Finite Group Databasebeta

← #10696789

Subgroups — #10696789

Order 768.

458 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 79 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#82949C211yesmaximal, characteristic
2#82939C211yesmaximal, characteristic
3#79321C211yesmaximal, characteristic, complemented
4C3×#1469C211yesmaximal, characteristic
5#70771C211yesmaximal, characteristic, complemented
6#70769C211yesmaximal, characteristic, complemented
7#67754C211yesmaximal, characteristic, complemented
8#10125—31nomaximal, Sylow
9C2×C3×#124C2211yescharacteristic
10#5470C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#431C2211yescharacteristic
14#5228—21no—
15#5225—21no—
16#5218—21no—
17#5215—21no—
18#5060—21no—
19#5056—22no—
20#5055—21no—
21#5055—21no—
22#5055—21no—
23#5054—22no—
24#4926C2211yescharacteristic
25C4×#208C2211yescharacteristic
26#4851—21no—
27#4850—21no—
28#4723—22no—
29#4703—21no—
30#4702—21no—
31C3×#323—22no—
32#4484—22no—
33#4456—22no—
34#3237—31no—
35#3215—31no—
36#3140—31no—
37#1497—31no—
38#1485—31no—
39#1469D311yescharacteristic
40#1291—31no—
41C22×C3×Q8D411yescharacteristic
42C22×C3×D4D411yescharacteristic
43C3×#124—21no—
44C3×#124—21no—
45C3×#124—21no—
46C3×#124—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#216—22no—
51C2×#214—22no—
52#926—21no—
53#926—21no—
54#924—21no—
55#924—21no—
56#924—21no—
57#924—21no—
58C2×#208—21no—
59#870—21no—
60#870—21no—
61#869D411yescharacteristic
62#869D411yescharacteristic
63#869D411yescharacteristic
64#869D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8D412yes—
73C3×#97—21no—
74C3×C4×C8—21no—
75#835—22no—
76#835—22no—
77#835—22no—
78#832—22no—
79#832—22no—
80#832—22no—
81#830—21no—
82#818—22no—
83#817—22no—
84#816—22no—
85#816—22no—
86C4⋊(C3⋊C8)D412yes—
87#803—21no—
88#803—21no—
89C4×C3⋊C8D412yes—
90#533—31no—
91#533—31no—
92C2×#124C2×D311yescharacteristic
93#501—61no—
94#497—61no—
95#491—61no—
96#489—61no—
97#489—61no—
98#489—61no—
99#488—62no—
100#485—62no—
101#477—61no—
102#475—61no—
103#464—62no—
104#451—61no—
105#450—61no—
106#432—31no—
107#432C2×D311yescharacteristic
108#431C2×D311yescharacteristic
109#419—61no—
110#418—61no—
111C4×M16—31no—
112Q8⋊C8—62no—
113Q8⋊C8—62no—
114#323—22no—
115C24×C3—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—42no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—42no—
122C22×C3×C4—21no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C2×Dic6—21no—
127C2×Dic6—21no—
128C2×Dic6—21no—
129C2×Dic6—21no—
130C2×C3×C8—22no—
131C2×C3×C8—41no—
132C3×C22⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140#217—41no—
141#217—41no—
142#216—82no—
143#214—82no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156C4×Dic3—41no—
157C4×Dic3—41no—
158#208—42no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—22no—
161C22×Q8D1211yescharacteristic
162C22×D4D1211yescharacteristic
163C2×Q16—62no—
164C2×SD16—62no—
165C2×M16—61no—
166C4⋊Q8—31no—
167C4⋊Q8—31no—
168C4⋊Q8—31no—
169C4⋊Q8—31no—
170#125—61no—
171#125—61no—
172#124—21no—
173#124—21no—
174#124—21no—
175#124—21no—
176C22⋊Q8—61no—
177C22⋊Q8—61no—
178C22⋊Q8—61no—
179C22⋊Q8—61no—
180C4×Q8—61no—
181C4×Q8—61no—
182C4×Q8—61no—
183C4×Q8—61no—
184#117—61no—
185#117—61no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C42C22×D311yescharacteristic
189#107—62no—
190#106—62no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—32no—
194#103—62no—
195#103—62no—
196#103—62no—
197#103—62no—
198#103—62no—
199#102—62no—
200#102—62no—
201#102—62no—
202C22⋊C8—21no—
203C22⋊C8—61no—
204C22⋊C8#6712yes—
205#97—21no—
206#97—61no—
207#97—61no—
208C4×C8—21no—
209C4×C8—32no—
210C23×C3—42no—
211C23×C3—41no—
212C23×C3—81no—
213C23×C3—81no—
214C23×C3#12111yescharacteristic
215C3×Q8—42no—
216C3×Q8—42no—
217C3×D4—42no—
218C3×D4—42no—
219C2×C3×C4—22no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—81no—
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12711yescharacteristic
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234Dic6—42no—
235Dic6—42no—
236Dic6—42no—
237Dic6—42no—
238C3×C8—42no—
239C3×C8—81no—
240C3⋊C8—42no—
241C3⋊C8—42no—
242C24—21no—
243C2×Q8—21no—
244C2×Q8—21no—
245C2×Q8—42no—
246C2×Q8—61no—
247C2×Q8—61no—
248C2×Q8—61no—
249C2×Q8—61no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—42no—
253C22×C4—21no—
254C22×C4C2×D1211yescharacteristic
255C22×C4#23511yescharacteristic
256C22×C4C2×#6711yescharacteristic
257Q16—242no—
258SD16—242no—
259M16—122no—
260C2×C8—22no—
261C2×C8—41no—
262C2×C8—62no—
263C2×C8—62no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—121no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—121no—
281C22⋊C4—121no—
282C42—121no—
283C42—121no—
284C42D3×D411yescharacteristic
285C42D3×D411yescharacteristic
286C42D3×D411yescharacteristic
287C42D3×D411yescharacteristic
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#45512yes—
303Dic3—81no—
304Dic3—81no—
305Dic3—81no—
306Dic3—81no—
307C23—42no—
308C23—41no—
309C23—81no—
310C23—81no—
311C23#93011yescharacteristic
312Q8—42no—
313Q8—42no—
314Q8—122no—
315Q8—122no—
316Q8—122no—
317Q8—122no—
318D4—42no—
319D4—42no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—81no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4#88211yescharacteristic
330C2×C4#88211yescharacteristic
331C2×C4#88411yescharacteristic
332C2×C4#88411yescharacteristic
333C2×C4#93711yescharacteristic
334C2×C4#94011yescharacteristic
335C8—42no—
336C8—81no—
337C8—122no—
338C8—122no—
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#496011yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—81no—
358C4—241no—
359C4—241no—
360C4—241no—
361C4—241no—
362C4#474612yes—
363C3#1012511yescharacteristic, complemented, Sylow
364C2—41no—
365C2—161no—
366C2#7001311yescharacteristic
367C2#7005311yescharacteristic
368C2#7008411yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1469C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#1469D3
hypocenterC3#10125
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1469

Complements

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

NH ≅ G/Ntype
3. #79321C2semidirect
5. #70771C2semidirect
6. #70769C2semidirect
7. #67754C2semidirect
363. C3#10125semidirect

Maximal subgroups

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

subgroupindex
#829492
#829392
#793212
C3×#14692
#707712
#707692
#677542
#101253