Finite Group Databasebeta

← #10704652

Subgroups — #10704652

Order 768.

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

normal subgroups only

This group has 596 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#76293C211yesmaximal, characteristic
2#68145C212yesmaximal, complemented
3C3×#1318C211yesmaximal, characteristic, complemented
4#67640C211yesmaximal, characteristic, complemented
5#67453C212yesmaximal, complemented
6#11529—31nomaximal, Sylow
7C2×#940C411yescharacteristic
8C2×D4×Dic3C411yescharacteristic
9C2×#893C2211yescharacteristic
10C2×#67⋊C4—22no—
11C3×#390C2211yescharacteristic
12#4975C2211yescharacteristic
13#4957—22no—
14#4939C411yescharacteristic
15C42×Dic3C411yescharacteristic
16#4726—22no—
17C3×#324C2212yes—
18#4528C2212yes—
19#4468—22no—
20C4×#127—31no—
21#1465—32no—
22#1325—32no—
23#1318—31no—
24#1318D311yescharacteristic, complemented
25C22×C3×D4C2×C411yescharacteristic
26C22×#67—22no—
27C23×Dic3—21no—
28C22×D12C2×C411yescharacteristic
29C22×C4×D3D412yes—
30C2×C3×C4⋊C4C2311yescharacteristic
31C2×C3×C22⋊C4—22no—
32C2×C3×C42C2×C411yescharacteristic
33C2×#217—21no—
34#940—22no—
35#940—22no—
36D4×Dic3—41no—
37D4×Dic3—41no—
38C2×(C2×D3)⋊C4—22no—
39C2×(C2×D3)⋊C4D412yes—
40C2×C4⋊Dic3C2×C411yescharacteristic
41C2×Dic3⋊C4—22no—
42C2×C4×Dic3—21no—
43C2×C4×Dic3—21no—
44C2×C4×Dic3C2×C411yescharacteristic
45C2×C4×Dic3C2×C411yescharacteristic
46C2×ΓL(1,𝔽25)—42no—
47#893—41no—
48#893—41no—
49#67⋊C4—48no—
50C3×C22⋊C8—22no—
51#830—22no—
52(C2×D3)⋊C8—42no—
53C2×#127—31no—
54C2×C4×D4—31no—
55C2×C4×D4—31no—
56C2×C4×D4—62no—
57#407—62no—
58#390—31no—
59#390—62no—
60#390C2×D311yescharacteristic
61#384—31no—
62C43—31no—
63#324—32no—
64#324C2×D312yes—
65#323—62no—
66C24×C3—21no—
67C23×D3C22⋊C412yes—
68C2×C3×D4—21no—
69C2×C3×D4—41no—
70C2×C3×D4—41no—
71C2×C3×D4M1612yescomplemented
72C22×C3×C4—21no—
73C22×C3×C4C22×C411yescharacteristic
74C22×C3×C4C2×D412yes—
75C2×#67—28no—
76C2×#67—44no—
77C2×#67—44no—
78C22×Dic3—21no—
79C22×Dic3—21no—
80C22×Dic3—44no—
81C22×Dic3—41no—
82C22×Dic3—41no—
83C22×Dic3C22⋊C411yescharacteristic
84C22×Dic3C22⋊C411yescharacteristic
85C2×D12—21no—
86C2×D12—21no—
87C2×D12—41no—
88C2×D12—41no—
89C2×C4×D3—22no—
90C2×C4×D3—22no—
91C2×C4×D3—44no—
92C2×C3×C8—42no—
93C3×C4⋊C4—41no—
94C3×C22⋊C4—44no—
95C3×C42—41no—
96#217—42no—
97(C2×D3)⋊C4—42no—
98(C2×D3)⋊C4—44no—
99C4⋊Dic3—41no—
100Dic3⋊C4—44no—
101C4×Dic3—24no—
102C4×Dic3—21no—
103C4×Dic3—21no—
104C4×Dic3—21no—
105C4×Dic3—42no—
106C4×Dic3M1612yescomplemented
107C2×C3⋊C8—42no—
108ΓL(1,𝔽25)—84no—
109C22×D4—31no—
110C22×D4—62no—
111C22×D4C4×D311yescharacteristic
112C23×C4—32no—
113C23×C4—61no—
114C2×M16—122no—
115#127—62no—
116#127—62no—
117C4×D4—128no—
118C4×D4—121no—
119C4×D4—121no—
120C4×D4—121no—
121C4×D4—121no—
122C2×C4⋊C4—31no—
123C2×C4⋊C4—62no—
124C2×C4⋊C4C22×D311yescharacteristic
125C2×C22⋊C4—22no—
126C2×C22⋊C4—32no—
127C2×C22⋊C4—62no—
128C2×C22⋊C4—61no—
129C2×C42—31no—
130C2×C42—31no—
131C2×C42—61no—
132C2×C42—61no—
133C2×C42C4×D311yescharacteristic
134C22⋊C8—22no—
135C22⋊C8—62no—
136C22⋊C8—122no—
137C23×C3—22no—
138C23×C3—42no—
139C23×C3—42no—
140C23×C3—41no—
141C23×C3—41no—
142C23×C3C2×C22⋊C411yescharacteristic
143C22×D3—22no—
144C22×D3—22no—
145C22×D3—22no—
146C22×D3—42no—
147C22×D3—42no—
148C3×D4—42no—
149C3×D4—42no—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4—41no—
155C2×C3×C4—41no—
156C2×C3×C4—42no—
157C2×C3×C4C2×M1611yescharacteristic
158C2×C3×C4C2×M1611yescharacteristic
159#67—416no—
160C2×Dic3—22no—
161C2×Dic3—22no—
162C2×Dic3—22no—
163C2×Dic3—22no—
164C2×Dic3—21no—
165C2×Dic3—41no—
166C2×Dic3—42no—
167C2×Dic3—41no—
168C2×Dic3—82no—
169C2×Dic3—82no—
170C2×Dic3#9911yescharacteristic
171C2×Dic3#9911yescharacteristic
172D12—42no—
173D12—42no—
174C4×D3—44no—
175C4×D3—82no—
176C3×C8—82no—
177C3⋊C8—82no—
178C24—21no—
179C24—32no—
180C2×D4—21no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4—68no—
184C2×D4—61no—
185C2×D4—61no—
186C2×D4—121no—
187C2×D4—121no—
188C2×D4—124no—
189C2×D4—124no—
190C2×D4ΓL(1,𝔽25)12yescomplemented
191C22×C4—21no—
192C22×C4—31no—
193C22×C4—31no—
194C22×C4—62no—
195C22×C4—62no—
196C22×C4—61no—
197C22×C4—61no—
198C22×C4—121no—
199C22×C4—121no—
200C22×C4—124no—
201C22×C4—124no—
202C22×C4C2×C4×D311yescharacteristic
203C22×C4D3×D412yes—
204M16—244no—
205C2×C8—42no—
206C2×C8—122no—
207C4⋊C4—41no—
208C4⋊C4—121no—
209C4⋊C4—124no—
210C22⋊C4—44no—
211C22⋊C4—122no—
212C22⋊C4—124no—
213C22⋊C4—122no—
214C42—32no—
215C42—41no—
216C42—61no—
217C42—61no—
218C42—64no—
219C42—61no—
220C42—122no—
221C22×C3—21no—
222C22×C3—21no—
223C22×C3—44no—
224C22×C3—42no—
225C22×C3—82no—
226C22×C3—81no—
227C22×C3—81no—
228C22×C3#40711yescharacteristic
229C22×C3C2×#9911yescharacteristic
230C22×C3#41311yescharacteristic
231C2×D3—44no—
232C2×D3—42no—
233C2×D3—42no—
234C2×D3—42no—
235C2×D3—82no—
236C3×C4—21no—
237C3×C4—21no—
238C3×C4—42no—
239C3×C4—42no—
240Dic3—24no—
241Dic3—21no—
242Dic3—41no—
243Dic3—81no—
244Dic3#35312yes—
245C23—22no—
246C23—42no—
247C23—42no—
248C23—41no—
249C23—41no—
250C23—62no—
251C23—62no—
252C23—62no—
253C23—122no—
254C23—122no—
255C23D3×C22⋊C411yescharacteristic
256D4—42no—
257D4—42no—
258D4—122no—
259D4—122no—
260D4—1216no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—31no—
266C2×C4—31no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—62no—
271C2×C4—62no—
272C2×C4—62no—
273C2×C4—62no—
274C2×C4—61no—
275C2×C4—124no—
276C2×C4—121no—
277C2×C4—122no—
278C2×C4—121no—
279C2×C4—242no—
280C2×C4—242no—
281C2×C4—242no—
282C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
283C2×C4D3×M1611yescharacteristic
284C8—82no—
285C8—242no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—82no—
289C2×C3#146511yescharacteristic
290C2×C3C2×#35311yescharacteristic
291C2×C3#213211yescharacteristic
292D3—82no—
293C22—21no—
294C22—21no—
295C22—44no—
296C22—42no—
297C22—82no—
298C22—81no—
299C22—81no—
300C22—124no—
301C22—122no—
302C22—122no—
303C22—122no—
304C22—242no—
305C22#472611yescharacteristic
306C22D3×#9911yescharacteristic
307C22#474511yescharacteristic
308C4—21no—
309C4—21no—
310C4—32no—
311C4—42no—
312C4—42no—
313C4—64no—
314C4—61no—
315C4—121no—
316C4—241no—
317C3#1152911yescharacteristic, complemented, Sylow
318C2—21no—
319C2—21no—
320C2—82no—
321C2—242no—
322C2#6812111yescharacteristic
323C2D3×#35311yescharacteristic
324C2#6916511yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1318C2
socleC22×C3C2×#99
prime radicalC2×#940C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1318D3
hypocenterC3#11529
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1318

Complements

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

NH ≅ G/Ntype
2. #68145C2semidirect
3. C3×#1318C2semidirect
4. #67640C2semidirect
5. #67453C2semidirect
24. #1318D3semidirect
71. C2×C3×D4M16semidirect
106. C4×Dic3M16semidirect
190. C2×D4ΓL(1,𝔽25)semidirect
317. C3#11529semidirect

Maximal subgroups

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

subgroupindex
#762932
#681452
C3×#13182
#676402
#674532
#115293