Finite Group Databasebeta

← #10692483

Subgroups — #10692483

Order 768.

424 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 103 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×#3236C211yesmaximal, characteristic
2#83024C211yesmaximal, characteristic
3#71841C211yesmaximal, characteristic
4#71837C211yesmaximal, characteristic
5#71735C211yesmaximal, characteristic
6#67752C211yesmaximal, characteristic
7#67749C211yesmaximal, characteristic
8#9380—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×#125C2211yescharacteristic
11C3×#501C412yescomplemented
12C3×#491C412yescomplemented
13C3×#484—21no—
14C3×#483—21no—
15C3×#479—22no—
16C3×#432C2211yescharacteristic
17#4984—21no—
18#4980—21no—
19#4980—21no—
20#4980—21no—
21#4980—21no—
22Dic3×C4⋊C4—21no—
23#4937C2211yescharacteristic
24C4×C4⋊Dic3C2211yescharacteristic
25#4926C2211yescharacteristic
26C4×#208C2211yescharacteristic
27#4543—22no—
28#4538—21no—
29#4538—21no—
30#4532—21no—
31#4532—21no—
32#3236D311yescharacteristic
33#2336—31no—
34#1986—31no—
35#1528—31no—
36#1515—31no—
37#1288—31no—
38#1285—31no—
39C3×C4⋊Q8C2×C411yescharacteristic
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×#125—21no—
42C3×#125—21no—
43C3×#125—21no—
44C3×#125C2×C412yes—
45C3×#125D412yes—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×#117—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C42C2311yescharacteristic
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C2×C4×Dic3—21no—
60C2×C4×Dic3—21no—
61C2×C4×Dic3D411yescharacteristic
62C2×C4×Dic3D411yescharacteristic
63C2×#208—21no—
64C3×#107—22no—
65C3×#106—22no—
66C3×C4⋊C8—21no—
67C3×#103—22no—
68C3×#103—22no—
69C3×C22⋊C8—21no—
70C3×#97C2×C412yes—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95D411yescharacteristic
79C3⋊#95D411yescharacteristic
80#830—21no—
81C4⋊(C3⋊C8)—21no—
82#803—21no—
83#803—21no—
84C4×C3⋊C8—21no—
85#533C2×D311yescharacteristic
86C2×#125C2×D311yescharacteristic
87#501Dic312yescomplemented
88#491Dic312yescomplemented
89#484—21no—
90#483—21no—
91#479—22no—
92#432—31no—
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—31no—
102C4×C4⋊C4—61no—
103#339—61no—
104#339—61no—
105#330—62no—
106#328—61no—
107#328—61no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3—21no—
118C22×Dic3—21no—
119C2×C3×C8—22no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4C22⋊C412yes—
133C3×C4⋊C4C22⋊C412yes—
134C3×C22⋊C4—41no—
135C3×C42—41no—
136C3×C42C22×C411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42#4111yescharacteristic
139C3×C42#4111yescharacteristic
140C4⋊Dic3—42no—
141C4⋊Dic3—42no—
142C4⋊Dic3—81no—
143C4×Dic3—42no—
144C4×Dic3—41no—
145C4×Dic3—41no—
146C4×Dic3—81no—
147#208—22no—
148#208—41no—
149C2×C3⋊C8—41no—
150C2×C3⋊C8—41no—
151C2×M16—61no—
152C4⋊Q8C2×Dic311yescharacteristic
153C4⋊Q8C2×Dic311yescharacteristic
154#125—21no—
155#125—21no—
156#125—21no—
157#125C2×Dic312yes—
158#125#6712yes—
159C22⋊Q8—21no—
160C22⋊Q8—21no—
161C4×Q8—21no—
162C4×Q8—21no—
163#117—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—61no—
167C2×C4⋊C4—61no—
168C2×C4⋊C4—61no—
169C2×C4⋊C4#6711yescharacteristic
170C2×C4⋊C4#6711yescharacteristic
171C2×C42—31no—
172C2×C42—31no—
173C2×C42—61no—
174C2×C42—61no—
175C2×C42C22×D311yescharacteristic
176#107—22no—
177#106—22no—
178C4⋊C8—21no—
179C4⋊C8—61no—
180#103—22no—
181#103—22no—
182C22⋊C8—21no—
183C22⋊C8—61no—
184#97—61no—
185#97—61no—
186#97C2×Dic312yes—
187C4×C8—61no—
188#95—31no—
189#95—31no—
190#95—61no—
191#95—61no—
192#95—61no—
193#95—61no—
194#95—61no—
195#95—61no—
196#95—61no—
197C23×C3#12011yescharacteristic
198C3×Q8—42no—
199C3×Q8—81no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—22no—
203C2×C3×C4—22no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4C2×C22⋊C411yescharacteristic
213C2×C3×C4C4×D411yescharacteristic
214C2×C3×C4C4×D411yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12311yescharacteristic
218C2×Dic3—22no—
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—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C3×C8—42no—
231C3⋊C8—42no—
232C3⋊C8—81no—
233C2×Q8—21no—
234C2×Q8—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4D3×D411yescharacteristic
242C22×C4D3×D411yescharacteristic
243C22×C4C2×#6711yescharacteristic
244M16—62no—
245M16—121no—
246C2×C8—22no—
247C2×C8—121no—
248C2×C8—121no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—122no—
262C4⋊C4—122no—
263C4⋊C4—241no—
264C4⋊C4#21712yes—
265C4⋊C4#21712yes—
266C22⋊C4—41no—
267C42—41no—
268C42—121no—
269C42—121no—
270C42—122no—
271C42—241no—
272C42#23711yescharacteristic
273C42#23711yescharacteristic
274C42C22×Dic311yescharacteristic
275C42C2×#6711yescharacteristic
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—42no—
289C3×C4—81no—
290C3×C4—81no—
291Dic3—42no—
292Dic3—81no—
293Dic3—81no—
294Dic3—81no—
295C23#93711yescharacteristic
296Q8—42no—
297Q8—81no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—62no—
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×C4D4×Dic311yescharacteristic
323C2×C4D4×Dic311yescharacteristic
324C2×C4#93511yescharacteristic
325C2×C4#93911yescharacteristic
326C2×C4#93911yescharacteristic
327C2×C4C2×#21711yescharacteristic
328C8—42no—
329C8—122no—
330C8—241no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3#188211yescharacteristic
334C2×C3#190611yescharacteristic
335C2×C3#191211yescharacteristic
336C22—21no—
337C22—21no—
338C22#520211yescharacteristic
339C22#520511yescharacteristic
340C22#522011yescharacteristic
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—42no—
348C4—42no—
349C4—81no—
350C4—81no—
351C4—122no—
352C4—241no—
353C4—241no—
354C4—241no—
355C3#938011yescharacteristic, complemented, Sylow
356C2—21no—
357C2—21no—
358C2#7186411yescharacteristic
359C2#7210011yescharacteristic
360C2#7211411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3236C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#3236D3
hypocenterC3#9380
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3236

Complements

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

NH ≅ G/Ntype
11. C3×#501C4semidirect
12. C3×#491C4semidirect
87. #501Dic3semidirect
88. #491Dic3semidirect
355. C3#9380semidirect

Maximal subgroups

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

subgroupindex
C3×#32362
#830242
#718412
#718372
#717352
#677522
#677492
#93803