Finite Group Databasebeta

← #10693067

Subgroups — #10693067

Order 768.

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

normal subgroups only

This group has 544 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×#2599C211yesmaximal, characteristic
2#82202C211yesmaximal, characteristic
3#71623C212yesmaximal, complemented
4#71572C211yesmaximal, characteristic
5#67633C212yesmaximal, complemented
6#9491—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×C3×#117C2211yescharacteristic
9C22×#208—21no—
10C3×#398C411yescharacteristic
11C3×#397C411yescharacteristic
12C3×#395C411yescharacteristic
13C3×#393C411yescharacteristic
14C3×#388—21no—
15C3×#387—21no—
16C3×#387—21no—
17C3×#385—21no—
18C3×#383C2211yescharacteristic
19#4997—24no—
20#4926C2212yes—
21C4×#208—21no—
22C4×#208C2212yes—
23#4535—24no—
24#4528—21no—
25#4525—21no—
26#2873—31no—
27#2599D311yescharacteristic
28#1491—32no—
29#1464—31no—
30#1288—32no—
31C22×C3×Q8C2×C411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C3×C4×Q8—22no—
34C3×C4×Q8—22no—
35C3×#117—22no—
36C3×#117—22no—
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⋊C4D411yescharacteristic
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4C2×C411yescharacteristic
46C2×C3×C22⋊C4D411yescharacteristic
47C2×C3×C42—21no—
48C2×C3×C42D411yescharacteristic
49C2×C3×C42C2311yescharacteristic
50C2×#208—22no—
51C2×#208—24no—
52C2×#208—24no—
53C2×#208—22no—
54C22×C3⋊C8—22no—
55C3×#95—21no—
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#803—22no—
68C4×C3⋊C8—22no—
69C4×C3⋊C8D414yes—
70C22×M16—61no—
71C2×C4×Q8C2×D311yescharacteristic
72C2×#117C2×D311yescharacteristic
73#432—32no—
74#405—64no—
75C4×M16—32no—
76C4×M16—61no—
77#398Dic311yescharacteristic
78#397Dic311yescharacteristic
79#395Dic311yescharacteristic
80#393Dic311yescharacteristic
81#388—21no—
82#387—21no—
83#387—21no—
84#385—21no—
85#383C2×D311yescharacteristic
86Q8⋊C8—64no—
87#324—61no—
88#323—61no—
89C24×C3C22⋊C411yescharacteristic
90C2×C3×Q8—22no—
91C2×C3×Q8—42no—
92C22×C3×C4—22no—
93C22×C3×C4—22no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—21no—
100C22×C3×C4C22⋊C411yescharacteristic
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4C22⋊C411yescharacteristic
103C22×C3×C4C22×C411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C3×C4⋊C4—22no—
107C3×C4⋊C4—22no—
108C3×C4⋊C4—22no—
109C3×C4⋊C4—42no—
110C3×C4⋊C4—42no—
111C3×C4⋊C4—41no—
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—
128C22×Q8C2×Dic311yescharacteristic
129C23×C4#6711yescharacteristic
130C2×M16—62no—
131C2×M16—64no—
132C2×M16—64no—
133C2×M16—62no—
134C22×C8—62no—
135C4×Q8—22no—
136C4×Q8—22no—
137#117—22no—
138#117—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4C2×Dic311yescharacteristic
143C2×C4⋊C4C2×Dic311yescharacteristic
144C2×C4⋊C4#6711yescharacteristic
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4C2×Dic311yescharacteristic
148C2×C22⋊C4#6711yescharacteristic
149C2×C42—21no—
150C2×C42#6711yescharacteristic
151C2×C42C22×D311yescharacteristic
152C4⋊C8—62no—
153C22⋊C8—62no—
154#97—62no—
155#97—62no—
156#97—62no—
157C4×C8—34no—
158C4×C8—62no—
159#95—21no—
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
173C3×Q8—24no—
174C3×Q8—42no—
175C2×C3×C4—22no—
176C2×C3×C4—22no—
177C2×C3×C4—24no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—22no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
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—41no—
199C2×C3×C4—41no—
200C2×C3×C4#10412yes—
201C2×C3×C4C4×D412yes—
202C2×C3×C4#12011yescharacteristic
203C2×C3×C4#12112yes—
204C2×C3×C4#12311yescharacteristic
205C3⋊C8—28no—
206C3⋊C8—82no—
207C24#21711yescharacteristic
208C2×Q8—22no—
209C2×Q8—42no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4#21711yescharacteristic
219C22×C4#21711yescharacteristic
220C22×C4#21711yescharacteristic
221C22×C4C22×Dic311yescharacteristic
222C22×C4C2×#6711yescharacteristic
223C22×C4C2×#6711yescharacteristic
224M16—68no—
225M16—64no—
226M16—64no—
227C2×C8—68no—
228C2×C8—64no—
229C2×C8—122no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C22⋊C4—22no—
238C22⋊C4—22no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C42—21no—
242C42—21no—
243C42—42no—
244C42D3×D412yes—
245C42#23712yes—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—22no—
249C22×C3—42no—
250C22×C3—41no—
251C22×C3—41no—
252C22×C3—81no—
253C22×C3#38611yescharacteristic
254C22×C3C2×#10411yescharacteristic
255C22×C3C2×#10411yescharacteristic
256C3×C4—22no—
257C3×C4—21no—
258C3×C4—22no—
259C3×C4—21no—
260C3×C4—22no—
261C3×C4—41no—
262C3×C4—42no—
263C3×C4—41no—
264C3×C4—81no—
265C3×C4—81no—
266C3×C4—81no—
267C23—21no—
268C23—21no—
269C23—41no—
270C23—41no—
271C23#83712yes—
272C23C2×#21711yescharacteristic
273Q8—24no—
274Q8—42no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
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—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4#83712yes—
301C2×C4D4×Dic312yes—
302C2×C4#93511yescharacteristic
303C2×C4#93711yescharacteristic
304C2×C4#93912yes—
305C8—68no—
306C8—242no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—42no—
310C2×C3#189111yescharacteristic
311C2×C3#189611yescharacteristic
312C2×C3#190911yescharacteristic
313C22—21no—
314C22—21no—
315C22—22no—
316C22—42no—
317C22—41no—
318C22—41no—
319C22—81no—
320C22#522011yescharacteristic
321C22C2×#83711yescharacteristic
322C22C2×#83711yescharacteristic
323C4—21no—
324C4—22no—
325C4—21no—
326C4—22no—
327C4—22no—
328C4—42no—
329C4—41no—
330C4—41no—
331C4—81no—
332C4—81no—
333C4—81no—
334C3#949111yescharacteristic, complemented, Sylow
335C2—21no—
336C2—21no—
337C2—42no—
338C2#7171111yescharacteristic
339C2#7171411yescharacteristic
340C2#7186511yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2599C2
socleC22×C3#386
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2599D3
hypocenterC3#9491
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2599

Complements

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

NH ≅ G/Ntype
3. #71623C2semidirect
5. #67633C2semidirect
334. C3#9491semidirect

Maximal subgroups

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

subgroupindex
C3×#25992
#822022
#716232
#715722
#676332
#94913