Finite Group Databasebeta

← #10693157

Subgroups — #10693157

Order 768.

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

normal subgroups only

This group has 524 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×#2487C211yesmaximal, characteristic
2#82302C211yesmaximal, characteristic
3#71602C211yesmaximal, characteristic
4#71562C212yesmaximal, complemented
5#67626C212yesmaximal, complemented
6#9509—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C22×C3×C4⋊C4C2211yescharacteristic
9C3×#400—21no—
10C3×#399—21no—
11C3×#389—21no—
12C3×#386C412yes—
13C3×#385C412yes—
14C3×#380—21no—
15C3×#380C2211yescharacteristic
16#5207C2212yes—
17#5207C2212yes—
18C2×#830—21no—
19#5011—24no—
20#4926—21no—
21#4530—24no—
22#4528—21no—
23#4525—21no—
24#2973—31no—
25#2487D311yescharacteristic
26#1821—32no—
27#1465—31no—
28#1305—32no—
29C22×C3×D4C2×C411yescharacteristic
30C23×C3×C4C2×C412yes—
31C23×C3×C4D411yescharacteristic
32C23×C3×C4C2311yescharacteristic
33C3×C4×D4—22no—
34C3×C4×D4—22no—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—22no—
39C2×C3×C4⋊C4—22no—
40C2×C3×C4⋊C4—22no—
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4C2×C411yescharacteristic
44C2×C3×C22⋊C4C2×C412yes—
45C2×C3×C22⋊C4D411yescharacteristic
46C2×C3×C42D411yescharacteristic
47C2×#208—22no—
48C2×#208D414yes—
49C22×C3⋊C8—22no—
50C3×#95—22no—
51C3×#95—22no—
52C3×#95—21no—
53#830—22no—
54#830—22no—
55#830—22no—
56#830—22no—
57#830—22no—
58C4⋊(C3⋊C8)—22no—
59#803—22no—
60C2×C4×D4C2×D311yescharacteristic
61C22×C4⋊C4C2×D311yescharacteristic
62#435—64no—
63#432—61no—
64#407—32no—
65#407—32no—
66C2×C22⋊C8—61no—
67#400—21no—
68#399—21no—
69#389—21no—
70#386Dic312yes—
71#385Dic312yes—
72#380—21no—
73#380C2×D311yescharacteristic
74#343—64no—
75#324—61no—
76#323—61no—
77C24×C3C22⋊C411yescharacteristic
78C24×C3C22×C411yescharacteristic
79C2×C3×D4—22no—
80C2×C3×D4—22no—
81C2×C3×D4—41no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—22no—
85C22×C3×C4—22no—
86C22×C3×C4—24no—
87C22×C3×C4—42no—
88C22×C3×C4—42no—
89C22×C3×C4—82no—
90C22×C3×C4C22⋊C411yescharacteristic
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C2×D412yes—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4#4112yes—
97C3×C4⋊C4—22no—
98C3×C4⋊C4—42no—
99C3×C4⋊C4—41no—
100C3×C4⋊C4—42no—
101C3×C4⋊C4—41no—
102C3×C4⋊C4—82no—
103C3×C22⋊C4—22no—
104C3×C22⋊C4—22no—
105C3×C22⋊C4—42no—
106C3×C22⋊C4—82no—
107C3×C42—21no—
108C3×C42—21no—
109#208—44no—
110#208—44no—
111C2×C3⋊C8—24no—
112C2×C3⋊C8—42no—
113C2×C3⋊C8—82no—
114C22×D4C2×Dic311yescharacteristic
115C23×C4C2×Dic312yes—
116C23×C4#6711yescharacteristic
117C23×C4C22×D311yescharacteristic
118C2×M16—34no—
119C2×M16—62no—
120C22×C8—62no—
121C4×D4—22no—
122C4×D4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4#6711yescharacteristic
130C2×C22⋊C4—22no—
131C2×C22⋊C4C2×Dic311yescharacteristic
132C2×C22⋊C4C2×Dic312yes—
133C2×C22⋊C4#6711yescharacteristic
134C2×C42#6711yescharacteristic
135C4⋊C8—62no—
136C22⋊C8—62no—
137C22⋊C8—62no—
138C22⋊C8—62no—
139C22⋊C8—62no—
140C22⋊C8—62no—
141#97—62no—
142#95—22no—
143#95—22no—
144#95—21no—
145C23×C3—21no—
146C23×C3—21no—
147C23×C3—21no—
148C23×C3—21no—
149C23×C3—41no—
150C23×C3—41no—
151C23×C3—41no—
152C23×C3—41no—
153C23×C3#10112yes—
154C23×C3C2×C22⋊C411yescharacteristic
155C23×C3C4×D412yes—
156C3×D4—44no—
157C2×C3×C4—24no—
158C2×C3×C4—22no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—41no—
165C2×C3×C4—42no—
166C2×C3×C4—42no—
167C2×C3×C4—41no—
168C2×C3×C4—41no—
169C2×C3×C4—42no—
170C2×C3×C4—42no—
171C2×C3×C4—42no—
172C2×C3×C4—42no—
173C2×C3×C4—42no—
174C2×C3×C4—42no—
175C2×C3×C4—81no—
176C2×C3×C4—84no—
177C2×C3×C4#12011yescharacteristic
178C2×C3×C4#12112yes—
179C2×C3×C4#12311yescharacteristic
180C3⋊C8—44no—
181C3⋊C8—82no—
182C24#21711yescharacteristic
183C24C22×Dic311yescharacteristic
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—41no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—24no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—82no—
195C22×C4#21711yescharacteristic
196C22×C4#21711yescharacteristic
197C22×C4#21711yescharacteristic
198C22×C4D3×D412yes—
199C22×C4#23712yes—
200C22×C4C2×#6711yescharacteristic
201C22×C4C2×#6711yescharacteristic
202M16—124no—
203M16—124no—
204C2×C8—64no—
205C2×C8—122no—
206C2×C8—242no—
207C4⋊C4—22no—
208C4⋊C4—42no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—42no—
212C4⋊C4—82no—
213C22⋊C4—22no—
214C22⋊C4—22no—
215C22⋊C4—42no—
216C22⋊C4—82no—
217C42—21no—
218C42—21no—
219C22×C3—21no—
220C22×C3—21no—
221C22×C3—22no—
222C22×C3—22no—
223C22×C3—22no—
224C22×C3—22no—
225C22×C3—42no—
226C22×C3—42no—
227C22×C3—42no—
228C22×C3—41no—
229C22×C3—41no—
230C22×C3—81no—
231C22×C3#38611yescharacteristic
232C22×C3C2×#10111yescharacteristic
233C22×C3#41311yescharacteristic
234C3×C4—22no—
235C3×C4—21no—
236C3×C4—41no—
237C3×C4—41no—
238C3×C4—41no—
239C3×C4—82no—
240C3×C4—82no—
241C3×C4#45812yes—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23#83612yes—
251C23D4×Dic312yes—
252C23C2×#21711yescharacteristic
253D4—44no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—81no—
273C2×C4—84no—
274C2×C4#93511yescharacteristic
275C2×C4#93711yescharacteristic
276C2×C4#93912yes—
277C8—124no—
278C8—242no—
279C2×C3—21no—
280C2×C3—21no—
281C2×C3—42no—
282C2×C3—42no—
283C2×C3#189011yescharacteristic
284C2×C3#190111yescharacteristic
285C2×C3#191411yescharacteristic
286C22—21no—
287C22—21no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—81no—
298C22#522011yescharacteristic
299C22C2×#83611yescharacteristic
300C22#523711yescharacteristic
301C4—22no—
302C4—21no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—82no—
307C4—82no—
308C4#505312yes—
309C3#950911yescharacteristic, complemented, Sylow
310C2—21no—
311C2—21no—
312C2—42no—
313C2—42no—
314C2#7169811yescharacteristic
315C2#7170411yescharacteristic
316C2#7185511yescharacteristic
subgroupG/H
centerC22#5220
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2487C2
socleC22×C3#386
prime radicalC22×C3×D4C2×C4
cosocleC23×C3×C4C23
hypercenter#2487D3
hypocenterC3#9509
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2487

Complements

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

NH ≅ G/Ntype
4. #71562C2semidirect
5. #67626C2semidirect
309. C3#9509semidirect

Maximal subgroups

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

subgroupindex
C3×#24872
#823022
#716022
#715622
#676262
#95093