Finite Group Databasebeta

← #10818096

Subgroups — #10818096

Order 768.

840 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5204C211yesmaximal, characteristic
2#83965C211yesmaximal, characteristic
3#82571C211yesmaximal, characteristic
4#82561C211yesmaximal, characteristic
5#78948C211yesmaximal, characteristic, complemented
6C3×#1882C211yesmaximal, characteristic
7#70370C211yesmaximal, characteristic
8#70365C211yesmaximal, characteristic
9#70338C211yesmaximal, characteristic, complemented
10#69678C211yesmaximal, characteristic, complemented
11#69672C211yesmaximal, characteristic, complemented
12#68984C211yesmaximal, characteristic, complemented
13#68982C211yesmaximal, characteristic, complemented
14#68764C211yesmaximal, characteristic, complemented
15#68742C211yesmaximal, characteristic, complemented
16#23047—31nomaximal, Sylow
17C2×C3×#137C2211yescharacteristic
18C3×#520C2211yescharacteristic
19#5827C2211yescharacteristic
20C2×#951C2211yescharacteristic
21C2×#948C2211yescharacteristic
22#5812C2211yescharacteristic
23C22×#216C2211yescharacteristic
24#5516C2211yescharacteristic
25C2×C3×#104C2211yescharacteristic
26C3×#419C2211yescharacteristic
27C3×#391C2211yescharacteristic
28#5233C2211yescharacteristic
29C2×#836C2211yescharacteristic
30C2×#835C2211yescharacteristic
31#5204C418yescomplemented
32#5184—21no—
33Dic3×Q16—21no—
34#5164—21no—
35Dic3×SD16—21no—
36C2×#824C2211yescharacteristic
37C2×#822C2211yescharacteristic
38#5122C2211yescharacteristic
39#5122C2211yescharacteristic
40#5121C2211yescharacteristic
41#5120C2211yescharacteristic
42#5030—21no—
43C4×#216—21no—
44#5018—21no—
45C4×#214—21no—
46#4998—21no—
47#4998C2211yescharacteristic
48#4997C2211yescharacteristic
49#4995—21no—
50#4995C2211yescharacteristic
51C2×#810C2211yescharacteristic
52#4964—21no—
53#4964C2211yescharacteristic
54#4964C2211yescharacteristic
55C2×#808—21no—
56C2×#808C2211yescharacteristic
57#4929C2211yescharacteristic
58C2×#803C2211yescharacteristic
59#4790—21no—
60#4790—21no—
61#4789—21no—
62#4789—21no—
63C3×#343C2211yescharacteristic
64C3×#339C2211yescharacteristic
65#4558C2211yescharacteristic
66#4554C2211yescharacteristic
67#4523C2211yescharacteristic
68#4523C2211yescharacteristic
69C2×#473—31no—
70#2950—31no—
71#2947—31no—
72#2947—31no—
73#2912—31no—
74#1971—31no—
75#1951—31no—
76#1930—31no—
77#1930—31no—
78#1882—31no—
79#1882D311yescharacteristic
80#1857—31no—
81#1857—31no—
82#1773—31no—
83#1765—31no—
84C2×C3×#41C2311yescharacteristic
85C22×C3×Q8C2311yescharacteristic
86C22×Dic6C2311yescharacteristic
87C3×#137—22no—
88C3×#137C2×C414yes—
89C2×C3×Q16—21no—
90C2×C3×SD16—21no—
91C2×C3×M16C2311yescharacteristic
92C2×C3×M16C2311yescharacteristic
93C3×C4×Q8—21no—
94C3×C4×D4—21no—
95C3×C4×D4—41no—
96C3×#117—21no—
97C3×#117C2311yescharacteristic
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101#951—22no—
102#951C2×C414yes—
103#948—22no—
104#948C2×C414yes—
105Q8×Dic3—21no—
106Q8×Dic3—21no—
107Q8×Dic3—21no—
108Q8×Dic3—21no—
109Q8×Dic3—21no—
110C2×#216—21no—
111C2×#216—21no—
112C2×#216—21no—
113C2×#216—22no—
114C2×#216C2×C414yescomplemented
115D4×Dic3—21no—
116D4×Dic3—41no—
117C2×#214—21no—
118#926—21no—
119#926—41no—
120#926C2311yescharacteristic
121#926C2311yescharacteristic
122C2×C4⋊Dic3—21no—
123C2×C4⋊Dic3C2311yescharacteristic
124C2×#208—21no—
125C2×#208C2311yescharacteristic
126C2×#208C2311yescharacteristic
127C22×C3⋊C8—21no—
128C22×C3⋊C8C2311yescharacteristic
129C22×C3⋊C8C2311yescharacteristic
130#887—21no—
131#887—21no—
132#877—41no—
133C4×Dic6—21no—
134C4×Dic6—21no—
135C3×#104—22no—
136C3×#103—21no—
137C3×#103—21no—
138C3×#95—41no—
139#836C2×C414yescomplemented
140#835—21no—
141#835—21no—
142#835D411yescharacteristic
143#835D411yescharacteristic
144#832D411yescharacteristic
145#832D411yescharacteristic
146C3⋊#95—41no—
147#824C2×C414yescomplemented
148#822C2×C414yescomplemented
149#818—21no—
150#817—21no—
151#816—21no—
152#816—21no—
153#815—21no—
154#815—21no—
155C8×Dic3—21no—
156C8×Dic3—21no—
157#810—21no—
158#810—21no—
159#808—21no—
160#808—21no—
161#808—21no—
162#808—21no—
163#808—21no—
164#808D411yescharacteristic
165#808D411yescharacteristic
166#807—21no—
167#807—21no—
168#807—21no—
169#807D411yescharacteristic
170#807D411yescharacteristic
171#803—21no—
172#803—22no—
173C4×C3⋊C8—21no—
174C2×#137—31no—
175C2×#137C2×D311yescharacteristic
176C22×Q16—31no—
177C2×#131—31no—
178#520—31no—
179#520C2×D311yescharacteristic
180#519—31no—
181#519—31no—
182#473—38no—
183#441—61no—
184#441—61no—
185#441—61no—
186#441—61no—
187#440—61no—
188#440—61no—
189C4×Q16—61no—
190C4×Q16—61no—
191C4×Q16—61no—
192C4×Q16—61no—
193C4×SD16—61no—
194C4×SD16—61no—
195C2×#108—31no—
196#428—31no—
197#428—31no—
198#428—31no—
199#428—31no—
200#428—61no—
201#428—61no—
202#427—31no—
203#427—61no—
204C2×#106—31no—
205C2×#106—61no—
206C2×#104C2×D311yescharacteristic
207#419—31no—
208#419C2×D311yescharacteristic
209#416—31no—
210C2×#103—31no—
211C2×#103—31no—
212C2×#101—31no—
213C2×#101—31no—
214#405—31no—
215#405—31no—
216#405—31no—
217C2×#97—31no—
218#391—31no—
219#391C2×D311yescharacteristic
220#343—31no—
221#343C2×D311yescharacteristic
222#341—31no—
223#339—31no—
224#339—31no—
225#339C2×D311yescharacteristic
226C3×#41—21no—
227C3×#41—21no—
228C3×#41C22×C412yes—
229C3×#41#4112yes—
230C2×C3×Q8—21no—
231C2×C3×Q8—21no—
232C2×C3×Q8—21no—
233C2×C3×Q8—21no—
234C2×C3×Q8—21no—
235C2×C3×Q8C22×C412yes—
236C2×C3×Q8C2×D411yescharacteristic
237C2×C3×D4—21no—
238C2×C3×D4C2×D411yescharacteristic
239C22×C3×C4—21no—
240C22×C3×C4—21no—
241C22×C3×C4—21no—
242C22×C3×C4—21no—
243C22×C3×C4—41no—
244C22×C3×C4C2411yescharacteristic
245C22×Dic3—21no—
246C22×Dic3—21no—
247C22×Dic3—41no—
248C2×Dic6—21no—
249C2×Dic6—21no—
250C2×Dic6—21no—
251C2×Dic6—21no—
252C2×Dic6—21no—
253C2×Dic6C22×C412yes—
254C3×Q16—24no—
255C3×SD16—24no—
256C3×M16—21no—
257C3×M16—21no—
258C3×M16—21no—
259C3×M16C22×C412yes—
260C2×C3×C8—21no—
261C2×C3×C8—21no—
262C3×C4⋊C4—21no—
263C3×C4⋊C4—21no—
264C3×C4⋊C4—41no—
265C3×C4⋊C4C2×D411yescharacteristic
266C3×C4⋊C4C2×D411yescharacteristic
267C3×C22⋊C4—21no—
268C3×C22⋊C4—41no—
269C3×C42—21no—
270C3×C42—21no—
271C3×C42—21no—
272C3×C42—41no—
273#217—21no—
274#217—21no—
275#217—41no—
276#216—24no—
277#216—24no—
278#216—24no—
279#214—24no—
280C4⋊Dic3—21no—
281C4⋊Dic3—21no—
282C4⋊Dic3—41no—
283C4⋊Dic3—41no—
284C4⋊Dic3—41no—
285C4⋊Dic3C2×D411yescharacteristic
286C4⋊Dic3C2×D411yescharacteristic
287C4⋊Dic3C2×D411yescharacteristic
288C4⋊Dic3C2×D411yescharacteristic
289Dic3⋊C4—41no—
290Dic3⋊C4—41no—
291C4×Dic3—21no—
292C4×Dic3—21no—
293C4×Dic3—41no—
294C4×Dic3—41no—
295C4×Dic3—41no—
296C4×Dic3—41no—
297C4×Dic3—41no—
298#208—21no—
299#208—21no—
300#208—24no—
301#208C22×C412yes—
302#208C22×C412yes—
303C2×C3⋊C8—21no—
304C2×C3⋊C8—21no—
305C2×C3⋊C8—21no—
306C2×C3⋊C8—21no—
307C2×C3⋊C8—24no—
308C2×C3⋊C8C22×C412yes—
309C2×C3⋊C8C2×D411yescharacteristic
310C2×C3⋊C8C2×D411yescharacteristic
311C2×C3⋊C8C2×D411yescharacteristic
312C2×C3⋊C8C2×D411yescharacteristic
313C2×C3⋊C8#4112yes—
314C2×#41C22×D311yescharacteristic
315C22×Q8—31no—
316C22×Q8C22×D311yescharacteristic
317#137—22no—
318#137—34no—
319#137—62no—
320#137C4×D314yes—
321C2×Q16—21no—
322C2×Q16—34no—
323C2×Q16—62no—
324C2×Q16—61no—
325C2×Q16—61no—
326C2×Q16—61no—
327C2×SD16—21no—
328C2×SD16—61no—
329#131—34no—
330#131—62no—
331C2×M16—31no—
332C2×M16—31no—
333C2×M16—61no—
334C2×M16C22×D311yescharacteristic
335C2×M16C22×D311yescharacteristic
336C22×C8—31no—
337C22×C8—31no—
338C22×C8—61no—
339C4×Q8—21no—
340C4×Q8—61no—
341C4×Q8—61no—
342C4×Q8—61no—
343C4×Q8—61no—
344C4×Q8—61no—
345C4×Q8—61no—
346C4×Q8—61no—
347C4×Q8—61no—
348C4×Q8—61no—
349C4×D4—21no—
350C4×D4—41no—
351C4×D4—61no—
352C4×D4—121no—
353#117—21no—
354#117—31no—
355#117—31no—
356#117—61no—
357#117—121no—
358#117—121no—
359#117C22×D311yescharacteristic
360C2×C4⋊C4—21no—
361C2×C4⋊C4—31no—
362C2×C4⋊C4—61no—
363C2×C4⋊C4C22×D311yescharacteristic
364C2×C42C22×D311yescharacteristic
365#108—34no—
366#107—31no—
367#107—31no—
368#107—61no—
369#107—61no—
370#107—61no—
371#107—61no—
372#106—31no—
373#106—31no—
374#106—61no—
375#106—61no—
376#106—61no—
377#106—61no—
378#106—61no—
379#106—61no—
380#104—22no—
381#103—21no—
382#103—21no—
383#103—31no—
384#103—31no—
385#103—61no—
386#103—61no—
387#103—61no—
388#103—61no—
389#103—61no—
390#103—61no—
391#102—31no—
392#102—31no—
393#101—34no—
394#101—34no—
395#97—62no—
396#97—61no—
397#97—61no—
398#97—61no—
399C4×C8—61no—
400C4×C8—61no—
401C4×C8—61no—
402#95—41no—
403#95—121no—
404C23×C3—21no—
405C23×C3H(5,𝔽2)11yescharacteristic
406C3×Q8—21no—
407C3×Q8—22no—
408C3×Q8—22no—
409C3×Q8—41no—
410C3×Q8—41no—
411C3×Q8C4×D412yes—
412C3×D4—22no—
413C3×D4—22no—
414C3×D4—21no—
415C3×D4C4×D412yes—
416C2×C3×C4—21no—
417C2×C3×C4—22no—
418C2×C3×C4—22no—
419C2×C3×C4—22no—
420C2×C3×C4—21no—
421C2×C3×C4—21no—
422C2×C3×C4—41no—
423C2×C3×C4—41no—
424C2×C3×C4—41no—
425C2×C3×C4—41no—
426C2×C3×C4—41no—
427C2×C3×C4—41no—
428C2×C3×C4—81no—
429C2×C3×C4—161no—
430C2×C3×C4C23×C411yescharacteristic
431C2×C3×C4C22×D411yescharacteristic
432C2×C3×C4C22×D411yescharacteristic
433C2×C3×C4C2×#4111yescharacteristic
434C2×C3×C4C2×#4111yescharacteristic
435C2×C3×C42−1+411yescharacteristic
436C2×Dic3—22no—
437C2×Dic3—21no—
438C2×Dic3—21no—
439C2×Dic3—21no—
440C2×Dic3—21no—
441C2×Dic3—41no—
442C2×Dic3—41no—
443C2×Dic3—41no—
444C2×Dic3—81no—
445C2×Dic3—161no—
446Dic6—22no—
447Dic6—22no—
448Dic6—41no—
449Dic6—41no—
450C3×C8—22no—
451C3×C8—41no—
452C3⋊C8—22no—
453C3⋊C8—22no—
454C3⋊C8—21no—
455C3⋊C8—21no—
456C3⋊C8C4×D412yes—
457C3⋊C8C4×D412yes—
458#41—21no—
459#41—21no—
460#41C2×C4×D312yes—
461#41#23712yes—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—32no—
468C2×Q8—61no—
469C2×Q8—61no—
470C2×Q8—61no—
471C2×Q8—61no—
472C2×Q8—61no—
473C2×Q8C2×C4×D312yes—
474C2×Q8D3×D411yescharacteristic
475C2×D4—21no—
476C2×D4D3×D411yescharacteristic
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—41no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4—121no—
485C22×C4C23×D311yescharacteristic
486Q16—24no—
487Q16—64no—
488Q16—64no—
489Q16—64no—
490SD16—24no—
491SD16—64no—
492M16—21no—
493M16—21no—
494M16—21no—
495M16—32no—
496M16—32no—
497M16—64no—
498M16—61no—
499M16—61no—
500M16C2×C4×D312yes—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—32no—
504C2×C8—31no—
505C2×C8—31no—
506C2×C8—32no—
507C2×C8—31no—
508C2×C8—31no—
509C2×C8—64no—
510C2×C8—61no—
511C2×C8—61no—
512C2×C8—61no—
513C2×C8—61no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—31no—
517C4⋊C4—31no—
518C4⋊C4—31no—
519C4⋊C4—31no—
520C4⋊C4—41no—
521C4⋊C4—61no—
522C4⋊C4—61no—
523C4⋊C4—121no—
524C4⋊C4—121no—
525C4⋊C4—121no—
526C4⋊C4—121no—
527C4⋊C4—121no—
528C4⋊C4D3×D411yescharacteristic
529C4⋊C4D3×D411yescharacteristic
530C22⋊C4—21no—
531C22⋊C4—41no—
532C22⋊C4—61no—
533C22⋊C4—61no—
534C22⋊C4—121no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—41no—
539C42—61no—
540C42—61no—
541C42—121no—
542C42—121no—
543C42—121no—
544C42—121no—
545C42—121no—
546C22×C3—21no—
547C22×C3—21no—
548C22×C3—22no—
549C22×C3—41no—
550C22×C3—41no—
551C22×C3#52011yescharacteristic
552C22×C3D4211yescharacteristic
553C22×C3#55311yescharacteristic
554C3×C4—22no—
555C3×C4—42no—
556C3×C4—41no—
557C3×C4—41no—
558C3×C4—81no—
559C3×C4—81no—
560C3×C4C2×C4×D411yescharacteristic
561C3×C4C2×C4×D411yescharacteristic
562C3×C4#54711yescharacteristic
563C3×C4#54811yescharacteristic
564Dic3—42no—
565Dic3—41no—
566Dic3—41no—
567Dic3—41no—
568Dic3—41no—
569Dic3—81no—
570C23—21no—
571C23#100411yescharacteristic
572Q8—21no—
573Q8—22no—
574Q8—22no—
575Q8—41no—
576Q8—41no—
577Q8—62no—
578Q8—62no—
579Q8—121no—
580Q8—121no—
581Q8#67⋊C412yes—
582D4—22no—
583D4—22no—
584D4—21no—
585D4#67⋊C412yes—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—62no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—81no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—161no—
608C2×C4—241no—
609C2×C4—481no—
610C2×C4C22×C4×D311yescharacteristic
611C2×C4C2×D3×D411yescharacteristic
612C2×C4C2×D3×D411yescharacteristic
613C2×C4C2×#23711yescharacteristic
614C2×C4D3×#4111yescharacteristic
615C2×C4#101011yescharacteristic
616C8—22no—
617C8—32no—
618C8—32no—
619C8—41no—
620C8—61no—
621C8—61no—
622C8—62no—
623C8—62no—
624C2×C3—21no—
625C2×C3—21no—
626C2×C3—42no—
627C2×C3#236111yescharacteristic
628C2×C3#328911yescharacteristic
629C2×C3#328911yescharacteristic
630C22—21no—
631C22—21no—
632C22—22no—
633C22—41no—
634C22—41no—
635C22#554611yescharacteristic
636C22#559011yescharacteristic
637C22#560111yescharacteristic
638C4—22no—
639C4—41no—
640C4—41no—
641C4—42no—
642C4—81no—
643C4—81no—
644C4—121no—
645C4—121no—
646C4—121no—
647C4—121no—
648C4—122no—
649C4—241no—
650C4C2×#67⋊C411yescharacteristic
651C4C4×D3×D411yescharacteristic
652C4#559811yescharacteristic
653C4#565511yescharacteristic
654C3#2304711yescharacteristic, complemented, Sylow
655C2—21no—
656C2—21no—
657C2—42no—
658C2#7467611yescharacteristic
659C2#8029511yescharacteristic
660C2#8030111yescharacteristic
subgroupG/H
centerC22#5590
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1882C2
socleC22×C3D42
prime radicalC2×C3×#41C23
cosocleC22×C3×C4C24
hypercenter#1882D3
hypocenterC3#23047
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1882

Complements

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

NH ≅ G/Ntype
5. #78948C2semidirect
9. #70338C2semidirect
10. #69678C2semidirect
11. #69672C2semidirect
12. #68984C2semidirect
13. #68982C2semidirect
14. #68764C2semidirect
15. #68742C2semidirect
31. #5204C4semidirect
114. C2×#216C2×C4semidirect
139. #836C2×C4semidirect
147. #824C2×C4semidirect
148. #822C2×C4semidirect
654. C3#23047semidirect

Maximal subgroups

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

subgroupindex
C2×#52042
#839652
#825712
#825612
#789482
C3×#18822
#703702
#703652
#703382
#696782
#696722
#689842
#689822
#687642
#687422
#230473