Finite Group Databasebeta

← #10818118

Subgroups — #10818118

Order 768.

866 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5203C211yesmaximal, characteristic
2#83965C211yesmaximal, characteristic, complemented
3#82564C211yesmaximal, characteristic
4#82562C211yesmaximal, characteristic, complemented
5#78947C211yesmaximal, characteristic, complemented
6C3×#1882C211yesmaximal, characteristic, complemented
7#70370C211yesmaximal, characteristic, complemented
8#70366C211yesmaximal, characteristic
9#70338C211yesmaximal, characteristic, complemented
10#69679C211yesmaximal, characteristic, complemented
11#69672C211yesmaximal, characteristic, complemented
12#68985C211yesmaximal, characteristic, complemented
13#68983C211yesmaximal, characteristic, complemented
14#68758C211yesmaximal, characteristic, complemented
15#68742C211yesmaximal, characteristic, complemented
16#23049—31nomaximal, Sylow
17C2×C3×#137C2211yescharacteristic
18C3×#520C2211yescharacteristic
19#5827C2211yescharacteristic
20C2×#949C2211yescharacteristic
21C2×#948C2211yescharacteristic
22#5812C2211yescharacteristic
23C22×#215C2211yescharacteristic
24#5521C2211yescharacteristic
25C2×C3×#104C2211yescharacteristic
26C3×#419C2211yescharacteristic
27C3×#391C2211yescharacteristic
28#5233C2211yescharacteristic
29C2×#836C2211yescharacteristic
30C2×#835C2211yescharacteristic
31#5203C418yescomplemented
32#5184—21no—
33Dic3×Q16—21no—
34#5164—21no—
35Dic3×SD16—21no—
36#5131C2211yescharacteristic
37C2×#823C2211yescharacteristic
38C2×#822C2211yescharacteristic
39#5122C2211yescharacteristic
40#5122C2211yescharacteristic
41#5121C2211yescharacteristic
42#5026—21no—
43C4×#215—21no—
44#5014—21no—
45C4×#213—21no—
46#4998—21no—
47#4998C2211yescharacteristic
48#4997C2211yescharacteristic
49#4995—21no—
50#4995C2211yescharacteristic
51C2×#809C2211yescharacteristic
52#4964—21no—
53#4964C2211yescharacteristic
54#4964C2211yescharacteristic
55C2×#807—21no—
56C2×#807C2211yescharacteristic
57#4939C2211yescharacteristic
58C2×#803C2211yescharacteristic
59#4817—21no—
60#4817—21no—
61#4788—21no—
62#4788—21no—
63C3×#343C2211yescharacteristic
64C3×#339C2211yescharacteristic
65#4558C2211yescharacteristic
66#4554C2211yescharacteristic
67#4523C2211yescharacteristic
68#4523C2211yescharacteristic
69C2×#471—31no—
70#2949—31no—
71#2947—31no—
72#2946—31no—
73#2913—31no—
74#1973—31no—
75#1952—31no—
76#1931—31no—
77#1930—31no—
78#1883—31no—
79#1882D311yescharacteristic, complemented
80#1857—31no—
81#1856—31no—
82#1773—31no—
83#1765—31no—
84C2×C3×#41C2311yescharacteristic
85C22×C3×Q8C2311yescharacteristic
86C22×D12C2311yescharacteristic
87C3×#137—22no—
88C3×#137C2×C414yescomplemented
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#949—22no—
102#949C2×C414yes—
103#948—22no—
104#948C2×C414yescomplemented
105Q8×Dic3—21no—
106Q8×Dic3—21no—
107Q8×Dic3—21no—
108Q8×Dic3—21no—
109Q8×Dic3—21no—
110C2×#215—21no—
111C2×#215—21no—
112C2×#215—21no—
113C2×#215—22no—
114C2×#215C2×C414yescomplemented
115D4×Dic3—21no—
116D4×Dic3—41no—
117C2×#213—21no—
118C2×(C2×D3)⋊C4—41no—
119#926—21no—
120#926—41no—
121#926C2311yescharacteristic
122#926C2311yescharacteristic
123C2×C4⋊Dic3—21no—
124C2×C4⋊Dic3C2311yescharacteristic
125C2×#208—21no—
126C2×#208C2311yescharacteristic
127C2×#208C2311yescharacteristic
128C22×C3⋊C8—21no—
129C22×C3⋊C8C2311yescharacteristic
130C22×C3⋊C8C2311yescharacteristic
131#893—21no—
132#893—21no—
133D3×C22⋊C4—41no—
134C4×D12—21no—
135C4×D12—21no—
136C3×#104—22no—
137C3×#103—21no—
138C3×#103—21no—
139C3×#95—41no—
140#836C2×C414yescomplemented
141#835—21no—
142#835—21no—
143#835D411yescharacteristic
144#835D411yescharacteristic
145#832D411yescharacteristic
146#832D411yescharacteristic
147#823C2×C414yescomplemented
148#822C2×C414yescomplemented
149#821—21no—
150#821—21no—
151#818—21no—
152#817—21no—
153#815—21no—
154#815—21no—
155C8×Dic3—21no—
156C8×Dic3—21no—
157#809—21no—
158#809—21no—
159#808—21no—
160#808—21no—
161#808—21no—
162#808D411yescharacteristic
163#808D411yescharacteristic
164#807—21no—
165#807—21no—
166#807—21no—
167#807—21no—
168#807—21no—
169#807D411yescharacteristic
170#807D411yescharacteristic
171#803—21no—
172#803—22no—
173C4×C3⋊C8—21no—
174C2×#137C2×D311yescharacteristic
175C2×AGL(1,ℤ/8)—31no—
176C22×SD16—31no—
177C2×#131—31no—
178#520—31no—
179#520C2×D311yescharacteristic
180#519—31no—
181#518—31no—
182#471—38no—
183#442—61no—
184#441—61no—
185#440—61no—
186#440—61no—
187#440—61no—
188#440—61no—
189C4×Q16—61no—
190C4×SD16—61no—
191C4×SD16—61no—
192C4×SD16—61no—
193C4×SD16—61no—
194C4×D8—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×#107—31no—
205C2×#107—61no—
206C2×#104C2×D311yescharacteristic
207#419C2×D311yescharacteristic
208#418—31no—
209#416—31no—
210C2×#103—31no—
211C2×#102—31no—
212C2×#101—31no—
213C2×#100—31no—
214#405—31no—
215#405—31no—
216#405—31no—
217C2×#97—31no—
218#391C2×D311yescharacteristic
219#390—31no—
220#343—31no—
221#343C2×D311yescharacteristic
222#341—31no—
223#339—31no—
224#339—31no—
225#339C2×D311yescharacteristic
226C23×D3—21no—
227C3×#41—21no—
228C3×#41—21no—
229C3×#41C22×C412yes—
230C3×#41#4112yes—
231C2×C3×Q8—21no—
232C2×C3×Q8—21no—
233C2×C3×Q8—21no—
234C2×C3×Q8—21no—
235C2×C3×Q8—21no—
236C2×C3×Q8C22×C412yes—
237C2×C3×Q8C2×D411yescharacteristic
238C2×C3×D4—21no—
239C2×C3×D4C2×D411yescharacteristic
240C22×C3×C4—21no—
241C22×C3×C4—21no—
242C22×C3×C4—21no—
243C22×C3×C4—21no—
244C22×C3×C4—41no—
245C22×C3×C4C2411yescharacteristic
246C22×Dic3—21no—
247C22×Dic3—41no—
248C2×D12—21no—
249C2×D12—21no—
250C2×D12—21no—
251C2×D12—21no—
252C2×D12—21no—
253C2×D12C22×C412yes—
254C2×C4×D3—41no—
255C2×C4×D3—41no—
256C3×Q16—24no—
257C3×SD16—24no—
258C3×M16—21no—
259C3×M16—21no—
260C3×M16—21no—
261C3×M16C22×C412yes—
262C2×C3×C8—21no—
263C2×C3×C8—21no—
264C3×C4⋊C4—21no—
265C3×C4⋊C4—21no—
266C3×C4⋊C4—41no—
267C3×C4⋊C4C2×D411yescharacteristic
268C3×C4⋊C4C2×D411yescharacteristic
269C3×C22⋊C4—21no—
270C3×C22⋊C4—41no—
271C3×C42—21no—
272C3×C42—21no—
273C3×C42—21no—
274C3×C42—41no—
275#217—21no—
276#217—21no—
277#217—41no—
278#215—24no—
279#215—24no—
280#215—24no—
281#213—24no—
282(C2×D3)⋊C4—41no—
283(C2×D3)⋊C4—41no—
284(C2×D3)⋊C4—82no—
285C4⋊Dic3—21no—
286C4⋊Dic3—21no—
287C4⋊Dic3—41no—
288C4⋊Dic3—41no—
289C4⋊Dic3—41no—
290C4⋊Dic3C2×D411yescharacteristic
291C4⋊Dic3C2×D411yescharacteristic
292C4⋊Dic3C2×D411yescharacteristic
293C4⋊Dic3C2×D411yescharacteristic
294C4×Dic3—21no—
295C4×Dic3—21no—
296C4×Dic3—41no—
297C4×Dic3—41no—
298C4×Dic3—41no—
299#208—21no—
300#208—21no—
301#208—24no—
302#208C22×C412yes—
303#208C22×C412yes—
304C2×C3⋊C8—21no—
305C2×C3⋊C8—21no—
306C2×C3⋊C8—21no—
307C2×C3⋊C8—21no—
308C2×C3⋊C8—24no—
309C2×C3⋊C8C22×C412yes—
310C2×C3⋊C8C2×D411yescharacteristic
311C2×C3⋊C8C2×D411yescharacteristic
312C2×C3⋊C8C2×D411yescharacteristic
313C2×C3⋊C8C2×D411yescharacteristic
314C2×C3⋊C8#4112yes—
315C2×#41C22×D311yescharacteristic
316C22×Q8C22×D311yescharacteristic
317C22×D4—31no—
318#137—22no—
319#137C4×D314yescomplemented
320AGL(1,ℤ/8)—34no—
321AGL(1,ℤ/8)—62no—
322C2×Q16—21no—
323C2×SD16—21no—
324C2×SD16—34no—
325C2×SD16—62no—
326C2×SD16—61no—
327C2×SD16—61no—
328C2×SD16—61no—
329C2×D8—61no—
330#131—34no—
331#131—62no—
332C2×M16—31no—
333C2×M16—31no—
334C2×M16—61no—
335C2×M16C22×D311yescharacteristic
336C2×M16C22×D311yescharacteristic
337C22×C8—31no—
338C22×C8—31no—
339C22×C8—61no—
340C4×Q8—21no—
341C4×Q8—61no—
342C4×Q8—61no—
343C4×Q8—61no—
344C4×Q8—61no—
345C4×Q8—61no—
346C4×D4—21no—
347C4×D4—41no—
348C4×D4—61no—
349C4×D4—61no—
350C4×D4—61no—
351C4×D4—61no—
352C4×D4—61no—
353C4×D4—121no—
354#117—21no—
355#117—31no—
356#117—31no—
357#117—61no—
358#117—121no—
359#117C22×D311yescharacteristic
360C2×C4⋊C4—21no—
361C2×C4⋊C4—31no—
362C2×C4⋊C4—61no—
363C2×C4⋊C4C22×D311yescharacteristic
364C2×C22⋊C4—121no—
365C2×C22⋊C4—121no—
366C2×C42C22×D311yescharacteristic
367#108—34no—
368#107—31no—
369#107—31no—
370#107—61no—
371#107—61no—
372#107—61no—
373#107—61no—
374#107—61no—
375#107—61no—
376#106—31no—
377#106—31no—
378#106—61no—
379#106—61no—
380#106—61no—
381#106—61no—
382#104—22no—
383#103—21no—
384#103—21no—
385#103—31no—
386#103—31no—
387#103—61no—
388#103—61no—
389#102—31no—
390#102—31no—
391#102—61no—
392#102—61no—
393#102—61no—
394#102—61no—
395#101—34no—
396#100—34no—
397#97—62no—
398#97—61no—
399#97—61no—
400#97—61no—
401C4×C8—61no—
402C4×C8—61no—
403C4×C8—61no—
404#95—41no—
405C23×C3—21no—
406C23×C3H(5,𝔽2)11yescharacteristic
407C22×D3—22no—
408C22×D3—41no—
409C22×D3—41no—
410C22×D3—81no—
411C22×D3—81no—
412C3×Q8—21no—
413C3×Q8—22no—
414C3×Q8—22no—
415C3×Q8—41no—
416C3×Q8—41no—
417C3×Q8C4×D412yes—
418C3×D4—22no—
419C3×D4—22no—
420C3×D4—21no—
421C3×D4C4×D412yes—
422C2×C3×C4—21no—
423C2×C3×C4—22no—
424C2×C3×C4—22no—
425C2×C3×C4—22no—
426C2×C3×C4—21no—
427C2×C3×C4—21no—
428C2×C3×C4—41no—
429C2×C3×C4—41no—
430C2×C3×C4—41no—
431C2×C3×C4—41no—
432C2×C3×C4—41no—
433C2×C3×C4—41no—
434C2×C3×C4—81no—
435C2×C3×C4—161no—
436C2×C3×C4C23×C411yescharacteristic
437C2×C3×C4C22×D411yescharacteristic
438C2×C3×C4C22×D411yescharacteristic
439C2×C3×C4C2×#4111yescharacteristic
440C2×C3×C4C2×#4111yescharacteristic
441C2×C3×C42−1+411yescharacteristic
442C2×Dic3—21no—
443C2×Dic3—21no—
444C2×Dic3—21no—
445C2×Dic3—21no—
446C2×Dic3—41no—
447C2×Dic3—81no—
448C2×Dic3—161no—
449D12—22no—
450D12—22no—
451D12—41no—
452D12—41no—
453C4×D3—161no—
454C4×D3—161no—
455C3×C8—22no—
456C3×C8—41no—
457C3⋊C8—22no—
458C3⋊C8—22no—
459C3⋊C8—21no—
460C3⋊C8—21no—
461C3⋊C8C4×D412yes—
462C3⋊C8C4×D412yes—
463C24—61no—
464#41—21no—
465#41—21no—
466#41C2×C4×D312yes—
467#41#23712yes—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8C2×C4×D312yes—
474C2×Q8D3×D411yescharacteristic
475C2×D4—21no—
476C2×D4—32no—
477C2×D4—61no—
478C2×D4—61no—
479C2×D4—61no—
480C2×D4—61no—
481C2×D4—61no—
482C2×D4D3×D411yescharacteristic
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—41no—
488C22×C4—61no—
489C22×C4—121no—
490C22×C4—121no—
491C22×C4—121no—
492C22×C4C23×D311yescharacteristic
493Q16—24no—
494SD16—24no—
495SD16—64no—
496SD16—64no—
497SD16—64no—
498D8—64no—
499M16—21no—
500M16—21no—
501M16—21no—
502M16—32no—
503M16—32no—
504M16—64no—
505M16—61no—
506M16—61no—
507M16C2×C4×D312yes—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—32no—
511C2×C8—31no—
512C2×C8—31no—
513C2×C8—32no—
514C2×C8—31no—
515C2×C8—31no—
516C2×C8—64no—
517C2×C8—61no—
518C2×C8—61no—
519C2×C8—61no—
520C2×C8—61no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—31no—
524C4⋊C4—31no—
525C4⋊C4—31no—
526C4⋊C4—31no—
527C4⋊C4—41no—
528C4⋊C4—61no—
529C4⋊C4—61no—
530C4⋊C4—121no—
531C4⋊C4—121no—
532C4⋊C4—121no—
533C4⋊C4D3×D411yescharacteristic
534C4⋊C4D3×D411yescharacteristic
535C22⋊C4—21no—
536C22⋊C4—41no—
537C22⋊C4—61no—
538C22⋊C4—61no—
539C22⋊C4—121no—
540C22⋊C4—121no—
541C22⋊C4—121no—
542C22⋊C4—242no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—41no—
547C42—61no—
548C42—61no—
549C42—121no—
550C42—121no—
551C42—121no—
552C22×C3—21no—
553C22×C3—21no—
554C22×C3—22no—
555C22×C3—41no—
556C22×C3—41no—
557C22×C3#52011yescharacteristic
558C22×C3D4211yescharacteristic
559C22×C3#55311yescharacteristic
560C2×D3—42no—
561C2×D3—81no—
562C2×D3—81no—
563C2×D3—82no—
564C2×D3—161no—
565C3×C4—22no—
566C3×C4—42no—
567C3×C4—41no—
568C3×C4—41no—
569C3×C4—81no—
570C3×C4—81no—
571C3×C4C2×C4×D411yescharacteristic
572C3×C4C2×C4×D411yescharacteristic
573C3×C4#54711yescharacteristic
574C3×C4#54811yescharacteristic
575Dic3—41no—
576Dic3—41no—
577Dic3—41no—
578Dic3—41no—
579Dic3—81no—
580C23—21no—
581C23—62no—
582C23—121no—
583C23—121no—
584C23—241no—
585C23—241no—
586C23#100411yescharacteristic
587Q8—21no—
588Q8—22no—
589Q8—22no—
590Q8—41no—
591Q8—41no—
592Q8#67⋊C412yes—
593D4—22no—
594D4—22no—
595D4—21no—
596D4—62no—
597D4—62no—
598D4—121no—
599D4—121no—
600D4#67⋊C412yes—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—81no—
618C2×C4—121no—
619C2×C4—161no—
620C2×C4—241no—
621C2×C4—481no—
622C2×C4—481no—
623C2×C4—481no—
624C2×C4C22×C4×D311yescharacteristic
625C2×C4C2×D3×D411yescharacteristic
626C2×C4C2×D3×D411yescharacteristic
627C2×C4C2×#23711yescharacteristic
628C2×C4D3×#4111yescharacteristic
629C2×C4#101011yescharacteristic
630C8—22no—
631C8—32no—
632C8—32no—
633C8—41no—
634C8—61no—
635C8—61no—
636C8—62no—
637C8—62no—
638C2×C3—21no—
639C2×C3—21no—
640C2×C3—42no—
641C2×C3#236111yescharacteristic
642C2×C3#328811yescharacteristic
643C2×C3#328911yescharacteristic
644D3—82no—
645C22—21no—
646C22—21no—
647C22—22no—
648C22—41no—
649C22—41no—
650C22—122no—
651C22—242no—
652C22—241no—
653C22—241no—
654C22—481no—
655C22#554611yescharacteristic
656C22#559011yescharacteristic
657C22#560111yescharacteristic
658C4—22no—
659C4—41no—
660C4—41no—
661C4—42no—
662C4—81no—
663C4—81no—
664C4—121no—
665C4—121no—
666C4—121no—
667C4—121no—
668C4—241no—
669C4C2×#67⋊C411yescharacteristic
670C4C4×D3×D411yescharacteristic
671C4#559811yescharacteristic
672C4#565511yescharacteristic
673C3#2304911yescharacteristic, complemented, Sylow
674C2—21no—
675C2—21no—
676C2—42no—
677C2—242no—
678C2#7467611yescharacteristic
679C2#8029411yescharacteristic
680C2#8030111yescharacteristic
subgroupG/H
centerC22#5590
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1882C2
socleC22×C3D42
prime radicalC2×#949C22
cosocleC22×C3×C4C24
hypercenter#1882D3
hypocenterC3#23049
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
2. #83965C2semidirect
4. #82562C2semidirect
5. #78947C2semidirect
6. C3×#1882C2semidirect
7. #70370C2semidirect
9. #70338C2semidirect
10. #69679C2semidirect
11. #69672C2semidirect
12. #68985C2semidirect
13. #68983C2semidirect
14. #68758C2semidirect
15. #68742C2semidirect
31. #5203C4semidirect
79. #1882D3semidirect
88. C3×#137C2×C4semidirect
104. #948C2×C4semidirect
114. C2×#215C2×C4semidirect
140. #836C2×C4semidirect
147. #823C2×C4semidirect
148. #822C2×C4semidirect
319. #137C4×D3semidirect
673. C3#23049semidirect

Maximal subgroups

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

subgroupindex
C2×#52032
#839652
#825642
#825622
#789472
C3×#18822
#703702
#703662
#703382
#696792
#696722
#689852
#689832
#687582
#687422
#230493