Finite Group Databasebeta

← #10818129

Subgroups — #10818129

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×C3×#471C211yesmaximal, characteristic
2#83965C211yesmaximal, characteristic, complemented
3#83930C211yesmaximal, characteristic
4#83843C211yesmaximal, characteristic, complemented
5#83785C211yesmaximal, characteristic, complemented
6#72100C211yesmaximal, characteristic, complemented
7#72088C211yesmaximal, characteristic
8#72050C211yesmaximal, characteristic, complemented
9#72009C211yesmaximal, characteristic, complemented
10#70703C211yesmaximal, characteristic, complemented
11#70701C211yesmaximal, characteristic, complemented
12#70106C211yesmaximal, characteristic, complemented
13#70102C211yesmaximal, characteristic, complemented
14#70072C211yesmaximal, characteristic, complemented
15#70068C211yesmaximal, characteristic, complemented
16#23049—31nomaximal, Sylow
17C2×C3×#137C2211yescharacteristic
18C2×C3×AGL(1,ℤ/8)C2211yescharacteristic
19C22×C3×SD16C2211yescharacteristic
20C2×C3×#131C2211yescharacteristic
21#5827C2211yescharacteristic
22#5827C2211yescharacteristic
23#5812C2211yescharacteristic
24#5798C2211yescharacteristic
25C3×#471C418yescomplemented
26C2×C3×#108C2211yescharacteristic
27C2×C3×#101C2211yescharacteristic
28C2×C3×#100C2211yescharacteristic
29C2×#837C2211yescharacteristic
30#5233C2211yescharacteristic
31#5229C2211yescharacteristic
32#5225C2211yescharacteristic
33C2×#835C2211yescharacteristic
34#5221C2211yescharacteristic
35#5215C2211yescharacteristic
36C2×#832C2211yescharacteristic
37#5184—21no—
38Dic3×Q16—21no—
39#5164—21no—
40#5164—21no—
41#5164—21no—
42#5164—21no—
43Dic3×SD16—21no—
44Dic3×SD16—21no—
45Dic3×SD16—21no—
46Dic3×SD16—21no—
47#5152—21no—
48Dic3×D8—21no—
49#5122C2211yescharacteristic
50#5122C2211yescharacteristic
51#5121—21no—
52#5121—21no—
53#5121C2211yescharacteristic
54#5121C2211yescharacteristic
55#5121C2211yescharacteristic
56#5121C2211yescharacteristic
57#5106—21no—
58#5106C2211yescharacteristic
59C2×#818—21no—
60C2×#818C2211yescharacteristic
61#5101C2211yescharacteristic
62C2×#815C2211yescharacteristic
63#4558C2211yescharacteristic
64#4555C2211yescharacteristic
65#4555C2211yescharacteristic
66#4555C2211yescharacteristic
67#4554C2211yescharacteristic
68#4554C2211yescharacteristic
69C2×#471D311yescharacteristic
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#1882—31no—
80#1857—31no—
81#1856—31no—
82#1773—31no—
83#1765—31no—
84C2×C3×#41C2311yescharacteristic
85C22×C3×Q8C2311yescharacteristic
86C22×C3×D4C2311yescharacteristic
87C3×#137—22no—
88C3×#137C2×C414yescomplemented
89C3×AGL(1,ℤ/8)—22no—
90C3×AGL(1,ℤ/8)C2×C414yes—
91C2×C3×Q16—21no—
92C2×C3×SD16—21no—
93C2×C3×SD16—21no—
94C2×C3×SD16—21no—
95C2×C3×SD16—21no—
96C2×C3×SD16—22no—
97C2×C3×SD16C2×C414yescomplemented
98C2×C3×D8—21no—
99C3×#131—22no—
100C3×#131C2×C414yescomplemented
101C2×C3×M16—21no—
102C2×C3×M16C2311yescharacteristic
103C2×C3×M16C2311yescharacteristic
104C2×C3×M16C2311yescharacteristic
105C22×C3×C8—21no—
106C22×C3×C8C2311yescharacteristic
107C2×#217—41no—
108C2×#217—41no—
109Q8×Dic3—21no—
110Q8×Dic3—21no—
111Q8×Dic3—21no—
112Q8×Dic3—21no—
113Q8×Dic3—21no—
114Q8×Dic3—21no—
115D4×Dic3—21no—
116D4×Dic3—21no—
117D4×Dic3—21no—
118D4×Dic3—21no—
119D4×Dic3—21no—
120D4×Dic3—21no—
121D4×Dic3—41no—
122D4×Dic3—41no—
123#926—21no—
124#926—21no—
125#926—41no—
126#926C2311yescharacteristic
127#926C2311yescharacteristic
128#926C2311yescharacteristic
129C2×C4⋊Dic3—21no—
130C2×C4⋊Dic3—21no—
131C2×C4⋊Dic3C2311yescharacteristic
132C2×C4⋊Dic3C2311yescharacteristic
133C2×C4×Dic3C2311yescharacteristic
134C2×#208C2311yescharacteristic
135C22×C3⋊C8C2311yescharacteristic
136C3×#108C2×C414yescomplemented
137C3×#101C2×C414yescomplemented
138C3×#100C2×C414yescomplemented
139#837—22no—
140#835—21no—
141#835—21no—
142#835—21no—
143#835—21no—
144#835D411yescharacteristic
145#835D411yescharacteristic
146#832—21no—
147#832—21no—
148#832—21no—
149#832—21no—
150#832D411yescharacteristic
151#832D411yescharacteristic
152C3⋊#95—41no—
153#818—21no—
154#818—21no—
155#818—21no—
156#818—21no—
157#818—21no—
158#818—21no—
159#818D411yescharacteristic
160#818D411yescharacteristic
161#817—21no—
162#817—21no—
163#817—21no—
164#817—21no—
165#817D411yescharacteristic
166#817D411yescharacteristic
167#815—21no—
168#815—22no—
169#815—21no—
170#815—21no—
171C8×Dic3—21no—
172C8×Dic3—21no—
173C8×Dic3—21no—
174C2×#137C2×D311yescharacteristic
175C2×AGL(1,ℤ/8)C2×D311yescharacteristic
176C22×SD16C2×D311yescharacteristic
177C2×#131C2×D311yescharacteristic
178#520—31no—
179#520—31no—
180#519—31no—
181#518—31no—
182#471Dic318yescomplemented
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×#108C2×D311yescharacteristic
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×#104—31no—
207#419—31no—
208#418—31no—
209#416—31no—
210C2×#103—31no—
211C2×#102—31no—
212C2×#101C2×D311yescharacteristic
213C2×#100C2×D311yescharacteristic
214#405—31no—
215#405—31no—
216#405—31no—
217C2×#97—31no—
218#391—31no—
219#390—31no—
220#343—31no—
221#343—31no—
222#341—31no—
223#339—31no—
224#339—31no—
225#339—31no—
226C24×C3—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×D4—21no—
240C2×C3×D4—21no—
241C2×C3×D4—21no—
242C2×C3×D4—21no—
243C2×C3×D4—21no—
244C2×C3×D4C22×C412yes—
245C2×C3×D4C2×D411yescharacteristic
246C22×C3×C4—21no—
247C22×C3×C4—21no—
248C22×C3×C4C2411yescharacteristic
249C22×Dic3—21no—
250C22×Dic3—21no—
251C22×Dic3—21no—
252C22×Dic3—41no—
253C22×Dic3—41no—
254C22×Dic3—41no—
255C22×Dic3—41no—
256C3×Q16—24no—
257C3×SD16—24no—
258C3×SD16—24no—
259C3×SD16—24no—
260C3×SD16—24no—
261C3×D8—24no—
262C3×M16—21no—
263C3×M16—21no—
264C3×M16—21no—
265C3×M16—24no—
266C3×M16C22×C412yes—
267C3×M16C22×C412yes—
268C3×M16C22×C412yes—
269C2×C3×C8—21no—
270C2×C3×C8—21no—
271C2×C3×C8—21no—
272C2×C3×C8—24no—
273C2×C3×C8C22×C412yes—
274C2×C3×C8C2×D411yescharacteristic
275C2×C3×C8C2×D411yescharacteristic
276C2×C3×C8#4112yes—
277#217—21no—
278#217—21no—
279#217—21no—
280#217—41no—
281#217—41no—
282#217—41no—
283#217—41no—
284#217—82no—
285C4⋊Dic3—21no—
286C4⋊Dic3—21no—
287C4⋊Dic3—21no—
288C4⋊Dic3—21no—
289C4⋊Dic3—41no—
290C4⋊Dic3—41no—
291C4⋊Dic3—41no—
292C4⋊Dic3—41no—
293C4⋊Dic3C2×D411yescharacteristic
294C4⋊Dic3C2×D411yescharacteristic
295C4⋊Dic3C2×D411yescharacteristic
296C4⋊Dic3C2×D411yescharacteristic
297C4⋊Dic3C2×D411yescharacteristic
298C4⋊Dic3C2×D411yescharacteristic
299C4×Dic3—21no—
300C4×Dic3—21no—
301C4×Dic3—21no—
302C4×Dic3—21no—
303C4×Dic3—21no—
304C4×Dic3—41no—
305C4×Dic3—41no—
306C4×Dic3—41no—
307C4×Dic3—41no—
308#208—21no—
309#208—21no—
310C2×C3⋊C8—21no—
311C2×C3⋊C8—21no—
312C2×C3⋊C8—21no—
313C2×C3⋊C8C2×D411yescharacteristic
314C2×C3⋊C8C2×D411yescharacteristic
315C2×#41C22×D311yescharacteristic
316C22×Q8C22×D311yescharacteristic
317C22×D4C22×D311yescharacteristic
318#137—22no—
319#137C2×Dic314yescomplemented
320AGL(1,ℤ/8)—22no—
321AGL(1,ℤ/8)C2×Dic314yes—
322C2×Q16—21no—
323C2×SD16—21no—
324C2×SD16—21no—
325C2×SD16—21no—
326C2×SD16—21no—
327C2×SD16—22no—
328C2×SD16C2×Dic314yescomplemented
329C2×D8—21no—
330#131—22no—
331#131C2×Dic314yescomplemented
332C2×M16—21no—
333C2×M16—31no—
334C2×M16C22×D311yescharacteristic
335C2×M16C22×D311yescharacteristic
336C2×M16C22×D311yescharacteristic
337C22×C8—21no—
338C22×C8—31no—
339C22×C8C22×D311yescharacteristic
340C4×Q8—61no—
341C4×Q8—61no—
342C4×Q8—61no—
343C4×Q8—61no—
344C4×Q8—61no—
345C4×Q8—61no—
346C4×D4—61no—
347C4×D4—61no—
348C4×D4—61no—
349C4×D4—61no—
350C4×D4—61no—
351C4×D4—61no—
352C4×D4—121no—
353C4×D4—121no—
354#117—31no—
355#117—31no—
356#117—31no—
357#117—61no—
358#117—61no—
359#117—121no—
360C2×C4⋊C4—31no—
361C2×C4⋊C4—31no—
362C2×C4⋊C4—61no—
363C2×C4⋊C4—61no—
364C2×C22⋊C4—121no—
365C2×C22⋊C4—121no—
366C2×C42—31no—
367#108C2×Dic314yescomplemented
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—62no—
383#103—31no—
384#103—31no—
385#103—61no—
386#103—61no—
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#101C2×Dic314yescomplemented
396#100C2×Dic314yescomplemented
397#97—61no—
398#97—61no—
399#97—61no—
400#97—62no—
401C4×C8—61no—
402C4×C8—61no—
403C4×C8—61no—
404#95—121no—
405C23×C3—21no—
406C23×C3—22no—
407C23×C3—41no—
408C23×C3—41no—
409C23×C3—81no—
410C23×C3—81no—
411C23×C3H(5,𝔽2)11yescharacteristic
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—22no—
421C3×D4—22no—
422C3×D4—21no—
423C3×D4—41no—
424C3×D4—41no—
425C3×D4C4×D412yes—
426C2×C3×C4—21no—
427C2×C3×C4—22no—
428C2×C3×C4—22no—
429C2×C3×C4—22no—
430C2×C3×C4—41no—
431C2×C3×C4—41no—
432C2×C3×C4C23×C411yescharacteristic
433C2×C3×C4C22×D411yescharacteristic
434C2×C3×C4C22×D411yescharacteristic
435C2×C3×C4C2×#4111yescharacteristic
436C2×C3×C4C2×#4111yescharacteristic
437C2×C3×C42−1+411yescharacteristic
438C2×Dic3—21no—
439C2×Dic3—21no—
440C2×Dic3—21no—
441C2×Dic3—21no—
442C2×Dic3—21no—
443C2×Dic3—21no—
444C2×Dic3—41no—
445C2×Dic3—41no—
446C2×Dic3—41no—
447C2×Dic3—41no—
448C2×Dic3—41no—
449C2×Dic3—81no—
450C2×Dic3—81no—
451C2×Dic3—161no—
452C2×Dic3—161no—
453C2×Dic3—161no—
454C2×Dic3—161no—
455C3×C8—22no—
456C3×C8—22no—
457C3×C8—22no—
458C3×C8C4×D412yes—
459C3×C8C4×D412yes—
460C3⋊C8—21no—
461C3⋊C8—21no—
462C3⋊C8—41no—
463C24—21no—
464#41—21no—
465#41—21no—
466#41#23712yes—
467#41C22×Dic312yes—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8D3×D411yescharacteristic
474C2×Q8C22×Dic312yes—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4D3×D411yescharacteristic
482C2×D4C22×Dic312yes—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—61no—
486C22×C4—61no—
487C22×C4—61no—
488C22×C4—121no—
489C22×C4—121no—
490C22×C4—121no—
491C22×C4—121no—
492C22×C4C23×D311yescharacteristic
493Q16—24no—
494SD16—24no—
495SD16—24no—
496SD16—24no—
497SD16—24no—
498D8—24no—
499M16—21no—
500M16—21no—
501M16—21no—
502M16—24no—
503M16—61no—
504M16—61no—
505M16C22×Dic312yes—
506M16C22×Dic312yes—
507M16C22×Dic312yes—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—24no—
512C2×C8—31no—
513C2×C8—31no—
514C2×C8—61no—
515C2×C8—61no—
516C2×C8—61no—
517C2×C8D3×D411yescharacteristic
518C2×C8D3×D411yescharacteristic
519C2×C8#23712yes—
520C2×C8C22×Dic312yes—
521C4⋊C4—31no—
522C4⋊C4—31no—
523C4⋊C4—31no—
524C4⋊C4—31no—
525C4⋊C4—31no—
526C4⋊C4—31no—
527C4⋊C4—61no—
528C4⋊C4—61no—
529C4⋊C4—61no—
530C4⋊C4—61no—
531C4⋊C4—121no—
532C4⋊C4—121no—
533C4⋊C4—121no—
534C4⋊C4—121no—
535C22⋊C4—61no—
536C22⋊C4—61no—
537C22⋊C4—61no—
538C22⋊C4—121no—
539C22⋊C4—121no—
540C22⋊C4—121no—
541C22⋊C4—121no—
542C22⋊C4—242no—
543C42—61no—
544C42—61no—
545C42—61no—
546C42—61no—
547C42—61no—
548C42—121no—
549C42—121no—
550C42—121no—
551C42—121no—
552C22×C3—21no—
553C22×C3—21no—
554C22×C3—22no—
555C22×C3—41no—
556C22×C3—41no—
557C22×C3—42no—
558C22×C3—82no—
559C22×C3—81no—
560C22×C3—81no—
561C22×C3—161no—
562C22×C3#52011yescharacteristic
563C22×C3D4211yescharacteristic
564C22×C3#55311yescharacteristic
565C3×C4—22no—
566C3×C4—42no—
567C3×C4C2×C4×D411yescharacteristic
568C3×C4C2×C4×D411yescharacteristic
569C3×C4#54711yescharacteristic
570C3×C4#54811yescharacteristic
571Dic3—41no—
572Dic3—41no—
573Dic3—41no—
574Dic3—41no—
575Dic3—41no—
576Dic3—41no—
577Dic3—81no—
578Dic3—81no—
579Dic3—81no—
580C23—21no—
581C23—22no—
582C23—41no—
583C23—41no—
584C23—81no—
585C23—81no—
586C23#100911yescharacteristic
587Q8—21no—
588Q8—22no—
589Q8—22no—
590Q8—41no—
591Q8—41no—
592Q8D4×Dic312yes—
593D4—22no—
594D4—22no—
595D4—22no—
596D4—22no—
597D4—21no—
598D4—41no—
599D4—41no—
600D4D4×Dic312yes—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—241no—
619C2×C4—241no—
620C2×C4—481no—
621C2×C4—481no—
622C2×C4—481no—
623C2×C4—481no—
624C2×C4C2×D3×D411yescharacteristic
625C2×C4C2×D3×D411yescharacteristic
626C2×C4C2×#23711yescharacteristic
627C2×C4C2×#23711yescharacteristic
628C2×C4#101011yescharacteristic
629C2×C4C23×Dic311yescharacteristic
630C8—22no—
631C8—22no—
632C8—22no—
633C8—61no—
634C8—61no—
635C8—121no—
636C8D4×Dic312yes—
637C8D4×Dic312yes—
638C2×C3—21no—
639C2×C3—21no—
640C2×C3—42no—
641C2×C3—82no—
642C2×C3#236111yescharacteristic
643C2×C3#328811yescharacteristic
644C2×C3#328911yescharacteristic
645C22—21no—
646C22—21no—
647C22—22no—
648C22—41no—
649C22—41no—
650C22—42no—
651C22—82no—
652C22—81no—
653C22—81no—
654C22—161no—
655C22#560911yescharacteristic
656C22#561811yescharacteristic
657C22#582711yescharacteristic
658C4—22no—
659C4—42no—
660C4—121no—
661C4—121no—
662C4—121no—
663C4—121no—
664C4—121no—
665C4—121no—
666C4—241no—
667C4—241no—
668C4—241no—
669C4#559811yescharacteristic
670C4#561111yescharacteristic
671C4C2×D4×Dic311yescharacteristic
672C4C2×D4×Dic311yescharacteristic
673C3#2304911yescharacteristic, complemented, Sylow
674C2—21no—
675C2—21no—
676C2—42no—
677C2—82no—
678C2#8093211yescharacteristic
679C2#8093311yescharacteristic
680C2#8263611yescharacteristic
subgroupG/H
centerC22#5609
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#471C2
socleC22×C3D42
prime radicalC2×C3×AGL(1,ℤ/8)C22
cosocleC22×C3×C4C24
hypercenterC2×#471D3
hypocenterC3#23049
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#471

Complements

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

NH ≅ G/Ntype
2. #83965C2semidirect
4. #83843C2semidirect
5. #83785C2semidirect
6. #72100C2semidirect
8. #72050C2semidirect
9. #72009C2semidirect
10. #70703C2semidirect
11. #70701C2semidirect
12. #70106C2semidirect
13. #70102C2semidirect
14. #70072C2semidirect
15. #70068C2semidirect
25. C3×#471C4semidirect
88. C3×#137C2×C4semidirect
97. C2×C3×SD16C2×C4semidirect
100. C3×#131C2×C4semidirect
136. C3×#108C2×C4semidirect
137. C3×#101C2×C4semidirect
138. C3×#100C2×C4semidirect
182. #471Dic3semidirect
319. #137C2×Dic3semidirect
328. C2×SD16C2×Dic3semidirect
331. #131C2×Dic3semidirect
367. #108C2×Dic3semidirect
395. #101C2×Dic3semidirect
396. #100C2×Dic3semidirect
673. C3#23049semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4712
#839652
#839302
#838432
#837852
#721002
#720882
#720502
#720092
#707032
#707012
#701062
#701022
#700722
#700682
#230493