Finite Group Databasebeta

← #10690112

Subgroups — #10690112

Order 768.

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

normal subgroups only

This group has 474 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×#5081C211yesmaximal, characteristic
2#74161C211yesmaximal, characteristic
3C3×#1993C211yesmaximal, characteristic
4#71786C211yesmaximal, characteristic
5C2×#4465C211yesmaximal, characteristic
6#67659C211yesmaximal, characteristic
7#67630C211yesmaximal, characteristic
8#9000—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C2×C3×#97C2211yescharacteristic
12C3×#382C2211yescharacteristic
13C2×#835—21no—
14C2×#835—21no—
15#5081—22no—
16#5081C414yescomplemented
17#4978—21no—
18#4976—21no—
19C2×#808—21no—
20C2×#807—21no—
21#4936C2211yescharacteristic
22C2×#803C2211yescharacteristic
23C2×#803C2211yescharacteristic
24#4655—21no—
25#4654—21no—
26#4653—21no—
27#4647—21no—
28C3×#340—21no—
29C3×#340—21no—
30#4465—22no—
31C2×#501—31no—
32#2382—31no—
33#1993D311yescharacteristic
34#1960—31no—
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—42no—
70#835—42no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77#808—24no—
78#807—24no—
79#803—21no—
80#803—21no—
81#803—21no—
82#803C2×C412yes—
83C2×C4⋊Q8C2×D311yescharacteristic
84C2×#125—31no—
85#501—34no—
86#501—62no—
87C2×#107—61no—
88C2×#106—61no—
89C2×#103—61no—
90C2×#103—61no—
91C2×#97—31no—
92C2×#97—31no—
93C2×#97C2×D311yescharacteristic
94#389—61no—
95#389—61no—
96#387—61no—
97#387—61no—
98#387—61no—
99#387—61no—
100#382—31no—
101#382C2×D311yescharacteristic
102#340—21no—
103#340—21no—
104#330—62no—
105C2×C3×Q8—22no—
106C2×C3×Q8—41no—
107C2×C3×Q8—41no—
108C2×C3×Q8—41no—
109C2×C3×Q8—41no—
110C2×C3×Q8—41no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4—41no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—21no—
121C2×C3×C8—22no—
122C2×C3×C8—41no—
123C2×C3×C8—41no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C42C22×C411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42#4111yescharacteristic
133C3×C42#4111yescharacteristic
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136C4⋊Dic3—41no—
137C4⋊Dic3—41no—
138C4⋊Dic3C22⋊C412yes—
139C4⋊Dic3C22⋊C412yes—
140Dic3⋊C4—22no—
141Dic3⋊C4—22no—
142Dic3⋊C4—22no—
143Dic3⋊C4—22no—
144Dic3⋊C4—81no—
145Dic3⋊C4—81no—
146C4×Dic3—81no—
147C4×Dic3—81no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—41no—
154C22×Q8—21no—
155C22×C8—21no—
156C22×C8—61no—
157C22×C8—61no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8C4×D312yes—
162#125—32no—
163#125—32no—
164#125—61no—
165#125—61no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—31no—
169C2×C4⋊C4—31no—
170C2×C4⋊C4—31no—
171C2×C4⋊C4—31no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4#6711yescharacteristic
177C2×C4⋊C4#6711yescharacteristic
178C2×C42—61no—
179C2×C42—61no—
180C2×C42C22×D311yescharacteristic
181#107—64no—
182#106—64no—
183#103—122no—
184#103—122no—
185#97—21no—
186#97—21no—
187#97—32no—
188#97—61no—
189#97—61no—
190#97—61no—
191#95—21no—
192#95—21no—
193#95—61no—
194#95—61no—
195#95—61no—
196#95—61no—
197#95—61no—
198#95—61no—
199C23×C3#12011yescharacteristic
200C3×Q8—42no—
201C3×Q8—42no—
202C3×Q8—81no—
203C3×Q8—81no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—42no—
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—81no—
218C2×C3×C4—81no—
219C2×C3×C4C2×C22⋊C411yescharacteristic
220C2×C3×C4C4×D411yescharacteristic
221C2×C3×C4C4×D411yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12311yescharacteristic
225C2×Dic3—22no—
226C2×Dic3—22no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C3×C8—42no—
238C3⋊C8—24no—
239C3⋊C8—42no—
240C2×Q8—22no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4D3×D411yescharacteristic
254C22×C4D3×D411yescharacteristic
255C22×C4C2×#6711yescharacteristic
256C2×C8—22no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—62no—
260C2×C8—62no—
261C2×C8—62no—
262C2×C8—62no—
263C2×C8—121no—
264C2×C8—121no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—32no—
268C4⋊C4—32no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—62no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—62no—
277C4⋊C4—62no—
278C4⋊C4—62no—
279C4⋊C4—121no—
280C4⋊C4—121no—
281C4⋊C4—241no—
282C4⋊C4—241no—
283C42—241no—
284C42—241no—
285C42C2×C4×D311yescharacteristic
286C42#23511yescharacteristic
287C42D3×D411yescharacteristic
288C42#23711yescharacteristic
289C22×C3#38611yescharacteristic
290C22×C3#45311yescharacteristic
291C22×C3#45411yescharacteristic
292C22×C3#45511yescharacteristic
293C22×C3#45611yescharacteristic
294C22×C3#45611yescharacteristic
295C22×C3#45611yescharacteristic
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—82no—
303C3×C4—81no—
304C3×C4—81no—
305Dic3—42no—
306Dic3—42no—
307Dic3—81no—
308Dic3—81no—
309C23#93711yescharacteristic
310Q8—42no—
311Q8—42no—
312Q8—81no—
313Q8—81no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—62no—
328C2×C4—62no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—121no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4D3×C22⋊C411yescharacteristic
342C2×C4#67⋊C411yescharacteristic
343C2×C4#88311yescharacteristic
344C2×C4#88411yescharacteristic
345C2×C4C4×#6711yescharacteristic
346C2×C4#93811yescharacteristic
347C8—42no—
348C8—64no—
349C8—122no—
350C2×C3#166011yescharacteristic
351C2×C3#166111yescharacteristic
352C2×C3#166311yescharacteristic
353C2×C3#166411yescharacteristic
354C2×C3#188211yescharacteristic
355C2×C3#188311yescharacteristic
356C2×C3#188611yescharacteristic
357C22#495611yescharacteristic
358C22#506111yescharacteristic
359C22#509511yescharacteristic
360C22#519811yescharacteristic
361C22#520111yescharacteristic
362C22#520511yescharacteristic
363C22#520511yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—81no—
371C4—81no—
372C4—82no—
373C4—122no—
374C4—122no—
375C4—241no—
376C4—241no—
377C3#900011yescharacteristic, complemented, Sylow
378C2#6969311yescharacteristic
379C2#7032811yescharacteristic
380C2#7034511yescharacteristic
381C2#7124511yescharacteristic
382C2#7124711yescharacteristic
383C2#7139111yescharacteristic
384C2#7139311yescharacteristic
subgroupG/H
centerC23#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1993C2
socle, prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#1993D3
hypocenterC3#9000
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. #5081C4semidirect
377. C3#9000semidirect

Maximal subgroups

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

subgroupindex
C2×#50812
#741612
C3×#19932
#717862
C2×#44652
#676592
#676302
#90003