Finite Group Databasebeta

← #10693142

Subgroups — #10693142

Order 768.

682 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C3×#2409C211yesmaximal, characteristic
2#82304C211yesmaximal, characteristic
3#72133C212yesmaximal, complemented
4#71602C211yesmaximal, characteristic
5#67626C212yesmaximal, complemented
6#9506—31nomaximal, Sylow
7C23×C3×Q8C2211yescharacteristic
8C2×C3×C22⋊Q8—21no—
9C2×C3×C4×D4C2211yescharacteristic
10C3×#395—21no—
11C3×#393—21no—
12C3×#391—21no—
13C3×#386C412yes—
14C3×#385C412yes—
15C3×#380C2211yescharacteristic
16C2×#836—24no—
17#5207—21no—
18#5207C2212yes—
19#5207C2212yes—
20#5012—24no—
21#4925—21no—
22#4528—21no—
23#4525—21no—
24#2975—31no—
25#2409D311yescharacteristic
26#1784—32no—
27#1465—31no—
28#1305—32no—
29C22×C3×Q8—22no—
30C22×C3×Q8—21no—
31C22×C3×Q8—21no—
32C22×C3×Q8—22no—
33C22×C3×Q8—44no—
34C22×C3×D4C2×C411yescharacteristic
35C23×C3×C4C2×C412yes—
36C23×C3×C4D411yescharacteristic
37C23×C3×C4C2311yescharacteristic
38C3×C22⋊Q8—44no—
39C3×C4×D4—22no—
40C3×C4×D4—22no—
41C2×C3×C4⋊C4—22no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4D411yescharacteristic
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4C2×C411yescharacteristic
46C2×C3×C22⋊C4C2×C412yes—
47C2×C3×C22⋊C4D411yescharacteristic
48C2×C3×C42D411yescharacteristic
49C2×#208—22no—
50C2×#208—22no—
51C2×#208D414yes—
52C3×#95—22no—
53C3×#95—22no—
54C3×#95—21no—
55#836—48no—
56#830—22no—
57#830—22no—
58#830—22no—
59#830—22no—
60#830—22no—
61C4⋊(C3⋊C8)—22no—
62C4×C3⋊C8—22no—
63C23×Q8C2×D311yescharacteristic
64C2×C22⋊Q8—21no—
65C2×C4×D4C2×D311yescharacteristic
66#436—64no—
67#431—61no—
68C2×#101—64no—
69#407—32no—
70#407—32no—
71#407—61no—
72#395—21no—
73#393—21no—
74#391—21no—
75#386Dic312yes—
76#385Dic312yes—
77#380C2×D311yescharacteristic
78#324—61no—
79#323—61no—
80C24×C3C22⋊C411yescharacteristic
81C24×C3C22×C411yescharacteristic
82C2×C3×Q8—28no—
83C2×C3×Q8—24no—
84C2×C3×Q8—44no—
85C2×C3×Q8—42no—
86C2×C3×Q8—44no—
87C2×C3×Q8—48no—
88C2×C3×Q8—41no—
89C2×C3×Q8—42no—
90C2×C3×Q8—41no—
91C2×C3×D4—22no—
92C2×C3×D4—22no—
93C2×C3×D4—41no—
94C22×C3×C4—22no—
95C22×C3×C4—24no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—22no—
99C22×C3×C4—44no—
100C22×C3×C4—42no—
101C22×C3×C4—42no—
102C22×C3×C4C22⋊C411yescharacteristic
103C22×C3×C4C22⋊C411yescharacteristic
104C22×C3×C4C22⋊C411yescharacteristic
105C22×C3×C4C2×D412yes—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4112yes—
109C3×C4⋊C4—22no—
110C3×C4⋊C4—42no—
111C3×C4⋊C4—44no—
112C3×C22⋊C4—22no—
113C3×C22⋊C4—22no—
114C3×C22⋊C4—44no—
115C3×C22⋊C4—42no—
116C3×C42—21no—
117C3×C42—21no—
118#208—44no—
119#208—44no—
120#208—82no—
121C2×C3⋊C8—24no—
122C2×C3⋊C8—42no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—44no—
128C22×D4C2×Dic311yescharacteristic
129C23×C4C2×Dic312yes—
130C23×C4#6711yescharacteristic
131C23×C4C22×D311yescharacteristic
132C2×M16—34no—
133C2×M16—62no—
134C2×M16—62no—
135C22⋊Q8—44no—
136C4×D4—22no—
137C4×D4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4#6711yescharacteristic
141C2×C22⋊C4—22no—
142C2×C22⋊C4C2×Dic311yescharacteristic
143C2×C22⋊C4C2×Dic312yes—
144C2×C22⋊C4#6711yescharacteristic
145C2×C42#6711yescharacteristic
146C4⋊C8—62no—
147#101—128no—
148C22⋊C8—62no—
149C22⋊C8—62no—
150C22⋊C8—62no—
151C22⋊C8—62no—
152C22⋊C8—62no—
153C4×C8—62no—
154#95—22no—
155#95—22no—
156#95—21no—
157C23×C3—21no—
158C23×C3—21no—
159C23×C3—21no—
160C23×C3—21no—
161C23×C3—21no—
162C23×C3—41no—
163C23×C3—41no—
164C23×C3—41no—
165C23×C3—41no—
166C23×C3C2×C22⋊C411yescharacteristic
167C23×C3C4×D412yes—
168C3×Q8—48no—
169C3×Q8—44no—
170C3×Q8—44no—
171C3×D4—44no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—24no—
175C2×C3×C4—22no—
176C2×C3×C4—24no—
177C2×C3×C4—28no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—41no—
181C2×C3×C4—42no—
182C2×C3×C4—42no—
183C2×C3×C4—44no—
184C2×C3×C4—42no—
185C2×C3×C4—42no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—42no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4—81no—
192C2×C3×C4#10112yescomplemented
193C2×C3×C4#12011yescharacteristic
194C2×C3×C4#12112yes—
195C2×C3×C4#12311yescharacteristic
196C3⋊C8—44no—
197C3⋊C8—82no—
198C24#21711yescharacteristic
199C24C22×Dic311yescharacteristic
200C2×Q8—28no—
201C2×Q8—24no—
202C2×Q8—42no—
203C2×Q8—44no—
204C2×Q8—44no—
205C2×Q8—48no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—42no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—41no—
212C22×C4—24no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—44no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4#21711yescharacteristic
221C22×C4#21711yescharacteristic
222C22×C4#21711yescharacteristic
223C22×C4D3×D412yes—
224C22×C4#23712yes—
225C22×C4C2×#6711yescharacteristic
226C22×C4C2×#6711yescharacteristic
227M16—124no—
228M16—124no—
229M16—242no—
230C2×C8—64no—
231C2×C8—122no—
232C4⋊C4—22no—
233C4⋊C4—42no—
234C4⋊C4—44no—
235C22⋊C4—22no—
236C22⋊C4—22no—
237C22⋊C4—44no—
238C22⋊C4—42no—
239C42—21no—
240C42—21no—
241C22×C3—21no—
242C22×C3—21no—
243C22×C3—22no—
244C22×C3—22no—
245C22×C3—22no—
246C22×C3—42no—
247C22×C3—42no—
248C22×C3—42no—
249C22×C3—41no—
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—22no—
259C3×C4—44no—
260C3×C4—41no—
261C3×C4—41no—
262C3×C4—82no—
263C3×C4B(4,𝔽2)12yes—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23D4×Dic312yes—
274C23C2×#21711yescharacteristic
275Q8—48no—
276Q8—44no—
277Q8—44no—
278D4—44no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—28no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—81no—
299C2×C4#83612yescomplemented
300C2×C4#93511yescharacteristic
301C2×C4#93711yescharacteristic
302C2×C4#93912yes—
303C8—124no—
304C8—242no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—42no—
308C2×C3—81no—
309C2×C3#188911yescharacteristic
310C2×C3#190111yescharacteristic
311C2×C3#191511yescharacteristic
312C22—21no—
313C22—21no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—81no—
324C22#522011yescharacteristic
325C22C2×#83611yescharacteristic
326C22#523711yescharacteristic
327C4—22no—
328C4—22no—
329C4—21no—
330C4—41no—
331C4—44no—
332C4—41no—
333C4—82no—
334C4#503212yes—
335C3#950611yescharacteristic, complemented, Sylow
336C2—21no—
337C2—21no—
338C2—42no—
339C2—81no—
340C2#7166911yescharacteristic
341C2#7169811yescharacteristic
342C2#7186611yescharacteristic
subgroupG/H
centerC22#5220
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2409C2
socleC22×C3#386
prime radicalC22×C3×D4C2×C4
cosocleC23×C3×C4C23
hypercenter#2409D3
hypocenterC3#9506
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2409

Complements

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

NH ≅ G/Ntype
3. #72133C2semidirect
5. #67626C2semidirect
192. C2×C3×C4#101semidirect
299. C2×C4#836semidirect
335. C3#9506semidirect

Maximal subgroups

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

subgroupindex
C3×#24092
#823042
#721332
#716022
#676262
#95063