Finite Group Databasebeta

← #10692644

Subgroups — #10692644

Order 768.

526 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 93 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#82785C211yesmaximal, characteristic
2#73445C211yesmaximal, characteristic
3C3×#1508C211yesmaximal, characteristic
4#71688C211yesmaximal, characteristic
5#69971C211yesmaximal, characteristic
6#67683C211yesmaximal, characteristic
7#67618C211yesmaximal, characteristic
8#9412—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×Dic3⋊C4C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×C4×C22⋊C4C2211yescharacteristic
13#5230C412yescomplemented
14C4×#217C2211yescharacteristic
15C4×#217C2211yescharacteristic
16#5207C2211yescharacteristic
17#4983C412yescomplemented
18#4976—22no—
19#4946—21no—
20#4946—21no—
21#4943—21no—
22#4942—21no—
23#4933—22no—
24#4652—22no—
25C3×#328—22no—
26#4538—22no—
27#4469—22no—
28#2583—31no—
29#2313—31no—
30#1508—31no—
31#1508D311yescharacteristic
32#1463—31no—
33#1297—31no—
34#1297—31no—
35C22×C3×Q8C2×C411yescharacteristic
36C23×C3×C4C2311yescharacteristic
37C23×Dic3D411yescharacteristic
38C23×Dic3D411yescharacteristic
39C2×C3×M16—21no—
40C3×C22⋊Q8—21no—
41C3×C22⋊Q8—21no—
42C3×C22⋊Q8—21no—
43C3×C22⋊Q8—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4C2×C411yescharacteristic
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C42—21no—
49C2×#217—21no—
50C2×#217—21no—
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4—22no—
56C2×Dic3⋊C4—22no—
57C2×Dic3⋊C4—22no—
58C2×Dic3⋊C4C2×C412yes—
59C2×Dic3⋊C4D412yes—
60C2×C4×Dic3—21no—
61C2×C4×Dic3—21no—
62C2×#208—21no—
63C3×C22⋊C8—21no—
64C3×C22⋊C8—21no—
65C3×#95D412yes—
66C3⋊#95—22no—
67C3⋊#95—24no—
68C3⋊#95—22no—
69C3⋊#95—22no—
70C3⋊#95C2×C412yes—
71C3⋊#95D412yes—
72#830—21no—
73#830—21no—
74C2×C22⋊Q8C2×D311yescharacteristic
75C22×C4⋊C4—31no—
76#407—31no—
77#407C2×D311yescharacteristic
78#398—32no—
79#395—32no—
80#389—62no—
81#387—62no—
82#387—62no—
83#385—61no—
84#385—61no—
85#381—61no—
86#380—61no—
87C4×C22⋊C4—31no—
88C4×C22⋊C4—31no—
89C4×C22⋊C4C2×D311yescharacteristic
90#328—22no—
91#328—62no—
92#324—62no—
93C24×C3C2×D411yescharacteristic
94C2×C3×Q8—21no—
95C2×C3×Q8—21no—
96C2×C3×Q8—81no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—22no—
100C22×C3×C4—81no—
101C22×C3×C4C22×C411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4#4111yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C22×Dic3—22no—
108C22×Dic3—21no—
109C22×Dic3—21no—
110C22×Dic3—21no—
111C22×Dic3—21no—
112C22×Dic3—22no—
113C22×Dic3—42no—
114C22×Dic3—81no—
115C22×Dic3C22⋊C412yes—
116C22×Dic3C22⋊C412yes—
117C3×M16—81no—
118C2×C3×C8—41no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C22⋊C4—42no—
124C3×C22⋊C4—41no—
125C3×C22⋊C4—41no—
126C3×C42—81no—
127#217—24no—
128#217—42no—
129Dic3⋊C4—22no—
130Dic3⋊C4—22no—
131Dic3⋊C4—22no—
132Dic3⋊C4—22no—
133Dic3⋊C4—44no—
134Dic3⋊C4—82no—
135Dic3⋊C4—82no—
136C4×Dic3—81no—
137C4×Dic3—81no—
138#208—81no—
139C2×C3⋊C8—41no—
140C22×Q8C4×D311yescharacteristic
141C23×C4—31no—
142C23×C4—31no—
143C23×C4C22×D311yescharacteristic
144C2×M16—21no—
145C2×M16—61no—
146C22⋊Q8—21no—
147C22⋊Q8—21no—
148C22⋊Q8—21no—
149C22⋊Q8—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—32no—
152C2×C4⋊C4—32no—
153C2×C4⋊C4—61no—
154C2×C4⋊C4—61no—
155C2×C4⋊C4—61no—
156C2×C4⋊C4—61no—
157C2×C4⋊C4—62no—
158C2×C4⋊C4—62no—
159C2×C4⋊C4—62no—
160C2×C4⋊C4C4×D311yescharacteristic
161C2×C22⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—61no—
164C2×C22⋊C4—61no—
165C2×C42—21no—
166C2×C42—61no—
167C2×C42—61no—
168C22⋊C8—21no—
169C22⋊C8—21no—
170C22⋊C8—61no—
171C22⋊C8—61no—
172#95—32no—
173#95—32no—
174#95—62no—
175#95—62no—
176#95—62no—
177#95—64no—
178#95#6712yes—
179C23×C3—22no—
180C23×C3—22no—
181C23×C3—41no—
182C23×C3C2×C22⋊C411yescharacteristic
183C23×C3#12011yescharacteristic
184C23×C3#12311yescharacteristic
185C3×Q8—81no—
186C3×Q8—81no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—42no—
194C2×C3×C4—42no—
195C2×C3×C4—42no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—42no—
199C2×C3×C4—42no—
200C2×C3×C4C4×D411yescharacteristic
201C2×C3×C4C4×D411yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×Dic3—24no—
205C2×Dic3—22no—
206C2×Dic3—22no—
207C2×Dic3—22no—
208C2×Dic3—22no—
209C2×Dic3—42no—
210C2×Dic3—42no—
211C2×Dic3—42no—
212C2×Dic3—42no—
213C2×Dic3—42no—
214C2×Dic3—42no—
215C2×Dic3—42no—
216C2×Dic3—42no—
217C2×Dic3—42no—
218C2×Dic3—81no—
219C3×C8—81no—
220C3⋊C8—81no—
221C24D3×D411yescharacteristic
222C2×Q8—21no—
223C2×Q8—21no—
224C2×Q8—81no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—32no—
229C22×C4—32no—
230C22×C4—62no—
231C22×C4—61no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4—62no—
236C22×C4—81no—
237C22×C4—122no—
238C22×C4—241no—
239C22×C4C2×C4×D311yescharacteristic
240C22×C4#23511yescharacteristic
241C22×C4D3×D411yescharacteristic
242C22×C4D3×D411yescharacteristic
243C22×C4#23711yescharacteristic
244C22×C4C2×#6711yescharacteristic
245M16—81no—
246M16—241no—
247C2×C8—41no—
248C2×C8—121no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—62no—
254C4⋊C4—62no—
255C4⋊C4—62no—
256C4⋊C4—62no—
257C4⋊C4—124no—
258C4⋊C4—242no—
259C4⋊C4—242no—
260C22⋊C4—42no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—64no—
264C22⋊C4—122no—
265C42—81no—
266C42—241no—
267C42—241no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—44no—
276C22×C3#38611yescharacteristic
277C22×C3#42112yes—
278C22×C3#45612yes—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—82no—
284C3×C4—81no—
285C3×C4—81no—
286Dic3—44no—
287Dic3—42no—
288Dic3—42no—
289Dic3—82no—
290C23—22no—
291C23—22no—
292C23—41no—
293C23D3×C22⋊C411yescharacteristic
294C23#88311yescharacteristic
295C23#93711yescharacteristic
296Q8—81no—
297Q8—81no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—64no—
312C2×C4—62no—
313C2×C4—62no—
314C2×C4—62no—
315C2×C4—62no—
316C2×C4—122no—
317C2×C4—122no—
318C2×C4—122no—
319C2×C4—122no—
320C2×C4—122no—
321C2×C4—122no—
322C2×C4—122no—
323C2×C4—122no—
324C2×C4—122no—
325C2×C4—241no—
326C2×C4#67⋊C411yescharacteristic
327C2×C4#88411yescharacteristic
328C2×C4C4×#6711yescharacteristic
329C2×C4#93811yescharacteristic
330C8—81no—
331C8—241no—
332C2×C3—22no—
333C2×C3—22no—
334C2×C3—41no—
335C2×C3#188611yescharacteristic
336C2×C3#189711yescharacteristic
337C2×C3#191311yescharacteristic
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—44no—
346C22#482112yes—
347C22#495611yescharacteristic
348C22#520512yes—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—82no—
354C4—81no—
355C4—81no—
356C4—124no—
357C4—122no—
358C4—122no—
359C4—242no—
360C3#941211yescharacteristic, complemented, Sylow
361C2—22no—
362C2—22no—
363C2—41no—
364C2#7003411yescharacteristic
365C2#7034511yescharacteristic
366C2#7038311yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1508C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1508D3
hypocenterC3#9412
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1508

Complements

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

NH ≅ G/Ntype
13. #5230C4semidirect
17. #4983C4semidirect
360. C3#9412semidirect

Maximal subgroups

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

subgroupindex
#827852
#734452
C3×#15082
#716882
#699712
#676832
#676182
#94123