Finite Group Databasebeta

← #10704582

Subgroups — #10704582

Order 768.

490 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 81 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
1D3×#382C211yesmaximal, characteristic
2#68144C212yesmaximal
3C3×#1324C211yesmaximal, characteristic, complemented
4#67632C211yesmaximal, characteristic, complemented
5#67452C212yesmaximal, complemented
6#11511—31nomaximal, Sylow
7C2×D3×C4⋊C4C411yescharacteristic
8C2×D3×C4⋊C4C2211yescharacteristic
9C2×C42×D3C411yescharacteristic
10C3×#382C2211yescharacteristic
11#4977C411yescharacteristic
12#4977C2211yescharacteristic
13#4936C411yescharacteristic
14#4726—22no—
15D3×#95—22no—
16#4650—22no—
17C3×#324C2212yes—
18#4528C2212yes—
19#4469—22no—
20C2×#382—31no—
21#1463—32no—
22#1324—31no—
23#1324—32no—
24#1324D311yescharacteristic, complemented
25C22×C4×D3—21no—
26C22×C4×D3—21no—
27C22×C4×D3C2×C411yescharacteristic
28C22×C4×D3D412yes—
29C2×C3×C4⋊C4C2×C411yescharacteristic
30C2×C3×C4⋊C4C2311yescharacteristic
31C2×C3×C42C2×C411yescharacteristic
32C2×C4⋊Dic3C2×C411yescharacteristic
33C2×C4⋊Dic3C2×C411yescharacteristic
34C2×Dic3⋊C4—21no—
35C2×Dic3⋊C4D412yes—
36C2×C4×Dic3—21no—
37C2×C4×Dic3C2×C411yescharacteristic
38C2×ΓL(1,𝔽25)—42no—
39D3×C4⋊C4—41no—
40D3×C4⋊C4—41no—
41D3×C4⋊C4—41no—
42D3×C4⋊C4—41no—
43C42×D3—22no—
44C42×D3—22no—
45C3×C22⋊C8—22no—
46C3×#95—22no—
47C3⋊#95—22no—
48C3⋊#95—22no—
49C3⋊#95—22no—
50#830—22no—
51(C2×D3)⋊C8—42no—
52C22×C4⋊C4—31no—
53C22×C4⋊C4—31no—
54C22×C42—31no—
55#407—62no—
56#382—31no—
57#382—31no—
58#382—31no—
59#382—62no—
60#382C2×D311yescharacteristic
61C2×#95—62no—
62#324—32no—
63#324—62no—
64#324C2×D312yes—
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—21no—
75C2×C4×D3—21no—
76C2×C4×D3—21no—
77C2×C4×D3—21no—
78C2×C4×D3—22no—
79C2×C4×D3—24no—
80C2×C4×D3—22no—
81C2×C4×D3—22no—
82C2×C4×D3—42no—
83C2×C4×D3—42no—
84C2×C4×D3—44no—
85C2×C4×D3—44no—
86C2×C4×D3—41no—
87C2×C4×D3—41no—
88C2×C3×C8—42no—
89C3×C4⋊C4—41no—
90C3×C4⋊C4—41no—
91C3×C42—21no—
92C3×C42M1612yescomplemented
93C4⋊Dic3—41no—
94C4⋊Dic3—41no—
95Dic3⋊C4—42no—
96Dic3⋊C4—42no—
97C4×Dic3—24no—
98C4×Dic3—21no—
99C4×Dic3M1612yescomplemented
100C2×C3⋊C8—42no—
101ΓL(1,𝔽25)—84no—
102C23×C4—31no—
103C23×C4—32no—
104C23×C4—61no—
105C23×C4—61no—
106C2×M16—122no—
107C2×C4⋊C4—31no—
108C2×C4⋊C4—32no—
109C2×C4⋊C4—31no—
110C2×C4⋊C4—61no—
111C2×C4⋊C4—121no—
112C2×C4⋊C4—121no—
113C2×C4⋊C4—121no—
114C2×C4⋊C4—121no—
115C2×C4⋊C4C4×D311yescharacteristic
116C2×C4⋊C4C22×D311yescharacteristic
117C2×C42—31no—
118C2×C42—62no—
119C2×C42—61no—
120C2×C42—62no—
121C2×C42C4×D311yescharacteristic
122C22⋊C8—22no—
123C22⋊C8—62no—
124C22⋊C8—122no—
125#95—22no—
126#95—62no—
127#95—62no—
128#95—62no—
129C23×C3C2×C22⋊C411yescharacteristic
130C22×D3—21no—
131C22×D3—21no—
132C22×D3—22no—
133C22×D3—22no—
134C22×D3#9911yescharacteristic
135C22×D3#9911yescharacteristic
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—41no—
141C2×C3×C4—41no—
142C2×C3×C4—42no—
143C2×C3×C4—42no—
144C2×C3×C4—41no—
145C2×C3×C4C2×M1611yescharacteristic
146C2×C3×C4C2×M1611yescharacteristic
147C2×Dic3—21no—
148C2×Dic3—21no—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3—41no—
152C2×Dic3—41no—
153C2×Dic3—42no—
154C2×Dic3—42no—
155C2×Dic3—41no—
156C2×Dic3#10112yes—
157C4×D3—22no—
158C4×D3—22no—
159C4×D3—22no—
160C4×D3—22no—
161C4×D3—48no—
162C4×D3—44no—
163C4×D3—82no—
164C4×D3—82no—
165C4×D3—82no—
166C3×C8—82no—
167C3⋊C8—82no—
168C24—31no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—31no—
172C22×C4—32no—
173C22×C4—62no—
174C22×C4—61no—
175C22×C4—61no—
176C22×C4—62no—
177C22×C4—64no—
178C22×C4—62no—
179C22×C4—61no—
180C22×C4—61no—
181C22×C4—61no—
182C22×C4—61no—
183C22×C4—122no—
184C22×C4—122no—
185C22×C4—124no—
186C22×C4—121no—
187C22×C4—121no—
188C22×C4—124no—
189C22×C4C2×C4×D311yescharacteristic
190C22×C4D3×D412yes—
191M16—244no—
192C2×C8—42no—
193C2×C8—122no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—122no—
197C4⋊C4—121no—
198C4⋊C4—121no—
199C4⋊C4—122no—
200C42—21no—
201C42—32no—
202C42—64no—
203C42—61no—
204C42ΓL(1,𝔽25)12yescomplemented
205C22×C3—21no—
206C22×C3—21no—
207C22×C3#40711yescharacteristic
208C22×C3C2×#9911yescharacteristic
209C22×C3C2×#10111yescharacteristic
210C2×D3—24no—
211C2×D3—24no—
212C2×D3—21no—
213C2×D3—21no—
214C2×D3—21no—
215C2×D3—21no—
216C2×D3#35412yes—
217C2×D3#35512yes—
218C3×C4—21no—
219C3×C4—21no—
220C3×C4—42no—
221C3×C4—42no—
222C3×C4—81no—
223Dic3—22no—
224Dic3—42no—
225Dic3—42no—
226Dic3—81no—
227C23—31no—
228C23—31no—
229C23—62no—
230C23—62no—
231C23—61no—
232C23—61no—
233C23D3×C22⋊C411yescharacteristic
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—32no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—62no—
245C2×C4—62no—
246C2×C4—62no—
247C2×C4—62no—
248C2×C4—61no—
249C2×C4—61no—
250C2×C4—62no—
251C2×C4—62no—
252C2×C4—124no—
253C2×C4—128no—
254C2×C4—121no—
255C2×C4—121no—
256C2×C4—122no—
257C2×C4—122no—
258C2×C4—121no—
259C2×C4—242no—
260C2×C4—242no—
261C2×C4—242no—
262C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
263C2×C4D3×M1611yescharacteristic
264C8—82no—
265C8—242no—
266C2×C3—21no—
267C2×C3—21no—
268C2×C3#146311yescharacteristic
269C2×C3C2×#35411yescharacteristic
270C2×C3C2×#35511yescharacteristic
271D3—22no—
272D3—22no—
273C22—21no—
274C22—21no—
275C22—32no—
276C22—32no—
277C22—64no—
278C22—64no—
279C22—61no—
280C22—61no—
281C22—61no—
282C22—61no—
283C22#472611yescharacteristic
284C22D3×#9911yescharacteristic
285C22D3×#10111yescharacteristic
286C4—21no—
287C4—21no—
288C4—42no—
289C4—42no—
290C4—62no—
291C4—81no—
292C4—122no—
293C4—122no—
294C4—241no—
295C3#1151111yescharacteristic, complemented, Sylow
296C2—21no—
297C2—21no—
298C2—62no—
299C2—62no—
300C2#6814411yescharacteristic
301C2D3×#35411yescharacteristic
302C2D3×#35511yescharacteristic
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#11511
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. #67452C2semidirect
24. #1324D3semidirect
92. C3×C42M16semidirect
99. C4×Dic3M16semidirect
204. C42ΓL(1,𝔽25)semidirect
295. C3#11511semidirect

Maximal subgroups

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

subgroupindex
D3×#3822
#681442
C3×#13242
#676322
#674522
#115113