⋊C8—42no—
85C2×C3⋊C8—42no—
86C22×Q8—61no—
87C22×C8—61no—
88C22×C8—62no—
89C4⋊Q8—32no—
90C4⋊Q8—62no—
91C4⋊Q8—61no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—62no—
95C2×C4⋊C4#6712yes—
96C2×C4⋊C4#6712yes—
97C2×C4⋊C4#6712yes—
98C2×C42—22no—
99C2×C42C22×D311yescharacteristic
100C4⋊C8—62no—
101C4⋊C8—62no—
102#103—124no—
103C4×C8—62no—
104#95—22no—
105#95—24no—
106#95—22no—
107C23×C3#12011yescharacteristic
108C2×C3×C4—21no—
109C2×C3×C4—21no—
110C2×C3×C4—42no—
111C2×C3×C4—42no—
112C2×C3×C4—42no—
113C2×C3×C4—42no—
114C2×C3×C4—44no—
115C2×C3×C4—42no—
116C2×C3×C4—42no—
117C2×C3×C4—42no—
118C2×C3×C4C22⋊Q811yescharacteristic
119C2×C3×C4C22⋊Q812yes—
120C2×C3×C4#12411yescharacteristic
121C2×C3×C4#12412yes—
122C2×C3×C4#13512yes—
123C2×C3×C4#13512yes—
124C2×C3×C4#13712yes—
125C2×C3×C4#13712yes—
126C2×Dic3—42no—
127C2×Dic3—81no—
128C2×Dic3—81no—
129Dic6—44no—
130Dic6—82no—
131C3⋊C8—42no—
132C3⋊C8—82no—
133C2×Q8—62no—
134C2×Q8—122no—
135C2×Q8—122no—
136C2×Q8—121no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—121no—
141C22×C4C2×#6711yescharacteristic
142C22×C4C2×#6712yes—
143C2×C8—62no—
144C2×C8—121no—
145C2×C8—121no—
146C2×C8—122no—
147C2×C8—122no—
148C2×C8—122no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—64no—
157C4⋊C4—82no—
158C4⋊C4—122no—
159C42—42no—
160C42—42no—
161C42D3×Q811yescharacteristic
162C42#23912yes—
163C42#23911yescharacteristic
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—82no—
174C3×C4#47912yes—
175C3×C4#47912yes—
176C3×C4#48812yes—
177C3×C4#49112yes—
178Dic3—82no—
179C23#95311yescharacteristic
180Q8—124no—
181Q8—242no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—44no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—122no—
193C2×C4—241no—
194C2×C4—241no—
195C2×C4#94611yescharacteristic
196C2×C4#94612yes—
197C2×C4#94711yescharacteristic
198C2×C4#94712yes—
199C2×C4#95012yes—
200C2×C4#95012yes—
201C2×C4#95112yes—
202C2×C4#95112yes—
203C8—122no—
204C8—242no—
205C2×C3#163011yescharacteristic
206C2×C3#163411yescharacteristic
207C2×C3#163812yes—
208C2×C3#202612yes—
209C2×C3#202711yescharacteristic
210C22#523111yescharacteristic
211C22#523912yes—
212C22#523912yes—
213C22#524111yescharacteristic
214C22#524511yescharacteristic
215C4—21no—
216C4—21no—
217C4—82no—
218C4—82no—
219C4—82no—
220C4—242no—
221C4#506312yes—
222C4#506312yes—
223C4#506812yes—
224C4#506912yes—
225C3#1035211yescharacteristic, complemented, Sylow
226C2#7174412yes—
227C2#7190311yescharacteristic
228C2#7192811yescharacteristic
229C2#7193212yes—
230C2#7194411yescharacteristic
subgroupG/H
centerC23#953
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2571C2
socle, prime radicalC23×C3#120
hypercenter#2571D3
hypocenterC3#10352
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2571

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
C3×#25712
#695792
#694892
#676342
#676192
#103523