Finite Group Databasebeta

← #10690128

Subgroups — #10690128

Order 768.

458 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C2×#5082C211yesmaximal, characteristic
2#74161C211yesmaximal, characteristic
3C3×#1993C211yesmaximal, characteristic
4#71788C211yesmaximal, characteristic
5C2×#4465C211yesmaximal, characteristic
6#67659C211yesmaximal, characteristic
7#67630C211yesmaximal, characteristic
8#9004—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C2×C3×#97C2211yescharacteristic
12C3×#382C2211yescharacteristic
13C2×#835—21no—
14C2×#835—21no—
15#5082—22no—
16#5082C414yescomplemented
17#4978—21no—
18#4976—21no—
19#4936C2211yescharacteristic
20C2×#803C2211yescharacteristic
21C2×#803C2211yescharacteristic
22#4655—21no—
23#4654—21no—
24#4653—21no—
25#4647—21no—
26C3×#340—21no—
27C3×#340—21no—
28#4529—21no—
29#4529—21no—
30#4465—22no—
31C2×#491—31no—
32#2382—31no—
33#1993—31no—
34#1993D311yescharacteristic
35C2×#330—31no—
36#1295—31no—
37#1295—31no—
38C22×C3×Q8—21no—
39C22×C3×C8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8C2×C412yes—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C4⋊C4D411yescharacteristic
48C2×C3×C42C2311yescharacteristic
49C2×C4⋊Dic3D411yescharacteristic
50C2×C4⋊Dic3D411yescharacteristic
51C2×C4⋊Dic3D411yescharacteristic
52C2×C4⋊Dic3D411yescharacteristic
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4—21no—
56C2×Dic3⋊C4—21no—
57C2×C4×Dic3—21no—
58C2×C4×Dic3—21no—
59C22×C3⋊C8—21no—
60C22×C3⋊C8—21no—
61#870—21no—
62#870—21no—
63#870C2×C412yes—
64#870D412yes—
65C3×#97—21no—
66C3×#97—21no—
67C3×#95—21no—
68C3×#95—21no—
69#835—24no—
70#835—24no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77#803—21no—
78#803—21no—
79#803—21no—
80#803C2×C412yes—
81C2×C4⋊Q8C2×D311yescharacteristic
82C2×#125—31no—
83#491—34no—
84#491—62no—
85C2×#103—61no—
86C2×#103—61no—
87C2×#97—31no—
88C2×#97—31no—
89C2×#97C2×D311yescharacteristic
90#389—61no—
91#389—61no—
92#387—61no—
93#387—61no—
94#387—61no—
95#387—61no—
96#382—31no—
97#382C2×D311yescharacteristic
98#340—21no—
99#340—21no—
100#340—61no—
101#340—61no—
102#330—62no—
103C2×C3×Q8—22no—
104C2×C3×Q8—42no—
105C2×C3×Q8—42no—
106C2×C3×Q8—41no—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4—41no—
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×C3×C8—22no—
118C2×C3×C8—41no—
119C2×C3×C8—41no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C42C22×C411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42#4111yescharacteristic
129C3×C42#4111yescharacteristic
130C4⋊Dic3—21no—
131C4⋊Dic3—21no—
132C4⋊Dic3—41no—
133C4⋊Dic3—41no—
134C4⋊Dic3C22⋊C412yes—
135C4⋊Dic3C22⋊C412yes—
136Dic3⋊C4—22no—
137Dic3⋊C4—22no—
138Dic3⋊C4—22no—
139Dic3⋊C4—22no—
140Dic3⋊C4—81no—
141Dic3⋊C4—81no—
142C4×Dic3—81no—
143C4×Dic3—81no—
144C2×C3⋊C8—22no—
145C2×C3⋊C8—22no—
146C2×C3⋊C8—41no—
147C2×C3⋊C8—41no—
148C2×C3⋊C8—41no—
149C2×C3⋊C8—41no—
150C22×Q8—21no—
151C22×C8—21no—
152C22×C8—61no—
153C22×C8—61no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8C4×D312yes—
158#125—32no—
159#125—32no—
160#125—61no—
161#125—61no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—31no—
165C2×C4⋊C4—31no—
166C2×C4⋊C4—31no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—61no—
169C2×C4⋊C4—61no—
170C2×C4⋊C4—61no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4#6711yescharacteristic
173C2×C4⋊C4#6711yescharacteristic
174C2×C42—61no—
175C2×C42—61no—
176C2×C42C22×D311yescharacteristic
177#103—64no—
178#103—64no—
179#97—21no—
180#97—21no—
181#97—32no—
182#97—61no—
183#97—61no—
184#97—61no—
185#95—21no—
186#95—21no—
187#95—61no—
188#95—61no—
189#95—61no—
190#95—61no—
191#95—61no—
192#95—61no—
193C23×C3#12011yescharacteristic
194C3×Q8—44no—
195C3×Q8—82no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—42no—
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—81no—
210C2×C3×C4—81no—
211C2×C3×C4C2×C22⋊C411yescharacteristic
212C2×C3×C4C4×D411yescharacteristic
213C2×C3×C4C4×D411yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12311yescharacteristic
217C2×Dic3—22no—
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—
229C3×C8—42no—
230C3⋊C8—42no—
231C3⋊C8—42no—
232C2×Q8—22no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—41no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—41no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4—61no—
243C22×C4D3×D411yescharacteristic
244C22×C4D3×D411yescharacteristic
245C22×C4C2×#6711yescharacteristic
246C2×C8—22no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—62no—
250C2×C8—62no—
251C2×C8—121no—
252C2×C8—121no—
253C2×C8—121no—
254C2×C8—121no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—32no—
258C4⋊C4—32no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—62no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—62no—
267C4⋊C4—62no—
268C4⋊C4—62no—
269C4⋊C4—121no—
270C4⋊C4—121no—
271C4⋊C4—241no—
272C4⋊C4—241no—
273C42—241no—
274C42—241no—
275C42C2×C4×D311yescharacteristic
276C42#23511yescharacteristic
277C42D3×D411yescharacteristic
278C42#23711yescharacteristic
279C22×C3#38611yescharacteristic
280C22×C3#45411yescharacteristic
281C22×C3#45411yescharacteristic
282C22×C3#45511yescharacteristic
283C22×C3#45511yescharacteristic
284C22×C3#45611yescharacteristic
285C22×C3#45611yescharacteristic
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—82no—
293C3×C4—81no—
294C3×C4—81no—
295Dic3—42no—
296Dic3—42no—
297Dic3—81no—
298Dic3—81no—
299C23#93711yescharacteristic
300Q8—44no—
301Q8—82no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—62no—
316C2×C4—62no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4D3×C22⋊C411yescharacteristic
330C2×C4#67⋊C411yescharacteristic
331C2×C4#88311yescharacteristic
332C2×C4#88411yescharacteristic
333C2×C4C4×#6711yescharacteristic
334C2×C4#93811yescharacteristic
335C8—42no—
336C8—122no—
337C8—122no—
338C2×C3#166211yescharacteristic
339C2×C3#166311yescharacteristic
340C2×C3#166411yescharacteristic
341C2×C3#166411yescharacteristic
342C2×C3#188211yescharacteristic
343C2×C3#188211yescharacteristic
344C2×C3#188811yescharacteristic
345C22#495611yescharacteristic
346C22#506111yescharacteristic
347C22#509511yescharacteristic
348C22#520111yescharacteristic
349C22#520111yescharacteristic
350C22#520211yescharacteristic
351C22#520511yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—81no—
359C4—81no—
360C4—82no—
361C4—122no—
362C4—122no—
363C4—241no—
364C4—241no—
365C3#900411yescharacteristic, complemented, Sylow
366C2#6969311yescharacteristic
367C2#7032511yescharacteristic
368C2#7034811yescharacteristic
369C2#7124611yescharacteristic
370C2#7124711yescharacteristic
371C2#7139311yescharacteristic
372C2#7139411yescharacteristic
subgroupG/H
centerC23#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1993C2
socle, prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#1993D3
hypocenterC3#9004
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
16. #5082C4semidirect
365. C3#9004semidirect

Maximal subgroups

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

subgroupindex
C2×#50822
#741612
C3×#19932
#717882
C2×#44652
#676592
#676302
#90043