Finite Group Databasebeta

← #10795162

Subgroups — #10795162

Order 768.

1382 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 1382 conjugacy classes of subgroups — too many to draw. Only its 251 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×#5521C211yesmaximal, characteristic
2#78620C212yesmaximal, complemented
3C3×#1822C211yesmaximal, characteristic, complemented
4#71589C211yesmaximal, characteristic, complemented
5#69988C212yesmaximal
6#68994C214yesmaximal, complemented
7#68972C214yesmaximal, complemented
8#20729—31nomaximal, Sylow
9C23×D12C2211yescharacteristic
10C2×C3×#117C2211yescharacteristic
11C2×#926C2211yescharacteristic
12#5521—28no—
13C2×#893C2212yes—
14C2×#893C2212yes—
15C2×D3×C22⋊C4C414yes—
16C2×C4×D12C414yes—
17C3×#407C2212yes—
18#5207C2212yes—
19C2×#823C2218yes—
20#5118C2214yescomplemented
21Dic3×M16C2214yescomplemented
22#4720—28no—
23C3×#343C2214yescomplemented
24#4530C2214yescomplemented
25C2×#518—31no—
26#2921—32no—
27#1822—31no—
28#1822D311yescharacteristic, complemented
29#1800—34no—
30#1783—32no—
31C4×#100—34no—
32C24×D3C2×C412yes—
33C23×C3×C4C2311yescharacteristic
34C22×D12—24no—
35C22×D12—28no—
36C22×D12C2×C412yes—
37C22×D12C2312yes—
38C22×C4×D3C2×C414yes—
39C22×C4×D3C2×C414yes—
40C2×C3×M16C2314yes—
41C3×#117—24no—
42C2×C3×C4⋊C4C2312yes—
43C2×C3×C22⋊C4C2×C412yes—
44C2×C3×C42C2×C412yes—
45C2×#217C2×C412yes—
46C2×(C2×D3)⋊C4C2×C414yes—
47C2×(C2×D3)⋊C4C2×C414yes—
48#926D418yes—
49C2×C4⋊Dic3C2×C412yes—
50C2×C4×Dic3C2311yescharacteristic
51C2×C4×Dic3C2311yescharacteristic
52C2×#208C2314yes—
53#893—28no—
54#893—28no—
55D3×C22⋊C4—216no—
56C4×D12—216no—
57C3×C22⋊C8—24no—
58#830—24no—
59#823—216no—
60#815—24no—
61Dic3⋊C8—28no—
62C8×Dic3—24no—
63C23×D4—31no—
64#518—68no—
65C2×C4×D4—32no—
66C2×C4×D4—32no—
67C2×C4×D4—34no—
68C2×#117—31no—
69C2×#117C2×D311yescharacteristic
70C22×C22⋊C4—34no—
71#433—68no—
72#423—34no—
73C2×#100—38no—
74#407—32no—
75#407C2×D312yes—
76C4×M16—34no—
77#343—34no—
78#343C2×D314yescomplemented
79C24×C3C22×C411yescharacteristic
80C23×D3—24no—
81C23×D3—28no—
82C23×D3—84no—
83C23×D3C22×C414yes—
84C22×C3×C4—24no—
85C22×C3×C4C22×C411yescharacteristic
86C22×C3×C4C22×C414yes—
87C22×C3×C4C2×D414yes—
88C22×C3×C4C2411yescharacteristic
89C22×Dic3C22×C412yes—
90C22×Dic3C22×C412yes—
91C2×D12—232no—
92C2×D12—28no—
93C2×D12—24no—
94C2×D12—24no—
95C2×D12—24no—
96C2×D12#4118yes—
97C2×C4×D3—28no—
98C2×C4×D3—28no—
99C2×C4×D3—84no—
100C2×C4×D3—84no—
101C3×M16—28no—
102C2×C3×C8—24no—
103C3×C4⋊C4—24no—
104C3×C22⋊C4—24no—
105C3×C42—24no—
106#217C22⋊C418yes—
107(C2×D3)⋊C4—28no—
108(C2×D3)⋊C4—28no—
109C4⋊Dic3C22⋊C418yes—
110C4×Dic3C2×D414yes—
111C4×Dic3C2×D414yes—
112#208—28no—
113C2×C3⋊C8—24no—
114C25—32no—
115C22×D4—32no—
116C22×D4—32no—
117C22×D4—68no—
118C22×D4—64no—
119C23×C4—34no—
120C23×C4—34no—
121C23×C4C22×D311yescharacteristic
122C2×M16—34no—
123C2×M16C22×D314yes—
124C4×D4—616no—
125C4×D4—68no—
126C4×D4—68no—
127#117—24no—
128#117—38no—
129C2×C4⋊C4—32no—
130C2×C4⋊C4C22×D312yes—
131C2×C22⋊C4—32no—
132C2×C22⋊C4—34no—
133C2×C22⋊C4—34no—
134C2×C22⋊C4—616no—
135C2×C22⋊C4C4×D312yes—
136C2×C42—31no—
137C2×C42—31no—
138C2×C42C4×D312yes—
139C4⋊C8—68no—
140#100—616no—
141C22⋊C8—24no—
142C22⋊C8—64no—
143#97—64no—
144C4×C8—64no—
145C23×C3—22no—
146C23×C3—42no—
147C23×C3C2×C22⋊C412yes—
148C23×C3C23×C411yescharacteristic
149C22×D3—24no—
150C22×D3—24no—
151C22×D3—216no—
152C22×D3—44no—
153C22×D3—44no—
154C22×D3—88no—
155C22×D3—816no—
156C22×D3#11718yes—
157C2×C3×C4—22no—
158C2×C3×C4—28no—
159C2×C3×C4—24no—
160C2×C3×C4—44no—
161C2×C3×C4C2×C22⋊C412yes—
162C2×C3×C4C22×D412yes—
163C2×C3×C4C2×#4114yes—
164C2×Dic3—42no—
165C2×Dic3—42no—
166C2×Dic3C2×C22⋊C414yes—
167C2×Dic3C2×C22⋊C414yes—
168D12—216no—
169D12—216no—
170C4×D3—88no—
171C4×D3—88no—
172C3×C8—44no—
173C3⋊C8—44no—
174C24—34no—
175C24—68no—
176C24—64no—
177C24—244no—
178C24C2×C4×D311yescharacteristic
179C2×D4—38no—
180C2×D4—68no—
181C2×D4—632no—
182C2×D4—64no—
183C2×D4—64no—
184C2×D4—64no—
185C22×C4—24no—
186C22×C4—32no—
187C22×C4—32no—
188C22×C4—68no—
189C22×C4—68no—
190C22×C4—244no—
191C22×C4—244no—
192C22×C4C2×C4×D311yescharacteristic
193C22×C4C2×C4×D314yes—
194C22×C4D3×D414yes—
195C22×C4C23×D311yescharacteristic
196M16—28no—
197M16—68no—
198C2×C8—24no—
199C2×C8—64no—
200C4⋊C4—24no—
201C4⋊C4—38no—
202C22⋊C4—24no—
203C22⋊C4—38no—
204C22⋊C4—68no—
205C22⋊C4—68no—
206C42—24no—
207C42—34no—
208C42—34no—
209C22×C3—22no—
210C22×C3—24no—
211C22×C3—22no—
212C22×C3—44no—
213C22×C3C22×C22⋊C411yescharacteristic
214C22×C3C2×#11712yes—
215C2×D3—28no—
216C2×D3—48no—
217C2×D3—44no—
218C2×D3—44no—
219C2×D3—48no—
220C2×D3—816no—
221C3×C4—22no—
222C3×C4—44no—
223C3×C4#52514yes—
224Dic3—24no—
225Dic3—24no—
226C23—22no—
227C23—38no—
228C23—42no—
229C23—616no—
230C23—64no—
231C23—64no—
232C23—124no—
233C23—124no—
234C23—248no—
235C23—2416no—
236C23D3×C22⋊C412yes—
237C23C22×C4×D311yescharacteristic
238D4—616no—
239D4—616no—
240C2×C4—22no—
241C2×C4—28no—
242C2×C4—24no—
243C2×C4—34no—
244C2×C4—34no—
245C2×C4—44no—
246C2×C4—122no—
247C2×C4—122no—
248C2×C4—248no—
249C2×C4—248no—
250C2×C4D3×C22⋊C412yes—
251C2×C4C2×D3×D412yes—
252C2×C4D3×#4114yes—
253C8—44no—
254C8—124no—
255C2×C3—22no—
256C2×C3—42no—
257C2×C3#228011yescharacteristic
258C2×C3#288812yes—
259D3—48no—
260C22—22no—
261C22—22no—
262C22—24no—
263C22—44no—
264C22—68no—
265C22—128no—
266C22—128no—
267C22—124no—
268C22—124no—
269C22—2416no—
270C22C2×D3×C22⋊C411yescharacteristic
271C22D3×#11712yes—
272C4—22no—
273C4—44no—
274C4—64no—
275C4—64no—
276C4#565614yes—
277C3#2072911yescharacteristic, complemented, Sylow
278C2—22no—
279C2—42no—
280C2—128no—
281C2#7405011yescharacteristic
282C2#7871412yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1822C2
socleC22×C3C22×C22⋊C4
prime radicalC23×D12C22
cosocleC22×C3×C4C24
hypercenter#1822D3
hypocenterC3#20729
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1822

Complements

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

NH ≅ G/Ntype
2. #78620C2semidirect
3. C3×#1822C2semidirect
4. #71589C2semidirect
6. #68994C2semidirect
7. #68972C2semidirect
20. #5118C22semidirect
21. Dic3×M16C22semidirect
23. C3×#343C22semidirect
24. #4530C22semidirect
28. #1822D3semidirect
78. #343C2×D3semidirect
277. C3#20729semidirect

Maximal subgroups

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

subgroupindex
C2×#55212
#786202
C3×#18222
#715892
#699882
#689942
#689722
#207293