Finite Group Databasebeta

← #10705476

Subgroups — #10705476

Order 768.

448 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2628C211yesmaximal, characteristic
2#71592C211yesmaximal, characteristic
3#71571C211yesmaximal, characteristic
4#67646C214yesmaximal
5#11717—31nomaximal, Sylow
6C2×C3×C4⋊Q8C411yescharacteristic, complemented
7C2×C3×#117C2211yescharacteristic
8C2×#926—21no—
9C3×#400—21no—
10C3×#399—21no—
11C3×#395C412yescomplemented
12C3×#393C2212yes—
13C3×#389—21no—
14C3×#382C411yescharacteristic, complemented
15C3×#380—21no—
16#4925—21no—
17#4528—21no—
18#4526—21no—
19#4526C2212yes—
20#4525C2212yes—
21#2628D311yescharacteristic
22#1740—31no—
23#1458—31no—
24#1351—34no—
25C22×C3×Q8C2×C412yes—
26C23×C3×C4D411yescharacteristic
27C3×C4⋊Q8—41no—
28C3×C4⋊Q8—41no—
29C3×#117—41no—
30C3×#117—41no—
31C2×C3×C4⋊C4—21no—
32C2×C3×C4⋊C4—21no—
33C2×C3×C4⋊C4—21no—
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4C2×C411yescharacteristic
36C2×C3×C4⋊C4D411yescharacteristic
37C2×C3×C22⋊C4—22no—
38C2×C3×C22⋊C4D411yescharacteristic
39C2×C3×C22⋊C4C2311yescharacteristic
40C2×C3×C42C2×C411yescharacteristic
41C2×C3×C42D411yescharacteristic
42C2×#217—22no—
43#926—24no—
44#926—42no—
45C2×C4⋊Dic3—41no—
46C2×C4×Dic3—41no—
47C3×#95—22no—
48C3×#95—21no—
49C3×#95—22no—
50C3×#95C2×C412yes—
51#830—22no—
52C4⋊(C3⋊C8)—41no—
53C4×C3⋊C8—41no—
54C2×C4⋊Q8Dic311yescharacteristic, complemented
55C2×#117—61no—
56C2×#117C2×D311yescharacteristic
57#431—61no—
58#400—21no—
59#399—21no—
60#395Dic312yescomplemented
61#393C2×D312yes—
62#389—21no—
63#382Dic311yescharacteristic, complemented
64#380—21no—
65#342—32no—
66#342—61no—
67#324—61no—
68#323—32no—
69C24×C3C2×D411yescharacteristic
70C2×C3×Q8—42no—
71C2×C3×Q8—82no—
72C22×C3×C4—21no—
73C22×C3×C4—21no—
74C22×C3×C4—22no—
75C22×C3×C4—22no—
76C22×C3×C4—22no—
77C22×C3×C4C22⋊C411yescharacteristic
78C22×C3×C4C22⋊C411yescharacteristic
79C22×C3×C4C22⋊C411yescharacteristic
80C22×C3×C4C22⋊C411yescharacteristic
81C22×C3×C4SD1612yes—
82C22×C3×C4Q1612yes—
83C22×C3×C4C22×C411yescharacteristic
84C22×C3×C4C2×D411yescharacteristic
85C22×Dic3—42no—
86C3×C4⋊C4—41no—
87C3×C4⋊C4—41no—
88C3×C4⋊C4—42no—
89C3×C4⋊C4—41no—
90C3×C4⋊C4—41no—
91C3×C4⋊C4—42no—
92C3×C4⋊C4—82no—
93C3×C22⋊C4—41no—
94C3×C22⋊C4—41no—
95C3×C22⋊C4—82no—
96C3×C42—21no—
97C3×C42—21no—
98C3×C42—21no—
99C3×C42—21no—
100#217—24no—
101#217—42no—
102C4⋊Dic3—44no—
103C4×Dic3—44no—
104C2×C3⋊C8—42no—
105C22×Q8C2×Dic312yes—
106C23×C4#6711yescharacteristic
107C4⋊Q8—41no—
108C4⋊Q8—41no—
109#117—41no—
110#117—41no—
111#117—64no—
112#117—122no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—121no—
118C2×C4⋊C4C2×Dic311yescharacteristic
119C2×C4⋊C4#6711yescharacteristic
120C2×C22⋊C4—22no—
121C2×C22⋊C4—62no—
122C2×C22⋊C4#6711yescharacteristic
123C2×C22⋊C4C22×D311yescharacteristic
124C2×C42—121no—
125C2×C42C2×Dic311yescharacteristic
126C2×C42#6711yescharacteristic
127C4⋊C8—121no—
128C22⋊C8—62no—
129C4×C8—121no—
130#95—22no—
131#95—21no—
132#95—22no—
133#95C2×Dic312yes—
134C23×C3—21no—
135C23×C3—21no—
136C23×C3—41no—
137C23×C3—41no—
138C23×C3C2×C22⋊C411yescharacteristic
139C23×C3C2×SD1611yescharacteristic
140C23×C3C2×Q1611yescharacteristic
141C3×Q8—84no—
142C2×C3×C4—21no—
143C2×C3×C4—21no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—22no—
147C2×C3×C4—22no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
152C2×C3×C4—41no—
153C2×C3×C4—41no—
154C2×C3×C4—41no—
155C2×C3×C4—41no—
156C2×C3×C4—42no—
157C2×C3×C4—42no—
158C2×C3×C4—42no—
159C2×C3×C4—42no—
160C2×C3×C4—42no—
161C2×C3×C4—42no—
162C2×C3×C4#9911yescharacteristic
163C2×C3×C4#9911yescharacteristic
164C2×C3×C4#10312yes—
165C2×C3×C4#10312yes—
166C2×C3×C4#10412yes—
167C2×Dic3—44no—
168C2×Dic3—82no—
169C2×Dic3—82no—
170C3⋊C8—82no—
171C24C2×#6711yescharacteristic
172C2×Q8—42no—
173C2×Q8—82no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—122no—
180C22×C4#21512yes—
181C22×C4#21612yes—
182C22×C4#21711yescharacteristic
183C22×C4#21711yescharacteristic
184C22×C4#21711yescharacteristic
185C22×C4#21711yescharacteristic
186C22×C4C22×Dic311yescharacteristic
187C22×C4C2×#6711yescharacteristic
188C2×C8—122no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—42no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—82no—
196C4⋊C4—124no—
197C22⋊C4—41no—
198C22⋊C4—41no—
199C22⋊C4—64no—
200C22⋊C4—82no—
201C22⋊C4—122no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—124no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—41no—
212C22×C3—41no—
213C22×C3—41no—
214C22×C3—41no—
215C22×C3—81no—
216C22×C3C2×#9911yescharacteristic
217C22×C3C2×#10311yescharacteristic
218C22×C3C2×#10411yescharacteristic
219C3×C4—22no—
220C3×C4—41no—
221C3×C4—41no—
222C3×C4—41no—
223C3×C4—41no—
224C3×C4—82no—
225C3×C4—82no—
226C3×C4#32812yes—
227C3×C4#32812yes—
228Dic3—84no—
229C23—21no—
230C23—21no—
231C23—41no—
232C23—41no—
233C23C2×#21511yescharacteristic
234C23C2×#21611yescharacteristic
235C23C2×#21711yescharacteristic
236Q8—84no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—124no—
258C2×C4—242no—
259C2×C4—242no—
260C2×C4#83411yescharacteristic
261C2×C4#83411yescharacteristic
262C2×C4#83512yes—
263C2×C4#83512yes—
264C2×C4#83712yes—
265C8—242no—
266C2×C3—21no—
267C2×C3—21no—
268C2×C3—41no—
269C2×C3—41no—
270C2×C3C2×#32811yescharacteristic
271C2×C3#212211yescharacteristic
272C2×C3#212911yescharacteristic
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—81no—
282C22C2×#83411yescharacteristic
283C22C2×#83511yescharacteristic
284C22C2×#83711yescharacteristic
285C4—22no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—82no—
291C4—82no—
292C4—244no—
293C4#453812yes—
294C4#453812yes—
295C3#1171711yescharacteristic, complemented, Sylow
296C2—21no—
297C2—21no—
298C2—41no—
299C2—41no—
300C2C2×#453811yescharacteristic
301C2#7185911yescharacteristic
302C2#7187111yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC2×C22⋊C4C22×D3
FittingC3×#2628C2
socleC22×C3C2×#99
prime radicalC24×C3C2×D4
cosocleC2×C3×C22⋊C4C23
hypercenter#2628D3
hypocenterC3#11717
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2628

Complements

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

NH ≅ G/Ntype
6. C2×C3×C4⋊Q8C4semidirect
11. C3×#395C4semidirect
14. C3×#382C4semidirect
54. C2×C4⋊Q8Dic3semidirect
60. #395Dic3semidirect
63. #382Dic3semidirect
295. C3#11717semidirect

Maximal subgroups

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

subgroupindex
C3×#26282
#715922
#715712
#676462
#117173