Finite Group Databasebeta

← #10692480

Subgroups — #10692480

Order 768.

424 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 89 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#79320C211yesmaximal, characteristic
2#73630C211yesmaximal, characteristic
3#70837C211yesmaximal, characteristic
4#70812C211yesmaximal, characteristic
5#69493C211yesmaximal, characteristic
6C3×#1285C211yesmaximal, characteristic
7#67752C211yesmaximal, characteristic
8#9380—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#4978—21no—
15#4976—21no—
16#4937C2211yescharacteristic
17#4933—21no—
18C4×Dic3⋊C4—21no—
19C4×#208C2211yescharacteristic
20#4829—22no—
21#4722—21no—
22#4721—21no—
23#4703C412yescomplemented
24#4702C412yescomplemented
25#4654—21no—
26#4653—21no—
27#4523—21no—
28#4523—21no—
29#4467—21no—
30#4467—21no—
31#4465—22no—
32#3236—31no—
33#2336—31no—
34#1986—31no—
35#1528—31no—
36#1515—31no—
37#1288—31no—
38#1285D311yescharacteristic
39C2×C3×C4⋊C4—21no—
40C2×C3×C42D411yescharacteristic
41C2×C3×C42D411yescharacteristic
42C2×C3×C42C2311yescharacteristic
43#926—21no—
44C2×C4⋊Dic3D411yescharacteristic
45C2×C4⋊Dic3D411yescharacteristic
46#924—21no—
47#924—21no—
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×C4×Dic3—21no—
53C2×C4×Dic3—21no—
54C2×#208—21no—
55#870—21no—
56#870—21no—
57#870—21no—
58#870C2×C412yes—
59#870D412yes—
60#869C2×C411yescharacteristic
61#869C2×C411yescharacteristic
62C4×Dic6—21no—
63C4×Dic6—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×#97—21no—
69C3×#97C2×C412yes—
70C3×#95—21no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95D411yescharacteristic
78C3⋊#95D411yescharacteristic
79#818—22no—
80#817—22no—
81#816—22no—
82#816—22no—
83#803—21no—
84C4×C3⋊C8—21no—
85#533—31no—
86C2×#125—31no—
87#501—32no—
88#491—32no—
89#484—61no—
90#483—61no—
91#479—62no—
92#432C2×D311yescharacteristic
93#432C2×D311yescharacteristic
94C4×M16—31no—
95#389—61no—
96#387—61no—
97#387—61no—
98#387—61no—
99#387—61no—
100#383—31no—
101C4×C4⋊C4—61no—
102C4×C4⋊C4C2×D311yescharacteristic
103#339—61no—
104#339—61no—
105#330—62no—
106#328—61no—
107#328—61no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×Dic3—21no—
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3—21no—
117C2×Dic6—21no—
118C2×Dic6—21no—
119C2×C3×C8—22no—
120C2×C3×C8—41no—
121C3×C4⋊C4—42no—
122C3×C42—41no—
123C3×C42—41no—
124C3×C42C22×C411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42#4111yescharacteristic
127C3×C42#4111yescharacteristic
128#217—41no—
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131C4⋊Dic3—21no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3C22⋊C412yes—
136C4⋊Dic3C22⋊C412yes—
137Dic3⋊C4—22no—
138Dic3⋊C4—22no—
139Dic3⋊C4—22no—
140Dic3⋊C4—22no—
141Dic3⋊C4—42no—
142Dic3⋊C4—41no—
143Dic3⋊C4—41no—
144Dic3⋊C4—81no—
145C4×Dic3—42no—
146C4×Dic3—41no—
147C4×Dic3—81no—
148#208—22no—
149#208—41no—
150C2×C3⋊C8—41no—
151C2×M16—61no—
152C4⋊Q8—31no—
153C4⋊Q8—31no—
154#125—32no—
155#125—32no—
156#125—61no—
157#125—61no—
158#125—61no—
159C22⋊Q8—61no—
160C22⋊Q8—61no—
161C4×Q8—61no—
162C4×Q8—61no—
163#117—61no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—31no—
166C2×C4⋊C4—31no—
167C2×C4⋊C4—61no—
168C2×C4⋊C4—61no—
169C2×C4⋊C4—61no—
170C2×C4⋊C4—61no—
171C2×C42—61no—
172C2×C42—61no—
173C2×C42D1211yescharacteristic
174C2×C42D1211yescharacteristic
175C2×C42C22×D311yescharacteristic
176#107—62no—
177#106—62no—
178C4⋊C8—21no—
179C4⋊C8—21no—
180#103—62no—
181#103—62no—
182C22⋊C8—21no—
183C22⋊C8—21no—
184#97—21no—
185#97—61no—
186#97C4×D312yes—
187C4×C8—61no—
188#95—21no—
189#95—31no—
190#95—31no—
191#95—61no—
192#95—61no—
193#95—61no—
194#95—61no—
195#95—61no—
196#95—61no—
197C23×C3#12011yescharacteristic
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—22no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4C2×C22⋊C411yescharacteristic
209C2×C3×C4C4×D411yescharacteristic
210C2×C3×C4C4×D411yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12311yescharacteristic
214C2×Dic3—22no—
215C2×Dic3—22no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228Dic6—42no—
229Dic6—81no—
230C3×C8—42no—
231C3×C8—81no—
232C3⋊C8—42no—
233C2×Q8—61no—
234C2×Q8—61no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4C2×D1211yescharacteristic
242C22×C4D3×D411yescharacteristic
243C22×C4D3×D411yescharacteristic
244M16—62no—
245M16—121no—
246C2×C8—22no—
247C2×C8—41no—
248C2×C8—121no—
249C4⋊C4—32no—
250C4⋊C4—32no—
251C4⋊C4—42no—
252C4⋊C4—62no—
253C4⋊C4—62no—
254C4⋊C4—62no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—62no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—122no—
263C4⋊C4—121no—
264C4⋊C4—121no—
265C4⋊C4—241no—
266C22⋊C4—121no—
267C42—41no—
268C42—41no—
269C42—122no—
270C42—121no—
271C42—241no—
272C42C2×C4×D311yescharacteristic
273C42#23511yescharacteristic
274C42D3×D411yescharacteristic
275C42#23911yescharacteristic
276C22×C3—21no—
277C22×C3—21no—
278C22×C3#38611yescharacteristic
279C22×C3#45511yescharacteristic
280C22×C3#45611yescharacteristic
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—42no—
288C3×C4—81no—
289Dic3—42no—
290Dic3—42no—
291Dic3—81no—
292Dic3—81no—
293Dic3—81no—
294Dic3—81no—
295C23#88211yescharacteristic
296Q8—122no—
297Q8—241no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—62no—
309C2×C4—62no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4C4×D1211yescharacteristic
323C2×C4D3×C22⋊C411yescharacteristic
324C2×C4#88411yescharacteristic
325C2×C4#89311yescharacteristic
326C2×C4#89411yescharacteristic
327C2×C4#89511yescharacteristic
328C8—42no—
329C8—81no—
330C8—122no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3#188211yescharacteristic
334C2×C3#190611yescharacteristic
335C2×C3#191211yescharacteristic
336C22—21no—
337C22—21no—
338C22#466511yescharacteristic
339C22#474611yescharacteristic
340C22#475211yescharacteristic
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—42no—
348C4—81no—
349C4—122no—
350C4—122no—
351C4—241no—
352C4—241no—
353C4—241no—
354C4—241no—
355C3#938011yescharacteristic, complemented, Sylow
356C2—21no—
357C2—21no—
358C2#6870811yescharacteristic
359C2#6877811yescharacteristic
360C2#6900411yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1285C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#1285D3
hypocenterC3#9380
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1285

Complements

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

NH ≅ G/Ntype
23. #4703C4semidirect
24. #4702C4semidirect
355. C3#9380semidirect

Maximal subgroups

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

subgroupindex
#793202
#736302
#708372
#708122
#694932
C3×#12852
#677522
#93803