Finite Group Databasebeta

← #10799054

Subgroups — #10799054

Order 768.

1566 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#950C211yesmaximal, characteristic
2C3×#2311C211yesmaximal, characteristic, complemented
3C2×#5232C211yesmaximal, characteristic, complemented
4#82270C211yesmaximal, characteristic, complemented
5C2×#4997C211yesmaximal, characteristic, complemented
6C2×#4966C211yesmaximal, characteristic, complemented
7C22×#807C211yesmaximal, characteristic, complemented
8#69679C212yesmaximal, complemented
9#69677C212yesmaximal, complemented
10#69649C212yesmaximal, complemented
11#69647C212yesmaximal, complemented
12#21230—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C2×#950—24no—
18C2×#950—24no—
19C2×#950C418yescomplemented
20C22×#216C2211yescharacteristic
21C22×#215C2211yescharacteristic
22C22×#214C2211yescharacteristic
23C22×#213C2211yescharacteristic
24C22×C4⋊Dic3C2211yescharacteristic
25C22×#208C2211yescharacteristic
26C23×C3⋊C8C2211yescharacteristic
27C3×#391C2212yes—
28C3×#390C2212yes—
29C3×#386—22no—
30C3×#380—22no—
31#5232—24no—
32C2×#835C2212yes—
33C2×#832C2212yes—
34#4997—24no—
35#4987—22no—
36C2×#810C2212yes—
37#4966—24no—
38C2×#809C2212yes—
39C2×#807—24no—
40C2×#807C2212yes—
41C2×#807C2212yes—
42#4958—22no—
43#4939C2212yes—
44#4929C2212yes—
45C2×#803C2212yes—
46C2×C4×C3⋊C8C2212yes—
47C22×#135—31no—
48C22×#107—31no—
49C2×#417—31no—
50C2×#417—31no—
51C2×#405—31no—
52#2311—31no—
53#2311D311yescharacteristic, complemented
54#1973—32no—
55#1973—32no—
56#1970—32no—
57#1969—32no—
58C2×C3×#41—22no—
59C2×C3×#41—22no—
60C2×C3×#41—22no—
61C2×C3×#41—22no—
62C2×C3×#41—22no—
63C2×C3×#41C2×C414yes—
64C22×C3×Q8C2311yescharacteristic
65C22×C3×D4—21no—
66C22×C3×D4C2311yescharacteristic
67C23×C3×C4—21no—
68C23×C3×C4—21no—
69C23×C3×C4C2311yescharacteristic
70C22×#67—21no—
71C23×Dic3—21no—
72C2×#235—22no—
73C2×#235—22no—
74C2×#235—22no—
75C2×#235—22no—
76C2×#235—22no—
77C2×#235C2×C414yes—
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×Dic6C2311yescharacteristic
81C3×#117—22no—
82C3×#117—22no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4D412yes—
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C22⋊C4—24no—
89C2×C3×C22⋊C4D412yes—
90C2×C3×C42C2312yes—
91#950—232no—
92C2×#216—22no—
93C2×#216—22no—
94C2×#216C2×C414yescomplemented
95C2×#215—22no—
96C2×#215—22no—
97C2×#215C2×C414yescomplemented
98C2×#214—22no—
99C2×#214—22no—
100C2×#214C2×C414yescomplemented
101C2×#213—22no—
102C2×#213—22no—
103C2×#213C2×C414yescomplemented
104C2×(C2×D3)⋊C4—24no—
105C2×C4⋊Dic3—22no—
106C2×C4⋊Dic3—22no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3C2312yes—
109C2×#208—22no—
110C2×#208—22no—
111C2×#208D414yes—
112C22×C3⋊C8C2×C414yes—
113C22×C3⋊C8D414yes—
114C22×C3⋊C8D414yes—
115C22×C3⋊C8C2312yes—
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C3×#95—24no—
119#835—24no—
120#832—24no—
121C3⋊#95—24no—
122#810—24no—
123#809—24no—
124#807—24no—
125#807—24no—
126#803—24no—
127C4×C3⋊C8—24no—
128C22×#41—31no—
129C22×#41C2×D311yescharacteristic
130C2×#135—38no—
131C2×#135—64no—
132C2×#135—64no—
133C22×Q16—31no—
134C22×SD16—31no—
135C22×SD16—31no—
136C22×D8—31no—
137C22×M16—31no—
138C23×C8—31no—
139C2×#117C2×D311yescharacteristic
140C22×C4⋊C4—31no—
141C22×C4⋊C4C2×D311yescharacteristic
142C2×#107—32no—
143C2×#107—32no—
144C2×#107—64no—
145#417—64no—
146#417—64no—
147C2×#103—32no—
148C2×#103—32no—
149C2×#102—32no—
150C2×#102—32no—
151#405—64no—
152C2×#97—32no—
153C2×C4×C8—32no—
154#391—32no—
155#391C2×D312yes—
156#390—32no—
157#390C2×D312yes—
158#386—22no—
159#386—62no—
160#380—22no—
161#380—62no—
162C24×C3—21no—
163C24×C3C2×D411yescharacteristic
164C23×D3—21no—
165C3×#41—28no—
166C3×#41—28no—
167C3×#41—44no—
168C3×#41—44no—
169C2×C3×Q8—21no—
170C2×C3×Q8—21no—
171C2×C3×Q8—21no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8C22×C412yes—
175C2×C3×D4—24no—
176C2×C3×D4—21no—
177C2×C3×D4—21no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—42no—
182C2×C3×D4—42no—
183C2×C3×D4C22×C412yes—
184C22×C3×C4—21no—
185C22×C3×C4—24no—
186C22×C3×C4—22no—
187C22×C3×C4—42no—
188C22×C3×C4—42no—
189C22×C3×C4—42no—
190C22×C3×C4—82no—
191C22×C3×C4C22×C412yes—
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D414yes—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4#4112yes—
198C22×C3×C4#4112yes—
199C22×C3×C4C2411yescharacteristic
200C2×#67—24no—
201C2×#67—42no—
202C2×#67—42no—
203C22×Dic3—21no—
204C22×Dic3—22no—
205C22×Dic3—42no—
206C22×Dic3—82no—
207#235—28no—
208#235—28no—
209#235—44no—
210#235—44no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12C22×C412yes—
217C2×C4×D3—24no—
218C2×C4×D3—42no—
219C2×C4×D3—42no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6C22×C412yes—
226C3×C4⋊C4—24no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—42no—
230C3×C22⋊C4—24no—
231C3×C22⋊C4—84no—
232C3×C42—22no—
233C3×C42—22no—
234#216—28no—
235#215—28no—
236#214—28no—
237#213—28no—
238(C2×D3)⋊C4—84no—
239C4⋊Dic3—22no—
240C4⋊Dic3—22no—
241C4⋊Dic3—42no—
242#208—28no—
243C2×C3⋊C8—22no—
244C2×C3⋊C8—22no—
245C2×C3⋊C8C22⋊C4116yescomplemented
246C2×C3⋊C8C22×C412yes—
247C2×C3⋊C8C22×C412yes—
248C2×C3⋊C8C2×D414yes—
249C2×C3⋊C8C2×D412yes—
250C2×C3⋊C8C2×D412yes—
251C2×C3⋊C8C2×D412yes—
252C2×C3⋊C8C2×D412yes—
253C2×#41—22no—
254C2×#41—22no—
255C2×#41—22no—
256C2×#41—22no—
257C2×#41—22no—
258C2×#41—34no—
259C2×#41—62no—
260C2×#41—62no—
261C2×#41—62no—
262C2×#41—62no—
263C2×#41—62no—
264C2×#41C4×D314yes—
265C22×Q8—31no—
266C22×Q8C22×D311yescharacteristic
267C22×D4—21no—
268C22×D4—31no—
269C22×D4—61no—
270C22×D4C22×D311yescharacteristic
271C23×C4—21no—
272C23×C4—21no—
273C23×C4—61no—
274C23×C4—61no—
275C23×C4C22×D311yescharacteristic
276#135—632no—
277C2×Q16—34no—
278C2×Q16—62no—
279C2×Q16—62no—
280C2×SD16—34no—
281C2×SD16—34no—
282C2×SD16—62no—
283C2×SD16—62no—
284C2×SD16—62no—
285C2×SD16—62no—
286C2×D8—34no—
287C2×D8—62no—
288C2×D8—62no—
289C2×M16—34no—
290C2×M16—62no—
291C2×M16—62no—
292C22×C8—34no—
293C22×C8—34no—
294C22×C8—34no—
295C22×C8—31no—
296C22×C8—31no—
297C22×C8—32no—
298#117—22no—
299#117—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—32no—
304C2×C4⋊C4—62no—
305C2×C4⋊C4—62no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4#6712yes—
308C2×C4⋊C4C22×D312yes—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—64no—
311C2×C22⋊C4#6712yes—
312C2×C42C22×D312yes—
313#107—64no—
314#107—64no—
315#103—64no—
316#103—64no—
317#102—64no—
318#102—64no—
319#97—64no—
320C4×C8—64no—
321#95—24no—
322#95—64no—
323C23×C3—22no—
324C23×C3—22no—
325C23×C3—22no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—42no—
329C23×C3—42no—
330C23×C3C4×D412yes—
331C23×C3#12112yes—
332C23×C3#12712yes—
333C23×C3C22×D411yescharacteristic
334C22×D3—22no—
335C22×D3—41no—
336C22×D3—41no—
337C22×D3—42no—
338C22×D3—42no—
339C3×Q8—22no—
340C3×Q8—22no—
341C3×Q8—41no—
342C3×Q8—41no—
343C3×D4—22no—
344C3×D4—22no—
345C3×D4—48no—
346C3×D4—41no—
347C3×D4—41no—
348C2×C3×C4—22no—
349C2×C3×C4—24no—
350C2×C3×C4—24no—
351C2×C3×C4—24no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—48no—
355C2×C3×C4—42no—
356C2×C3×C4—44no—
357C2×C3×C4—84no—
358C2×C3×C4C2×C22⋊C414yes—
359C2×C3×C4C4×D414yes—
360C2×C3×C4C4×D412yes—
361C2×C3×C4#12114yes—
362C2×C3×C4#12112yes—
363C2×C3×C4#12414yes—
364C2×C3×C4#12712yes—
365C2×C3×C4C23×C411yescharacteristic
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C22×D411yescharacteristic
368C2×C3×C4C22×D411yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371#67—48no—
372C2×Dic3—22no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—42no—
376C2×Dic3—44no—
377C2×Dic3—84no—
378D12—22no—
379D12—22no—
380D12—41no—
381D12—41no—
382C4×D3—48no—
383Dic6—22no—
384Dic6—22no—
385Dic6—41no—
386Dic6—41no—
387C3⋊C8—24no—
388C3⋊C8C2×C22⋊C414yes—
389C3⋊C8C2×C22⋊C414yes—
390C24—21no—
391C24—61no—
392C24C2×#6711yescharacteristic
393#41—28no—
394#41—28no—
395#41—44no—
396#41—44no—
397#41—68no—
398#41—68no—
399#41—124no—
400#41—124no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—32no—
407C2×Q8—61no—
408C2×Q8—61no—
409C2×Q8—61no—
410C2×Q8—61no—
411C2×Q8—61no—
412C2×Q8C2×C4×D312yes—
413C2×D4—24no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—32no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—64no—
423C2×D4—61no—
424C2×D4—61no—
425C2×D4—61no—
426C2×D4—61no—
427C2×D4—61no—
428C2×D4—122no—
429C2×D4—122no—
430C2×D4C2×C4×D312yes—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—61no—
438C22×C4—64no—
439C22×C4—62no—
440C22×C4—82no—
441C22×C4—122no—
442C22×C4—122no—
443C22×C4—122no—
444C22×C4—242no—
445C22×C4C2×C4×D312yes—
446C22×C4#23512yes—
447C22×C4D3×D412yes—
448C22×C4D3×D412yes—
449C22×C4D3×D412yes—
450C22×C4#23712yes—
451C22×C4C2×#6714yes—
452C22×C4C2×#6711yescharacteristic
453C22×C4C23×D311yescharacteristic
454Q16—68no—
455SD16—68no—
456SD16—68no—
457D8—68no—
458M16—68no—
459C2×C8—316no—
460C2×C8—32no—
461C2×C8—32no—
462C2×C8—32no—
463C2×C8—32no—
464C2×C8—32no—
465C2×C8—32no—
466C2×C8—34no—
467C2×C8—62no—
468C2×C8—62no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—42no—
473C4⋊C4—62no—
474C4⋊C4—62no—
475C4⋊C4—122no—
476C22⋊C4—24no—
477C22⋊C4—84no—
478C22⋊C4—244no—
479C42—22no—
480C42—22no—
481C22×C3—22no—
482C22×C3—22no—
483C22×C3—24no—
484C22×C3—24no—
485C22×C3—44no—
486C22×C3—42no—
487C22×C3—81no—
488C22×C3—81no—
489C22×C3—82no—
490C22×C3#39014yes—
491C22×C3C2×C4×D411yescharacteristic
492C22×C3C2×#12111yescharacteristic
493C22×C3C2×#12711yescharacteristic
494C22×C3#57612yes—
495C22×C3#57812yes—
496C2×D3—44no—
497C2×D3—42no—
498C2×D3—81no—
499C2×D3—81no—
500C2×D3—82no—
501C3×C4—42no—
502C3×C4—44no—
503C3×C4—82no—
504C3×C4#39014yes—
505C3×C4C22×C22⋊C411yescharacteristic
506C3×C4C2×C4×D411yescharacteristic
507C3×C4C2×#12111yescharacteristic
508C3×C4C2×#12411yescharacteristic
509Dic3—42no—
510Dic3—82no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—42no—
518C23—62no—
519C23—121no—
520C23—121no—
521C23—122no—
522C23—122no—
523C23C4×#6712yes—
524C23#93912yes—
525C23#94012yes—
526C23C22×#6711yescharacteristic
527Q8—22no—
528Q8—22no—
529Q8—41no—
530Q8—41no—
531Q8—62no—
532Q8—62no—
533Q8—121no—
534Q8—121no—
535D4—22no—
536D4—22no—
537D4—48no—
538D4—41no—
539D4—41no—
540D4—62no—
541D4—62no—
542D4—128no—
543D4—121no—
544D4—121no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—48no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—62no—
555C2×C4—84no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—128no—
559C2×C4—122no—
560C2×C4—124no—
561C2×C4—244no—
562C2×C4D3×C22⋊C414yes—
563C2×C4#67⋊C414yes—
564C2×C4#88414yes—
565C2×C4#88514yes—
566C2×C4C4×#6712yes—
567C2×C4#93912yes—
568C2×C4#94012yes—
569C2×C4C22×C4×D311yescharacteristic
570C2×C4C2×#23511yescharacteristic
571C2×C4C2×D3×D411yescharacteristic
572C2×C4C2×D3×D411yescharacteristic
573C2×C4C2×D3×D411yescharacteristic
574C2×C4C2×#23711yescharacteristic
575C8—34no—
576C8—34no—
577C8—64no—
578C2×C3—24no—
579C2×C3—82no—
580C2×C3C2×#39011yescharacteristic
581C2×C3#295112yes—
582C2×C3#306112yes—
583C2×C3#315511yescharacteristic
584C2×C3#315611yescharacteristic
585D3—82no—
586C22—24no—
587C22—22no—
588C22—22no—
589C22—24no—
590C22—44no—
591C22—42no—
592C22—81no—
593C22—81no—
594C22—82no—
595C22—124no—
596C22—122no—
597C22—241no—
598C22—241no—
599C22—242no—
600C22#495714yes—
601C22C2×C4×#6711yescharacteristic
602C22C2×#93911yescharacteristic
603C22C2×#94011yescharacteristic
604C22#583512yes—
605C22#583812yes—
606C4—42no—
607C4—44no—
608C4—82no—
609C4—122no—
610C4—242no—
611C4#495714yes—
612C4C2×D3×C22⋊C411yescharacteristic
613C4C2×#67⋊C411yescharacteristic
614C4C2×#88411yescharacteristic
615C4C2×#88511yescharacteristic
616C3#2123011yescharacteristic, complemented, Sylow
617C2—24no—
618C2—82no—
619C2—242no—
620C2C2×#495711yescharacteristic
621C2#8255712yes—
622C2#8338211yescharacteristic
623C2#8340311yescharacteristic
624C2#8342112yes—
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2311C2
socleC23×C3C22×D4
prime radicalC22×#950C2
cosocleC22×C3×C4C24
hypercenter#2311D3
hypocenterC3#21230
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. C3×#2311C2semidirect
3. C2×#5232C2semidirect
4. #82270C2semidirect
5. C2×#4997C2semidirect
6. C2×#4966C2semidirect
7. C22×#807C2semidirect
8. #69679C2semidirect
9. #69677C2semidirect
10. #69649C2semidirect
11. #69647C2semidirect
19. C2×#950C4semidirect
53. #2311D3semidirect
94. C2×#216C2×C4semidirect
97. C2×#215C2×C4semidirect
100. C2×#214C2×C4semidirect
103. C2×#213C2×C4semidirect
245. C2×C3⋊C8C22⋊C4semidirect
616. C3#21230semidirect

Maximal subgroups

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

subgroupindex
C22×#9502
C3×#23112
C2×#52322
#822702
C2×#49972
C2×#49662
C22×#8072
#696792
#696772
#696492
#696472
#212303