Finite Group Databasebeta

← #10705656

Subgroups — #10705656

Order 768.

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

normal subgroups only

This group has 444 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×#1740C211yesmaximal, characteristic
3#71572C211yesmaximal, characteristic
4#67491C212yesmaximal
5#67490C212yesmaximal
6#11756—31nomaximal, Sylow
7C2×C3×#117—21no—
8C2×C3×#117C2211yescharacteristic
9C2×C4×Dic6C411yescharacteristic, complemented
10#5210—21no—
11#4981C411yescharacteristic, complemented
12#4979C411yescharacteristic, complemented
13#4978—21no—
14#4953C2211yescharacteristic
15#4947C2211yescharacteristic
16#4944—21no—
17#4937C411yescharacteristic, complemented
18#4933—21no—
19#4926—21no—
20C3×#342—21no—
21C3×#342C2212yes—
22#4528—21no—
23#4525C2212yes—
24#2599—31no—
25#1740D311yescharacteristic
26#1464—31no—
27#1351—32no—
28#1344—32no—
29C23×C3×C4D411yescharacteristic
30C22×Dic6C2×C411yescharacteristic
31C3×#117—24no—
32C3×#117—41no—
33C3×#117—41no—
34C3×#117—42no—
35C2×C3×C4⋊C4—41no—
36C2×C3×C4⋊C4C2×C411yescharacteristic
37C2×C3×C4⋊C4D411yescharacteristic
38C2×C3×C22⋊C4—22no—
39C2×C3×C22⋊C4D411yescharacteristic
40C2×C3×C22⋊C4C2311yescharacteristic
41C2×C3×C42—41no—
42C2×C3×C42C2×C411yescharacteristic
43C2×C3×C42D411yescharacteristic
44C2×#217—21no—
45C2×#217—21no—
46C2×C4⋊Dic3C2×C411yescharacteristic
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×C4×Dic3—21no—
51C4×Dic6—41no—
52C4×Dic6—41no—
53C3⋊#95—21no—
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—
64C2×C4×Q8—31no—
65C2×#117—21no—
66C2×#117C2×D311yescharacteristic
67#432—61no—
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#342—21no—
78#342C2×D312yes—
79#324—61no—
80#323—32no—
81C24×C3C2×D411yescharacteristic
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—22no—
87C22×C3×C4—42no—
88C22×C3×C4C22⋊C411yescharacteristic
89C22×C3×C4C22⋊C411yescharacteristic
90C22×C3×C4C22⋊C411yescharacteristic
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C22×C411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C22×Dic3—21no—
95C22×Dic3—21no—
96C22×Dic3—21no—
97C22×Dic3—21no—
98C2×Dic6—41no—
99C2×Dic6—42no—
100C3×C4⋊C4—41no—
101C3×C4⋊C4—41no—
102C3×C4⋊C4—44no—
103C3×C22⋊C4—24no—
104C3×C22⋊C4—41no—
105C3×C22⋊C4—41no—
106C3×C22⋊C4—42no—
107C3×C42—21no—
108C3×C42—21no—
109C3×C42—21no—
110C3×C42—21no—
111C3×C42—44no—
112#217—42no—
113#217—41no—
114#217—41no—
115C4⋊Dic3—41no—
116Dic3⋊C4—42no—
117Dic3⋊C4—41no—
118Dic3⋊C4—41no—
119Dic3⋊C4—42no—
120C4×Dic3—42no—
121C2×C3⋊C8—42no—
122C22×Q8—31no—
123C23×C4#6711yescharacteristic
124C4×Q8—121no—
125C4×Q8—121no—
126#117—24no—
127#117—41no—
128#117—41no—
129#117—42no—
130C2×C4⋊C4—31no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—61no—
133C2×C4⋊C4—61no—
134C2×C4⋊C4—61no—
135C2×C4⋊C4C4×D311yescharacteristic
136C2×C4⋊C4D1211yescharacteristic
137C2×C22⋊C4—22no—
138C2×C22⋊C4—61no—
139C2×C22⋊C4—61no—
140C2×C22⋊C4#6711yescharacteristic
141C2×C22⋊C4C22×D311yescharacteristic
142C2×C42—41no—
143C2×C42—61no—
144C2×C42C4×D311yescharacteristic
145C2×C42D1211yescharacteristic
146C4⋊C8—121no—
147C22⋊C8—62no—
148#97—121no—
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×C3C2×C22⋊C411yescharacteristic
162C23×C3AGL(1,ℤ/8)11yescharacteristic
163C23×C3#13711yescharacteristic
164C2×C3×C4—22no—
165C2×C3×C4—22no—
166C2×C3×C4—22no—
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—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—44no—
181C2×C3×C4—82no—
182C2×C3×C4—82no—
183C2×C3×C4#9911yescharacteristic
184C2×C3×C4#9911yescharacteristic
185C2×C3×C4#10412yes—
186C2×Dic3—22no—
187C2×Dic3—41no—
188C2×Dic3—41no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—41no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198Dic6—42no—
199Dic6—42no—
200C3⋊C8—82no—
201C24C2×#6711yescharacteristic
202C2×Q8—121no—
203C2×Q8—122no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—42no—
210C22×C4—61no—
211C22×C4—61no—
212C22×C4—61no—
213C22×C4—61no—
214C22×C4(C2×D3)⋊C411yescharacteristic
215C22×C4(C2×D3)⋊C411yescharacteristic
216C22×C4(C2×D3)⋊C411yescharacteristic
217C22×C4(C2×D3)⋊C411yescharacteristic
218C22×C4C2×C4×D311yescharacteristic
219C22×C4C2×D1211yescharacteristic
220C2×C8—122no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—44no—
224C4⋊C4—121no—
225C4⋊C4—122no—
226C4⋊C4—122no—
227C4⋊C4—121no—
228C4⋊C4—121no—
229C22⋊C4—24no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—42no—
233C22⋊C4—122no—
234C22⋊C4—121no—
235C22⋊C4—121no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—44no—
241C42—122no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—41no—
247C22×C3—41no—
248C22×C3—41no—
249C22×C3—41no—
250C22×C3—81no—
251C22×C3C2×#9911yescharacteristic
252C22×C3#41711yescharacteristic
253C22×C3C2×#10411yescharacteristic
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—22no—
257C3×C4—41no—
258C3×C4—41no—
259C3×C4—41no—
260C3×C4—41no—
261C3×C4—84no—
262Dic3—42no—
263Dic3—81no—
264Dic3—81no—
265Dic3—81no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23C2×(C2×D3)⋊C411yescharacteristic
271C23#93211yescharacteristic
272C23#94211yescharacteristic
273Q8—122no—
274Q8—122no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—44no—
292C2×C4—62no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4—121no—
298C2×C4—121no—
299C2×C4—121no—
300C2×C4—121no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4#80512yes—
307C2×C4#80611yescharacteristic
308C2×C4#80611yescharacteristic
309C8—242no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—41no—
313C2×C3—41no—
314C2×C3#150411yescharacteristic
315C2×C3#212011yescharacteristic
316C2×C3#212911yescharacteristic
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—81no—
326C22C2×#80511yescharacteristic
327C22C2×#80611yescharacteristic
328C22#496611yescharacteristic
329C4—22no—
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—84no—
337C4—122no—
338C4—241no—
339C4—241no—
340C4—241no—
341C3#1175611yescharacteristic, complemented, Sylow
342C2—21no—
343C2—21no—
344C2—41no—
345C2—41no—
346C2#6939811yescharacteristic
347C2#7005611yescharacteristic
348C2#7005911yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1740C2
socleC22×C3C2×#99
prime radicalC24×C3C2×D4
cosocleC2×C3×C22⋊C4C23
hypercenter#1740D3
hypocenterC3#11756
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1740

Complements

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

NH ≅ G/Ntype
9. C2×C4×Dic6C4semidirect
11. #4981C4semidirect
12. #4979C4semidirect
17. #4937C4semidirect
341. C3#11756semidirect

Maximal subgroups

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

subgroupindex
#822582
C3×#17402
#715722
#674912
#674902
#117563