Finite Group Databasebeta

← #10704583

Subgroups — #10704583

Order 768.

490 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 95 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×#4977C211yesmaximal, characteristic
2C3×#1463C211yesmaximal, characteristic
3#71552C211yesmaximal, characteristic
4#67452C214yesmaximal, complemented
5#11511—31nomaximal, Sylow
6C22×C3×C4⋊C4C2211yescharacteristic
7C22×C4⋊Dic3C411yescharacteristic
8C22×C4×Dic3C411yescharacteristic
9C3×#407—21no—
10C2×C3⋊#95—21no—
11C2×C3⋊#95—21no—
12#5207—21no—
13#4977—21no—
14#4977—21no—
15#4977C412yes—
16#4977C2212yes—
17C3×#324—21no—
18C3×#324C2212yes—
19#4528—21no—
20#4528C2212yes—
21C2×#382—31no—
22#1463—31no—
23#1463D311yescharacteristic
24#1324—34no—
25C23×C3×C4C2×C411yescharacteristic
26C23×C3×C4D411yescharacteristic
27C23×C3×C4D411yescharacteristic
28C23×Dic3—21no—
29C23×Dic3—21no—
30C2×C3×M16—41no—
31C2×C3×C4⋊C4—41no—
32C2×C3×C4⋊C4—41no—
33C2×C3×C4⋊C4C2×C411yescharacteristic
34C2×C3×C4⋊C4D411yescharacteristic
35C2×C3×C4⋊C4D411yescharacteristic
36C2×C3×C4⋊C4C2311yescharacteristic
37C2×C4⋊Dic3—21no—
38C2×C4⋊Dic3—41no—
39C2×C4⋊Dic3—41no—
40C2×C4⋊Dic3C2×C412yes—
41C2×C4×Dic3—21no—
42C2×C4×Dic3—24no—
43C2×C4×Dic3C2×C412yes—
44C2×#208—41no—
45C3×C22⋊C8—22no—
46C3×C22⋊C8—41no—
47C3⋊#95—22no—
48C3⋊#95—22no—
49C3⋊#95—22no—
50C3⋊#95—22no—
51#830—22no—
52#830—41no—
53C22×C4⋊C4—31no—
54C22×C4⋊C4C2×D311yescharacteristic
55C22×C42—31no—
56#407—21no—
57#407—61no—
58#382—32no—
59#382—32no—
60#382—61no—
61#382—61no—
62C2×#95—61no—
63C2×#95—61no—
64#324—21no—
65#324—32no—
66#324—61no—
67#324C2×D312yes—
68C24×C3C22⋊C411yescharacteristic
69C22×C3×C4—21no—
70C22×C3×C4—21no—
71C22×C3×C4—22no—
72C22×C3×C4—22no—
73C22×C3×C4—21no—
74C22×C3×C4—21no—
75C22×C3×C4—21no—
76C22×C3×C4—21no—
77C22×C3×C4—42no—
78C22×C3×C4—42no—
79C22×C3×C4C22⋊C411yescharacteristic
80C22×C3×C4C22⋊C411yescharacteristic
81C22×C3×C4C22⋊C411yescharacteristic
82C22×C3×C4C22×C411yescharacteristic
83C22×C3×C4C2×D411yescharacteristic
84C22×C3×C4C2×D411yescharacteristic
85C22×Dic3—24no—
86C22×Dic3—22no—
87C22×Dic3—22no—
88C22×Dic3—42no—
89C22×Dic3—42no—
90C22×Dic3—44no—
91C22×Dic3—41no—
92C22×Dic3—41no—
93C3×M16—82no—
94C2×C3×C8—42no—
95C3×C4⋊C4—41no—
96C3×C4⋊C4—41no—
97C3×C4⋊C4—41no—
98C3×C4⋊C4—41no—
99C4⋊Dic3—42no—
100C4⋊Dic3—42no—
101C4×Dic3—24no—
102C4×Dic3—22no—
103C4×Dic3M1614yescomplemented
104#208—82no—
105C2×C3⋊C8—42no—
106C23×C4—61no—
107C23×C4—61no—
108C23×C4C4×D311yescharacteristic
109C23×C4D1211yescharacteristic
110C23×C4#6711yescharacteristic
111C2×M16—41no—
112C2×M16—121no—
113C2×C4⋊C4—32no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—61no—
117C2×C4⋊C4—121no—
118C2×C4⋊C4—121no—
119C2×C4⋊C4C4×D311yescharacteristic
120C2×C4⋊C4D1211yescharacteristic
121C2×C4⋊C4#6711yescharacteristic
122C2×C4⋊C4C22×D311yescharacteristic
123C2×C42—32no—
124C2×C42—64no—
125C2×C42—61no—
126C22⋊C8—22no—
127C22⋊C8—41no—
128C22⋊C8—62no—
129C22⋊C8—121no—
130#95—62no—
131#95—62no—
132#95—62no—
133#95—62no—
134C23×C3—22no—
135C23×C3—22no—
136C23×C3—21no—
137C23×C3—21no—
138C23×C3#9911yescharacteristic
139C23×C3#9911yescharacteristic
140C23×C3C2×C22⋊C411yescharacteristic
141C2×C3×C4—24no—
142C2×C3×C4—24no—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—21no—
151C2×C3×C4—42no—
152C2×C3×C4—42no—
153C2×C3×C4—41no—
154C2×C3×C4—41no—
155C2×C3×C4—41no—
156C2×C3×C4—41no—
157C2×C3×C4—81no—
158C2×C3×C4—81no—
159C2×C3×C4#10112yes—
160C2×C3×C4C2×M1612yes—
161C2×Dic3—24no—
162C2×Dic3—48no—
163C2×Dic3—42no—
164C2×Dic3—42no—
165C2×Dic3—44no—
166C2×Dic3—42no—
167C2×Dic3—82no—
168C2×Dic3—82no—
169C3×C8—82no—
170C3⋊C8—82no—
171C24(C2×D3)⋊C411yescharacteristic
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—64no—
183C22×C4—62no—
184C22×C4—62no—
185C22×C4—122no—
186C22×C4—122no—
187C22×C4—124no—
188C22×C4—121no—
189C22×C4—121no—
190C22×C4(C2×D3)⋊C411yescharacteristic
191C22×C4(C2×D3)⋊C411yescharacteristic
192C22×C4(C2×D3)⋊C411yescharacteristic
193C22×C4C2×C4×D311yescharacteristic
194C22×C4C2×D1211yescharacteristic
195C22×C4C2×#6711yescharacteristic
196M16—82no—
197M16—242no—
198C2×C8—42no—
199C2×C8—122no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—122no—
205C4⋊C4—122no—
206C42—34no—
207C42—64no—
208C42—62no—
209C22×C3—24no—
210C22×C3—24no—
211C22×C3—21no—
212C22×C3—21no—
213C22×C3—21no—
214C22×C3—21no—
215C22×C3—21no—
216C22×C3—21no—
217C22×C3#35412yes—
218C22×C3#35512yes—
219C22×C3#40711yescharacteristic
220C22×C3C2×#9911yescharacteristic
221C22×C3C2×#10111yescharacteristic
222C3×C4—22no—
223C3×C4—22no—
224C3×C4—41no—
225C3×C4—41no—
226C3×C4—41no—
227C3×C4—41no—
228Dic3—44no—
229Dic3—82no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—22no—
234C23#80611yescharacteristic
235C23#80611yescharacteristic
236C23C2×(C2×D3)⋊C411yescharacteristic
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—64no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—128no—
257C2×C4—122no—
258C2×C4—122no—
259C2×C4—124no—
260C2×C4—122no—
261C2×C4—242no—
262C2×C4—242no—
263C2×C4#82412yes—
264C2×C4D3×M1612yes—
265C8—82no—
266C8—242no—
267C2×C3—21no—
268C2×C3—22no—
269C2×C3—22no—
270C2×C3—21no—
271C2×C3#146311yescharacteristic
272C2×C3C2×#35411yescharacteristic
273C2×C3C2×#35511yescharacteristic
274C22—24no—
275C22—24no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22#447312yes—
283C22#447712yes—
284C22C2×#80611yescharacteristic
285C22#512611yescharacteristic
286C22C2×#82411yescharacteristic
287C4—22no—
288C4—22no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—124no—
294C4—242no—
295C3#1151111yescharacteristic, complemented, Sylow
296C2—21no—
297C2—21no—
298C2—22no—
299C2—22no—
300C2#6997111yescharacteristic
301C2C2×#447311yescharacteristic
302C2C2×#447711yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1463C2
socleC22×C3C2×#99
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1463D3
hypocenterC3#11511
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
4. #67452C2semidirect
103. C4×Dic3M16semidirect
295. C3#11511semidirect

Maximal subgroups

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

subgroupindex
C2×#49772
C3×#14632
#715522
#674522
#115113