Finite Group Databasebeta

← #10705713

Subgroups — #10705713

Order 768.

420 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 91 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#71587C211yesmaximal, characteristic
3#71572C211yesmaximal, characteristic
4#67652C212yesmaximal, complemented
5#67650C212yesmaximal, complemented
6#11767—31nomaximal, Sylow
7C2×C3×#125C411yescharacteristic
8C2×C3×#117C2211yescharacteristic
9C22×#208—21no—
10C3×#400C2211yescharacteristic
11C3×#399C411yescharacteristic
12C3×#398C2211yescharacteristic
13C3×#397C411yescharacteristic
14C3×#389—21no—
15C3×#388—21no—
16C3×#387—21no—
17C3×#385—21no—
18C3×#376C411yescharacteristic
19#4926—21no—
20#4530—21no—
21#4530C2212yes—
22#4528C2212yes—
23#4525—21no—
24#2611D311yescharacteristic
25#1742—31no—
26#1464—31no—
27#1353—32no—
28#1346—32no—
29C23×C3×C4D411yescharacteristic
30C3×#125—41no—
31C3×#125—41no—
32C3×#117—41no—
33C3×#117—41no—
34C2×C3×C4⋊C4—21no—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4C2×C411yescharacteristic
39C2×C3×C4⋊C4C2×C411yescharacteristic
40C2×C3×C4⋊C4D411yescharacteristic
41C2×C3×C4⋊C4C2311yescharacteristic
42C2×C3×C22⋊C4—21no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4C2×C411yescharacteristic
45C2×C3×C22⋊C4D411yescharacteristic
46C2×C3×C42—21no—
47C2×C3×C42C2×C411yescharacteristic
48C2×C3×C42D411yescharacteristic
49C2×#208—24no—
50C2×#208—22no—
51C2×#208—41no—
52C2×#208—42no—
53C22×C3⋊C8—41no—
54C3×#95—21no—
55C3×#95—21no—
56C3×#95—21no—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95C2×C411yescharacteristic
60C3×#95C2×C411yescharacteristic
61#830—22no—
62C4⋊(C3⋊C8)—41no—
63#803—41no—
64C22×M16—61no—
65C2×#125Dic311yescharacteristic
66C2×#117C2×D311yescharacteristic
67#432—61no—
68#400C2×D311yescharacteristic
69#399Dic311yescharacteristic
70#398C2×D311yescharacteristic
71#397Dic311yescharacteristic
72#389—21no—
73#388—21no—
74#387—21no—
75#385—21no—
76#376Dic311yescharacteristic
77#343—32no—
78#343—61no—
79#324—32no—
80#323—61no—
81C24×C3C22⋊C411yescharacteristic
82C22×C3×C4—22no—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C22⋊C411yescharacteristic
94C22×C3×C4C22×C411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C3×C4⋊C4—41no—
98C3×C4⋊C4—41no—
99C3×C4⋊C4—41no—
100C3×C4⋊C4—41no—
101C3×C4⋊C4—42no—
102C3×C4⋊C4—42no—
103C3×C4⋊C4—41no—
104C3×C4⋊C4—41no—
105C3×C4⋊C4—41no—
106C3×C4⋊C4—41no—
107C3×C22⋊C4—41no—
108C3×C22⋊C4—41no—
109C3×C22⋊C4—42no—
110C3×C22⋊C4—42no—
111C3×C42—21no—
112C3×C42—21no—
113C3×C42—21no—
114C3×C42—21no—
115C3×C42—42no—
116#208—24no—
117#208—44no—
118#208—42no—
119C2×C3⋊C8—44no—
120C2×C3⋊C8—42no—
121C2×C3⋊C8—42no—
122C23×C4#6711yescharacteristic
123C2×M16—64no—
124C2×M16—62no—
125C2×M16—121no—
126C2×M16—122no—
127C22×C8—121no—
128#125—41no—
129#125—41no—
130#117—41no—
131#117—41no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4C2×Dic311yescharacteristic
137C2×C4⋊C4C2×Dic311yescharacteristic
138C2×C4⋊C4#6711yescharacteristic
139C2×C4⋊C4C22×D311yescharacteristic
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4C2×Dic311yescharacteristic
143C2×C22⋊C4#6711yescharacteristic
144C2×C42—21no—
145C2×C42C2×Dic311yescharacteristic
146C2×C42#6711yescharacteristic
147C4⋊C8—121no—
148C22⋊C8—62no—
149#97—121no—
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—41no—
160C23×C3—41no—
161C23×C3#10412yes—
162C23×C3C2×C22⋊C411yescharacteristic
163C2×C3×C4—22no—
164C2×C3×C4—22no—
165C2×C3×C4—22no—
166C2×C3×C4—22no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—22no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—41no—
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×C4—41no—
191C2×C3×C4#9911yescharacteristic
192C2×C3×C4#9911yescharacteristic
193C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
194C2×C3×C4#13711yescharacteristic
195C3⋊C8—44no—
196C3⋊C8—82no—
197C24#21711yescharacteristic
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4#21711yescharacteristic
208C22×C4#21711yescharacteristic
209C22×C4#21711yescharacteristic
210C22×C4C22×Dic311yescharacteristic
211C22×C4C2×#6711yescharacteristic
212C22×C4C2×#6711yescharacteristic
213M16—64no—
214M16—124no—
215M16—122no—
216C2×C8—124no—
217C2×C8—122no—
218C2×C8—122no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—41no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—42no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—22no—
241C22×C3—42no—
242C22×C3—41no—
243C22×C3—41no—
244C22×C3—81no—
245C22×C3C2×#9911yescharacteristic
246C22×C3#41711yescharacteristic
247C22×C3C2×#10411yescharacteristic
248C3×C4—21no—
249C3×C4—21no—
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—42no—
253C3×C4—41no—
254C3×C4—41no—
255C3×C4—41no—
256C3×C4—41no—
257C3×C4—81no—
258C3×C4—81no—
259C3×C4—81no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23#83712yes—
265C23C2×#21711yescharacteristic
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
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×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4#83411yescharacteristic
295C2×C4#83411yescharacteristic
296C2×C4#94911yescharacteristic
297C2×C4#95111yescharacteristic
298C8—124no—
299C8—242no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3—42no—
303C2×C3#150411yescharacteristic
304C2×C3#212311yescharacteristic
305C2×C3#213211yescharacteristic
306C22—21no—
307C22—21no—
308C22—22no—
309C22—42no—
310C22—41no—
311C22—41no—
312C22—81no—
313C22C2×#83411yescharacteristic
314C22#523211yescharacteristic
315C22C2×#83711yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—42no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—81no—
326C4—81no—
327C4—81no—
328C3#1176711yescharacteristic, complemented, Sylow
329C2—21no—
330C2—21no—
331C2—42no—
332C2#7170811yescharacteristic
333C2#7182011yescharacteristic
334C2#7187011yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#2611C2
socleC22×C3C2×#99
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C4⋊C4C23
hypercenter#2611D3
hypocenterC3#11767
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2611

Complements

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

NH ≅ G/Ntype
4. #67652C2semidirect
5. #67650C2semidirect
328. C3#11767semidirect

Maximal subgroups

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

subgroupindex
C3×#26112
#715872
#715722
#676522
#676502
#117673