Finite Group Databasebeta

← #10705657

Subgroups — #10705657

Order 768.

444 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 71 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#76805C211yesmaximal, characteristic
2#68951C211yesmaximal, characteristic
3#68102C211yesmaximal, characteristic
4C3×#1351C211yesmaximal, characteristic
5#67646C211yesmaximal, characteristic
6#67491C211yesmaximal, characteristic
7#67450C211yesmaximal, characteristic
8#11756—31nomaximal, Sylow
9C2×Q8×Dic3C411yescharacteristic, complemented
10C2×#877—21no—
11C2×#877C2211yescharacteristic
12C3×#393C2211yescharacteristic
13#5230C411yescharacteristic, complemented
14#4953C2211yescharacteristic
15#4950—21no—
16#4945—21no—
17#4719—21no—
18#4661C411yescharacteristic, complemented
19#4655—21no—
20#4654—21no—
21#4651C411yescharacteristic, complemented
22C3×#342C2211yescharacteristic
23C3×#323C2211yescharacteristic
24#4527—21no—
25#4526C2211yescharacteristic
26#4525C2211yescharacteristic
27#4469—21no—
28#2599—31no—
29#1740—31no—
30#1464—31no—
31#1351—31no—
32#1351D311yescharacteristic
33#1344—31no—
34#1344—31no—
35C22×C3×Q8C2×C411yescharacteristic
36C23×Dic3D411yescharacteristic
37C2×C3×C22⋊C4—21no—
38C2×C3×C22⋊C4—21no—
39C2×C3×C22⋊C4C2311yescharacteristic
40C2×#217—21no—
41C2×#217—21no—
42C2×#217D411yescharacteristic
43Q8×Dic3—41no—
44Q8×Dic3—41no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3C2×C411yescharacteristic
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—41no—
50C2×Dic3⋊C4C2×C411yescharacteristic
51C2×Dic3⋊C4D411yescharacteristic
52C2×C4×Dic3—21no—
53C2×C4×Dic3—41no—
54C2×C4×Dic3C2×C411yescharacteristic
55C2×C4×Dic3D411yescharacteristic
56#877—24no—
57#877—41no—
58#877—41no—
59#877—42no—
60C3×C22⋊C8—21no—
61C3×#95—21no—
62C3×#95C2×C411yescharacteristic
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95C2×C411yescharacteristic
69#830—21no—
70#815—41no—
71Dic3⋊C8—41no—
72C2×C4×Q8—31no—
73C2×#117—31no—
74C2×#117—61no—
75#432—61no—
76#398—31no—
77#397—31no—
78#395—31no—
79#393C2×D311yescharacteristic
80#388—61no—
81#387—61no—
82#387—61no—
83#385—61no—
84#383—31no—
85#342—31no—
86#342—61no—
87#342C2×D311yescharacteristic
88#324—61no—
89#323—31no—
90#323C2×D311yescharacteristic
91C24×C3C2×D411yescharacteristic
92C2×C3×Q8—42no—
93C2×C3×Q8—41no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—41no—
97C22×C3×C4C22×C411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C22×Dic3—21no—
103C22×Dic3—22no—
104C22×Dic3—21no—
105C22×Dic3—21no—
106C22×Dic3—41no—
107C22×Dic3C22⋊C411yescharacteristic
108C22×Dic3C22⋊C411yescharacteristic
109C22×Dic3C22⋊C411yescharacteristic
110C22×Dic3C22⋊C411yescharacteristic
111C2×C3×C8—41no—
112C3×C22⋊C4—22no—
113C3×C22⋊C4—41no—
114C3×C22⋊C4—42no—
115C3×C22⋊C4—41no—
116#217—22no—
117#217—41no—
118#217—41no—
119#217—41no—
120#217—41no—
121C4⋊Dic3—41no—
122C4⋊Dic3—41no—
123C4⋊Dic3—42no—
124C4⋊Dic3—41no—
125Dic3⋊C4—41no—
126Dic3⋊C4—41no—
127Dic3⋊C4—44no—
128Dic3⋊C4—42no—
129C4×Dic3—21no—
130C4×Dic3—21no—
131C4×Dic3—21no—
132C4×Dic3—21no—
133C4×Dic3—44no—
134C4×Dic3—42no—
135C2×C3⋊C8—41no—
136C22×Q8C4×D311yescharacteristic
137C23×C4—31no—
138C4×Q8—121no—
139C4×Q8—121no—
140#117—64no—
141#117—121no—
142#117—121no—
143#117—122no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4—31no—
146C2×C4⋊C4—31no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—121no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—31no—
154C2×C22⋊C4—61no—
155C2×C22⋊C4—61no—
156C2×C22⋊C4C22×D311yescharacteristic
157C2×C42—31no—
158C2×C42—31no—
159C2×C42—61no—
160C2×C42—121no—
161C4⋊C8—121no—
162C22⋊C8—21no—
163C22⋊C8—61no—
164#97—121no—
165#95—21no—
166#95—31no—
167#95—61no—
168#95—61no—
169#95—61no—
170#95—61no—
171#95—61no—
172#95C4×D311yescharacteristic
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—41no—
176C23×C3—41no—
177C23×C3C2×C22⋊C411yescharacteristic
178C23×C3AGL(1,ℤ/8)11yescharacteristic
179C23×C3#13711yescharacteristic
180C3×Q8—42no—
181C3×Q8—42no—
182C2×C3×C4—22no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—42no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—81no—
194C2×C3×C4—81no—
195C2×Dic3—22no—
196C2×Dic3—22no—
197C2×Dic3—22no—
198C2×Dic3—22no—
199C2×Dic3—21no—
200C2×Dic3—21no—
201C2×Dic3—21no—
202C2×Dic3—21no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—42no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—81no—
215C2×Dic3—81no—
216C2×Dic3#9911yescharacteristic
217C2×Dic3#9911yescharacteristic
218C2×Dic3#10412yes—
219C3×C8—81no—
220C3⋊C8—81no—
221C24D3×D411yescharacteristic
222C2×Q8—42no—
223C2×Q8—41no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—31no—
227C22×C4—31no—
228C22×C4—31no—
229C22×C4—31no—
230C22×C4—41no—
231C22×C4—61no—
232C22×C4—61no—
233C22×C4—62no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4—61no—
237C22×C4—61no—
238C22×C4—121no—
239C22×C4C2×C4×D311yescharacteristic
240C22×C4D3×D411yescharacteristic
241C2×C8—41no—
242C2×C8—121no—
243C4⋊C4—121no—
244C4⋊C4—121no—
245C4⋊C4—121no—
246C4⋊C4—122no—
247C4⋊C4—121no—
248C4⋊C4—121no—
249C4⋊C4—122no—
250C4⋊C4—124no—
251C22⋊C4—22no—
252C22⋊C4—41no—
253C22⋊C4—42no—
254C22⋊C4—41no—
255C22⋊C4—62no—
256C22⋊C4—121no—
257C22⋊C4—121no—
258C22⋊C4—121no—
259C22⋊C4—121no—
260C42—61no—
261C42—61no—
262C42—61no—
263C42—61no—
264C42—122no—
265C42—124no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—81no—
275C22×C3C2×#9911yescharacteristic
276C22×C3#41711yescharacteristic
277C22×C3C2×#10411yescharacteristic
278C3×C4—42no—
279C3×C4—41no—
280C3×C4—41no—
281C3×C4—81no—
282C3×C4—82no—
283Dic3—21no—
284Dic3—21no—
285Dic3—22no—
286Dic3—41no—
287Dic3—41no—
288Dic3—81no—
289Dic3—82no—
290Dic3—81no—
291C23—21no—
292C23—21no—
293C23—41no—
294C23—41no—
295C23D3×C22⋊C411yescharacteristic
296C23#91411yescharacteristic
297C23#91811yescharacteristic
298Q8—42no—
299Q8—42no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—31no—
304C2×C4—31no—
305C2×C4—32no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—62no—
315C2×C4—62no—
316C2×C4—62no—
317C2×C4—62no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—61no—
323C2×C4—61no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—122no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—241no—
336C2×C4—241no—
337C8—81no—
338C8—241no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—41no—
342C2×C3—41no—
343C2×C3#150411yescharacteristic
344C2×C3#212011yescharacteristic
345C2×C3#212911yescharacteristic
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—81no—
355C22D3×#9911yescharacteristic
356C22#480311yescharacteristic
357C22D3×#10411yescharacteristic
358C4—42no—
359C4—41no—
360C4—41no—
361C4—61no—
362C4—61no—
363C4—62no—
364C4—81no—
365C4—82no—
366C4—121no—
367C4—121no—
368C4—241no—
369C4—242no—
370C4—241no—
371C3#1175611yescharacteristic, complemented, Sylow
372C2—21no—
373C2—21no—
374C2—41no—
375C2—41no—
376C2#6833411yescharacteristic
377C2#6913611yescharacteristic
378C2#6915911yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC22×C3×C4C22×C4
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1351C2
socleC22×C3C2×#99
prime radicalC24×C3C2×D4
cosocleC2×C3×C22⋊C4C23
hypercenter#1351D3
hypocenterC3#11756
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1351

Complements

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

NH ≅ G/Ntype
9. C2×Q8×Dic3C4semidirect
13. #5230C4semidirect
18. #4661C4semidirect
21. #4651C4semidirect
371. C3#11756semidirect

Maximal subgroups

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

subgroupindex
#768052
#689512
#681022
C3×#13512
#676462
#674912
#674502
#117563