Finite Group Databasebeta

← #10692528

Subgroups — #10692528

Order 768.

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

normal subgroups only

This group has 532 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×#2611C211yesmaximal, characteristic
2#82202C211yesmaximal, characteristic
3#71735C212yesmaximal, complemented
4#71572C211yesmaximal, characteristic
5#67615C212yesmaximal, complemented
6#9388—31nomaximal, Sylow
7C2×C3×#125C2211yescharacteristic
8C2×C3×#117C2211yescharacteristic
9C22×#208—21no—
10C3×#400C411yescharacteristic
11C3×#399C411yescharacteristic
12C3×#398C411yescharacteristic
13C3×#397C411yescharacteristic
14C3×#389—21no—
15C3×#388—21no—
16C3×#387—21no—
17C3×#385—21no—
18C3×#376C2211yescharacteristic
19#4997—24no—
20#4926C2212yes—
21C4×#208—21no—
22C4×#208C2212yes—
23#4543—24no—
24#4528—21no—
25#4525—21no—
26#2873—31no—
27#2611D311yescharacteristic
28#1528—32no—
29#1464—31no—
30#1300—32no—
31C23×C3×C4D411yescharacteristic
32C3×#125—22no—
33C3×#125—22no—
34C3×#117—22no—
35C3×#117—22no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4C2×C411yescharacteristic
41C2×C3×C4⋊C4C2×C411yescharacteristic
42C2×C3×C4⋊C4C2×C411yescharacteristic
43C2×C3×C4⋊C4D411yescharacteristic
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4C2×C411yescharacteristic
47C2×C3×C22⋊C4D411yescharacteristic
48C2×C3×C42—21no—
49C2×C3×C42D411yescharacteristic
50C2×C3×C42C2311yescharacteristic
51C2×#208—22no—
52C2×#208—24no—
53C2×#208—24no—
54C2×#208—22no—
55C22×C3⋊C8—22no—
56C3×#95—21no—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95—21no—
60C3×#95—21no—
61C3×#95C2×C411yescharacteristic
62C3×#95C2×C411yescharacteristic
63#830—22no—
64C4⋊(C3⋊C8)—22no—
65#803—22no—
66#803—22no—
67#803D414yes—
68C4×C3⋊C8—22no—
69C4×C3⋊C8—22no—
70C22×M16—61no—
71C2×#125C2×D311yescharacteristic
72C2×#117C2×D311yescharacteristic
73#432—32no—
74#405—64no—
75C4×M16—32no—
76C4×M16—61no—
77#400Dic311yescharacteristic
78#399Dic311yescharacteristic
79#398Dic311yescharacteristic
80#397Dic311yescharacteristic
81#389—21no—
82#388—21no—
83#387—21no—
84#385—21no—
85#376C2×D311yescharacteristic
86#330—64no—
87#324—61no—
88#323—61no—
89C24×C3C22⋊C411yescharacteristic
90C22×C3×C4—22no—
91C22×C3×C4—22no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4C22⋊C411yescharacteristic
99C22×C3×C4C22⋊C411yescharacteristic
100C22×C3×C4C22⋊C411yescharacteristic
101C22×C3×C4C22×C411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C3×C4⋊C4—22no—
105C3×C4⋊C4—22no—
106C3×C4⋊C4—22no—
107C3×C4⋊C4—22no—
108C3×C4⋊C4—42no—
109C3×C4⋊C4—42no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—42no—
112C3×C4⋊C4—41no—
113C3×C22⋊C4—22no—
114C3×C22⋊C4—22no—
115C3×C22⋊C4—42no—
116C3×C22⋊C4—42no—
117C3×C42—21no—
118C3×C42—21no—
119C3×C42—42no—
120C3×C42C2×D412yes—
121C3×C42#4112yes—
122#208—28no—
123#208—24no—
124#208—24no—
125C2×C3⋊C8—28no—
126C2×C3⋊C8—24no—
127C2×C3⋊C8—42no—
128C23×C4#6711yescharacteristic
129C2×M16—62no—
130C2×M16—64no—
131C2×M16—64no—
132C2×M16—62no—
133C22×C8—62no—
134#125—22no—
135#125—22no—
136#117—22no—
137#117—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4C2×Dic311yescharacteristic
143C2×C4⋊C4C2×Dic311yescharacteristic
144C2×C4⋊C4C2×Dic311yescharacteristic
145C2×C4⋊C4#6711yescharacteristic
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4C2×Dic311yescharacteristic
149C2×C22⋊C4#6711yescharacteristic
150C2×C42—21no—
151C2×C42#6711yescharacteristic
152C2×C42C22×D311yescharacteristic
153C4⋊C8—62no—
154C22⋊C8—62no—
155#97—34no—
156#97—62no—
157#97—62no—
158C4×C8—62no—
159C4×C8—62no—
160#95—21no—
161#95—21no—
162#95—21no—
163#95—21no—
164#95—21no—
165#95C2×Dic311yescharacteristic
166#95C2×Dic311yescharacteristic
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—41no—
170C23×C3—41no—
171C23×C3#10412yes—
172C23×C3C2×C22⋊C411yescharacteristic
173C2×C3×C4—22no—
174C2×C3×C4—22no—
175C2×C3×C4—24no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—22no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4#10412yes—
199C2×C3×C4C4×D412yes—
200C2×C3×C4#12011yescharacteristic
201C2×C3×C4#12112yes—
202C2×C3×C4#12311yescharacteristic
203C3⋊C8—28no—
204C3⋊C8—82no—
205C24#21711yescharacteristic
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4#21711yescharacteristic
215C22×C4#21711yescharacteristic
216C22×C4#21711yescharacteristic
217C22×C4C22×Dic311yescharacteristic
218C22×C4C2×#6711yescharacteristic
219C22×C4C2×#6711yescharacteristic
220M16—68no—
221M16—64no—
222M16—64no—
223C2×C8—68no—
224C2×C8—64no—
225C2×C8—122no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C22⋊C4—22no—
236C22⋊C4—22no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C42—21no—
240C42—21no—
241C42—42no—
242C42D3×D412yes—
243C42#23712yes—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—22no—
247C22×C3—42no—
248C22×C3—41no—
249C22×C3—41no—
250C22×C3—81no—
251C22×C3#38611yescharacteristic
252C22×C3C2×#10411yescharacteristic
253C22×C3C2×#10411yescharacteristic
254C3×C4—22no—
255C3×C4—21no—
256C3×C4—22no—
257C3×C4—21no—
258C3×C4—22no—
259C3×C4—42no—
260C3×C4—41no—
261C3×C4—41no—
262C3×C4—81no—
263C3×C4—81no—
264C3×C4—81no—
265C23—21no—
266C23—21no—
267C23—41no—
268C23—41no—
269C23#83712yes—
270C23C2×#21711yescharacteristic
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4#83712yes—
297C2×C4D4×Dic312yes—
298C2×C4#93511yescharacteristic
299C2×C4#93711yescharacteristic
300C2×C4#93912yes—
301C8—68no—
302C8—242no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—42no—
306C2×C3#188411yescharacteristic
307C2×C3#189611yescharacteristic
308C2×C3#191511yescharacteristic
309C22—21no—
310C22—21no—
311C22—22no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22#522011yescharacteristic
317C22C2×#83711yescharacteristic
318C22C2×#83711yescharacteristic
319C4—21no—
320C4—22no—
321C4—21no—
322C4—22no—
323C4—22no—
324C4—42no—
325C4—41no—
326C4—41no—
327C4—81no—
328C4—81no—
329C4—81no—
330C3#938811yescharacteristic, complemented, Sylow
331C2—21no—
332C2—21no—
333C2—42no—
334C2#7171111yescharacteristic
335C2#7181711yescharacteristic
336C2#7186611yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2611C2
socleC22×C3#386
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2611D3
hypocenterC3#9388
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2611

Complements

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

NH ≅ G/Ntype
3. #71735C2semidirect
5. #67615C2semidirect
330. C3#9388semidirect

Maximal subgroups

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

subgroupindex
C3×#26112
#822022
#717352
#715722
#676152
#93883