Finite Group Databasebeta

← #10799101

Subgroups — #10799101

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×#932C211yesmaximal, characteristic
2C3×#2309C211yesmaximal, characteristic, complemented
3C22×#832C211yesmaximal, characteristic, complemented
4#82270C211yesmaximal, characteristic, complemented
5C2×#5002C211yesmaximal, characteristic, complemented
6C2×#4995C211yesmaximal, characteristic, complemented
7C2×#4997C211yesmaximal, characteristic, complemented
8#69649C212yesmaximal, complemented
9#69648C212yesmaximal, complemented
10#69647C212yesmaximal, complemented
11#69646C212yesmaximal, complemented
12#21235—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C22×#214C2212yes—
18C2×#932—24no—
19C2×#932—24no—
20C2×#932C418yescomplemented
21C22×#213C2212yes—
22C22×C4⋊Dic3C2211yescharacteristic
23C22×#208C2211yescharacteristic
24C23×C3⋊C8C2211yescharacteristic
25C3×#390—22no—
26C3×#390C2212yes—
27C3×#390C2212yes—
28C3×#379—22no—
29C2×#832—24no—
30C2×#832C2212yes—
31C2×#832C2212yes—
32#5002—24no—
33#4997—24no—
34#4995—24no—
35#4987—22no—
36C2×#810C2211yescharacteristic, complemented
37C2×#810C2211yescharacteristic, complemented
38C2×#809C2211yescharacteristic, complemented
39C2×#809C2211yescharacteristic, complemented
40C2×#808C2212yescomplemented
41C2×#807C2212yescomplemented
42#4958—22no—
43#4939C2212yescomplemented
44#4929C2212yescomplemented
45C2×#803C2212yescomplemented
46C2×C4×C3⋊C8C2212yescomplemented
47C22×AGL(1,ℤ/8)—31no—
48C2×#428—31no—
49C2×#416—31no—
50C22×#102—31no—
51C2×#405—31no—
52#2311—31no—
53#2309D311yescharacteristic, complemented
54#1973—32no—
55#1972—32no—
56#1970—32no—
57#1968—32no—
58C25×C3—21no—
59C22×C3×D4—22no—
60C22×C3×D4—22no—
61C22×C3×D4—22no—
62C22×C3×D4—21no—
63C22×C3×D4—22no—
64C22×C3×D4—22no—
65C22×C3×D4C2×C414yes—
66C22×C3×D4C2312yes—
67C23×C3×C4C2311yescharacteristic
68C22×#67—21no—
69C23×Dic3—21no—
70C2×#235—22no—
71C2×#235—22no—
72C2×#235—22no—
73C2×#235—22no—
74C2×#235—22no—
75C2×#235C2×C414yes—
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×Dic6C2311yescharacteristic
79C3×#117—22no—
80C3×#117—22no—
81C3×#117—22no—
82C3×#117—22no—
83C2×C3×C4⋊C4D412yes—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—24no—
88C2×C3×C22⋊C4—24no—
89C2×C3×C22⋊C4D412yes—
90C2×C3×C42—21no—
91C2×C3×C42C2312yes—
92C2×#214—24no—
93C2×#214—24no—
94C2×#214C2×C418yescomplemented
95#932—232no—
96C2×#213—24no—
97C2×#213—24no—
98C2×#213C2×C418yescomplemented
99C2×(C2×D3)⋊C4—24no—
100C2×C4⋊Dic3—22no—
101C2×C4⋊Dic3—22no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3C2312yes—
104C2×#208C2×C414yes—
105C2×#208D414yes—
106C2×#208D414yes—
107C22×C3⋊C8—22no—
108C22×C3⋊C8—22no—
109C22×C3⋊C8D414yes—
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2312yes—
113#832—24no—
114#832—24no—
115C3⋊#95—24no—
116#810—22no—
117#810—22no—
118#809—22no—
119#809—22no—
120#808—24no—
121#807—24no—
122#803—24no—
123C4×C3⋊C8—24no—
124C22×#41—31no—
125C23×D4C2×D311yescharacteristic
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×C8—31no—
133C2×#117C2×D311yescharacteristic
134C2×#117C2×D311yescharacteristic
135C22×C4⋊C4—31no—
136#428—64no—
137C2×#107—32no—
138C2×#106—32no—
139#416—64no—
140C2×#103—31no—
141C2×#103—31no—
142C2×#102—32no—
143C2×#102—32no—
144C2×#102—31no—
145C2×#102—31no—
146C2×#102—64no—
147#405—64no—
148C2×#97—32no—
149C2×C4×C8—32no—
150#391—32no—
151#390—22no—
152#390—32no—
153#390C2×D312yes—
154#390C2×D312yes—
155#386—62no—
156#380—62no—
157#379—22no—
158C24×C3—22no—
159C24×C3—24no—
160C24×C3—42no—
161C24×C3—42no—
162C24×C3—82no—
163C24×C3—82no—
164C24×C3C2×D411yescharacteristic
165C23×D3—21no—
166C2×C3×D4—28no—
167C2×C3×D4—28no—
168C2×C3×D4—24no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—44no—
175C2×C3×D4—42no—
176C2×C3×D4—44no—
177C2×C3×D4—42no—
178C2×C3×D4C22×C414yes—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4C22×C412yes—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D414yes—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4#4112yes—
188C22×C3×C4#4112yes—
189C22×C3×C4C2411yescharacteristic
190C2×#67—24no—
191C2×#67—42no—
192C2×#67—42no—
193C22×Dic3—21no—
194C22×Dic3—22no—
195C22×Dic3—42no—
196C22×Dic3—82no—
197#235—28no—
198#235—28no—
199#235—44no—
200#235—44no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12C22×C412yes—
207C2×C4×D3—24no—
208C2×C4×D3—42no—
209C2×C4×D3—42no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6C22×C412yes—
216C3×C4⋊C4—24no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C22⋊C4—24no—
222C3×C22⋊C4—42no—
223C3×C22⋊C4—84no—
224C3×C22⋊C4—84no—
225C3×C42—22no—
226C3×C42—22no—
227C3×C42—42no—
228#214—216no—
229#213—216no—
230(C2×D3)⋊C4—84no—
231C4⋊Dic3—22no—
232C4⋊Dic3—22no—
233C4⋊Dic3—42no—
234#208C22⋊C4116yescomplemented
235C2×C3⋊C8—28no—
236C2×C3⋊C8—21no—
237C2×C3⋊C8—21no—
238C2×C3⋊C8—21no—
239C2×C3⋊C8—21no—
240C2×C3⋊C8C22×C414yes—
241C2×C3⋊C8C2×D412yes—
242C2×C3⋊C8C2×D412yes—
243C2×C3⋊C8C2×D414yes—
244C2×C3⋊C8C2×D414yes—
245C25—21no—
246C2×#41—34no—
247C2×#41—62no—
248C2×#41—62no—
249C2×#41—62no—
250C2×#41—62no—
251C2×#41—62no—
252C22×Q8—31no—
253C22×D4—22no—
254C22×D4—22no—
255C22×D4—22no—
256C22×D4—22no—
257C22×D4—22no—
258C22×D4—21no—
259C22×D4—31no—
260C22×D4—61no—
261C22×D4C4×D314yes—
262C22×D4C22×D312yes—
263C23×C4—61no—
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—32no—
277C22×C8—34no—
278C22×C8—31no—
279C22×C8—31no—
280C22×C8—62no—
281C22×C8—62no—
282#117—22no—
283#117—22no—
284#117—22no—
285#117—22no—
286C2×C4⋊C4—32no—
287C2×C4⋊C4—62no—
288C2×C4⋊C4—62no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4#6712yes—
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C22⋊C4—21no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—64no—
297C2×C22⋊C4#6712yes—
298C2×C42—21no—
299C2×C42C22×D312yes—
300#107—64no—
301#106—64no—
302#103—62no—
303#103—62no—
304#102—62no—
305#102—62no—
306#102—64no—
307#102—64no—
308#97—64no—
309C4×C8—64no—
310#95—64no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—24no—
314C23×C3—42no—
315C23×C3—42no—
316C23×C3—44no—
317C23×C3—44no—
318C23×C3—48no—
319C23×C3—88no—
320C23×C3—84no—
321C23×C3—84no—
322C23×C3—88no—
323C23×C3C4×D412yes—
324C23×C3#12112yes—
325C23×C3#12712yes—
326C23×C3C22×D411yescharacteristic
327C22×D3—22no—
328C22×D3—41no—
329C22×D3—41no—
330C22×D3—42no—
331C22×D3—42no—
332C3×D4—24no—
333C3×D4—24no—
334C3×D4—42no—
335C3×D4—42no—
336C3×D4—48no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—24no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4C2×C22⋊C414yes—
345C2×C3×C4C4×D414yes—
346C2×C3×C4C4×D412yes—
347C2×C3×C4#12114yes—
348C2×C3×C4#12112yes—
349C2×C3×C4#12414yes—
350C2×C3×C4#12712yes—
351C2×C3×C4C23×C411yescharacteristic
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357#67—48no—
358C2×Dic3—22no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—42no—
362C2×Dic3—44no—
363C2×Dic3—84no—
364D12—22no—
365D12—22no—
366D12—41no—
367D12—41no—
368C4×D3—48no—
369Dic6—22no—
370Dic6—22no—
371Dic6—41no—
372Dic6—41no—
373C3⋊C8—22no—
374C3⋊C8—22no—
375C3⋊C8C2×C22⋊C418yes—
376C24—22no—
377C24—24no—
378C24—42no—
379C24—42no—
380C24—61no—
381C24—82no—
382C24—82no—
383C24C2×#6711yescharacteristic
384#41—68no—
385#41—68no—
386#41—124no—
387#41—124no—
388C2×Q8—32no—
389C2×Q8—61no—
390C2×Q8—61no—
391C2×Q8—61no—
392C2×Q8—61no—
393C2×Q8—61no—
394C2×D4—28no—
395C2×D4—28no—
396C2×D4—24no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—32no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—64no—
408C2×D4—61no—
409C2×D4—61no—
410C2×D4—61no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4—122no—
414C2×D4—122no—
415C2×D4C2×C4×D314yes—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—61no—
419C22×C4—64no—
420C22×C4—62no—
421C22×C4—122no—
422C22×C4—122no—
423C22×C4—122no—
424C22×C4—242no—
425C22×C4C2×C4×D312yes—
426C22×C4#23512yes—
427C22×C4D3×D412yes—
428C22×C4D3×D412yes—
429C22×C4D3×D412yes—
430C22×C4#23712yes—
431C22×C4C2×#6714yes—
432C22×C4C2×#6711yescharacteristic
433C22×C4C23×D311yescharacteristic
434SD16—616no—
435D8—616no—
436M16—316no—
437C2×C8—34no—
438C2×C8—34no—
439C2×C8—34no—
440C2×C8—32no—
441C2×C8—32no—
442C2×C8—68no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—61no—
447C4⋊C4—24no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—62no—
453C4⋊C4—62no—
454C4⋊C4—122no—
455C22⋊C4—24no—
456C22⋊C4—42no—
457C22⋊C4—84no—
458C22⋊C4—84no—
459C22⋊C4—244no—
460C42—22no—
461C42—22no—
462C42—42no—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—24no—
466C22×C3—24no—
467C22×C3—48no—
468C22×C3—44no—
469C22×C3—816no—
470C22×C3—82no—
471C22×C3—82no—
472C22×C3—84no—
473C22×C3#39014yes—
474C22×C3C2×C4×D411yescharacteristic
475C22×C3C2×#12111yescharacteristic
476C22×C3C2×#12711yescharacteristic
477C22×C3#57612yes—
478C22×C3#57712yes—
479C2×D3—44no—
480C2×D3—42no—
481C2×D3—81no—
482C2×D3—81no—
483C2×D3—82no—
484C3×C4—44no—
485C3×C4—81no—
486C3×C4—81no—
487C3×C4#39014yes—
488C3×C4C22×C22⋊C411yescharacteristic
489C3×C4C2×C4×D411yescharacteristic
490C3×C4C2×#12111yescharacteristic
491C3×C4C2×#12411yescharacteristic
492Dic3—42no—
493Dic3—82no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—48no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—44no—
502C23—62no—
503C23—88no—
504C23—88no—
505C23—84no—
506C23—84no—
507C23—121no—
508C23—121no—
509C23—122no—
510C23—122no—
511C23C4×#6712yes—
512C23#93912yes—
513C23#94012yes—
514C23C22×#6711yescharacteristic
515Q8—62no—
516Q8—62no—
517Q8—121no—
518Q8—121no—
519D4—24no—
520D4—24no—
521D4—48no—
522D4—42no—
523D4—42no—
524D4—62no—
525D4—62no—
526D4—128no—
527D4—121no—
528D4—121no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—62no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—128no—
540C2×C4—122no—
541C2×C4—124no—
542C2×C4—244no—
543C2×C4D3×C22⋊C414yes—
544C2×C4#67⋊C414yes—
545C2×C4#88414yes—
546C2×C4#88514yes—
547C2×C4C4×#6712yes—
548C2×C4#93912yes—
549C2×C4#94012yes—
550C2×C4C22×C4×D311yescharacteristic
551C2×C4C2×#23511yescharacteristic
552C2×C4C2×D3×D411yescharacteristic
553C2×C4C2×D3×D411yescharacteristic
554C2×C4C2×D3×D411yescharacteristic
555C2×C4C2×#23711yescharacteristic
556C8—38no—
557C8—62no—
558C8—62no—
559C2×C3—24no—
560C2×C3—84no—
561C2×C3C2×#39011yescharacteristic
562C2×C3#294612yes—
563C2×C3#306212yes—
564C2×C3#315411yescharacteristic
565C2×C3#315511yescharacteristic
566D3—82no—
567C22—24no—
568C22—24no—
569C22—22no—
570C22—22no—
571C22—48no—
572C22—44no—
573C22—816no—
574C22—84no—
575C22—82no—
576C22—82no—
577C22—124no—
578C22—122no—
579C22—242no—
580C22—241no—
581C22—241no—
582C22#495714yes—
583C22C2×C4×#6711yescharacteristic
584C22C2×#93911yescharacteristic
585C22C2×#94011yescharacteristic
586C22#583512yes—
587C22#583612yes—
588C4—44no—
589C4—81no—
590C4—81no—
591C4—122no—
592C4—242no—
593C4#495714yes—
594C4C2×D3×C22⋊C411yescharacteristic
595C4C2×#67⋊C411yescharacteristic
596C4C2×#88411yescharacteristic
597C4C2×#88511yescharacteristic
598C3#2123511yescharacteristic, complemented, Sylow
599C2—24no—
600C2—84no—
601C2—242no—
602C2C2×#495711yescharacteristic
603C2#8253712yes—
604C2#8337912yes—
605C2#8338111yescharacteristic
606C2#8338211yescharacteristic
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2309C2
socleC23×C3C22×D4
prime radicalC22×#932C2
cosocleC22×C3×C4C24
hypercenter#2309D3
hypocenterC3#21235
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2309

Complements

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

NH ≅ G/Ntype
2. C3×#2309C2semidirect
3. C22×#832C2semidirect
4. #82270C2semidirect
5. C2×#5002C2semidirect
6. C2×#4995C2semidirect
7. C2×#4997C2semidirect
8. #69649C2semidirect
9. #69648C2semidirect
10. #69647C2semidirect
11. #69646C2semidirect
20. C2×#932C4semidirect
36. C2×#810C22semidirect
37. C2×#810C22semidirect
38. C2×#809C22semidirect
39. C2×#809C22semidirect
40. C2×#808C22semidirect
41. C2×#807C22semidirect
43. #4939C22semidirect
44. #4929C22semidirect
45. C2×#803C22semidirect
46. C2×C4×C3⋊C8C22semidirect
53. #2309D3semidirect
94. C2×#214C2×C4semidirect
98. C2×#213C2×C4semidirect
234. #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×#9322
C3×#23092
C22×#8322
#822702
C2×#50022
C2×#49952
C2×#49972
#696492
#696482
#696472
#696462
#212353