Finite Group Databasebeta

← #10682490

Subgroups — #10682490

Order 768.

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

normal subgroups only

This group has 456 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×#1785C211yesmaximal, characteristic
2#72139C211yesmaximal, characteristic
3#72137C211yesmaximal, characteristic
4#7319—31nomaximal, Sylow
5C22×C3×#41C2211yescharacteristic
6C2×C3×#101—21no—
7C2×C3×#100—21no—
8C3×#407C412yescomplemented
9C2×#836—21no—
10C2×#834—21no—
11C2×#834—21no—
12C2×#833—21no—
13#5211C412yes—
14#5207C412yescomplemented
15#1785—31no—
16#1785D311yescharacteristic
17#1782—31no—
18C2×C3×#41—42no—
19C2×C3×#41—42no—
20C2×C3×#41—42no—
21C22×C3×Q8D411yescharacteristic
22C22×C3×D4D411yescharacteristic
23C22×C3×D4D411yescharacteristic
24C22×C3×D4Q811yescharacteristic
25C23×C3×C4C2×C411yescharacteristic
26C23×C3×C4C2×C411yescharacteristic
27C23×C3×C4C2×C411yescharacteristic
28C2×C3×M16—22no—
29C2×#217—22no—
30C2×#208—22no—
31C3×#101—42no—
32C3×#100—42no—
33C3×C22⋊C8—42no—
34#836—42no—
35#834—42no—
36#834—42no—
37#833—42no—
38C3⋊#95—42no—
39#830—42no—
40C22×#41C2×D311yescharacteristic
41C2×#101—21no—
42C2×#101—61no—
43C2×#100—21no—
44C2×#100—61no—
45C2×#99—61no—
46C2×#99—61no—
47#407—32no—
48#407Dic312yescomplemented
49#381—32no—
50C24×C3C22⋊C411yescharacteristic
51C24×C3C4⋊C411yescharacteristic
52C24×C3C4⋊C411yescharacteristic
53C3×#41—416no—
54C2×C3×Q8—22no—
55C2×C3×Q8—22no—
56C2×C3×Q8—41no—
57C2×C3×D4—22no—
58C2×C3×D4—22no—
59C2×C3×D4—22no—
60C2×C3×D4—22no—
61C2×C3×D4—41no—
62C2×C3×D4—41no—
63C2×C3×D4—41no—
64C2×C3×D4—41no—
65C2×C3×D4—41no—
66C22×C3×C4—22no—
67C22×C3×C4—22no—
68C22×C3×C4—22no—
69C22×C3×C4—41no—
70C22×C3×C4—41no—
71C22×C3×C4—41no—
72C22×C3×C4—41no—
73C22×C3×C4—41no—
74C22×C3×C4—41no—
75C22×C3×C4C4211yescharacteristic
76C22×C3×C4C22⋊C411yescharacteristic
77C22×C3×C4C22⋊C411yescharacteristic
78C22×C3×C4C4⋊C411yescharacteristic
79C22×Dic3—42no—
80C3×M16—42no—
81C3×M16—42no—
82C2×C3×C8—42no—
83#217—42no—
84#217—42no—
85#208—42no—
86#208—42no—
87C2×C3⋊C8—42no—
88C2×#41—42no—
89C2×#41—42no—
90C2×#41—42no—
91C22×Q8#6711yescharacteristic
92C22×D4Dic611yescharacteristic
93C22×D4D1211yescharacteristic
94C22×D4#6711yescharacteristic
95C23×C4C4×D311yescharacteristic
96C23×C4C4×D311yescharacteristic
97C23×C4C2×Dic311yescharacteristic
98C2×M16—22no—
99C2×M16—62no—
100C2×C22⋊C4—62no—
101#101—42no—
102#101—122no—
103#100—42no—
104#100—122no—
105#99—122no—
106#99—122no—
107C22⋊C8—42no—
108C22⋊C8—122no—
109#95—122no—
110C23×C3—21no—
111C23×C3—21no—
112C23×C3—21no—
113C23×C3—21no—
114C23×C3—21no—
115C23×C3—21no—
116C23×C3—21no—
117C23×C3—21no—
118C23×C3—81no—
119C23×C3—81no—
120C23×C3—81no—
121C23×C3#9511yescharacteristic
122C23×C3#9912yes—
123C3×Q8—44no—
124C3×D4—44no—
125C3×D4—44no—
126C3×D4—44no—
127C2×C3×C4—21no—
128C2×C3×C4—21no—
129C2×C3×C4—21no—
130C2×C3×C4—21no—
131C2×C3×C4—21no—
132C2×C3×C4—21no—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—44no—
138C2×C3×C4—44no—
139C2×C3×C4—44no—
140C2×C3×C4#9912yes—
141C2×C3×C4#9912yes—
142C2×Dic3—82no—
143C2×Dic3—84no—
144C3×C8—82no—
145C3⋊C8—82no—
146C24Dic3⋊C411yescharacteristic
147C24C4⋊Dic311yescharacteristic
148C24(C2×D3)⋊C411yescharacteristic
149#41—416no—
150C2×Q8—22no—
151C2×Q8—22no—
152C2×Q8—41no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—22no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—41no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—122no—
172C22×C4C4×Dic311yescharacteristic
173C22×C4Dic3⋊C411yescharacteristic
174C22×C4(C2×D3)⋊C411yescharacteristic
175C22×C4#21711yescharacteristic
176M16—42no—
177M16—42no—
178M16—122no—
179M16—122no—
180C2×C8—42no—
181C2×C8—122no—
182C22⋊C4—122no—
183C22⋊C4—122no—
184C22×C3—22no—
185C22×C3—41no—
186C22×C3—42no—
187C22×C3—42no—
188C22×C3—42no—
189C22×C3—41no—
190C22×C3—41no—
191C22×C3—81no—
192C22×C3—81no—
193C22×C3—81no—
194C22×C3—81no—
195C22×C3—81no—
196C22×C3—81no—
197C22×C3#34211yescharacteristic
198C22×C3#34211yescharacteristic
199C22×C3#34211yescharacteristic
200C3×C4—22no—
201C3×C4—22no—
202C3×C4—41no—
203C3×C4—41no—
204Dic3—162no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—81no—
214C23—81no—
215C23—81no—
216C23#80612yes—
217C23C3⋊#9511yescharacteristic
218Q8—44no—
219D4—44no—
220D4—44no—
221D4—44no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—242no—
236C2×C4—244no—
237C2×C4#80612yes—
238C2×C4#83412yes—
239C8—82no—
240C8—242no—
241C2×C3—21no—
242C2×C3—21no—
243C2×C3—21no—
244C2×C3—42no—
245C2×C3—81no—
246C2×C3—81no—
247C2×C3#130411yescharacteristic
248C22—22no—
249C22—41no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—41no—
254C22—41no—
255C22—81no—
256C22—81no—
257C22—81no—
258C22—81no—
259C22—81no—
260C22—81no—
261C22#452611yescharacteristic
262C22#452711yescharacteristic
263C22#452711yescharacteristic
264C4—22no—
265C4—22no—
266C4—41no—
267C4—41no—
268C4—482no—
269C3#731911yescharacteristic, complemented, Sylow
270C2—21no—
271C2—21no—
272C2—21no—
273C2—42no—
274C2—81no—
275C2—81no—
276C2C3⋊#130411yescharacteristic
subgroupG/H
centerC2C3⋊#1304
commutatorC22×C3×C4C42
FrattiniC22×#41C2×D3
FittingC3×#1785C2
socleC2×C3#1304
prime radical, cosocleC22×C3×#41C22
hypercenter#1785D3
hypocenterC3#7319
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ #1785

Complements

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

NH ≅ G/Ntype
8. C3×#407C4semidirect
14. #5207C4semidirect
48. #407Dic3semidirect
269. C3#7319semidirect

Maximal subgroups

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

subgroupindex
C3×#17852
#721392
#721372
#73193