Finite Group Databasebeta

← #10704588

Subgroups — #10704588

Order 768.

446 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 77 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#74148C211yesmaximal, characteristic
2#68126C212yesmaximal
3C3×#1324C211yesmaximal, characteristic, complemented
4#67632C211yesmaximal, characteristic, complemented
5#67453C212yesmaximal, complemented
6#11513—31nomaximal, Sylow
7C2×#892C411yescharacteristic
8C2×#892C2211yescharacteristic
9C2×C42×D3C411yescharacteristic
10C3×#382C2211yescharacteristic
11#4975C411yescharacteristic
12#4975C2211yescharacteristic
13#4936C411yescharacteristic
14#4719—22no—
15#4667—22no—
16#4649—22no—
17C3×#324C2212yes—
18#4528C2212yes—
19#4468—22no—
20#2285—31no—
21#1464—32no—
22#1325—32no—
23#1324—31no—
24#1324D311yescharacteristic, complemented
25C22×C4×D3—21no—
26C22×C4×D3C2×C411yescharacteristic
27C2×C3×C4⋊C4C2×C411yescharacteristic
28C2×C3×C4⋊C4C2311yescharacteristic
29C2×C3×C42C2×C411yescharacteristic
30C2×(C2×D3)⋊C4—21no—
31C2×(C2×D3)⋊C4D412yes—
32C2×C4⋊Dic3C2×C411yescharacteristic
33C2×C4⋊Dic3C2×C411yescharacteristic
34C2×Dic3⋊C4—22no—
35C2×Dic3⋊C4—22no—
36C2×C4×Dic3—21no—
37C2×C4×Dic3—21no—
38C2×C4×Dic3C2×C411yescharacteristic
39C2×C4×Dic3D412yes—
40#892—41no—
41#892—41no—
42#892—41no—
43#892—41no—
44C42×D3—22no—
45C42×D3—22no—
46C3×C22⋊C8—22no—
47C3×#95—22no—
48C3⋊#95—22no—
49#830—22no—
50#815—42no—
51Dic3⋊C8—42no—
52C2×#117—31no—
53C2×#117—31no—
54C22×C42—31no—
55#432—62no—
56#384—31no—
57#384—31no—
58#382—31no—
59#382C2×D311yescharacteristic
60#380—62no—
61C4×C4⋊C4—62no—
62#324—32no—
63#324C2×D312yes—
64#323—62no—
65C23×D3C22⋊C411yescharacteristic
66C22×C3×C4—21no—
67C22×C3×C4—21no—
68C22×C3×C4C22×C411yescharacteristic
69C22×C3×C4C2×D412yes—
70C22×Dic3—21no—
71C22×Dic3—21no—
72C22×Dic3C22⋊C411yescharacteristic
73C22×Dic3C22⋊C412yes—
74C2×C4×D3—22no—
75C2×C4×D3—21no—
76C2×C4×D3—21no—
77C2×C4×D3—21no—
78C2×C4×D3—21no—
79C2×C4×D3—24no—
80C2×C4×D3—42no—
81C2×C4×D3—42no—
82C2×C3×C8—42no—
83C3×C4⋊C4—41no—
84C3×C4⋊C4—41no—
85C3×C42—21no—
86C3×C42M1612yes—
87(C2×D3)⋊C4—42no—
88(C2×D3)⋊C4—42no—
89C4⋊Dic3—41no—
90C4⋊Dic3—41no—
91Dic3⋊C4—44no—
92Dic3⋊C4—44no—
93C4×Dic3—24no—
94C4×Dic3—21no—
95C4×Dic3—22no—
96C4×Dic3—22no—
97C4×Dic3—41no—
98C4×Dic3—41no—
99C4×Dic3M1612yes—
100C2×C3⋊C8—42no—
101C23×C4—31no—
102C23×C4—61no—
103#117—121no—
104#117—121no—
105#117—121no—
106#117—121no—
107C2×C4⋊C4—31no—
108C2×C4⋊C4—31no—
109C2×C4⋊C4—62no—
110C2×C4⋊C4—62no—
111C2×C4⋊C4C4×D311yescharacteristic
112C2×C4⋊C4C22×D311yescharacteristic
113C2×C22⋊C4—32no—
114C2×C22⋊C4—61no—
115C2×C42—31no—
116C2×C42—32no—
117C2×C42—62no—
118C2×C42—61no—
119C2×C42—62no—
120C2×C42—61no—
121C2×C42C4×D311yescharacteristic
122C4⋊C8—122no—
123C22⋊C8—22no—
124C22⋊C8—62no—
125#97—122no—
126#95—22no—
127#95—62no—
128C23×C3C2×C22⋊C411yescharacteristic
129C22×D3—21no—
130C22×D3—21no—
131C22×D3—41no—
132C22×D3—41no—
133C22×D3#10112yes—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—41no—
139C2×C3×C4—41no—
140C2×C3×C4—42no—
141C2×C3×C4—42no—
142C2×C3×C4—41no—
143C2×C3×C4C2×M1611yescharacteristic
144C2×C3×C4C2×M1611yescharacteristic
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—21no—
148C2×Dic3—22no—
149C2×Dic3—21no—
150C2×Dic3—22no—
151C2×Dic3—42no—
152C2×Dic3—42no—
153C2×Dic3—41no—
154C2×Dic3#9911yescharacteristic
155C2×Dic3#9911yescharacteristic
156C4×D3—22no—
157C4×D3—22no—
158C4×D3—22no—
159C4×D3—22no—
160C4×D3—48no—
161C3×C8—82no—
162C3⋊C8—82no—
163C24—31no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—31no—
167C22×C4—32no—
168C22×C4—61no—
169C22×C4—61no—
170C22×C4—62no—
171C22×C4—64no—
172C22×C4—61no—
173C22×C4—61no—
174C22×C4—61no—
175C22×C4—61no—
176C22×C4—122no—
177C22×C4—122no—
178C22×C4C2×C4×D311yescharacteristic
179C22×C4D3×D412yes—
180C2×C8—42no—
181C2×C8—122no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—121no—
185C4⋊C4—121no—
186C4⋊C4—124no—
187C4⋊C4—124no—
188C22⋊C4—122no—
189C22⋊C4—122no—
190C42—21no—
191C42—32no—
192C42—62no—
193C42—62no—
194C42—64no—
195C42—61no—
196C42—121no—
197C42—121no—
198C42ΓL(1,𝔽25)12yes—
199C22×C3—21no—
200C22×C3—21no—
201C22×C3#40711yescharacteristic
202C22×C3C2×#9911yescharacteristic
203C22×C3C2×#10111yescharacteristic
204C2×D3—22no—
205C2×D3—42no—
206C2×D3—41no—
207C2×D3—41no—
208C2×D3—42no—
209C3×C4—21no—
210C3×C4—21no—
211C3×C4—42no—
212C3×C4—42no—
213C3×C4—81no—
214Dic3—24no—
215Dic3—21no—
216Dic3—21no—
217Dic3—42no—
218Dic3—81no—
219C23—32no—
220C23—61no—
221C23—61no—
222C23—121no—
223C23—121no—
224C23D3×C22⋊C411yescharacteristic
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—31no—
230C2×C4—31no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—62no—
237C2×C4—62no—
238C2×C4—62no—
239C2×C4—62no—
240C2×C4—62no—
241C2×C4—62no—
242C2×C4—62no—
243C2×C4—61no—
244C2×C4—61no—
245C2×C4—62no—
246C2×C4—128no—
247C2×C4—122no—
248C2×C4—122no—
249C2×C4—121no—
250C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
251C2×C4D3×M1611yescharacteristic
252C8—82no—
253C8—242no—
254C2×C3—21no—
255C2×C3—21no—
256C2×C3#146311yescharacteristic
257C2×C3#212611yescharacteristic
258C2×C3#213211yescharacteristic
259D3—42no—
260C22—21no—
261C22—21no—
262C22—62no—
263C22—122no—
264C22—122no—
265C22—121no—
266C22—121no—
267C22#472611yescharacteristic
268C22D3×#9911yescharacteristic
269C22D3×#10111yescharacteristic
270C4—21no—
271C4—21no—
272C4—42no—
273C4—42no—
274C4—64no—
275C4—61no—
276C4—61no—
277C4—81no—
278C4—122no—
279C4—241no—
280C3#1151311yescharacteristic, complemented, Sylow
281C2—21no—
282C2—21no—
283C2—122no—
284C2#6814411yescharacteristic
285C2#6915811yescharacteristic
286C2#6916511yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1324C2
socleC22×C3C2×#99
prime radicalC23×D3C22⋊C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1324D3
hypocenterC3#11513
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1324

Complements

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

NH ≅ G/Ntype
3. C3×#1324C2semidirect
4. #67632C2semidirect
5. #67453C2semidirect
24. #1324D3semidirect
280. C3#11513semidirect

Maximal subgroups

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

subgroupindex
#741482
#681262
C3×#13242
#676322
#674532
#115133