Finite Group Databasebeta

← #10696788

Subgroups — #10696788

Order 768.

458 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 81 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×#3215C211yesmaximal, characteristic
2#83372C211yesmaximal, characteristic, complemented
3#82939C211yesmaximal, characteristic
4#71825C211yesmaximal, characteristic, complemented
5#71720C211yesmaximal, characteristic
6#70803C211yesmaximal, characteristic, complemented
7#67753C211yesmaximal, characteristic, complemented
8#10125—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×#124C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#497—21no—
13C3×#489—21no—
14C3×#464—22no—
15C3×#451—21no—
16C3×#450—21no—
17C3×#432C2211yescharacteristic
18#5225—21no—
19#5215—21no—
20#5094—21no—
21#5091—21no—
22#5069—21no—
23#5064—21no—
24#5056—22no—
25#5055—21no—
26#5055—21no—
27#4996—22no—
28#4926C2211yescharacteristic
29#4925C2211yescharacteristic
30C4×#208C2211yescharacteristic
31#4535—22no—
32#4525—22no—
33#4484—22no—
34#3237—31no—
35#3215D311yescharacteristic
36#3140—31no—
37#1497—31no—
38#1485—31no—
39#1469—31no—
40#1291—31no—
41C22×C3×Q8D411yescharacteristic
42C22×C3×D4D411yescharacteristic
43C2×C3×Q16—22no—
44C2×C3×SD16—22no—
45C3×C4⋊Q8D411yescharacteristic
46C3×C4⋊Q8D411yescharacteristic
47C3×#125—21no—
48C3×#124—21no—
49C3×#124—21no—
50C3×#124—21no—
51C3×#124—21no—
52C3×C22⋊Q8—21no—
53C3×C22⋊Q8—21no—
54C3×C4×Q8—21no—
55C3×C4×Q8—21no—
56C3×#117—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C42C2311yescharacteristic
60#926—21no—
61#924—21no—
62#924—21no—
63C2×#208—21no—
64#870—21no—
65#869D411yescharacteristic
66#869D411yescharacteristic
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×C4⋊C8D412yes—
70C3×#103—22no—
71C3×#102—22no—
72C3×C22⋊C8—21no—
73C3×#97—21no—
74#835—22no—
75#835—22no—
76#832—22no—
77#832—22no—
78#830—21no—
79#830D412yes—
80#810—22no—
81#810—22no—
82#808—22no—
83#807—22no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86#803—21no—
87#803—21no—
88C4×C3⋊C8—21no—
89C4×C3⋊C8D412yes—
90#533—31no—
91#533C2×D311yescharacteristic
92C2×#124C2×D311yescharacteristic
93#501—61no—
94#497—21no—
95#491—61no—
96#489—21no—
97#489—61no—
98#489—61no—
99#488—62no—
100#485—62no—
101#477—61no—
102#475—61no—
103#464—22no—
104#451—21no—
105#450—21no—
106#432—31no—
107#432C2×D311yescharacteristic
108#431—31no—
109#419—61no—
110#418—61no—
111C4×M16—31no—
112Q8⋊C8—62no—
113Q8⋊C8—62no—
114#323—62no—
115C24×C3—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—42no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—42no—
124C22×C3×C4—21no—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4#4111yescharacteristic
128C2×Dic6—21no—
129C2×Dic6—21no—
130C3×Q16—82no—
131C3×SD16—82no—
132C2×C3×C8—22no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C42—41no—
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149#217—41no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156C4×Dic3—41no—
157#208—42no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—41no—
161C22×Q8#6711yescharacteristic
162C22×D4#6711yescharacteristic
163C2×Q16—22no—
164C2×SD16—22no—
165C2×M16—61no—
166C4⋊Q8—31no—
167C4⋊Q8—31no—
168C4⋊Q8#6711yescharacteristic
169C4⋊Q8#6711yescharacteristic
170#125—21no—
171#125—61no—
172#124—21no—
173#124—21no—
174#124—21no—
175#124—21no—
176C22⋊Q8—21no—
177C22⋊Q8—21no—
178C22⋊Q8—61no—
179C22⋊Q8—61no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—61no—
183C4×Q8—61no—
184#117—21no—
185#117—61no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C42C22×D311yescharacteristic
189#107—62no—
190#106—62no—
191C4⋊C8—61no—
192C4⋊C8—61no—
193C4⋊C8#6712yes—
194#103—22no—
195#103—62no—
196#103—62no—
197#103—62no—
198#103—62no—
199#102—22no—
200#102—62no—
201#102—62no—
202C22⋊C8—21no—
203C22⋊C8—32no—
204C22⋊C8—61no—
205#97—21no—
206#97—61no—
207#97—61no—
208C4×C8—32no—
209C4×C8—61no—
210C23×C3—42no—
211C23×C3—41no—
212C23×C3—81no—
213C23×C3—81no—
214C23×C3#12111yescharacteristic
215C3×Q8—42no—
216C3×Q8—42no—
217C3×Q8—42no—
218C3×Q8—42no—
219C3×D4—42no—
220C3×D4—42no—
221C2×C3×C4—22no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—81no—
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12011yescharacteristic
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12711yescharacteristic
234C2×Dic3—41no—
235C2×Dic3—41no—
236Dic6—42no—
237Dic6—42no—
238C3×C8—42no—
239C3⋊C8—42no—
240C3⋊C8—42no—
241C3⋊C8—81no—
242C24—21no—
243C2×Q8—21no—
244C2×Q8—21no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—42no—
248C2×Q8—61no—
249C2×Q8—61no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—42no—
253C22×C4—21no—
254C22×C4D3×D411yescharacteristic
255C22×C4#23711yescharacteristic
256C22×C4C2×#6711yescharacteristic
257Q16—82no—
258SD16—82no—
259M16—122no—
260C2×C8—22no—
261C2×C8—62no—
262C2×C8—62no—
263C2×C8—121no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—121no—
282C42—41no—
283C42—121no—
284C42D3×D411yescharacteristic
285C42D3×D411yescharacteristic
286C42C2×#6711yescharacteristic
287C42C2×#6711yescharacteristic
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—81no—
292C22×C3—81no—
293C22×C3—81no—
294C22×C3—162no—
295C22×C3#39211yescharacteristic
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—81no—
302C3×C4—81no—
303C3×C4—81no—
304C3×C4#45512yes—
305Dic3—81no—
306Dic3—81no—
307C23—42no—
308C23—41no—
309C23—81no—
310C23—81no—
311C23#93911yescharacteristic
312Q8—42no—
313Q8—42no—
314Q8—42no—
315Q8—42no—
316Q8—122no—
317Q8—122no—
318D4—42no—
319D4—42no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—81no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4#93711yescharacteristic
330C2×C4#93711yescharacteristic
331C2×C4#93811yescharacteristic
332C2×C4#93911yescharacteristic
333C2×C4#94011yescharacteristic
334C2×C4#95311yescharacteristic
335C8—42no—
336C8—122no—
337C8—122no—
338C8—241no—
339C2×C3—41no—
340C2×C3—161no—
341C2×C3#200411yescharacteristic
342C2×C3#200911yescharacteristic
343C2×C3#201711yescharacteristic
344C22—21no—
345C22—21no—
346C22—21no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—162no—
351C22#522311yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—81no—
358C4—81no—
359C4—81no—
360C4—241no—
361C4—241no—
362C4#524212yes—
363C3#1012511yescharacteristic, complemented, Sylow
364C2—41no—
365C2—161no—
366C2#7185711yescharacteristic
367C2#7211111yescharacteristic
368C2#7212111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3215C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#3215D3
hypocenterC3#10125
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3215

Complements

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

NH ≅ G/Ntype
2. #83372C2semidirect
4. #71825C2semidirect
6. #70803C2semidirect
7. #67753C2semidirect
363. C3#10125semidirect

Maximal subgroups

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

subgroupindex
C3×#32152
#833722
#829392
#718252
#717202
#708032
#677532
#101253