Finite Group Databasebeta

← #10799099

Subgroups — #10799099

Order 768.

1678 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#949C211yesmaximal, characteristic
2C2×C3×#416C211yesmaximal, characteristic, complemented
3#84087C211yesmaximal, characteristic, complemented
4C22×#821C211yesmaximal, characteristic, complemented
5C2×#5101C211yesmaximal, characteristic, complemented
6#82269C211yesmaximal, characteristic, complemented
7C2×#4995C211yesmaximal, characteristic, complemented
8#70260C212yesmaximal, complemented
9#70257C212yesmaximal, complemented
10#70191C212yesmaximal, complemented
11#70190C212yesmaximal, complemented
12#21235—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×D12C2211yescharacteristic
15C23×C3×C8C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C2×#949—24no—
18C2×#949—24no—
19C2×#949C418yescomplemented
20C22×#215C2212yes—
21C22×#213C2212yes—
22C2×#926C2211yescharacteristic
23C22×C4⋊Dic3C2211yescharacteristic
24C22×#208C2211yescharacteristic
25C3×#416—24no—
26C2×C3×#103C2211yescharacteristic, complemented
27C2×C3×#103C2211yescharacteristic, complemented
28C2×C3×#102C2211yescharacteristic, complemented
29C2×C3×#102C2211yescharacteristic, complemented
30#5229C2212yescomplemented
31#5221C2212yescomplemented
32#5220—22no—
33#5212—22no—
34C2×#821—24no—
35C2×#821C2212yes—
36C2×#821C2212yes—
37#5101—24no—
38C2×#815C2212yescomplemented
39C2×C8×Dic3C2212yescomplemented
40#4995—24no—
41#4988C2212yes—
42#4988C2212yes—
43C2×#808C2212yescomplemented
44C2×#807C2212yescomplemented
45#4955—22no—
46#4939—22no—
47C22×AGL(1,ℤ/8)—31no—
48C2×#428—31no—
49C2×#416D311yescharacteristic, complemented
50C22×#102—31no—
51C2×#405—31no—
52#2311—31no—
53#2309—31no—
54#1973—32no—
55#1972—32no—
56#1970—32no—
57#1968—32no—
58C24×D3—21no—
59C2×C3×#41—22no—
60C2×C3×#41—22no—
61C2×C3×#41—22no—
62C2×C3×#41—22no—
63C2×C3×#41—22no—
64C2×C3×#41C2×C414yes—
65C22×C3×Q8C2311yescharacteristic
66C22×C3×D4—21no—
67C22×C3×D4C2311yescharacteristic
68C23×C3×C4—21no—
69C23×C3×C4C2311yescharacteristic
70C23×Dic3—21no—
71C22×D12—22no—
72C22×D12—22no—
73C22×D12—22no—
74C22×D12—22no—
75C22×D12—21no—
76C22×D12—22no—
77C22×D12C2×C414yes—
78C22×D12C2312yes—
79C22×C3×C8—22no—
80C22×C3×C8—22no—
81C22×C3×C8D414yes—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×#117—22no—
85C3×#117—22no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C22⋊C4—21no—
89C2×C3×C42—21no—
90C2×#217—24no—
91C2×#217D412yes—
92#949—232no—
93C2×#215—24no—
94C2×#215—24no—
95C2×#215C2×C418yescomplemented
96C2×#213—24no—
97C2×#213—24no—
98C2×#213C2×C418yescomplemented
99C2×(C2×D3)⋊C4—24no—
100C2×(C2×D3)⋊C4—24no—
101#926—22no—
102#926—22no—
103C2×C4⋊Dic3—22no—
104C2×C4⋊Dic3—22no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3D412yes—
107C2×C4⋊Dic3C2312yes—
108C2×C4×Dic3C2312yes—
109C2×#208C2×C414yes—
110C2×#208D414yes—
111C2×#208D414yes—
112C22×C3⋊C8C2312yes—
113C3×#103—22no—
114C3×#103—22no—
115C3×#102—22no—
116C3×#102—22no—
117C3⋊#95—24no—
118#821—24no—
119#821—24no—
120#815—24no—
121C8×Dic3—24no—
122#808—24no—
123#807—24no—
124C22×#41C2×D311yescharacteristic
125C23×D4—31no—
126C2×AGL(1,ℤ/8)—38no—
127C2×AGL(1,ℤ/8)—64no—
128C2×AGL(1,ℤ/8)—64no—
129C22×SD16—32no—
130C22×D8—32no—
131C22×M16—31no—
132C23×C8C2×D311yescharacteristic
133C2×#117—31no—
134C2×#117C2×D311yescharacteristic
135C22×C4⋊C4—31no—
136#428—64no—
137C2×#107—32no—
138C2×#106—32no—
139#416—24no—
140C2×#103C2×D311yescharacteristic, complemented
141C2×#103C2×D311yescharacteristic, complemented
142C2×#102—32no—
143C2×#102—32no—
144C2×#102—64no—
145C2×#102C2×D311yescharacteristic, complemented
146C2×#102C2×D311yescharacteristic, complemented
147#405—64no—
148C2×#97—32no—
149C2×C4×C8—32no—
150#391—32no—
151#390—32no—
152#390—32no—
153#390—32no—
154#390—62no—
155#386—62no—
156#380—62no—
157#379—62no—
158C24×C3—21no—
159C24×C3C2×D411yescharacteristic
160C23×D3—22no—
161C23×D3—24no—
162C23×D3—42no—
163C23×D3—42no—
164C23×D3—82no—
165C23×D3—82no—
166C3×#41—28no—
167C3×#41—28no—
168C3×#41—44no—
169C3×#41—44no—
170C2×C3×Q8—21no—
171C2×C3×Q8—21no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8—21no—
175C2×C3×Q8C22×C412yes—
176C2×C3×D4—24no—
177C2×C3×D4—21no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—42no—
183C2×C3×D4—42no—
184C2×C3×D4C22×C412yes—
185C22×C3×C4—21no—
186C22×C3×C4—24no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—42no—
190C22×C3×C4—42no—
191C22×C3×C4C22×C412yes—
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D412yes—
196C22×C3×C4#4112yes—
197C22×C3×C4#4112yes—
198C22×C3×C4C2411yescharacteristic
199C22×Dic3—22no—
200C22×Dic3—42no—
201C22×Dic3—82no—
202C22×Dic3C2×D414yes—
203C2×D12—28no—
204C2×D12—28no—
205C2×D12—22no—
206C2×D12—22no—
207C2×D12—22no—
208C2×D12—22no—
209C2×D12—24no—
210C2×D12—22no—
211C2×D12—42no—
212C2×D12—44no—
213C2×D12—44no—
214C2×D12—42no—
215C2×D12C22×C414yes—
216C2×C3×C8—28no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8C2×D412yes—
222C2×C3×C8C2×D412yes—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C22⋊C4—42no—
228C3×C42—42no—
229#217—24no—
230#217—84no—
231#215—216no—
232#213—216no—
233(C2×D3)⋊C4—84no—
234(C2×D3)⋊C4—84no—
235C4⋊Dic3—24no—
236C4⋊Dic3—22no—
237C4⋊Dic3—22no—
238C4⋊Dic3—42no—
239C4×Dic3—22no—
240C4×Dic3—22no—
241#208C22⋊C4116yescomplemented
242C2×C3⋊C8C22×C414yes—
243C2×C3⋊C8C2×D414yes—
244C2×C3⋊C8C2×D414yes—
245C25—61no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41—22no—
251C2×#41C4×D314yes—
252C22×Q8C22×D311yescharacteristic
253C22×D4—21no—
254C22×D4—34no—
255C22×D4—32no—
256C22×D4—62no—
257C22×D4—62no—
258C22×D4—62no—
259C22×D4—61no—
260C22×D4—62no—
261C22×D4—62no—
262C22×D4C22×D311yescharacteristic
263C23×C4—21no—
264C23×C4—61no—
265C23×C4C22×D311yescharacteristic
266AGL(1,ℤ/8)—632no—
267C2×SD16—38no—
268C2×SD16—64no—
269C2×SD16—64no—
270C2×D8—38no—
271C2×D8—64no—
272C2×D8—64no—
273C2×M16—34no—
274C2×M16—34no—
275C2×M16—34no—
276C22×C8—22no—
277C22×C8—22no—
278C22×C8—32no—
279C22×C8#6714yes—
280C22×C8C22×D311yescharacteristic
281C22×C8C22×D311yescharacteristic
282#117—22no—
283#117—22no—
284#117—62no—
285#117—62no—
286C2×C4⋊C4—32no—
287C2×C4⋊C4—32no—
288C2×C4⋊C4—62no—
289C2×C4⋊C4—62no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C22⋊C4—21no—
294C2×C22⋊C4—32no—
295C2×C22⋊C4—64no—
296C2×C22⋊C4—64no—
297C2×C22⋊C4—64no—
298C2×C42—21no—
299C2×C42—32no—
300#107—64no—
301#106—64no—
302#103—22no—
303#103—22no—
304#102—22no—
305#102—22no—
306#102—64no—
307#102—64no—
308#97—64no—
309C4×C8—64no—
310#95—64no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3C4×D412yes—
319C23×C3#12112yes—
320C23×C3#12712yes—
321C23×C3C22×D411yescharacteristic
322C22×D3—24no—
323C22×D3—42no—
324C22×D3—42no—
325C22×D3—44no—
326C22×D3—44no—
327C22×D3—48no—
328C22×D3—88no—
329C22×D3—84no—
330C22×D3—84no—
331C22×D3—88no—
332C3×Q8—22no—
333C3×Q8—22no—
334C3×Q8—41no—
335C3×Q8—41no—
336C3×D4—22no—
337C3×D4—22no—
338C3×D4—48no—
339C3×D4—41no—
340C3×D4—41no—
341C2×C3×C4—22no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—48no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4C2×C22⋊C414yes—
350C2×C3×C4C4×D412yes—
351C2×C3×C4C4×D414yes—
352C2×C3×C4#12112yes—
353C2×C3×C4#12114yes—
354C2×C3×C4#12414yes—
355C2×C3×C4#12712yes—
356C2×C3×C4C23×C411yescharacteristic
357C2×C3×C4C22×D411yescharacteristic
358C2×C3×C4C22×D411yescharacteristic
359C2×C3×C4C22×D411yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×Dic3—24no—
363C2×Dic3—24no—
364C2×Dic3—24no—
365C2×Dic3—42no—
366C2×Dic3—44no—
367C2×Dic3—84no—
368D12—24no—
369D12—24no—
370D12—42no—
371D12—42no—
372D12—48no—
373C3×C8—22no—
374C3×C8—22no—
375C3⋊C8C2×C22⋊C418yes—
376C24—21no—
377C24—62no—
378C24—64no—
379C24—122no—
380C24—122no—
381C24—242no—
382C24—242no—
383C24D3×D411yescharacteristic
384#41—28no—
385#41—28no—
386#41—44no—
387#41—44no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8C2×C4×D312yes—
394C2×D4—24no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—34no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—68no—
404C2×D4—64no—
405C2×D4—68no—
406C2×D4—62no—
407C2×D4—62no—
408C2×D4—62no—
409C2×D4—62no—
410C2×D4—62no—
411C2×D4—122no—
412C2×D4—124no—
413C2×D4—124no—
414C2×D4—122no—
415C2×D4C2×C4×D312yes—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—34no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—62no—
424C22×C4—122no—
425C22×C4—242no—
426C22×C4C2×C4×D312yes—
427C22×C4#23512yes—
428C22×C4D3×D412yes—
429C22×C4D3×D412yes—
430C22×C4D3×D411yescharacteristic
431C22×C4#23712yes—
432C22×C4C2×#6712yes—
433C22×C4C23×D311yescharacteristic
434SD16—616no—
435D8—616no—
436M16—316no—
437C2×C8—28no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—34no—
443C2×C8—34no—
444C2×C8—34no—
445C2×C8C2×#6712yes—
446C2×C8C2×#6712yes—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—64no—
452C4⋊C4—62no—
453C4⋊C4—62no—
454C4⋊C4—122no—
455C22⋊C4—42no—
456C22⋊C4—64no—
457C22⋊C4—244no—
458C22⋊C4—244no—
459C22⋊C4—244no—
460C42—42no—
461C42—62no—
462C42—62no—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—24no—
466C22×C3—24no—
467C22×C3—44no—
468C22×C3—42no—
469C22×C3—81no—
470C22×C3—81no—
471C22×C3—82no—
472C22×C3#39014yes—
473C22×C3C2×C4×D411yescharacteristic
474C22×C3C2×#12111yescharacteristic
475C22×C3C2×#12711yescharacteristic
476C22×C3#57612yes—
477C22×C3#57712yes—
478C2×D3—48no—
479C2×D3—44no—
480C2×D3—816no—
481C2×D3—82no—
482C2×D3—82no—
483C2×D3—84no—
484C3×C4—42no—
485C3×C4—81no—
486C3×C4—81no—
487C3×C4#39014yes—
488C3×C4C22×C22⋊C411yescharacteristic
489C3×C4C2×C4×D411yescharacteristic
490C3×C4C2×#12111yescharacteristic
491C3×C4C2×#12411yescharacteristic
492Dic3—44no—
493Dic3—82no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—41no—
501C23—64no—
502C23—128no—
503C23—122no—
504C23—124no—
505C23—124no—
506C23—122no—
507C23—248no—
508C23—244no—
509C23—244no—
510C23—248no—
511C23#67⋊C412yes—
512C23#88412yes—
513C23#94012yes—
514C23C2×D3×D411yescharacteristic
515Q8—22no—
516Q8—22no—
517Q8—41no—
518Q8—41no—
519D4—22no—
520D4—22no—
521D4—48no—
522D4—41no—
523D4—41no—
524D4—64no—
525D4—64no—
526D4—128no—
527D4—122no—
528D4—122no—
529C2×C4—22no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—48no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—64no—
538C2×C4—64no—
539C2×C4—64no—
540C2×C4—124no—
541C2×C4—122no—
542C2×C4—244no—
543C2×C4D3×C22⋊C414yes—
544C2×C4#67⋊C412yes—
545C2×C4#88412yes—
546C2×C4#88514yes—
547C2×C4C4×#6714yes—
548C2×C4#93914yes—
549C2×C4#94012yes—
550C2×C4C22×C4×D311yescharacteristic
551C2×C4C2×#23511yescharacteristic
552C2×C4C2×D3×D411yescharacteristic
553C2×C4C2×D3×D411yescharacteristic
554C2×C4C2×#23711yescharacteristic
555C2×C4C22×#6711yescharacteristic
556C8—22no—
557C8—22no—
558C8—38no—
559C2×C3—24no—
560C2×C3—82no—
561C2×C3C2×#39011yescharacteristic
562C2×C3#294612yes—
563C2×C3#306212yes—
564C2×C3#315411yescharacteristic
565C2×C3#315511yescharacteristic
566D3—84no—
567C22—22no—
568C22—22no—
569C22—24no—
570C22—24no—
571C22—42no—
572C22—44no—
573C22—81no—
574C22—81no—
575C22—82no—
576C22—124no—
577C22—128no—
578C22—2416no—
579C22—242no—
580C22—242no—
581C22—244no—
582C22#495714yes—
583C22C2×#67⋊C411yescharacteristic
584C22C2×#88411yescharacteristic
585C22#576912yes—
586C22#577012yes—
587C22C2×#94011yescharacteristic
588C4—42no—
589C4—81no—
590C4—81no—
591C4—124no—
592C4—242no—
593C4#495714yes—
594C4C2×D3×C22⋊C411yescharacteristic
595C4C2×#88511yescharacteristic
596C4C2×C4×#6711yescharacteristic
597C4C2×#93911yescharacteristic
598C3#2123511yescharacteristic, complemented, Sylow
599C2—24no—
600C2—82no—
601C2—244no—
602C2#7882012yes—
603C2#7884012yes—
604C2C2×#495711yescharacteristic
605C2#8390511yescharacteristic
606C2#8390611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#416C2
socleC23×C3C22×D4
prime radicalC22×#949C2
cosocleC22×C3×C4C24
hypercenterC2×#416D3
hypocenterC3#21235
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
2. C2×C3×#416C2semidirect
3. #84087C2semidirect
4. C22×#821C2semidirect
5. C2×#5101C2semidirect
6. #82269C2semidirect
7. C2×#4995C2semidirect
8. #70260C2semidirect
9. #70257C2semidirect
10. #70191C2semidirect
11. #70190C2semidirect
19. C2×#949C4semidirect
26. C2×C3×#103C22semidirect
27. C2×C3×#103C22semidirect
28. C2×C3×#102C22semidirect
29. C2×C3×#102C22semidirect
30. #5229C22semidirect
31. #5221C22semidirect
38. C2×#815C22semidirect
39. C2×C8×Dic3C22semidirect
43. C2×#808C22semidirect
44. C2×#807C22semidirect
49. C2×#416D3semidirect
95. C2×#215C2×C4semidirect
98. C2×#213C2×C4semidirect
140. C2×#103C2×D3semidirect
141. C2×#103C2×D3semidirect
145. C2×#102C2×D3semidirect
146. C2×#102C2×D3semidirect
241. #208C22⋊C4semidirect
598. C3#21235semidirect

Maximal subgroups

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

subgroupindex
C22×#9492
C2×C3×#4162
#840872
C22×#8212
C2×#51012
#822692
C2×#49952
#702602
#702572
#701912
#701902
#212353