Finite Group Databasebeta

← #10693077

Subgroups — #10693077

Order 768.

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

normal subgroups only

This group has 458 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
1C3×#2599C211yesmaximal, characteristic
2#82209C211yesmaximal, characteristic
3#71623C212yesmaximal, complemented
4#71572C211yesmaximal, characteristic
5#67633C212yesmaximal, complemented
6#9493—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×C3×#117C2211yescharacteristic
9C3×#398C411yescharacteristic
10C3×#397C411yescharacteristic
11C3×#395C411yescharacteristic
12C3×#393C411yescharacteristic
13C3×#388—21no—
14C3×#387—21no—
15C3×#387—21no—
16C3×#385—21no—
17C3×#383C2211yescharacteristic
18#5207—21no—
19#5000—24no—
20#4926C2212yes—
21#4924—21no—
22C4×#208C2212yes—
23#4535—24no—
24#4528—21no—
25#4525—21no—
26#2921—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—
51C3×#95—21no—
52C3×#95—21no—
53C3×#95—21no—
54C3×#95—21no—
55C3×#95—21no—
56C3×#95—21no—
57C3×#95C2×C411yescharacteristic
58C3×#95C2×C411yescharacteristic
59#830—24no—
60#830—22no—
61C4⋊(C3⋊C8)—24no—
62C4⋊(C3⋊C8)—22no—
63#803—22no—
64#803—22no—
65C4×C3⋊C8D414yes—
66C2×C4×Q8C2×D311yescharacteristic
67C2×#117C2×D311yescharacteristic
68#433—64no—
69#432—32no—
70#423—61no—
71#407—61no—
72C4×M16—32no—
73#398Dic311yescharacteristic
74#397Dic311yescharacteristic
75#395Dic311yescharacteristic
76#393Dic311yescharacteristic
77#388—21no—
78#387—21no—
79#387—21no—
80#385—21no—
81#383C2×D311yescharacteristic
82Q8⋊C8—64no—
83#324—61no—
84#323—61no—
85C24×C3C22⋊C411yescharacteristic
86C2×C3×Q8—22no—
87C2×C3×Q8—42no—
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—41no—
95C22×C3×C4—41no—
96C22×C3×C4C22⋊C411yescharacteristic
97C22×C3×C4C22⋊C411yescharacteristic
98C22×C3×C4C22⋊C411yescharacteristic
99C22×C3×C4C22×C411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C3×C4⋊C4—22no—
103C3×C4⋊C4—22no—
104C3×C4⋊C4—22no—
105C3×C4⋊C4—42no—
106C3×C4⋊C4—42no—
107C3×C4⋊C4—41no—
108C3×C4⋊C4—41no—
109C3×C22⋊C4—22no—
110C3×C22⋊C4—22no—
111C3×C22⋊C4—42no—
112C3×C22⋊C4—42no—
113C3×C42—21no—
114C3×C42—21no—
115C3×C42—42no—
116C3×C42C2×D412yes—
117C3×C42#4112yes—
118#208—42no—
119#208—42no—
120C2×C3⋊C8—24no—
121C2×C3⋊C8—42no—
122C22×Q8C2×Dic311yescharacteristic
123C23×C4#6711yescharacteristic
124C2×M16—62no—
125C4×Q8—22no—
126C4×Q8—22no—
127#117—22no—
128#117—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4C2×Dic311yescharacteristic
133C2×C4⋊C4C2×Dic311yescharacteristic
134C2×C4⋊C4#6711yescharacteristic
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4C2×Dic311yescharacteristic
138C2×C22⋊C4#6711yescharacteristic
139C2×C42—21no—
140C2×C42#6711yescharacteristic
141C2×C42C22×D311yescharacteristic
142C4⋊C8—64no—
143C4⋊C8—62no—
144C22⋊C8—64no—
145C22⋊C8—62no—
146#97—62no—
147#97—62no—
148C4×C8—34no—
149#95—21no—
150#95—21no—
151#95—21no—
152#95—21no—
153#95—21no—
154#95—21no—
155#95C2×Dic311yescharacteristic
156#95C2×Dic311yescharacteristic
157C23×C3—21no—
158C23×C3—21no—
159C23×C3—21no—
160C23×C3—41no—
161C23×C3—41no—
162C23×C3C2×C22⋊C411yescharacteristic
163C3×Q8—24no—
164C3×Q8—42no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—22no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4—42no—
176C2×C3×C4—42no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
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×C4C4×D412yes—
191C2×C3×C4#12011yescharacteristic
192C2×C3×C4#12112yes—
193C2×C3×C4#12311yescharacteristic
194C3⋊C8—44no—
195C3⋊C8—82no—
196C24#21711yescharacteristic
197C2×Q8—22no—
198C2×Q8—42no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4#21711yescharacteristic
208C22×C4#21711yescharacteristic
209C22×C4#21711yescharacteristic
210C22×C4C22×Dic311yescharacteristic
211C22×C4C2×#6711yescharacteristic
212C22×C4C2×#6711yescharacteristic
213M16—122no—
214M16—122no—
215C2×C8—64no—
216C2×C8—122no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C22⋊C4—22no—
225C22⋊C4—22no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C42—21no—
229C42—21no—
230C42—42no—
231C42D3×D412yes—
232C42#23712yes—
233C22×C3—21no—
234C22×C3—21no—
235C22×C3—42no—
236C22×C3—41no—
237C22×C3—41no—
238C22×C3—41no—
239C22×C3—81no—
240C22×C3#38611yescharacteristic
241C22×C3#42111yescharacteristic
242C22×C3#42111yescharacteristic
243C3×C4—22no—
244C3×C4—21no—
245C3×C4—21no—
246C3×C4—22no—
247C3×C4—41no—
248C3×C4—42no—
249C3×C4—41no—
250C3×C4—41no—
251C3×C4—81no—
252C3×C4—81no—
253C3×C4—81no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—41no—
258C23—41no—
259C23C2×#21711yescharacteristic
260Q8—24no—
261Q8—42no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4D4×Dic312yes—
288C2×C4#93511yescharacteristic
289C2×C4#93711yescharacteristic
290C2×C4#93912yes—
291C8—124no—
292C8—242no—
293C2×C3—21no—
294C2×C3—21no—
295C2×C3—81no—
296C2×C3#189111yescharacteristic
297C2×C3#189711yescharacteristic
298C2×C3#190911yescharacteristic
299C22—21no—
300C22—21no—
301C22—42no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—81no—
306C22#522011yescharacteristic
307C22#523611yescharacteristic
308C22#523611yescharacteristic
309C4—21no—
310C4—21no—
311C4—22no—
312C4—22no—
313C4—41no—
314C4—42no—
315C4—41no—
316C4—41no—
317C4—81no—
318C4—81no—
319C4—81no—
320C3#949311yescharacteristic, complemented, Sylow
321C2—21no—
322C2—21no—
323C2—81no—
324C2#7171211yescharacteristic
325C2#7171411yescharacteristic
326C2#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#9493
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
320. C3#9493semidirect

Maximal subgroups

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

subgroupindex
C3×#25992
#822092
#716232
#715722
#676332
#94933