Finite Group Databasebeta

← #10710687

Subgroups — #10710687

Order 768.

1182 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 1182 conjugacy classes of subgroups — too many to draw. Only its 139 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×C3×#535C211yesmaximal, characteristic
2#72139C212yesmaximal, complemented
3C2×#4545C214yesmaximal, complemented
4#12697—31nomaximal, Sylow
5C22×C3×#41C2211yescharacteristic
6C23×C3×D4C411yescharacteristic, complemented
7C3×#535—28no—
8C2×C3×#127C2212yes—
9C2×C3×#124C412yescomplemented
10C2×C3×#121—22no—
11C2×C3×#120—22no—
12C2×C3×#117C411yescharacteristic, complemented
13C2×#834—22no—
14C2×#834C2214yescomplemented
15#5211—22no—
16#4545—28no—
17C2×#535D311yescharacteristic
18C2×#353—34no—
19#1782—32no—
20C25×C3—21no—
21C2×C3×#41—24no—
22C2×C3×#41—42no—
23C2×C3×#41—42no—
24C22×C3×Q8C2×C411yescharacteristic
25C22×C3×D4—24no—
26C22×C3×D4—24no—
27C22×C3×D4—21no—
28C22×C3×D4—42no—
29C22×C3×D4—42no—
30C22×C3×D4C2×C412yes—
31C22×C3×D4D412yes—
32C22×C3×D4C2311yescharacteristic
33C23×C3×C4C2×C411yescharacteristic
34C23×C3×C4D412yes—
35C3×#127—24no—
36C3×#127D418yes—
37C3×#124—24no—
38C3×#124—24no—
39C3×#121—216no—
40C3×#120—216no—
41C3×#117—24no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—24no—
45C2×C3×C42C2×C412yes—
46C2×#217—24no—
47#834—28no—
48#834—44no—
49C3⋊#95—44no—
50C22×#41C2×D311yescharacteristic
51C23×D4Dic311yescharacteristic, complemented
52#535—28no—
53C2×#127C2×D312yes—
54C2×#124Dic312yescomplemented
55C2×#121—22no—
56C2×#120—22no—
57C2×#117Dic311yescharacteristic, complemented
58C2×#99—34no—
59C2×#99—62no—
60#381—62no—
61#353—68no—
62C24×C3—24no—
63C24×C3—22no—
64C24×C3—42no—
65C24×C3—42no—
66C24×C3—84no—
67C24×C3C22⋊C411yescharacteristic
68C24×C3C2×D412yes—
69C3×#41—416no—
70C2×C3×Q8—22no—
71C2×C3×Q8—41no—
72C2×C3×Q8—41no—
73C2×C3×D4—24no—
74C2×C3×D4—24no—
75C2×C3×D4—24no—
76C2×C3×D4—216no—
77C2×C3×D4—22no—
78C2×C3×D4—42no—
79C2×C3×D4—41no—
80C2×C3×D4—41no—
81C2×C3×D4—42no—
82C2×C3×D4—42no—
83C2×C3×D4—416no—
84C2×C3×D4—42no—
85C2×C3×D4—42no—
86C2×C3×D4—88no—
87C2×C3×D4C22⋊C414yes—
88C2×C3×D4C2×D414yes—
89C22×C3×C4—22no—
90C22×C3×C4—22no—
91C22×C3×C4—22no—
92C22×C3×C4—22no—
93C22×C3×C4—24no—
94C22×C3×C4—24no—
95C22×C3×C4—42no—
96C22×C3×C4C22⋊C412yes—
97C22×C3×C4C22⋊C411yescharacteristic
98C22×C3×C4C22×C411yescharacteristic
99C22×Dic3—44no—
100C3×C4⋊C4—24no—
101C3×C22⋊C4—24no—
102C3×C22⋊C4—216no—
103C3×C42—22no—
104C3×C42C22⋊C414yes—
105#217—44no—
106#217—44no—
107C25—21no—
108C2×#41—24no—
109C2×#41—42no—
110C2×#41—42no—
111C22×Q8C2×Dic311yescharacteristic
112C22×D4—24no—
113C22×D4—24no—
114C22×D4—21no—
115C22×D4—42no—
116C22×D4—42no—
117C22×D4C2×Dic312yes—
118C22×D4#6712yes—
119C22×D4C22×D311yescharacteristic
120C23×C4C2×Dic311yescharacteristic
121C23×C4#6712yes—
122#127—24no—
123#127#6718yes—
124#124—24no—
125#124—24no—
126#121—216no—
127#120—216no—
128#117—24no—
129C2×C4⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—64no—
133C2×C42C2×Dic312yes—
134#99—68no—
135#99—124no—
136#95—124no—
137C23×C3—21no—
138C23×C3—22no—
139C23×C3—21no—
140C23×C3—22no—
141C23×C3—24no—
142C23×C3—42no—
143C23×C3—44no—
144C23×C3—48no—
145C23×C3—48no—
146C23×C3—42no—
147C23×C3—42no—
148C23×C3—88no—
149C23×C3—88no—
150C23×C3—88no—
151C23×C3#9912yes—
152C23×C3C2×C22⋊C411yescharacteristic
153C23×C3#12014yes—
154C3×Q8—44no—
155C3×D4—44no—
156C3×D4—44no—
157C3×D4—48no—
158C3×D4—44no—
159C3×D4—48no—
160C3×D4—816no—
161C2×C3×C4—28no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—24no—
167C2×C3×C4—22no—
168C2×C3×C4—22no—
169C2×C3×C4—22no—
170C2×C3×C4—48no—
171C2×C3×C4—44no—
172C2×C3×C4#9912yes—
173C2×C3×C4C2×C22⋊C412yes—
174C2×Dic3—84no—
175C2×Dic3—88no—
176C24—24no—
177C24—22no—
178C24—42no—
179C24—42no—
180C24—84no—
181C24#21711yescharacteristic
182C24C2×#6712yes—
183#41—416no—
184C2×Q8—22no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×D4—24no—
188C2×D4—24no—
189C2×D4—24no—
190C2×D4—216no—
191C2×D4—22no—
192C2×D4—42no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—416no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—88no—
201C2×D4#21714yes—
202C2×D4C2×#6714yes—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—42no—
210C22×C4—124no—
211C22×C4#21712yes—
212C22×C4#21711yescharacteristic
213C22×C4C22×Dic311yescharacteristic
214C4⋊C4—24no—
215C22⋊C4—24no—
216C22⋊C4—216no—
217C22⋊C4—124no—
218C22⋊C4—124no—
219C42—22no—
220C42#21714yes—
221C22×C3—22no—
222C22×C3—24no—
223C22×C3—22no—
224C22×C3—44no—
225C22×C3—42no—
226C22×C3—42no—
227C22×C3—48no—
228C22×C3—44no—
229C22×C3—42no—
230C22×C3—44no—
231C22×C3—82no—
232C22×C3—84no—
233C22×C3—816no—
234C22×C3—84no—
235C22×C3#37911yescharacteristic
236C22×C3C2×#9911yescharacteristic
237C22×C3C2×#9911yescharacteristic
238C3×C4—22no—
239C3×C4—22no—
240C3×C4—44no—
241C3×C4—42no—
242Dic3—164no—
243C23—22no—
244C23—21no—
245C23—21no—
246C23—22no—
247C23—24no—
248C23—42no—
249C23—44no—
250C23—48no—
251C23—48no—
252C23—42no—
253C23—42no—
254C23—88no—
255C23—88no—
256C23—88no—
257C23#83412yes—
258C23C2×#21711yescharacteristic
259C23#95314yes—
260Q8—44no—
261D4—44no—
262D4—44no—
263D4—48no—
264D4—44no—
265D4—48no—
266D4—816no—
267C2×C4—28no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—48no—
277C2×C4—44no—
278C2×C4—244no—
279C2×C4—248no—
280C2×C4#83412yes—
281C2×C4C2×#21712yes—
282C2×C3—21no—
283C2×C3—21no—
284C2×C3—44no—
285C2×C3—42no—
286C2×C3—84no—
287C2×C3#178111yescharacteristic
288C2×C3#212812yes—
289C22—22no—
290C22—22no—
291C22—24no—
292C22—44no—
293C22—42no—
294C22—42no—
295C22—48no—
296C22—44no—
297C22—42no—
298C22—44no—
299C22—82no—
300C22—84no—
301C22—816no—
302C22—84no—
303C22C2×#83411yescharacteristic
304C22C2×#83411yescharacteristic
305C22#524711yescharacteristic
306C4—22no—
307C4—22no—
308C4—44no—
309C4—42no—
310C4—484no—
311C3#1269711yescharacteristic, complemented, Sylow
312C2—21no—
313C2—21no—
314C2—44no—
315C2—42no—
316C2—84no—
317C2#7186112yes—
318C2#7213011yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC22×D4C22×D3
Fitting, prime radicalC2×C3×#535C2
socleC22×C3C2×#99
cosocleC22×C3×D4C23
hypercenterC2×#535D3
hypocenterC3#12697
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
2. #72139C2semidirect
3. C2×#4545C2semidirect
6. C23×C3×D4C4semidirect
9. C2×C3×#124C4semidirect
12. C2×C3×#117C4semidirect
14. C2×#834C22semidirect
51. C23×D4Dic3semidirect
54. C2×#124Dic3semidirect
57. C2×#117Dic3semidirect
311. C3#12697semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5352
#721392
C2×#45452
#126973