Finite Group Databasebeta

← #10705694

Subgroups — #10705694

Order 768.

428 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 83 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
1#82258C211yesmaximal, characteristic
2C3×#1464C211yesmaximal, characteristic
3#71587C211yesmaximal, characteristic
4#67458C212yesmaximal, complemented
5#67457C212yesmaximal, complemented
6#11763—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C22×#208—21no—
9C2×C4×Dic6C411yescharacteristic
10C3×#432—21no—
11#5210—21no—
12#4981C2211yescharacteristic
13#4979C2211yescharacteristic
14#4978—21no—
15#4953C411yescharacteristic
16#4947C411yescharacteristic
17#4944—21no—
18#4937C411yescharacteristic
19#4933—21no—
20C3×#324C2212yes—
21C3×#323—21no—
22#4530—21no—
23#4530C2212yes—
24#2599—31no—
25#1742—31no—
26#1464D311yescharacteristic
27#1354—32no—
28#1346—32no—
29C23×C3×C4D411yescharacteristic
30C22×Dic6C2×C411yescharacteristic
31C3×#117—41no—
32C3×#117—41no—
33C2×C3×C4⋊C4D411yescharacteristic
34C2×C3×C4⋊C4C2311yescharacteristic
35C2×C3×C22⋊C4C2×C411yescharacteristic
36C2×C3×C22⋊C4D411yescharacteristic
37C2×C3×C42C2×C411yescharacteristic
38C2×C3×C42D411yescharacteristic
39C2×#217—21no—
40C2×#217—21no—
41C2×C4⋊Dic3C2×C411yescharacteristic
42C2×Dic3⋊C4—21no—
43C2×Dic3⋊C4—21no—
44C2×Dic3⋊C4—21no—
45C2×C4×Dic3—21no—
46C2×#208—24no—
47C2×#208—22no—
48C2×#208—41no—
49C2×#208—42no—
50C22×C3⋊C8—41no—
51C4×Dic6—41no—
52C4×Dic6—41no—
53C3×C4⋊C8—41no—
54C3×C22⋊C8—22no—
55C3×#97—41no—
56C3⋊#95—21no—
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95C2×C411yescharacteristic
63C3⋊#95C2×C411yescharacteristic
64C22×M16—61no—
65C2×C4×Q8—31no—
66C2×#117C2×D311yescharacteristic
67#432—21no—
68#398—31no—
69#397—31no—
70#395—31no—
71#393—31no—
72#388—61no—
73#387—61no—
74#387—61no—
75#385—61no—
76#383—31no—
77#343—32no—
78#343—61no—
79#324C2×D312yes—
80#323—21no—
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×C4C22⋊C411yescharacteristic
88C22×C3×C4C22⋊C411yescharacteristic
89C22×C3×C4C22⋊C411yescharacteristic
90C22×C3×C4C22×C411yescharacteristic
91C22×C3×C4C2×D411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C22×Dic3—21no—
94C22×Dic3—21no—
95C22×Dic3—21no—
96C22×Dic3—21no—
97C2×Dic6—41no—
98C2×Dic6—42no—
99C2×C3×C8—42no—
100C3×C4⋊C4—41no—
101C3×C4⋊C4—41no—
102C3×C22⋊C4—41no—
103C3×C22⋊C4—41no—
104C3×C42—21no—
105C3×C42—21no—
106C3×C42—21no—
107C3×C42—21no—
108#217—42no—
109#217—42no—
110C4⋊Dic3—41no—
111Dic3⋊C4—42no—
112Dic3⋊C4—41no—
113Dic3⋊C4—41no—
114Dic3⋊C4—41no—
115Dic3⋊C4—41no—
116C4×Dic3—42no—
117#208—24no—
118#208—44no—
119#208—42no—
120C2×C3⋊C8—44no—
121C2×C3⋊C8—42no—
122C22×Q8—31no—
123C23×C4D1211yescharacteristic
124C2×M16—64no—
125C2×M16—62no—
126C2×M16—121no—
127C2×M16—122no—
128C22×C8—121no—
129C4×Q8—121no—
130C4×Q8—121no—
131#117—41no—
132#117—41no—
133C2×C4⋊C4—31no—
134C2×C4⋊C4—61no—
135C2×C4⋊C4—61no—
136C2×C4⋊C4—61no—
137C2×C4⋊C4D1211yescharacteristic
138C2×C4⋊C4C22×D311yescharacteristic
139C2×C22⋊C4—61no—
140C2×C22⋊C4—61no—
141C2×C22⋊C4C4×D311yescharacteristic
142C2×C22⋊C4#6711yescharacteristic
143C2×C42—61no—
144C2×C42C4×D311yescharacteristic
145C2×C42#6711yescharacteristic
146C4⋊C8—41no—
147C22⋊C8—22no—
148#97—41no—
149#95—31no—
150#95—31no—
151#95—61no—
152#95—61no—
153#95—61no—
154#95—61no—
155#95—61no—
156#95—61no—
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—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4#9911yescharacteristic
180C2×C3×C4#9911yescharacteristic
181C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
182C2×C3×C4#13711yescharacteristic
183C2×Dic3—22no—
184C2×Dic3—41no—
185C2×Dic3—41no—
186C2×Dic3—41no—
187C2×Dic3—41no—
188C2×Dic3—41no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—41no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195Dic6—42no—
196Dic6—42no—
197C3×C8—82no—
198C3⋊C8—44no—
199C24(C2×D3)⋊C411yescharacteristic
200C2×Q8—121no—
201C2×Q8—122no—
202C22×C4—22no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—61no—
208C22×C4—61no—
209C22×C4—61no—
210C22×C4—61no—
211C22×C4(C2×D3)⋊C411yescharacteristic
212C22×C4(C2×D3)⋊C411yescharacteristic
213C22×C4(C2×D3)⋊C411yescharacteristic
214C22×C4C2×C4×D311yescharacteristic
215C22×C4C2×D1211yescharacteristic
216C22×C4C2×#6711yescharacteristic
217M16—64no—
218M16—124no—
219M16—122no—
220C2×C8—42no—
221C2×C8—124no—
222C2×C8—122no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—121no—
226C4⋊C4—122no—
227C4⋊C4—121no—
228C4⋊C4—121no—
229C4⋊C4—121no—
230C4⋊C4—121no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—122no—
234C22⋊C4—122no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—122no—
240C22×C3—21no—
241C22×C3—21no—
242C22×C3—22no—
243C22×C3—42no—
244C22×C3—41no—
245C22×C3—41no—
246C22×C3—81no—
247C22×C3C2×#9911yescharacteristic
248C22×C3#41711yescharacteristic
249C22×C3C2×#10411yescharacteristic
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—21no—
253C3×C4—21no—
254C3×C4—41no—
255C3×C4—41no—
256C3×C4—41no—
257C3×C4—41no—
258Dic3—42no—
259Dic3—81no—
260Dic3—81no—
261Dic3—81no—
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23#82512yes—
267C23C2×(C2×D3)⋊C411yescharacteristic
268Q8—122no—
269Q8—122no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—62no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4—121no—
290C2×C4—121no—
291C2×C4—121no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4—121no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4—121no—
298C2×C4#80611yescharacteristic
299C2×C4#80611yescharacteristic
300C2×C4#90811yescharacteristic
301C2×C4#90911yescharacteristic
302C8—82no—
303C8—124no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—42no—
307C2×C3#150411yescharacteristic
308C2×C3#212011yescharacteristic
309C2×C3#213511yescharacteristic
310C22—21no—
311C22—21no—
312C22—22no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22C2×#80611yescharacteristic
318C22#513311yescharacteristic
319C22C2×#82511yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—122no—
329C4—241no—
330C4—241no—
331C4—241no—
332C3#1176311yescharacteristic, complemented, Sylow
333C2—21no—
334C2—21no—
335C2—42no—
336C2#6997611yescharacteristic
337C2#7005611yescharacteristic
338C2#7015011yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1464C2
socleC22×C3C2×#99
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1464D3
hypocenterC3#11763
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
4. #67458C2semidirect
5. #67457C2semidirect
332. C3#11763semidirect

Maximal subgroups

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

subgroupindex
#822582
C3×#14642
#715872
#674582
#674572
#117633