Finite Group Databasebeta

← #10685560

Subgroups — #10685560

Order 768.

914 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 155 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×#407C212yesmaximal
2#74050C211yesmaximal, characteristic
3C3×#1306C211yesmaximal, characteristic, complemented
4#67686C212yesmaximal, complemented
5#67629C211yesmaximal, characteristic, complemented
6#8115—31nomaximal, Sylow
7C23×C4×D3C2211yescharacteristic
8C2×D3×M16—22no—
9C2×#877C412yes—
10C3×#407C2212yes—
11C3×#381C2211yescharacteristic
12#5211C2211yescharacteristic
13#5207C2212yes—
14#5126—22no—
15#4955C412yes—
16#4726—22no—
17#4726C414yescomplemented
18D3×C22⋊C8—22no—
19D3×C22⋊C8C414yescomplemented
20#4667—22no—
21#4666—22no—
22C2×#407—32no—
23#2280—31no—
24#1306—32no—
25#1306—31no—
26#1306D311yescharacteristic, complemented
27C24×D3C2×C411yescharacteristic
28C23×C3×C4C2311yescharacteristic
29C23×Dic3C2×C411yescharacteristic
30C22×C4×D3—44no—
31C22×C4×D3C2×C412yes—
32C22×C4×D3C2×C412yes—
33C22×C4×D3D412yes—
34C22×C4×D3D412yes—
35C22×C4×D3D412yes—
36C22×C4×D3Q812yes—
37C2×C3×M16—22no—
38C2×C3×C22⋊C4C2×C412yes—
39C2×#217C2×C412yes—
40C2×(C2×D3)⋊C4—24no—
41C2×Dic3⋊C4—22no—
42C2×C4×Dic3—22no—
43C2×#208—22no—
44D3×M16—48no—
45C2×ΓL(1,𝔽25)—24no—
46C2×D3×C8—24no—
47#877—44no—
48C3×C22⋊C8—22no—
49C3×C22⋊C8C2×C414yes—
50C3×#95—22no—
51C3⋊#95—22no—
52#830—22no—
53#830C2×C414yes—
54(C2×D3)⋊C8—24no—
55(C2×D3)⋊C8—24no—
56C24×C4—31no—
57C22×M16—62no—
58C2×#117—32no—
59#407—32no—
60#407—34no—
61#407—62no—
62#407—62no—
63#407C2×D312yes—
64C2×C22⋊C8—34no—
65C2×C22⋊C8—62no—
66#381—31no—
67#381C2×D311yescharacteristic
68#380—62no—
69#379—32no—
70C4×C22⋊C4—62no—
71C24×C3C22×C411yescharacteristic
72C23×D3—44no—
73C23×D3—42no—
74C23×D3C4211yescharacteristic
75C23×D3C4211yescharacteristic
76C23×D3C22⋊C412yes—
77C23×D3C4⋊C412yes—
78C22×C3×C4—22no—
79C22×C3×C4—42no—
80C22×C3×C4C22×C412yes—
81C22×C3×C4C2×D412yes—
82C22×C3×C4C2×D411yescharacteristic
83C22×C3×C4C2×Q811yescharacteristic
84C22×Dic3—24no—
85C22×Dic3—22no—
86C22×Dic3C4212yes—
87C22×Dic3C22⋊C411yescharacteristic
88C22×Dic3C22⋊C411yescharacteristic
89C22×Dic3C4⋊C412yes—
90C2×C4×D3—28no—
91C2×C4×D3—44no—
92C2×C4×D3—44no—
93C2×C4×D3—44no—
94C2×C4×D3—44no—
95C2×C4×D3—44no—
96C2×C4×D3C22⋊C412yes—
97C2×C4×D3C22⋊C412yes—
98C2×C4×D3C22⋊C412yes—
99C2×C4×D3C22⋊C412yes—
100C2×C4×D3C4⋊C412yes—
101C2×C4×D3C4⋊C412yes—
102C2×C4×D3C4⋊C412yes—
103C2×C4×D3C4⋊C412yes—
104C3×M16—44no—
105C2×C3×C8—24no—
106C3×C22⋊C4—42no—
107#217—42no—
108(C2×D3)⋊C4—48no—
109Dic3⋊C4—44no—
110C4×Dic3—44no—
111#208—44no—
112C2×C3⋊C8—24no—
113ΓL(1,𝔽25)—48no—
114D3×C8—48no—
115C25—31no—
116C23×C4—31no—
117C23×C4—32no—
118C23×C4—32no—
119C23×C4—32no—
120C23×C4—32no—
121C23×C4—32no—
122C23×C4—32no—
123C23×C4—124no—
124C23×C4C22×D311yescharacteristic
125C2×M16—22no—
126C2×M16—64no—
127C2×M16—62no—
128C2×M16—128no—
129C22×C8—64no—
130#117—124no—
131C2×C4⋊C4—62no—
132C2×C22⋊C4—32no—
133C2×C22⋊C4—64no—
134C2×C22⋊C4C4×D312yes—
135C2×C42—62no—
136C22⋊C8—22no—
137C22⋊C8—34no—
138C22⋊C8—64no—
139C22⋊C8—64no—
140C22⋊C8—62no—
141C22⋊C8C4×D314yes—
142#95—22no—
143#95—62no—
144C23×C3—22no—
145C23×C3—42no—
146C23×C3C2×C4211yescharacteristic
147C23×C3C2×C22⋊C411yescharacteristic
148C23×C3C2×C4⋊C411yescharacteristic
149C22×D3—24no—
150C22×D3—24no—
151C22×D3—24no—
152C22×D3—24no—
153C22×D3—41no—
154C22×D3—41no—
155C22×D3—42no—
156C22×D3—42no—
157C22×D3—42no—
158C22×D3—42no—
159C22×D3—416no—
160C22×D3#9512yes—
161C22×D3#9512yes—
162C2×C3×C4—24no—
163C2×C3×C4—42no—
164C2×C3×C4—41no—
165C2×C3×C4—41no—
166C2×C3×C4—44no—
167C2×C3×C4—42no—
168C2×C3×C4C2×C22⋊C412yes—
169C2×C3×C4C2×C4⋊C412yes—
170C2×Dic3—22no—
171C2×Dic3—24no—
172C2×Dic3—24no—
173C2×Dic3—44no—
174C2×Dic3—42no—
175C2×Dic3#9512yes—
176C2×Dic3#9512yes—
177C2×Dic3#9914yes—
178C4×D3—24no—
179C4×D3—24no—
180C4×D3—24no—
181C4×D3—24no—
182C4×D3—42no—
183C4×D3—42no—
184C4×D3—42no—
185C4×D3—42no—
186C3×C8—44no—
187C3⋊C8—44no—
188C24—31no—
189C24—32no—
190C24—32no—
191C24—31no—
192C24—122no—
193C24—124no—
194C24C2×C4×D311yescharacteristic
195C22×C4—22no—
196C22×C4—31no—
197C22×C4—32no—
198C22×C4—31no—
199C22×C4—32no—
200C22×C4—32no—
201C22×C4—32no—
202C22×C4—32no—
203C22×C4—32no—
204C22×C4—32no—
205C22×C4—32no—
206C22×C4—32no—
207C22×C4—32no—
208C22×C4—42no—
209C22×C4—68no—
210C22×C4—64no—
211C22×C4—62no—
212C22×C4—124no—
213C22×C4—124no—
214C22×C4—124no—
215C22×C4—124no—
216C22×C4—124no—
217C22×C4C2×C4×D312yes—
218C22×C4D3×D412yes—
219C22×C4D3×D411yescharacteristic
220C22×C4D3×Q811yescharacteristic
221M16—44no—
222M16—128no—
223M16—124no—
224C2×C8—24no—
225C2×C8—64no—
226C2×C8—128no—
227C4⋊C4—124no—
228C22⋊C4—42no—
229C22⋊C4—128no—
230C22⋊C4—122no—
231C42—124no—
232C22×C3—22no—
233C22×C3—21no—
234C22×C3—21no—
235C22×C3—22no—
236C22×C3—22no—
237C22×C3—44no—
238C22×C3C2×#9511yescharacteristic
239C22×C3C2×#9912yes—
240C2×D3—22no—
241C2×D3—22no—
242C2×D3—22no—
243C2×D3—22no—
244C2×D3—22no—
245C2×D3—44no—
246C2×D3—48no—
247C2×D3—48no—
248C2×D3—44no—
249C2×D3#34414yes—
250C3×C4—22no—
251C3×C4—22no—
252C3×C4—82no—
253Dic3—22no—
254Dic3—82no—
255Dic3#34214yes—
256C23—22no—
257C23—32no—
258C23—32no—
259C23—42no—
260C23—64no—
261C23—64no—
262C23—64no—
263C23—64no—
264C23—121no—
265C23—121no—
266C23—122no—
267C23—122no—
268C23—122no—
269C23—122no—
270C23—1216no—
271C23C42×D311yescharacteristic
272C23D3×C22⋊C411yescharacteristic
273C23D3×C4⋊C411yescharacteristic
274C2×C4—24no—
275C2×C4—32no—
276C2×C4—34no—
277C2×C4—32no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—64no—
284C2×C4—64no—
285C2×C4—64no—
286C2×C4—64no—
287C2×C4—64no—
288C2×C4—64no—
289C2×C4—62no—
290C2×C4—122no—
291C2×C4—122no—
292C2×C4—122no—
293C2×C4—122no—
294C2×C4—124no—
295C2×C4—122no—
296C2×C4D3×C22⋊C412yes—
297C2×C4D3×C4⋊C412yes—
298C8—44no—
299C8—124no—
300C2×C3—22no—
301C2×C3—41no—
302C2×C3—41no—
303C2×C3C2×#34211yescharacteristic
304C2×C3C2×#34412yes—
305D3—24no—
306D3—42no—
307C22—22no—
308C22—21no—
309C22—21no—
310C22—22no—
311C22—22no—
312C22—34no—
313C22—44no—
314C22—62no—
315C22—62no—
316C22—62no—
317C22—62no—
318C22—62no—
319C22—128no—
320C22—124no—
321C22—128no—
322C22—124no—
323C22D3×#9511yescharacteristic
324C22D3×#9912yes—
325C4—22no—
326C4—22no—
327C4—34no—
328C4—62no—
329C4—82no—
330C4—242no—
331C3#811511yescharacteristic, complemented, Sylow
332C2—22no—
333C2—41no—
334C2—41no—
335C2—64no—
336C2—122no—
337C2D3×#34211yescharacteristic
338C2D3×#34412yes—
subgroupG/H
centerC22D3×#95
commutatorC23×C3C2×C42
FrattiniC23×C4C22×D3
FittingC3×#1306C2
socleC22×C3C2×#95
prime radicalC24×D3C2×C4
cosocleC23×C3×C4C23
hypercenter#1306D3
hypocenterC3#8115
LCSG ▷ C23×C3 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1306

Complements

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

NH ≅ G/Ntype
3. C3×#1306C2semidirect
4. #67686C2semidirect
5. #67629C2semidirect
17. #4726C4semidirect
19. D3×C22⋊C8C4semidirect
26. #1306D3semidirect
331. C3#8115semidirect

Maximal subgroups

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

subgroupindex
D3×#4072
#740502
C3×#13062
#676862
#676292
#81153