Finite Group Databasebeta

← #10693134

Subgroups — #10693134

Order 768.

684 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 103 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#82349C211yesmaximal, characteristic
2#82302C211yesmaximal, characteristic
3C3×#1465C211yesmaximal, characteristic
4#69990C212yesmaximal, complemented
5#67685C212yesmaximal, complemented
6#9505—31nomaximal, Sylow
7C23×Dic6C2211yescharacteristic
8C2×C3×C4×D4C2211yescharacteristic
9C2×#924—21no—
10C3×#407C2212yes—
11#5220C412yes—
12#5212C2211yescharacteristic
13#5207C2212yes—
14C2×#830—21no—
15C2×#824—24no—
16#5011—24no—
17#4979—21no—
18#4947—21no—
19#4946C412yes—
20#4929—21no—
21#4926—21no—
22C3×#324—21no—
23C3×#323—21no—
24#2973—31no—
25#2409—31no—
26#1784—32no—
27#1465D311yescharacteristic
28#1305—32no—
29C22×C3×D4C2×C411yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4C2311yescharacteristic
32C23×Dic3C2×C412yes—
33C22×Dic6—22no—
34C22×Dic6—21no—
35C22×Dic6—21no—
36C22×Dic6—22no—
37C22×Dic6—44no—
38C2×C3×M16—22no—
39C3×C4×D4—22no—
40C3×C4×D4—22no—
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C22⋊C4C2×C411yescharacteristic
43C2×C3×C22⋊C4D411yescharacteristic
44C2×C3×C42D411yescharacteristic
45C2×#217—22no—
46C2×#217C2×C412yes—
47C2×C4⋊Dic3—21no—
48#924—44no—
49C2×Dic3⋊C4—22no—
50C2×#208D414yes—
51C22×C3⋊C8—22no—
52C3×C22⋊C8—22no—
53C3×C22⋊C8—22no—
54C3⋊#95—22no—
55C3⋊#95—22no—
56C3⋊#95—21no—
57#830—22no—
58#830—22no—
59#830—22no—
60#824—48no—
61C4⋊(C3⋊C8)—22no—
62#803—22no—
63C23×Q8—31no—
64C2×C22⋊Q8—61no—
65C2×C4×D4C2×D311yescharacteristic
66#435—64no—
67#432—61no—
68C2×#101—64no—
69#407—32no—
70#407C2×D312yes—
71C2×C22⋊C8—61no—
72#395—61no—
73#393—61no—
74#391—61no—
75#386—32no—
76#385—32no—
77#380—31no—
78#324—21no—
79#323—21no—
80C24×C3C22⋊C411yescharacteristic
81C24×C3C22×C411yescharacteristic
82C2×C3×D4—22no—
83C2×C3×D4—22no—
84C2×C3×D4—41no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—42no—
88C22×C3×C4—42no—
89C22×C3×C4C22⋊C411yescharacteristic
90C22×C3×C4C22⋊C411yescharacteristic
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4C2×D412yes—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4#4112yes—
96C22×Dic3—22no—
97C22×Dic3—24no—
98C22×Dic3—22no—
99C22×Dic3—44no—
100C2×Dic6—28no—
101C2×Dic6—24no—
102C2×Dic6—44no—
103C2×Dic6—48no—
104C2×Dic6—44no—
105C2×Dic6—42no—
106C2×Dic6—41no—
107C2×Dic6—42no—
108C2×Dic6—41no—
109C3×M16—44no—
110C2×C3×C8—42no—
111C3×C4⋊C4—22no—
112C3×C22⋊C4—22no—
113C3×C22⋊C4—22no—
114C3×C42—21no—
115C3×C42—21no—
116#217—44no—
117#217—42no—
118C4⋊Dic3—42no—
119Dic3⋊C4—44no—
120#208—44no—
121C2×C3⋊C8—24no—
122C2×C3⋊C8—82no—
123C22×Q8—61no—
124C22×Q8—62no—
125C22×Q8—62no—
126C22×Q8—61no—
127C22×Q8—124no—
128C22×D4C4×D311yescharacteristic
129C23×C4—32no—
130C23×C4D1211yescharacteristic
131C23×C4C22×D311yescharacteristic
132C2×M16—22no—
133C2×M16—34no—
134C22×C8—62no—
135C22⋊Q8—124no—
136C4×D4—22no—
137C4×D4—22no—
138C2×C4⋊C4—62no—
139C2×C4⋊C4—61no—
140C2×C4⋊C4#6711yescharacteristic
141C2×C22⋊C4—32no—
142C2×C22⋊C4—62no—
143C2×C22⋊C4C4×D311yescharacteristic
144C2×C22⋊C4#6711yescharacteristic
145C2×C42D1211yescharacteristic
146C4⋊C8—62no—
147#101—128no—
148C22⋊C8—22no—
149C22⋊C8—22no—
150C22⋊C8—62no—
151C22⋊C8—62no—
152C22⋊C8—62no—
153#97—62no—
154#95—62no—
155#95—62no—
156#95—61no—
157C23×C3—21no—
158C23×C3—21no—
159C23×C3—21no—
160C23×C3—21no—
161C23×C3—41no—
162C23×C3—41no—
163C23×C3—41no—
164C23×C3—41no—
165C23×C3#10112yescomplemented
166C23×C3C2×C22⋊C411yescharacteristic
167C23×C3C4×D412yes—
168C3×D4—44no—
169C2×C3×C4—24no—
170C2×C3×C4—22no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—41no—
177C2×C3×C4—42no—
178C2×C3×C4—42no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—81no—
182C2×C3×C4#12011yescharacteristic
183C2×C3×C4#12112yes—
184C2×C3×C4#12311yescharacteristic
185C2×Dic3—24no—
186C2×Dic3—28no—
187C2×Dic3—42no—
188C2×Dic3—44no—
189C2×Dic3—42no—
190C2×Dic3—42no—
191C2×Dic3—42no—
192C2×Dic3—42no—
193Dic6—48no—
194Dic6—44no—
195Dic6—44no—
196C3×C8—82no—
197C3⋊C8—44no—
198C24(C2×D3)⋊C411yescharacteristic
199C24C2×C4×D311yescharacteristic
200C2×Q8—68no—
201C2×Q8—64no—
202C2×Q8—122no—
203C2×Q8—124no—
204C2×Q8—128no—
205C2×Q8—124no—
206C2×Q8—121no—
207C2×Q8—121no—
208C2×Q8—122no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—41no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—64no—
217C22×C4—62no—
218C22×C4—62no—
219C22×C4—124no—
220C22×C4(C2×D3)⋊C411yescharacteristic
221C22×C4(C2×D3)⋊C411yescharacteristic
222C22×C4(C2×D3)⋊C411yescharacteristic
223C22×C4C2×D1211yescharacteristic
224C22×C4D3×D412yes—
225C22×C4#23712yes—
226C22×C4C2×#6711yescharacteristic
227M16—44no—
228M16—124no—
229C2×C8—42no—
230C2×C8—64no—
231C2×C8—242no—
232C4⋊C4—22no—
233C4⋊C4—122no—
234C4⋊C4—124no—
235C22⋊C4—22no—
236C22⋊C4—22no—
237C22⋊C4—124no—
238C22⋊C4—122no—
239C42—21no—
240C42—21no—
241C22×C3—21no—
242C22×C3—21no—
243C22×C3—22no—
244C22×C3—22no—
245C22×C3—22no—
246C22×C3—22no—
247C22×C3—42no—
248C22×C3—42no—
249C22×C3—42no—
250C22×C3—41no—
251C22×C3—41no—
252C22×C3—81no—
253C22×C3#38611yescharacteristic
254C22×C3C2×#10111yescharacteristic
255C22×C3#41311yescharacteristic
256C3×C4—22no—
257C3×C4—21no—
258C3×C4—41no—
259C3×C4—41no—
260C3×C4—41no—
261C3×C4B(4,𝔽2)12yes—
262Dic3—44no—
263Dic3—82no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23#82412yescomplemented
273C23#67⋊C412yes—
274C23C2×(C2×D3)⋊C411yescharacteristic
275Q8—128no—
276Q8—124no—
277Q8—124no—
278D4—44no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—64no—
292C2×C4—68no—
293C2×C4—81no—
294C2×C4—122no—
295C2×C4—124no—
296C2×C4—122no—
297C2×C4—122no—
298C2×C4—122no—
299C2×C4—122no—
300C2×C4#88211yescharacteristic
301C2×C4#88611yescharacteristic
302C2×C4#93912yes—
303C8—82no—
304C8—124no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—42no—
308C2×C3—42no—
309C2×C3#188911yescharacteristic
310C2×C3#190111yescharacteristic
311C2×C3#191511yescharacteristic
312C22—21no—
313C22—21no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—41no—
322C22—41no—
323C22—81no—
324C22#495811yescharacteristic
325C22#513211yescharacteristic
326C22C2×#82411yescharacteristic
327C4—22no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—124no—
333C4—242no—
334C4#474112yes—
335C3#950511yescharacteristic, complemented, Sylow
336C2—21no—
337C2—21no—
338C2—42no—
339C2—42no—
340C2#6998611yescharacteristic
341C2#7002211yescharacteristic
342C2#7013811yescharacteristic
subgroupG/H
centerC22#4958
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1465C2
socleC22×C3#386
prime radicalC22×C3×D4C2×C4
cosocleC23×C3×C4C23
hypercenter#1465D3
hypocenterC3#9505
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1465

Complements

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

NH ≅ G/Ntype
4. #69990C2semidirect
5. #67685C2semidirect
165. C23×C3#101semidirect
272. C23#824semidirect
335. C3#9505semidirect

Maximal subgroups

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

subgroupindex
#823492
#823022
C3×#14652
#699902
#676852
#95053