Finite Group Databasebeta

← #10704585

Subgroups — #10704585

Order 768.

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

normal subgroups only

This group has 578 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×#384C211yesmaximal, characteristic
2#68144C212yesmaximal
3C3×#1325C211yesmaximal, characteristic, complemented
4#67635C211yesmaximal, characteristic, complemented
5#67453C212yesmaximal, complemented
6#11512—31nomaximal, Sylow
7C2×D3×C4⋊C4—22no—
8C2×D3×C4⋊C4C411yescharacteristic
9C2×D3×C4⋊C4C2211yescharacteristic
10C2×C42×D3C411yescharacteristic
11C3×#384C2211yescharacteristic
12#4975—22no—
13#4975C411yescharacteristic
14#4975C2211yescharacteristic
15#4935C411yescharacteristic
16#4726—22no—
17C3×#324C2212yes—
18#4528C2212yes—
19#4469—22no—
20C2×#384—31no—
21#1463—32no—
22#1325—31no—
23#1325—32no—
24#1325D311yescharacteristic, complemented
25C22×C4×D3—21no—
26C22×C4×D3—21no—
27C22×C4×D3C2×C411yescharacteristic
28C22×C4×D3D412yes—
29C2×C3×C4⋊C4—22no—
30C2×C3×C4⋊C4C2×C411yescharacteristic
31C2×C3×C4⋊C4C2311yescharacteristic
32C2×C3×C42C2×C411yescharacteristic
33C2×C4⋊Dic3—22no—
34C2×C4⋊Dic3C2×C411yescharacteristic
35C2×C4⋊Dic3C2×C411yescharacteristic
36C2×Dic3⋊C4—22no—
37C2×Dic3⋊C4—21no—
38C2×Dic3⋊C4—22no—
39C2×Dic3⋊C4D412yes—
40C2×C4×Dic3—21no—
41C2×C4×Dic3C2×C411yescharacteristic
42C2×ΓL(1,𝔽25)—42no—
43D3×C4⋊C4—41no—
44D3×C4⋊C4—41no—
45D3×C4⋊C4—41no—
46D3×C4⋊C4—41no—
47D3×C4⋊C4—48no—
48C42×D3—22no—
49C42×D3—22no—
50C3×C22⋊C8—22no—
51#830—22no—
52(C2×D3)⋊C8—42no—
53C22×C4⋊C4—31no—
54C22×C4⋊C4—31no—
55C22×C4⋊C4—62no—
56C22×C42—31no—
57#407—62no—
58#384—31no—
59#384—31no—
60#384—31no—
61#384—62no—
62#384C2×D311yescharacteristic
63#324—32no—
64#324—62no—
65#324C2×D312yes—
66C23×D3C22⋊C411yescharacteristic
67C22×C3×C4—21no—
68C22×C3×C4—21no—
69C22×C3×C4C22×C411yescharacteristic
70C22×C3×C4C2×D412yes—
71C22×Dic3—21no—
72C22×Dic3—21no—
73C22×Dic3C22⋊C411yescharacteristic
74C22×Dic3C22⋊C412yes—
75C2×C4×D3—24no—
76C2×C4×D3—24no—
77C2×C4×D3—24no—
78C2×C4×D3—21no—
79C2×C4×D3—21no—
80C2×C4×D3—21no—
81C2×C4×D3—21no—
82C2×C4×D3—22no—
83C2×C4×D3—22no—
84C2×C4×D3—22no—
85C2×C4×D3—44no—
86C2×C4×D3—44no—
87C2×C4×D3—41no—
88C2×C4×D3—41no—
89C2×C3×C8—42no—
90C3×C4⋊C4—41no—
91C3×C4⋊C4—41no—
92C3×C4⋊C4—44no—
93C3×C42—21no—
94C3×C42M1612yescomplemented
95C4⋊Dic3—41no—
96C4⋊Dic3—44no—
97C4⋊Dic3—41no—
98Dic3⋊C4—42no—
99Dic3⋊C4—44no—
100Dic3⋊C4—44no—
101Dic3⋊C4—42no—
102C4×Dic3—24no—
103C4×Dic3—21no—
104C4×Dic3M1612yescomplemented
105C2×C3⋊C8—42no—
106ΓL(1,𝔽25)—84no—
107C23×C4—31no—
108C23×C4—32no—
109C23×C4—61no—
110C23×C4—61no—
111C2×M16—122no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—31no—
114C2×C4⋊C4—32no—
115C2×C4⋊C4—31no—
116C2×C4⋊C4—61no—
117C2×C4⋊C4—62no—
118C2×C4⋊C4—62no—
119C2×C4⋊C4—62no—
120C2×C4⋊C4—121no—
121C2×C4⋊C4—121no—
122C2×C4⋊C4—121no—
123C2×C4⋊C4—121no—
124C2×C4⋊C4—128no—
125C2×C4⋊C4C4×D311yescharacteristic
126C2×C4⋊C4C22×D311yescharacteristic
127C2×C42—31no—
128C2×C42—62no—
129C2×C42—61no—
130C2×C42—62no—
131C2×C42C4×D311yescharacteristic
132C22⋊C8—22no—
133C22⋊C8—62no—
134C22⋊C8—122no—
135C23×C3C2×C22⋊C411yescharacteristic
136C22×D3—21no—
137C22×D3—21no—
138C22×D3—22no—
139C22×D3—22no—
140C22×D3#9911yescharacteristic
141C22×D3#9911yescharacteristic
142C2×C3×C4—22no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—22no—
148C2×C3×C4—42no—
149C2×C3×C4—42no—
150C2×C3×C4—41no—
151C2×C3×C4C2×M1611yescharacteristic
152C2×C3×C4C2×M1611yescharacteristic
153C2×Dic3—22no—
154C2×Dic3—22no—
155C2×Dic3—21no—
156C2×Dic3—22no—
157C2×Dic3—21no—
158C2×Dic3—22no—
159C2×Dic3—42no—
160C2×Dic3—42no—
161C2×Dic3—41no—
162C2×Dic3#10112yes—
163C4×D3—216no—
164C4×D3—22no—
165C4×D3—22no—
166C4×D3—22no—
167C4×D3—22no—
168C4×D3—44no—
169C4×D3—82no—
170C4×D3—82no—
171C4×D3—82no—
172C3×C8—82no—
173C3⋊C8—82no—
174C24—31no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—31no—
178C22×C4—32no—
179C22×C4—62no—
180C22×C4—61no—
181C22×C4—61no—
182C22×C4—64no—
183C22×C4—64no—
184C22×C4—64no—
185C22×C4—62no—
186C22×C4—62no—
187C22×C4—61no—
188C22×C4—61no—
189C22×C4—61no—
190C22×C4—61no—
191C22×C4—124no—
192C22×C4—121no—
193C22×C4—121no—
194C22×C4—124no—
195C22×C4C2×C4×D311yescharacteristic
196C22×C4D3×D412yes—
197M16—244no—
198C2×C8—42no—
199C2×C8—122no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—44no—
203C4⋊C4—122no—
204C4⋊C4—121no—
205C4⋊C4—121no—
206C4⋊C4—124no—
207C4⋊C4—124no—
208C4⋊C4—124no—
209C4⋊C4—122no—
210C42—21no—
211C42—32no—
212C42—64no—
213C42—61no—
214C42ΓL(1,𝔽25)12yescomplemented
215C22×C3—21no—
216C22×C3—21no—
217C22×C3#40711yescharacteristic
218C22×C3C2×#9911yescharacteristic
219C22×C3C2×#10111yescharacteristic
220C2×D3—24no—
221C2×D3—24no—
222C2×D3—21no—
223C2×D3—21no—
224C2×D3—21no—
225C2×D3—21no—
226C2×D3#35312yes—
227C2×D3#35612yes—
228C3×C4—24no—
229C3×C4—21no—
230C3×C4—21no—
231C3×C4—42no—
232C3×C4—81no—
233Dic3—24no—
234Dic3—22no—
235Dic3—42no—
236Dic3—81no—
237C23—31no—
238C23—31no—
239C23—62no—
240C23—62no—
241C23—61no—
242C23—61no—
243C23D3×C22⋊C411yescharacteristic
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—32no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—616no—
255C2×C4—62no—
256C2×C4—62no—
257C2×C4—62no—
258C2×C4—62no—
259C2×C4—62no—
260C2×C4—62no—
261C2×C4—62no—
262C2×C4—61no—
263C2×C4—61no—
264C2×C4—62no—
265C2×C4—124no—
266C2×C4—122no—
267C2×C4—122no—
268C2×C4—121no—
269C2×C4—242no—
270C2×C4—242no—
271C2×C4—242no—
272C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
273C2×C4D3×M1611yescharacteristic
274C8—82no—
275C8—242no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3#146311yescharacteristic
279C2×C3C2×#35311yescharacteristic
280C2×C3C2×#35611yescharacteristic
281D3—22no—
282D3—22no—
283C22—21no—
284C22—21no—
285C22—32no—
286C22—32no—
287C22—64no—
288C22—64no—
289C22—61no—
290C22—61no—
291C22—61no—
292C22—61no—
293C22#472611yescharacteristic
294C22D3×#9911yescharacteristic
295C22D3×#10111yescharacteristic
296C4—24no—
297C4—21no—
298C4—21no—
299C4—42no—
300C4—64no—
301C4—62no—
302C4—81no—
303C4—122no—
304C4—241no—
305C3#1151211yescharacteristic, complemented, Sylow
306C2—21no—
307C2—21no—
308C2—62no—
309C2—62no—
310C2#6814411yescharacteristic
311C2D3×#35311yescharacteristic
312C2D3×#35611yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1325C2
socleC22×C3C2×#99
prime radicalC23×D3C22⋊C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1325D3
hypocenterC3#11512
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1325

Complements

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

NH ≅ G/Ntype
3. C3×#1325C2semidirect
4. #67635C2semidirect
5. #67453C2semidirect
24. #1325D3semidirect
94. C3×C42M16semidirect
104. C4×Dic3M16semidirect
214. C42ΓL(1,𝔽25)semidirect
305. C3#11512semidirect

Maximal subgroups

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

subgroupindex
D3×#3842
#681442
C3×#13252
#676352
#674532
#115123