Finite Group Databasebeta

← #10943884

Subgroups — #10943884

Order 768.

4070 conjugacy classes of subgroups; only the 255 normal subgroups are shown.

normal subgroups only

This group has 4070 conjugacy classes of subgroups — too many to draw. Only its 255 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
1C22×#1004C211yesmaximal, characteristic, complemented
2C3×#2433C211yesmaximal, characteristic, complemented
3C2×#5077C214yesmaximal, complemented
4#82624C212yesmaximal, complemented
5#72129C211yesmaximal, characteristic, complemented
6#69662C214yesmaximal, complemented
7#69454C212yesmaximal, complemented
8#32888—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C23×#67—22no—
11C2×#1004—22no—
12C2×#1004—216no—
13C2×#1004—24no—
14C2×#1004C2214yes—
15C22×#237—22no—
16C22×D3×D4—22no—
17C22×#235C2212yes—
18C2×C3×#127C2212yescomplemented
19C2×C3×#121C2211yescharacteristic, complemented
20C2×C3×#120—22no—
21C2×#932C2214yescomplemented
22C3×#392—22no—
23C3×#390C2212yescomplemented
24C3×C4×C22⋊C4C2211yescharacteristic, complemented
25C2×#833C2214yescomplemented
26#5207C2212yescomplemented
27#5077—216no—
28#5040—22no—
29#5037—24no—
30#5036—22no—
31#4966C2214yescomplemented
32C2×#805C2218yescomplemented
33C22×H(5,𝔽2)—31no—
34C2×#453—34no—
35#3006—32no—
36#2433D311yescharacteristic, complemented
37#1792—34no—
38#1789—32no—
39#1783—31no—
40C25×C3D412yes—
41C24×D3—22no—
42C22×C3×D4—28no—
43C22×C3×D4—21no—
44C22×C3×D4—24no—
45C22×C3×D4—22no—
46C22×C3×D4—22no—
47C22×C3×D4D412yes—
48C22×C3×D4C2312yes—
49C22×C3×D4C2311yescharacteristic, complemented
50C23×C3×C4C2311yescharacteristic
51C22×#67—24no—
52C22×#67—232no—
53C22×#67—28no—
54C22×#67—22no—
55C22×#67—28no—
56C23×Dic3—22no—
57#1004—232no—
58#1004—232no—
59#1004—232no—
60#1004—28no—
61#1004—216no—
62#1004D4116yescomplemented
63C2×#237—28no—
64C2×#237—232no—
65C2×#237—28no—
66C2×D3×D4—232no—
67C2×D3×D4—28no—
68C2×D3×D4—28no—
69C2×#235—22no—
70C2×#235—22no—
71C2×#235—216no—
72C2×#235D414yescomplemented
73C2×#235C2314yes—
74C22×D12D412yescomplemented
75C22×C4×D3—22no—
76C22×Dic6D412yescomplemented
77C3×#127—28no—
78C3×#121—22no—
79C3×#121—22no—
80C3×#120—48no—
81C2×C3×C4⋊C4C2311yescharacteristic, complemented
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—24no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C42C2312yescomplemented
86C2×#214—24no—
87#932—216no—
88C2×#213—24no—
89C2×#208C2314yescomplemented
90C3×#95—21no—
91#833—28no—
92#830—22no—
93#830—22no—
94#810—24no—
95#809—24no—
96#805—216no—
97C2×H(5,𝔽2)—34no—
98C2×H(5,𝔽2)—64no—
99C2×H(5,𝔽2)—616no—
100C2×H(5,𝔽2)—62no—
101C22×#41—32no—
102C22×#41—62no—
103C23×D4—62no—
104C23×D4—62no—
105C23×D4C2×D311yescharacteristic
106C2×AGL(1,ℤ/8)—34no—
107C2×#127C2×D312yescomplemented
108C2×#121C2×D311yescharacteristic, complemented
109C2×#120—22no—
110#453—616no—
111#449—64no—
112#448—62no—
113#447—62no—
114C2×#104—38no—
115#417—34no—
116C2×#100—34no—
117#407—32no—
118#392—22no—
119#390C2×D312yescomplemented
120C4×C22⋊C4C2×D311yescharacteristic, complemented
121C24×C3—22no—
122C24×C3—28no—
123C24×C3—22no—
124C24×C3—21no—
125C24×C3—44no—
126C24×C3—82no—
127C24×C3C2×D411yescharacteristic
128C24×C3C2×D414yes—
129C23×D3—22no—
130C23×D3—216no—
131C23×D3—28no—
132C23×D3—84no—
133C23×D3—84no—
134C2×C3×D4—216no—
135C2×C3×D4—216no—
136C2×C3×D4—28no—
137C2×C3×D4—24no—
138C2×C3×D4—24no—
139C2×C3×D4—24no—
140C2×C3×D4—22no—
141C2×C3×D4—24no—
142C2×C3×D4—48no—
143C2×C3×D4—44no—
144C2×C3×D4—41no—
145C2×C3×D4—41no—
146C2×C3×D4—48no—
147C2×C3×D4—84no—
148C2×C3×D4C2×D418yes—
149C22×C3×C4—24no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2411yescharacteristic
156C2×#67—216no—
157C2×#67—232no—
158C2×#67—216no—
159C2×#67—264no—
160C2×#67—232no—
161C2×#67—24no—
162C2×#67—232no—
163C2×#67—24no—
164C22×Dic3—216no—
165C22×Dic3—28no—
166C22×Dic3—22no—
167#237—264no—
168#237—232no—
169#237—232no—
170D3×D4—264no—
171D3×D4—232no—
172D3×D4—232no—
173#235—216no—
174#235—216no—
175#235—28no—
176#235—28no—
177#235—28no—
178#235C2×D4116yes—
179C2×D12—22no—
180C2×D12—28no—
181C2×D12C2×D414yes—
182C2×C4×D3—216no—
183C2×C4×D3—24no—
184C2×C4×D3—24no—
185C2×Dic6—22no—
186C2×Dic6—28no—
187C2×Dic6C2×D414yes—
188C3×C4⋊C4—22no—
189C3×C22⋊C4—24no—
190C3×C22⋊C4—48no—
191C3×C22⋊C4—42no—
192C3×C42—24no—
193#214—48no—
194#213—48no—
195#208—28no—
196C2×C3⋊C8—24no—
197C25—62no—
198C25#6712yes—
199H(5,𝔽2)—316no—
200H(5,𝔽2)—616no—
201H(5,𝔽2)—632no—
202H(5,𝔽2)—632no—
203H(5,𝔽2)—632no—
204H(5,𝔽2)—68no—
205C2×#41—34no—
206C2×#41—34no—
207C2×#41—616no—
208C2×#41—68no—
209C2×#41—632no—
210C2×#41—68no—
211C2×#41—62no—
212C2×#41—62no—
213C22×Q8—32no—
214C22×D4—28no—
215C22×D4—21no—
216C22×D4—24no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—32no—
220C22×D4—68no—
221C22×D4—632no—
222C22×D4—68no—
223C22×D4—64no—
224C22×D4—632no—
225C22×D4—68no—
226C22×D4—62no—
227C22×D4—68no—
228C22×D4#6712yes—
229C22×D4C22×D312yes—
230C22×D4C22×D311yescharacteristic, complemented
231C23×C4—62no—
232C23×C4—62no—
233C23×C4C22×D311yescharacteristic
234AGL(1,ℤ/8)—616no—
235C2×SD16—64no—
236C2×D8—64no—
237C2×M16—34no—
238#127—28no—
239#121—22no—
240#121—22no—
241#120—48no—
242C2×C4⋊C4C22×D311yescharacteristic, complemented
243C2×C22⋊C4—21no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—21no—
246C2×C42C22×D312yescomplemented
247#104—616no—
248#103—64no—
249#102—64no—
250#100—68no—
251C22⋊C8—62no—
252C22⋊C8—62no—
253#95—21no—
254C23×C3—28no—
255C23×C3—28no—
256C23×C3—24no—
257C23×C3—22no—
258C23×C3—22no—
259C23×C3—24no—
260C23×C3—41no—
261C23×C3—44no—
262C23×C3—44no—
263C23×C3—44no—
264C23×C3—42no—
265C23×C3—42no—
266C23×C3—44no—
267C23×C3—41no—
268C23×C3—42no—
269C23×C3—82no—
270C23×C3—82no—
271C23×C3—816no—
272C23×C3#12011yescharacteristic
273C23×C3#12011yescharacteristic
274C23×C3#12018yes—
275C23×C3C22×D411yescharacteristic
276C22×D3—28no—
277C22×D3—24no—
278C22×D3—232no—
279C22×D3—44no—
280C22×D3—44no—
281C22×D3—832no—
282C22×D3—88no—
283C22×D3—88no—
284C3×D4—216no—
285C3×D4—216no—
286C3×D4—44no—
287C3×D4—816no—
288C3×D4—82no—
289C3×D4—82no—
290C2×C3×C4—22no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—41no—
298C2×C3×C4#12012yes—
299C2×C3×C4C22×D412yes—
300C2×C3×C4C22×D414yes—
301#67—216no—
302#67—264no—
303#67—216no—
304C2×Dic3—28no—
305C2×Dic3—232no—
306C2×Dic3—24no—
307D12—28no—
308D12—24no—
309D12#12018yescomplemented
310C4×D3—216no—
311C4×D3—216no—
312Dic6—28no—
313Dic6—24no—
314Dic6#12018yescomplemented
315C3⋊C8—44no—
316C24—22no—
317C24—28no—
318C24—22no—
319C24—21no—
320C24—44no—
321C24—616no—
322C24—68no—
323C24—62no—
324C24—82no—
325C24—244no—
326C24—244no—
327C24C2×#6711yescharacteristic
328C24C2×#6714yes—
329#41—316no—
330#41—68no—
331#41—68no—
332#41—616no—
333#41—616no—
334#41—632no—
335#41—664no—
336#41—632no—
337#41—68no—
338C2×Q8—34no—
339C2×Q8—68no—
340C2×Q8—62no—
341C2×D4—216no—
342C2×D4—216no—
343C2×D4—28no—
344C2×D4—24no—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—22no—
348C2×D4—24no—
349C2×D4—34no—
350C2×D4—48no—
351C2×D4—44no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—48no—
355C2×D4—68no—
356C2×D4—632no—
357C2×D4—664no—
358C2×D4—632no—
359C2×D4—632no—
360C2×D4—616no—
361C2×D4—616no—
362C2×D4—664no—
363C2×D4—632no—
364C2×D4—64no—
365C2×D4—632no—
366C2×D4—64no—
367C2×D4—62no—
368C2×D4—84no—
369C2×D4D3×D418yes—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—616no—
374C22×C4—616no—
375C22×C4—68no—
376C22×C4—62no—
377C22×C4—64no—
378C22×C4—64no—
379C22×C4D3×D412yes—
380C22×C4D3×D412yes—
381C22×C4C2×#6711yescharacteristic
382C22×C4C23×D311yescharacteristic
383SD16—128no—
384D8—128no—
385M16—68no—
386C2×C8—64no—
387C4⋊C4—22no—
388C22⋊C4—24no—
389C22⋊C4—48no—
390C22⋊C4—42no—
391C42—24no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—28no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—44no—
401C22×C3—48no—
402C22×C3—416no—
403C22×C3—44no—
404C22×C3—44no—
405C22×C3—84no—
406C22×C3—84no—
407C22×C3—88no—
408C22×C3—81no—
409C22×C3—81no—
410C22×C3—81no—
411C22×C3#45312yescomplemented
412C22×C3C2×#12012yes—
413C22×C3C2×#12011yescharacteristic
414C2×D3—28no—
415C2×D3—416no—
416C2×D3—44no—
417C2×D3—44no—
418C2×D3—832no—
419C3×C4—22no—
420C3×C4—44no—
421C3×C4—81no—
422C3×C4C2×#12014yes—
423Dic3—28no—
424C23—28no—
425C23—28no—
426C23—22no—
427C23—24no—
428C23—22no—
429C23—24no—
430C23—41no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—42no—
435C23—42no—
436C23—44no—
437C23—42no—
438C23—41no—
439C23—68no—
440C23—632no—
441C23—64no—
442C23—82no—
443C23—82no—
444C23—816no—
445C23—124no—
446C23—124no—
447C23—2432no—
448C23—248no—
449C23—248no—
450C23#93711yescharacteristic
451C23#93711yescharacteristic
452C23#93718yes—
453C23C22×#6711yescharacteristic
454Q8—38no—
455Q8—68no—
456Q8—64no—
457D4—216no—
458D4—216no—
459D4—38no—
460D4—44no—
461D4—68no—
462D4—616no—
463D4—664no—
464D4—616no—
465D4—64no—
466D4—816no—
467D4—82no—
468D4—82no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—616no—
478C2×C4—68no—
479C2×C4—632no—
480C2×C4—64no—
481C2×C4—616no—
482C2×C4#93712yes—
483C2×C4C2×D3×D414yes—
484C2×C4C2×D3×D412yes—
485C8—124no—
486C2×C3—22no—
487C2×C3—22no—
488C2×C3—48no—
489C2×C3—41no—
490C2×C3—161no—
491C2×C3#284611yescharacteristic
492C2×C3C2×#45311yescharacteristic
493C2×C3#301911yescharacteristic
494D3—48no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—28no—
501C22—21no—
502C22—21no—
503C22—44no—
504C22—48no—
505C22—416no—
506C22—44no—
507C22—44no—
508C22—68no—
509C22—84no—
510C22—84no—
511C22—88no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—1216no—
516C22—124no—
517C22—124no—
518C22—2432no—
519C22#507712yescomplemented
520C22C2×#93712yes—
521C22C2×#93711yescharacteristic
522C4—22no—
523C4—44no—
524C4—68no—
525C4—81no—
526C4D3×#12014yes—
527C3#3288811yescharacteristic, complemented, Sylow
528C2—22no—
529C2—22no—
530C2—41no—
531C2—48no—
532C2—128no—
533C2—161no—
534C2#8261211yescharacteristic
535C2C2×#507711yescharacteristic
536C2#8339411yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2433C2
socleC22×C3C2×#120
hypercenter#2433D3
hypocenterC3#32888
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2433

Complements

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

NH ≅ G/Ntype
1. C22×#1004C2semidirect
2. C3×#2433C2semidirect
3. C2×#5077C2semidirect
4. #82624C2semidirect
5. #72129C2semidirect
6. #69662C2semidirect
7. #69454C2semidirect
18. C2×C3×#127C22semidirect
19. C2×C3×#121C22semidirect
21. C2×#932C22semidirect
23. C3×#390C22semidirect
24. C3×C4×C22⋊C4C22semidirect
25. C2×#833C22semidirect
26. #5207C22semidirect
31. #4966C22semidirect
32. C2×#805C22semidirect
36. #2433D3semidirect
49. C22×C3×D4C23semidirect
62. #1004D4semidirect
72. C2×#235D4semidirect
74. C22×D12D4semidirect
76. C22×Dic6D4semidirect
81. C2×C3×C4⋊C4C23semidirect
85. C2×C3×C42C23semidirect
89. C2×#208C23semidirect
107. C2×#127C2×D3semidirect
108. C2×#121C2×D3semidirect
119. #390C2×D3semidirect
120. C4×C22⋊C4C2×D3semidirect
230. C22×D4C22×D3semidirect
242. C2×C4⋊C4C22×D3semidirect
246. C2×C42C22×D3semidirect
309. D12#120semidirect
314. Dic6#120semidirect
411. C22×C3#453semidirect
519. C22#5077semidirect
527. C3#32888semidirect

Maximal subgroups

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

subgroupindex
C22×#10042
C3×#24332
C2×#50772
#826242
#721292
#696622
#694542
#328883