Finite Group Databasebeta

← #10876311

Subgroups — #10876311

Order 768.

1124 conjugacy classes of subgroups; only the 231 normal subgroups are shown.

normal subgroups only

This group has 1124 conjugacy classes of subgroups — too many to draw. Only its 231 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×#5212C212yesmaximal
2C3×#1779C211yesmaximal, characteristic
3#69938C214yesmaximal, complemented
4#69927C214yesmaximal, complemented
5#69924C214yesmaximal, complemented
6#27600—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C22×#217—22no—
9C22×C4⋊Dic3C2212yes—
10C22×C4⋊Dic3C2212yes—
11C2×C3×C22⋊C8C2214yes—
12C2×C3×C22⋊C8C2212yes—
13#5212—22no—
14#5212—22no—
15#5212C2214yes—
16#5212C2214yes—
17C2×C3⋊#95—22no—
18C2×#818C2214yescomplemented
19#4551C2214yescomplemented
20#4551C2218yescomplemented
21C2×#380—32no—
22#2075—34no—
23#1852—34no—
24#1823—34no—
25#1779D311yescharacteristic
26C25×C3D411yescharacteristic
27C23×C3×C4—28no—
28C23×C3×C4—42no—
29C23×C3×C4D412yes—
30C23×C3×C4D411yescharacteristic
31C23×C3×C4C2311yescharacteristic
32C23×C3×C4C2312yes—
33C23×Dic3—22no—
34C23×Dic3—22no—
35C22×C3×C8C2314yes—
36C2×#217—24no—
37C2×#217—24no—
38C2×#217—48no—
39C2×C4⋊Dic3—28no—
40C2×C4⋊Dic3—22no—
41C2×C4⋊Dic3—22no—
42C2×C4⋊Dic3—44no—
43C2×C4⋊Dic3C2314yes—
44C2×C4⋊Dic3C2314yes—
45C3×C22⋊C8—28no—
46C3×C22⋊C8—42no—
47C3⋊#95—24no—
48C3⋊#95—24no—
49#818—44no—
50C24×C4C2×D311yescharacteristic
51C22×C4⋊C4—32no—
52C22×C4⋊C4—32no—
53C22×C22⋊C4—62no—
54C2×#107—34no—
55C2×C22⋊C8C2×D314yes—
56C2×C22⋊C8C2×D312yes—
57#380—34no—
58#380—34no—
59#380—62no—
60#380—62no—
61C2×#95—62no—
62#340—34no—
63#340—38no—
64C24×C3—24no—
65C24×C3—24no—
66C24×C3—41no—
67C24×C3—41no—
68C24×C3D814yes—
69C24×C3C2×D411yescharacteristic
70C24×C3C2×D412yes—
71C22×C3×C4—24no—
72C22×C3×C4—24no—
73C22×C3×C4—24no—
74C22×C3×C4—24no—
75C22×C3×C4—24no—
76C22×C3×C4—24no—
77C22×C3×C4—22no—
78C22×C3×C4—41no—
79C22×C3×C4—42no—
80C22×C3×C4—416no—
81C22×C3×C4C2×D411yescharacteristic
82C22×C3×C4C2×D412yes—
83C22×C3×C4#4118yes—
84C22×C3×C4C2411yescharacteristic
85C22×Dic3—24no—
86C22×Dic3—24no—
87C22×Dic3—44no—
88C22×Dic3—44no—
89C22×Dic3—44no—
90C22×Dic3—44no—
91C22×Dic3—44no—
92C22×Dic3—44no—
93C2×C3×C8—28no—
94C2×C3×C8—24no—
95#217—48no—
96#217—48no—
97C4⋊Dic3—28no—
98C4⋊Dic3—28no—
99C4⋊Dic3—44no—
100C4⋊Dic3—44no—
101C25D1211yescharacteristic
102C23×C4—28no—
103C23×C4—42no—
104C23×C4—62no—
105C23×C4—62no—
106C23×C4D1212yes—
107C23×C4D1211yescharacteristic
108C23×C4C22×D311yescharacteristic
109C23×C4C22×D312yes—
110C22×C8C22×D314yes—
111C2×C4⋊C4—34no—
112C2×C4⋊C4—34no—
113C2×C4⋊C4—68no—
114C2×C4⋊C4—62no—
115C2×C4⋊C4—62no—
116C2×C4⋊C4—124no—
117C2×C22⋊C4—64no—
118C2×C22⋊C4—64no—
119C2×C22⋊C4—128no—
120#107—124no—
121C22⋊C8—28no—
122C22⋊C8—42no—
123#95—64no—
124#95—64no—
125C23×C3—22no—
126C23×C3—22no—
127C23×C3—22no—
128C23×C3—22no—
129C23×C3—22no—
130C23×C3—21no—
131C23×C3—22no—
132C23×C3—21no—
133C23×C3—216no—
134C23×C3—44no—
135C23×C3—44no—
136C23×C3—48no—
137C23×C3—44no—
138C23×C3#12314yes—
139C23×C3C2×D812yes—
140C23×C3C2×D814yes—
141C23×C3#13714yes—
142C23×C3C22×D411yescharacteristic
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—216no—
146C2×C3×C4—22no—
147C2×C3×C4—21no—
148C2×C3×C4—44no—
149C2×C3×C4—48no—
150C2×C3×C4—44no—
151C2×C3×C4#12314yes—
152C2×C3×C4C2×#4114yes—
153C2×C3×C4H(5,𝔽2)12yes—
154C2×Dic3—44no—
155C2×Dic3—44no—
156C2×Dic3—44no—
157C2×Dic3—44no—
158C2×Dic3—44no—
159C2×Dic3—44no—
160C2×Dic3—88no—
161C2×Dic3—88no—
162C3×C8—44no—
163C24—24no—
164C24—24no—
165C24—41no—
166C24—41no—
167C24D2414yes—
168C24C2×D1211yescharacteristic
169C24C2×D1212yes—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—22no—
177C22×C4—41no—
178C22×C4—42no—
179C22×C4—416no—
180C22×C4—64no—
181C22×C4—64no—
182C22×C4—124no—
183C22×C4—124no—
184C22×C4—124no—
185C22×C4—124no—
186C22×C4—124no—
187C22×C4—124no—
188C22×C4C2×D1211yescharacteristic
189C22×C4C2×D1212yes—
190C22×C4#23718yes—
191C22×C4C23×D311yescharacteristic
192C2×C8—28no—
193C2×C8—24no—
194C4⋊C4—68no—
195C4⋊C4—68no—
196C4⋊C4—124no—
197C4⋊C4—124no—
198C22⋊C4—128no—
199C22⋊C4—128no—
200C22×C3—24no—
201C22×C3—24no—
202C22×C3—24no—
203C22×C3—24no—
204C22×C3—24no—
205C22×C3—24no—
206C22×C3—24no—
207C22×C3—22no—
208C22×C3—416no—
209C22×C3—42no—
210C22×C3—41no—
211C22×C3—41no—
212C22×C3#48018yes—
213C22×C3C2×#12312yes—
214C22×C3#53411yescharacteristic
215C22×C3C22×D811yescharacteristic
216C22×C3C2×#13712yes—
217C22×C3#57511yescharacteristic
218C3×C4—24no—
219C3×C4—41no—
220C3×C4#55114yes—
221Dic3—84no—
222Dic3—84no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—21no—
230C23—21no—
231C23—216no—
232C23—44no—
233C23—44no—
234C23—48no—
235C23—44no—
236C23#88614yes—
237C23C2×D2412yes—
238C23C2×D2414yes—
239C23#90914yes—
240C23C22×D1211yescharacteristic
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—216no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—44no—
247C2×C4—48no—
248C2×C4—44no—
249C2×C4—124no—
250C2×C4—124no—
251C2×C4—124no—
252C2×C4—124no—
253C2×C4—124no—
254C2×C4—124no—
255C2×C4—248no—
256C2×C4—248no—
257C2×C4#88614yes—
258C2×C4C2×#23714yes—
259C2×C4#100412yes—
260C8—44no—
261C2×C3—28no—
262C2×C3—42no—
263C2×C3#248311yescharacteristic
264C2×C3C2×#48012yes—
265C2×C3#309512yes—
266C2×C3#318611yescharacteristic
267C2×C3#319511yescharacteristic
268C22—24no—
269C22—24no—
270C22—24no—
271C22—24no—
272C22—24no—
273C22—24no—
274C22—22no—
275C22—24no—
276C22—416no—
277C22—42no—
278C22—41no—
279C22—41no—
280C22#473618yes—
281C22C2×#88612yes—
282C22#549311yescharacteristic
283C22C22×D2411yescharacteristic
284C22C2×#90912yes—
285C22#574811yescharacteristic
286C4—24no—
287C4—41no—
288C4—244no—
289C4—244no—
290C4#558714yes—
291C3#2760011yescharacteristic, complemented, Sylow
292C2—28no—
293C2—42no—
294C2#7383311yescharacteristic
295C2C2×#473612yes—
296C2#7861111yescharacteristic
297C2#7862512yes—
298C2#7864611yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1779C2
socleC23×C3C22×D4
prime radicalC25×C3D4
hypercenter#1779D3
hypocenterC3#27600
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
3. #69938C2semidirect
4. #69927C2semidirect
5. #69924C2semidirect
18. C2×#818C22semidirect
19. #4551C22semidirect
20. #4551C22semidirect
291. C3#27600semidirect

Maximal subgroups

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

subgroupindex
C2×#52122
C3×#17792
#699382
#699272
#699242
#276003