Finite Group Databasebeta

← #10692481

Subgroups — #10692481

Order 768.

424 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 99 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×#2336C211yesmaximal, characteristic
2#83021C211yesmaximal, characteristic
3#71735C211yesmaximal, characteristic
4#69566C211yesmaximal, characteristic
5#69493C211yesmaximal, characteristic
6#67633C211yesmaximal, characteristic
7#67621C211yesmaximal, characteristic
8#9380—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#389—21no—
12C3×#387—21no—
13C3×#387—21no—
14C3×#387—21no—
15C3×#387—21no—
16C3×#383C2211yescharacteristic
17C3×C4×C4⋊C4—21no—
18C3×C4×C4⋊C4C2211yescharacteristic
19#5069C412yescomplemented
20#5064C412yescomplemented
21#5063—22no—
22#4971—21no—
23#4970—21no—
24#4926C2211yescharacteristic
25#4926C2211yescharacteristic
26C4×#208C2211yescharacteristic
27#4543—22no—
28#4523—21no—
29#4523—21no—
30#4452—21no—
31#4452—21no—
32#3236—31no—
33#2336D311yescharacteristic
34#1986—31no—
35#1528—31no—
36#1515—31no—
37#1288—31no—
38#1285—31no—
39C3×#125—21no—
40C3×#125—21no—
41C3×#125C2×C412yes—
42C3×#125D412yes—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4D411yescharacteristic
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C42—21no—
51C2×C3×C42—21no—
52C2×C3×C42D411yescharacteristic
53C2×C3×C42D411yescharacteristic
54C2×C3×C42C2311yescharacteristic
55#926—21no—
56#924—21no—
57#924—21no—
58C2×#208—21no—
59#870—21no—
60#869C2×C411yescharacteristic
61#869C2×C411yescharacteristic
62C4×Dic6—21no—
63C4×Dic6—21no—
64C3×#95—21no—
65C3×#95—21no—
66C3×#95—21no—
67C3×#95—21no—
68C3×#95—21no—
69C3×#95—21no—
70C3×#95—21no—
71C3×#95D411yescharacteristic
72C3×#95D411yescharacteristic
73#830—21no—
74#830—21no—
75#810—22no—
76#810—22no—
77#808—22no—
78#807—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81#803—21no—
82#803—21no—
83#803C2×C412yes—
84C4×C3⋊C8—21no—
85#533—31no—
86C2×#125C2×D311yescharacteristic
87#501—32no—
88#491—32no—
89#484—61no—
90#483—61no—
91#479—62no—
92#432—31no—
93#432—31no—
94C4×M16—31no—
95#389—21no—
96#387—21no—
97#387—21no—
98#387—21no—
99#387—21no—
100#383C2×D311yescharacteristic
101C4×C4⋊C4—21no—
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×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C2×Dic6—21no—
118C2×Dic6—21no—
119C3×C4⋊C4—22no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—42no—
126C3×C4⋊C4—42no—
127C3×C4⋊C4—81no—
128C3×C4⋊C4C22⋊C412yes—
129C3×C4⋊C4C22⋊C412yes—
130C3×C42—41no—
131C3×C42—41no—
132C3×C42—42no—
133C3×C42—81no—
134C3×C42C22×C411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42#4111yescharacteristic
137C3×C42#4111yescharacteristic
138#217—41no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143Dic3⋊C4—41no—
144Dic3⋊C4—41no—
145C4×Dic3—41no—
146#208—22no—
147#208—41no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—41no—
150C2×C3⋊C8—41no—
151C2×M16—61no—
152C4⋊Q8—31no—
153C4⋊Q8—31no—
154#125—21no—
155#125—21no—
156#125—61no—
157#125C4×D312yes—
158#125D1212yes—
159C22⋊Q8—61no—
160C22⋊Q8—61no—
161C4×Q8—61no—
162C4×Q8—61no—
163#117—61no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4#6711yescharacteristic
170C2×C4⋊C4#6711yescharacteristic
171C2×C42—21no—
172C2×C42—21no—
173C2×C42#6711yescharacteristic
174C2×C42#6711yescharacteristic
175C2×C42C22×D311yescharacteristic
176#107—62no—
177#106—62no—
178C4⋊C8—61no—
179C4⋊C8—61no—
180#103—62no—
181#103—62no—
182C22⋊C8—61no—
183C22⋊C8—61no—
184#97—32no—
185#97—61no—
186#97—61no—
187C4×C8—61no—
188#95—21no—
189#95—21no—
190#95—21no—
191#95—21no—
192#95—21no—
193#95—21no—
194#95—21no—
195#95#6711yescharacteristic
196#95#6711yescharacteristic
197C23×C3#12011yescharacteristic
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—41no—
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×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4C2×C22⋊C411yescharacteristic
221C2×C3×C4C4×D411yescharacteristic
222C2×C3×C4C4×D411yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12311yescharacteristic
226C2×Dic3—41no—
227C2×Dic3—41no—
228Dic6—42no—
229Dic6—81no—
230C3⋊C8—42no—
231C3⋊C8—42no—
232C3⋊C8—81no—
233C2×Q8—61no—
234C2×Q8—61no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4C2×#6711yescharacteristic
242C22×C4C2×#6711yescharacteristic
243C22×C4C2×#6711yescharacteristic
244M16—62no—
245M16—121no—
246C2×C8—62no—
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—42no—
256C4⋊C4—42no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—81no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C4⋊C4(C2×D3)⋊C412yes—
265C4⋊C4(C2×D3)⋊C412yes—
266C22⋊C4—121no—
267C42—41no—
268C42—41no—
269C42—42no—
270C42—81no—
271C42—121no—
272C42C2×C4×D311yescharacteristic
273C42C2×D1211yescharacteristic
274C42#23511yescharacteristic
275C42#23511yescharacteristic
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—42no—
290C3×C4—81no—
291C3×C4—81no—
292C3×C4—81no—
293Dic3—81no—
294Dic3—81no—
295C23#95311yescharacteristic
296Q8—122no—
297Q8—241no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4C2×(C2×D3)⋊C411yescharacteristic
323C2×C4C4×#6711yescharacteristic
324C2×C4C4×#6711yescharacteristic
325C2×C4#92911yescharacteristic
326C2×C4#93011yescharacteristic
327C2×C4#93011yescharacteristic
328C8—122no—
329C8—122no—
330C8—241no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3#188211yescharacteristic
334C2×C3#190611yescharacteristic
335C2×C3#191211yescharacteristic
336C22—21no—
337C22—21no—
338C22#521311yescharacteristic
339C22#524211yescharacteristic
340C22#524611yescharacteristic
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—42no—
348C4—42no—
349C4—42no—
350C4—81no—
351C4—81no—
352C4—81no—
353C4—241no—
354C4—241no—
355C3#938011yescharacteristic, complemented, Sylow
356C2—21no—
357C2—21no—
358C2#7164511yescharacteristic
359C2#7166311yescharacteristic
360C2#7166611yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2336C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#2336D3
hypocenterC3#9380
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2336

Complements

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

NH ≅ G/Ntype
19. #5069C4semidirect
20. #5064C4semidirect
355. C3#9380semidirect

Maximal subgroups

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

subgroupindex
C3×#23362
#830212
#717352
#695662
#694932
#676332
#676212
#93803