Finite Group Databasebeta

← #10705477

Subgroups — #10705477

Order 768.

448 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 107 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
1#82264C211yesmaximal, characteristic
2C3×#1458C211yesmaximal, characteristic
3#71592C211yesmaximal, characteristic
4#67449C214yesmaximal
5#11717—31nomaximal, Sylow
6C2×C3×#117C2211yescharacteristic
7C2×#926—21no—
8C2×#869C411yescharacteristic, complemented
9C3×#431—21no—
10#5212—21no—
11#4986—21no—
12#4979C412yescomplemented
13#4949—21no—
14#4947C2212yes—
15#4936C411yescharacteristic, complemented
16#4928—21no—
17C3×#324—21no—
18C3×#323C2212yes—
19#4526—21no—
20#4526C2212yes—
21#2628—31no—
22#1740—31no—
23#1458D311yescharacteristic
24#1351—34no—
25C23×C3×C4D411yescharacteristic
26C22×Dic6C2×C412yes—
27C3×#117—41no—
28C3×#117—41no—
29C2×C3×C4⋊C4C2×C411yescharacteristic
30C2×C3×C4⋊C4D411yescharacteristic
31C2×C3×C22⋊C4D411yescharacteristic
32C2×C3×C22⋊C4C2311yescharacteristic
33C2×C3×C42C2×C411yescharacteristic
34C2×C3×C42D411yescharacteristic
35C2×#217—22no—
36C2×#217—22no—
37#926—24no—
38#926—42no—
39C2×C4⋊Dic3—21no—
40C2×C4⋊Dic3—21no—
41C2×C4⋊Dic3—21no—
42C2×C4⋊Dic3—41no—
43C2×Dic3⋊C4—22no—
44C2×C4×Dic3—41no—
45#869—41no—
46#869—41no—
47C3×C4⋊C8—41no—
48C3×C22⋊C8—22no—
49C3×C4×C8—41no—
50C3⋊#95—21no—
51C3⋊#95—22no—
52C3⋊#95—22no—
53C3⋊#95C2×C412yes—
54C2×C4⋊Q8—31no—
55C2×#117—61no—
56C2×#117C2×D311yescharacteristic
57#431—21no—
58#400—61no—
59#399—61no—
60#395—32no—
61#393—32no—
62#389—61no—
63#382—31no—
64#380—61no—
65#342—32no—
66#342—61no—
67#324—21no—
68#323C2×D312yes—
69C24×C3C2×D411yescharacteristic
70C22×C3×C4—21no—
71C22×C3×C4—21no—
72C22×C3×C4—22no—
73C22×C3×C4C22⋊C411yescharacteristic
74C22×C3×C4C22⋊C411yescharacteristic
75C22×C3×C4C22⋊C411yescharacteristic
76C22×C3×C4C22⋊C411yescharacteristic
77C22×C3×C4SD1612yes—
78C22×C3×C4Q1612yes—
79C22×C3×C4C22×C411yescharacteristic
80C22×C3×C4C2×D411yescharacteristic
81C22×Dic3—22no—
82C22×Dic3—22no—
83C22×Dic3—42no—
84C2×Dic6—42no—
85C2×Dic6—82no—
86C2×C3×C8—42no—
87C3×C4⋊C4—41no—
88C3×C4⋊C4—41no—
89C3×C22⋊C4—41no—
90C3×C22⋊C4—41no—
91C3×C42—21no—
92C3×C42—21no—
93C3×C42—21no—
94C3×C42—21no—
95#217—24no—
96#217—42no—
97#217—82no—
98C4⋊Dic3—42no—
99C4⋊Dic3—44no—
100C4⋊Dic3—41no—
101C4⋊Dic3—41no—
102C4⋊Dic3—42no—
103Dic3⋊C4—82no—
104C4×Dic3—44no—
105C22×Q8—32no—
106C23×C4D1211yescharacteristic
107C4⋊Q8—121no—
108C4⋊Q8—121no—
109#117—41no—
110#117—41no—
111#117—64no—
112#117—122no—
113C2×C4⋊C4—61no—
114C2×C4⋊C4—61no—
115C2×C4⋊C4—61no—
116C2×C4⋊C4—62no—
117C2×C4⋊C4—121no—
118C2×C4⋊C4C4×D311yescharacteristic
119C2×C4⋊C4#6711yescharacteristic
120C2×C22⋊C4—62no—
121C2×C22⋊C4—62no—
122C2×C22⋊C4D1211yescharacteristic
123C2×C22⋊C4C22×D311yescharacteristic
124C2×C42—121no—
125C2×C42C4×D311yescharacteristic
126C2×C42#6711yescharacteristic
127C4⋊C8—41no—
128C22⋊C8—22no—
129C4×C8—41no—
130#95—32no—
131#95—62no—
132#95—61no—
133#95—62no—
134C23×C3—21no—
135C23×C3—21no—
136C23×C3—41no—
137C23×C3—41no—
138C23×C3C2×C22⋊C411yescharacteristic
139C23×C3C2×SD1611yescharacteristic
140C23×C3C2×Q1611yescharacteristic
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—22no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—21no—
151C2×C3×C4—41no—
152C2×C3×C4—41no—
153C2×C3×C4—41no—
154C2×C3×C4—41no—
155C2×C3×C4#9911yescharacteristic
156C2×C3×C4#9911yescharacteristic
157C2×C3×C4#10312yes—
158C2×C3×C4#10312yes—
159C2×C3×C4#10412yes—
160C2×Dic3—44no—
161C2×Dic3—42no—
162C2×Dic3—42no—
163C2×Dic3—42no—
164C2×Dic3—42no—
165C2×Dic3—42no—
166C2×Dic3—42no—
167C2×Dic3—82no—
168C2×Dic3—82no—
169Dic6—84no—
170C3×C8—82no—
171C24C2×D1211yescharacteristic
172C2×Q8—122no—
173C2×Q8—242no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—62no—
178C22×C4—62no—
179C22×C4—122no—
180C22×C4#20412yes—
181C22×C4Dic1212yes—
182C22×C4(C2×D3)⋊C411yescharacteristic
183C22×C4(C2×D3)⋊C411yescharacteristic
184C22×C4(C2×D3)⋊C411yescharacteristic
185C22×C4(C2×D3)⋊C411yescharacteristic
186C22×C4C2×C4×D311yescharacteristic
187C22×C4C2×#6711yescharacteristic
188C2×C8—42no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—122no—
192C4⋊C4—124no—
193C4⋊C4—121no—
194C4⋊C4—121no—
195C4⋊C4—122no—
196C4⋊C4—242no—
197C22⋊C4—41no—
198C22⋊C4—41no—
199C22⋊C4—64no—
200C22⋊C4—122no—
201C22⋊C4—242no—
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#32812yes—
225C3×C4#32812yes—
226Dic3—82no—
227Dic3—84no—
228Dic3—82no—
229C23—21no—
230C23—21no—
231C23—41no—
232C23—41no—
233C23C2×#20411yescharacteristic
234C23C2×Dic1211yescharacteristic
235C23C2×(C2×D3)⋊C411yescharacteristic
236Q8—244no—
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—124no—
252C2×C4—122no—
253C2×C4—122no—
254C2×C4—122no—
255C2×C4—122no—
256C2×C4—122no—
257C2×C4—122no—
258C2×C4—242no—
259C2×C4—242no—
260C2×C4#80611yescharacteristic
261C2×C4#80611yescharacteristic
262C2×C4#81612yes—
263C2×C4#81612yes—
264C2×C4#82512yes—
265C8—82no—
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×#80611yescharacteristic
283C22C2×#81611yescharacteristic
284C22C2×#82511yescharacteristic
285C4—22no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—242no—
291C4—244no—
292C4—242no—
293C4#446712yes—
294C4#446712yes—
295C3#1171711yescharacteristic, complemented, Sylow
296C2—21no—
297C2—21no—
298C2—41no—
299C2—41no—
300C2C2×#446711yescharacteristic
301C2#7005911yescharacteristic
302C2#7014611yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1458C2
socleC22×C3C2×#99
prime radicalC24×C3C2×D4
cosocleC2×C3×C22⋊C4C23
hypercenter#1458D3
hypocenterC3#11717
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
8. C2×#869C4semidirect
12. #4979C4semidirect
15. #4936C4semidirect
295. C3#11717semidirect

Maximal subgroups

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

subgroupindex
#822642
C3×#14582
#715922
#674492
#117173