Finite Group Databasebeta

← #10682473

Subgroups — #10682473

Order 768.

650 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 67 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×#1781C211yesmaximal, characteristic
2#72130C212yesmaximal
3#7310—31nomaximal, Sylow
4C23×C3×D4C2211yescharacteristic
5C2×C3×#99—22no—
6C3×#379C412yescomplemented
7#5247—22no—
8C2×#834—22no—
9C2×#834C414yescomplemented
10#1781—32no—
11#1781D311yescharacteristic
12C25×C3C2×C411yescharacteristic
13C25×C3D411yescharacteristic
14C22×C3×D4—44no—
15C22×C3×D4—42no—
16C22×C3×D4C2×C412yes—
17C22×C3×D4D412yes—
18C23×C3×C4Q811yescharacteristic
19C2×C3×C22⋊C4—22no—
20C2×C3×C22⋊C4—42no—
21C2×#217—24no—
22C2×#217—44no—
23C3×#99—44no—
24#834—44no—
25#834—44no—
26C23×D4C2×D311yescharacteristic
27C2×#99—22no—
28C2×#99—34no—
29C2×#99—62no—
30#379—62no—
31#379Dic312yescomplemented
32C24×C3—22no—
33C24×C3—24no—
34C24×C3—42no—
35C24×C3—41no—
36C24×C3—82no—
37C24×C3—82no—
38C24×C3C4211yescharacteristic
39C24×C3C22⋊C412yes—
40C24×C3C22⋊C411yescharacteristic
41C24×C3C4⋊C411yescharacteristic
42C2×C3×D4—24no—
43C2×C3×D4—24no—
44C2×C3×D4—24no—
45C2×C3×D4—42no—
46C2×C3×D4—42no—
47C2×C3×D4—42no—
48C2×C3×D4—416no—
49C22×C3×C4—41no—
50C22×C3×C4—42no—
51C22×C3×C4—42no—
52C22×C3×C4C4⋊C412yes—
53C22×Dic3—44no—
54C3×C22⋊C4—44no—
55C3×C22⋊C4—84no—
56#217—44no—
57#217—44no—
58#217—88no—
59C25D1211yescharacteristic
60C25C2×Dic311yescharacteristic
61C22×D4—44no—
62C22×D4—42no—
63C22×D4C4×D312yes—
64C22×D4#6712yes—
65C23×C4Dic611yescharacteristic
66C2×C22⋊C4—22no—
67C2×C22⋊C4—42no—
68C2×C22⋊C4—64no—
69C2×C22⋊C4—124no—
70#99—44no—
71#99—124no—
72#99—124no—
73C23×C3—22no—
74C23×C3—21no—
75C23×C3—22no—
76C23×C3—22no—
77C23×C3—22no—
78C23×C3—21no—
79C23×C3—22no—
80C23×C3—42no—
81C23×C3—44no—
82C23×C3—48no—
83C23×C3—44no—
84C23×C3—44no—
85C23×C3—82no—
86C23×C3—84no—
87C23×C3—82no—
88C23×C3—81no—
89C23×C3—81no—
90C23×C3—82no—
91C23×C3—84no—
92C23×C3—82no—
93C23×C3—84no—
94C23×C3#9511yescharacteristic
95C23×C3#9914yes—
96C23×C3#9912yes—
97C3×D4—48no—
98C3×D4—48no—
99C2×C3×C4—22no—
100C2×C3×C4—22no—
101C2×C3×C4—22no—
102C2×C3×C4—44no—
103C2×C3×C4—82no—
104C2×C3×C4—84no—
105C2×Dic3—84no—
106C2×Dic3—88no—
107C24—24no—
108C24—22no—
109C24—41no—
110C24—42no—
111C24—82no—
112C24—82no—
113C24C4×Dic311yescharacteristic
114C24C4⋊Dic311yescharacteristic
115C24(C2×D3)⋊C412yes—
116C24#21711yescharacteristic
117C2×D4—24no—
118C2×D4—24no—
119C2×D4—24no—
120C2×D4—42no—
121C2×D4—42no—
122C2×D4—42no—
123C2×D4—416no—
124C22×C4—41no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—124no—
128C22×C4Dic3⋊C412yes—
129C22⋊C4—44no—
130C22⋊C4—84no—
131C22⋊C4—124no—
132C22⋊C4—124no—
133C22⋊C4—248no—
134C22×C3—24no—
135C22×C3—22no—
136C22×C3—24no—
137C22×C3—22no—
138C22×C3—41no—
139C22×C3—42no—
140C22×C3—44no—
141C22×C3—44no—
142C22×C3—41no—
143C22×C3—42no—
144C22×C3—42no—
145C22×C3—41no—
146C22×C3—41no—
147C22×C3—82no—
148C22×C3—82no—
149C22×C3—82no—
150C22×C3—88no—
151C22×C3—88no—
152C22×C3#34211yescharacteristic
153C22×C3#34212yes—
154C3×C4—42no—
155C3×C4—162no—
156Dic3—164no—
157C23—21no—
158C23—22no—
159C23—22no—
160C23—22no—
161C23—22no—
162C23—21no—
163C23—22no—
164C23—42no—
165C23—44no—
166C23—44no—
167C23—48no—
168C23—44no—
169C23—82no—
170C23—84no—
171C23—82no—
172C23—81no—
173C23—81no—
174C23—82no—
175C23—84no—
176C23—82no—
177C23—84no—
178C23#80614yes—
179C23C3⋊#9511yescharacteristic
180C23#83412yes—
181D4—48no—
182D4—48no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—44no—
187C2×C4—82no—
188C2×C4—84no—
189C2×C4—244no—
190C2×C4—248no—
191C2×C3—21no—
192C2×C3—22no—
193C2×C3—44no—
194C2×C3—42no—
195C2×C3—42no—
196C2×C3—81no—
197C2×C3#130411yescharacteristic
198C22—22no—
199C22—24no—
200C22—22no—
201C22—24no—
202C22—41no—
203C22—42no—
204C22—44no—
205C22—44no—
206C22—41no—
207C22—42no—
208C22—42no—
209C22—41no—
210C22—41no—
211C22—82no—
212C22—82no—
213C22—82no—
214C22—88no—
215C22—88no—
216C22#452611yescharacteristic
217C22#452712yes—
218C4—42no—
219C4—162no—
220C4—484no—
221C3#731011yescharacteristic, complemented, Sylow
222C2—21no—
223C2—22no—
224C2—42no—
225C2—44no—
226C2—42no—
227C2—81no—
228C2C3⋊#130411yescharacteristic
subgroupG/H
centerC2C3⋊#1304
commutatorC24×C3C42
FrattiniC23×D4C2×D3
FittingC3×#1781C2
socleC2×C3#1304
prime radical, cosocleC23×C3×D4C22
hypercenter#1781D3
hypocenterC3#7310
LCSG ▷ C24×C3 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ #1781

Complements

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

NH ≅ G/Ntype
6. C3×#379C4semidirect
9. C2×#834C4semidirect
31. #379Dic3semidirect
221. C3#7310semidirect

Maximal subgroups

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

subgroupindex
C3×#17812
#721302
#73103