Finite Group Databasebeta

← #10900876

Subgroups — #10900876

Order 768.

1452 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5525C211yesmaximal, characteristic
2C3×#2625C211yesmaximal, characteristic, complemented
3C2×#5047C211yesmaximal, characteristic, complemented
4C2×#5046C211yesmaximal, characteristic, complemented
5#71584C211yesmaximal, characteristic, complemented
6#69689C212yesmaximal, complemented
7#69688C212yesmaximal, complemented
8#69628C212yesmaximal, complemented
9#69627C212yesmaximal, complemented
10#69522C211yesmaximal, characteristic, complemented
11#69521C211yesmaximal, characteristic, complemented
12#29632—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C23×D12C2211yescharacteristic
15C2×C3×C22⋊Q8C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C22×#216C2211yescharacteristic, complemented
18C22×#215C2212yes—
19C22×#215C2211yescharacteristic, complemented
20#5525—24no—
21#5525—24no—
22C2×#895—21no—
23C2×#895—21no—
24C2×#882—22no—
25C2×#875C2212yes—
26C2×#874C2212yes—
27C3×#400—21no—
28C3×#398—21no—
29C3×#397—22no—
30C3×#391C2212yescomplemented
31C3×#389C2212yescomplemented
32C3×#380C2211yescharacteristic
33C2×#830C2211yescharacteristic
34C2×#830C2211yescharacteristic
35#5047—24no—
36#5046—24no—
37C2×#810C2211yescharacteristic, complemented
38C2×#810C2212yescomplemented
39C2×#809C2212yes—
40C2×#809C2212yes—
41C2×#809C2212yes—
42C2×#809C2211yescharacteristic, complemented
43C2×#809C2212yescomplemented
44#4524C2212yescomplemented
45#4524C2212yescomplemented
46C2×#535—31no—
47C2×#452—31no—
48C2×#450—31no—
49#2625D311yescharacteristic, complemented
50#2042—32no—
51#2041—32no—
52#2015—32no—
53#2014—32no—
54#1818—31no—
55#1788—31no—
56#1786—31no—
57C24×D3—21no—
58C22×C3×Q8C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×#67—21no—
61C2×#235—22no—
62C2×#235—42no—
63C2×#235—42no—
64C2×#235D414yes—
65C22×D12—21no—
66C22×D12—22no—
67C22×D12—22no—
68C22×D12—22no—
69C22×D12—42no—
70C22×D12—42no—
71C22×D12D414yes—
72C22×D12C2312yes—
73C22×D12C2311yescharacteristic
74C22×C4×D3—21no—
75C22×Dic6C2311yescharacteristic
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8—22no—
78C3×#117—22no—
79C3×#117—22no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4D411yescharacteristic
83C2×C3×C4⋊C4D411yescharacteristic
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4D412yes—
89C2×C3×C42C2312yes—
90C2×#216—22no—
91C2×#216—42no—
92C2×#215—24no—
93C2×#215—22no—
94C2×#215—42no—
95C2×#215—42no—
96C2×#215—42no—
97C2×(C2×D3)⋊C4—22no—
98C2×(C2×D3)⋊C4—22no—
99C22×C3⋊C8C2311yescharacteristic
100C22×C3⋊C8C2311yescharacteristic
101C22×C3⋊C8C2312yes—
102#895—28no—
103#895—28no—
104#882—216no—
105#875—24no—
106#875—24no—
107#874—24no—
108#874—24no—
109C3×#95—21no—
110C3×#95—22no—
111C3×#95—22no—
112C3×#95—22no—
113#830—22no—
114#830—22no—
115#810—22no—
116#810—42no—
117#809—24no—
118#809—22no—
119#809—42no—
120#809—42no—
121#809—42no—
122C22×#41—31no—
123C23×D4—31no—
124C22×Q16—31no—
125C22×SD16—32no—
126C22×SD16—31no—
127#535—64no—
128#535—64no—
129C2×#127—32no—
130C2×#124—32no—
131C2×C22⋊Q8C2×D311yescharacteristic
132C2×#121—61no—
133C2×#121—61no—
134C2×#120—62no—
135C2×#117C2×D311yescharacteristic
136#452—64no—
137#450—64no—
138C2×#103—31no—
139C2×#103—32no—
140C2×#102—32no—
141C2×#102—31no—
142C2×#102—32no—
143C2×#102—32no—
144C2×#102—32no—
145C2×C22⋊C8—31no—
146C2×C22⋊C8—31no—
147#400—21no—
148#398—21no—
149#397—22no—
150#391C2×D312yescomplemented
151#389C2×D312yescomplemented
152#380C2×D311yescharacteristic
153#336—32no—
154#336—32no—
155C24×C3C2×D411yescharacteristic
156C23×D3—22no—
157C23×D3—21no—
158C23×D3—24no—
159C23×D3—44no—
160C23×D3—81no—
161C23×D3—82no—
162C23×D3—81no—
163C2×C3×Q8—21no—
164C2×C3×Q8—21no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2411yescharacteristic
178C2×#67—24no—
179C2×#67—44no—
180C22×Dic3—21no—
181#235—44no—
182#235—44no—
183#235—48no—
184C2×D12—22no—
185C2×D12—22no—
186C2×D12—22no—
187C2×D12—22no—
188C2×D12—22no—
189C2×D12—22no—
190C2×D12—24no—
191C2×D12—21no—
192C2×D12—22no—
193C2×D12—28no—
194C2×D12—28no—
195C2×D12—48no—
196C2×D12—44no—
197C2×D12—44no—
198C2×D12—44no—
199C2×D12—82no—
200C2×D12—82no—
201C2×D12—82no—
202C2×D12—82no—
203C2×D12C2×D412yes—
204C2×D12C2×D414yes—
205C2×C4×D3—24no—
206C2×C4×D3—44no—
207C2×Dic6—22no—
208C2×Dic6—22no—
209C2×Dic6—21no—
210C2×Dic6C2×D412yes—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—42no—
218C3×C4⋊C4—82no—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—42no—
221C3×C22⋊C4—81no—
222C3×C42—22no—
223C3×C42—22no—
224#216—44no—
225#215—48no—
226#215—44no—
227(C2×D3)⋊C4—28no—
228(C2×D3)⋊C4—28no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—21no—
231C2×C3⋊C8—21no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—22no—
234C2×C3⋊C8—22no—
235C2×C3⋊C8—22no—
236C25—61no—
237C2×#41—34no—
238C2×#41—62no—
239C2×#41—122no—
240C2×#41—122no—
241C22×Q8—31no—
242C22×Q8C22×D311yescharacteristic
243C22×D4—34no—
244C22×D4—32no—
245C22×D4—31no—
246C22×D4—61no—
247C22×D4—62no—
248C22×D4—61no—
249C22×D4—62no—
250C22×D4—62no—
251C22×D4—122no—
252C22×D4—122no—
253C23×C4—61no—
254C23×C4C22×D311yescharacteristic
255C2×Q16—62no—
256C2×Q16—122no—
257C2×SD16—64no—
258C2×SD16—62no—
259C2×SD16—122no—
260C2×SD16—122no—
261C2×SD16—122no—
262C22×C8—32no—
263C22×C8—31no—
264C22×C8—31no—
265#127—64no—
266#127—64no—
267#124—64no—
268#124—64no—
269C22⋊Q8—22no—
270C22⋊Q8—22no—
271#121—68no—
272#121—68no—
273#120—616no—
274#117—22no—
275#117—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4#6711yescharacteristic
279C2×C4⋊C4#6711yescharacteristic
280C2×C4⋊C4C22×D311yescharacteristic
281C2×C4⋊C4C22×D312yes—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—62no—
285C2×C22⋊C4—62no—
286C2×C22⋊C4#6712yes—
287C2×C42C22×D312yes—
288#103—62no—
289#103—122no—
290#102—64no—
291#102—62no—
292#102—122no—
293#102—122no—
294#102—122no—
295C22⋊C8—62no—
296C22⋊C8—62no—
297#95—21no—
298#95—22no—
299#95—22no—
300#95—22no—
301C23×C3—22no—
302C23×C3—21no—
303C23×C3—22no—
304C23×C3—21no—
305C23×C3#12012yes—
306C23×C3C22×D411yescharacteristic
307C22×D3—24no—
308C22×D3—22no—
309C22×D3—48no—
310C22×D3—42no—
311C22×D3—42no—
312C22×D3—42no—
313C22×D3—41no—
314C22×D3—42no—
315C22×D3—42no—
316C22×D3—41no—
317C22×D3—42no—
318C22×D3—42no—
319C22×D3—42no—
320C22×D3—84no—
321C22×D3—84no—
322C22×D3—88no—
323C22×D3—88no—
324C3×Q8—41no—
325C3×Q8—42no—
326C3×Q8—41no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—24no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—41no—
343C2×C3×C4—42no—
344C2×C3×C4—41no—
345C2×C3×C4#12012yes—
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4H(5,𝔽2)12yes—
349C2×C3×C4H(5,𝔽2)12yes—
350#67—48no—
351C2×Dic3—22no—
352C2×Dic3—42no—
353D12—24no—
354D12—22no—
355D12—24no—
356D12—22no—
357D12—48no—
358D12—42no—
359D12—42no—
360D12—42no—
361D12—88no—
362D12—88no—
363C4×D3—48no—
364Dic6—22no—
365Dic6—22no—
366Dic6—42no—
367C3⋊C8—41no—
368C3⋊C8—42no—
369C3⋊C8—41no—
370C24—62no—
371C24—61no—
372C24—64no—
373C24—124no—
374C24—241no—
375C24—242no—
376C24—241no—
377C24C2×#6711yescharacteristic
378#41—124no—
379#41—124no—
380#41—128no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—32no—
386C2×Q8—61no—
387C2×Q8—62no—
388C2×Q8—62no—
389C2×D4—34no—
390C2×D4—32no—
391C2×D4—68no—
392C2×D4—68no—
393C2×D4—64no—
394C2×D4—64no—
395C2×D4—62no—
396C2×D4—61no—
397C2×D4—62no—
398C2×D4—62no—
399C2×D4—62no—
400C2×D4—62no—
401C2×D4—62no—
402C2×D4—62no—
403C2×D4—128no—
404C2×D4—124no—
405C2×D4—124no—
406C2×D4—124no—
407C2×D4—124no—
408C2×D4—242no—
409C2×D4—242no—
410C2×D4—242no—
411C2×D4—242no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—61no—
418C22×C4—64no—
419C22×C4—124no—
420C22×C4D3×D412yes—
421C22×C4D3×D412yes—
422C22×C4C2×#6712yes—
423C22×C4C2×#6712yes—
424C22×C4C2×#6711yescharacteristic
425C22×C4C23×D311yescharacteristic
426Q16—124no—
427SD16—128no—
428SD16—124no—
429C2×C8—62no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—62no—
433C2×C8—62no—
434C2×C8—62no—
435C2×C8—62no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—42no—
443C4⋊C4—82no—
444C22⋊C4—24no—
445C22⋊C4—42no—
446C22⋊C4—68no—
447C22⋊C4—68no—
448C22⋊C4—81no—
449C42—22no—
450C42—22no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3C2×#12011yescharacteristic
457C22×C3#53512yes—
458C22×C3#57611yescharacteristic
459C22×C3#57711yescharacteristic
460C22×C3#57712yes—
461C2×D3—42no—
462C2×D3—44no—
463C2×D3—42no—
464C2×D3—44no—
465C2×D3—42no—
466C2×D3—44no—
467C2×D3—816no—
468C2×D3—82no—
469C2×D3—82no—
470C2×D3—82no—
471C2×D3—82no—
472C2×D3—82no—
473C2×D3—82no—
474C3×C4—22no—
475C3×C4—42no—
476C3×C4—42no—
477C3×C4—81no—
478C3×C4—82no—
479C3×C4—81no—
480C3×C4#53411yescharacteristic
481C3×C4#53411yescharacteristic
482C3×C4#53412yes—
483Dic3—42no—
484C23—22no—
485C23—21no—
486C23—22no—
487C23—21no—
488C23—64no—
489C23—62no—
490C23—128no—
491C23—122no—
492C23—122no—
493C23—122no—
494C23—121no—
495C23—122no—
496C23—122no—
497C23—121no—
498C23—122no—
499C23—122no—
500C23—122no—
501C23—244no—
502C23—244no—
503C23—248no—
504C23—248no—
505C23#93712yes—
506C23C22×#6711yescharacteristic
507Q8—42no—
508Q8—41no—
509Q8—41no—
510Q8—62no—
511Q8—62no—
512Q8—122no—
513D4—64no—
514D4—62no—
515D4—64no—
516D4—62no—
517D4—128no—
518D4—128no—
519D4—122no—
520D4—122no—
521D4—122no—
522D4—248no—
523D4—248no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—62no—
543C2×C4—128no—
544C2×C4—122no—
545C2×C4#93712yes—
546C2×C4C2×D3×D411yescharacteristic
547C2×C4C2×D3×D411yescharacteristic
548C2×C4#100912yes—
549C2×C4#100912yes—
550C8—122no—
551C8—121no—
552C8—121no—
553C2×C3—42no—
554C2×C3#261311yescharacteristic
555C2×C3#300911yescharacteristic
556C2×C3#301211yescharacteristic
557C2×C3#322212yes—
558C2×C3#322412yes—
559D3—82no—
560D3—84no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—42no—
565C22—42no—
566C22—122no—
567C22—124no—
568C22—122no—
569C22—124no—
570C22—122no—
571C22—124no—
572C22—2416no—
573C22—242no—
574C22—242no—
575C22—242no—
576C22—242no—
577C22—242no—
578C22—242no—
579C22C2×#93711yescharacteristic
580C22#583012yes—
581C22#583511yescharacteristic
582C22#583712yes—
583C22#583711yescharacteristic
584C4—22no—
585C4—42no—
586C4—42no—
587C4—82no—
588C4—81no—
589C4—81no—
590C4—122no—
591C4#565312yes—
592C4#565311yescharacteristic
593C4#565311yescharacteristic
594C3#2963211yescharacteristic, complemented, Sylow
595C2—42no—
596C2—244no—
597C2—242no—
598C2#8290311yescharacteristic
599C2#8340411yescharacteristic
600C2#8342411yescharacteristic
601C2#8344812yes—
602C2#8346312yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2625C2
socleC23×C3C22×D4
prime radicalC2×#5525C2
hypercenter#2625D3
hypocenterC3#29632
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2625

Complements

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

NH ≅ G/Ntype
2. C3×#2625C2semidirect
3. C2×#5047C2semidirect
4. C2×#5046C2semidirect
5. #71584C2semidirect
6. #69689C2semidirect
7. #69688C2semidirect
8. #69628C2semidirect
9. #69627C2semidirect
10. #69522C2semidirect
11. #69521C2semidirect
17. C22×#216C22semidirect
19. C22×#215C22semidirect
30. C3×#391C22semidirect
31. C3×#389C22semidirect
37. C2×#810C22semidirect
38. C2×#810C22semidirect
42. C2×#809C22semidirect
43. C2×#809C22semidirect
44. #4524C22semidirect
45. #4524C22semidirect
49. #2625D3semidirect
150. #391C2×D3semidirect
151. #389C2×D3semidirect
594. C3#29632semidirect

Maximal subgroups

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

subgroupindex
C2×#55252
C3×#26252
C2×#50472
C2×#50462
#715842
#696892
#696882
#696282
#696272
#695222
#695212
#296323