Finite Group Databasebeta

← #10818119

Subgroups — #10818119

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
1C3×#2949C211yesmaximal, characteristic, complemented
2C2×#5177C211yesmaximal, characteristic
3#82547C211yesmaximal, characteristic, complemented
4#82537C211yesmaximal, characteristic
5#79179C211yesmaximal, characteristic, complemented
6#72050C211yesmaximal, characteristic, complemented
7#71658C211yesmaximal, characteristic, complemented
8#71647C211yesmaximal, characteristic, complemented
9#70660C211yesmaximal, characteristic, complemented
10#70348C211yesmaximal, characteristic, complemented
11#70338C211yesmaximal, characteristic, complemented
12#70328C211yesmaximal, characteristic
13#70318C211yesmaximal, characteristic, complemented
14#69894C211yesmaximal, characteristic, complemented
15#69888C211yesmaximal, characteristic, complemented
16#23049—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C3×#519C2211yescharacteristic
19C3×#518C2211yescharacteristic
20C2×#942C2211yescharacteristic
21C2×#932C2211yescharacteristic
22C2×#901C2211yescharacteristic
23#5523C2211yescharacteristic
24#5523C2211yescharacteristic
25C3×#440—21no—
26C3×#440—21no—
27C3×C4×SD16—21no—
28C3×C4×SD16—21no—
29C3×#427C2211yescharacteristic
30C3×#419C2211yescharacteristic
31C3×#418C2211yescharacteristic
32C3×#405C2211yescharacteristic
33#5229C2211yescharacteristic
34C2×#836C2211yescharacteristic
35C2×#835C2211yescharacteristic
36#5221C2211yescharacteristic
37C2×#833C2211yescharacteristic
38C2×#832C2211yescharacteristic
39#5177C418yescomplemented
40C2×#825C2211yescharacteristic
41C2×#819C2211yescharacteristic
42C2×#818C2211yescharacteristic
43C2×#815C2211yescharacteristic
44#5030—21no—
45C4×#216—21no—
46#5026—21no—
47C4×#215—21no—
48#5018—21no—
49C4×#214—21no—
50#5014—21no—
51C4×#213—21no—
52#5002C2211yescharacteristic
53#4997C2211yescharacteristic
54#4997C2211yescharacteristic
55#4995C2211yescharacteristic
56#4995C2211yescharacteristic
57#4995C2211yescharacteristic
58#4995C2211yescharacteristic
59#4880—21no—
60D3×#107—21no—
61#4861—21no—
62#4861—21no—
63#4558C2211yescharacteristic
64#4554C2211yescharacteristic
65#4554C2211yescharacteristic
66#4532C2211yescharacteristic
67#4532C2211yescharacteristic
68#4532C2211yescharacteristic
69C2×#471—31no—
70#2949D311yescharacteristic, complemented
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—
84C22×C3×Q8C2311yescharacteristic
85C22×C3×D4C2311yescharacteristic
86C2×#235C2311yescharacteristic
87C2×C3×SD16—21no—
88C2×C3×SD16—21no—
89C2×C3×SD16—22no—
90C2×C3×SD16C2×C414yescomplemented
91C2×C3×M16C2311yescharacteristic
92C22×C3×C8C2311yescharacteristic
93C3×C4×Q8—21no—
94C3×C4×Q8—21no—
95C3×C4×Q8—21no—
96C3×C4×Q8—21no—
97C3×C4×D4—21no—
98C3×C4×D4—21no—
99C3×C4×D4—21no—
100C3×C4×D4—21no—
101C3×#117—41no—
102C3×#117C2311yescharacteristic
103C3×#117C2311yescharacteristic
104C3×#117C2311yescharacteristic
105C2×C3×C22⋊C4—41no—
106C2×#217—41no—
107C2×#216—21no—
108#942—22no—
109#942C2×C414yescomplemented
110C2×#215—21no—
111C2×#214—21no—
112#932—22no—
113#932C2×C414yes—
114C2×#213—21no—
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4×Dic3C2311yescharacteristic
118C2×#208C2311yescharacteristic
119C2×#208C2311yescharacteristic
120C2×#208C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122#901—22no—
123#901C2×C414yescomplemented
124C2×ΓL(1,𝔽25)—21no—
125C2×D3×C8—21no—
126#892—21no—
127#892—21no—
128D3×C4⋊C4—21no—
129D3×C4⋊C4—21no—
130#67⋊C4—41no—
131#67⋊C4—41no—
132C4×D12—21no—
133C4×D12—21no—
134C4×Dic6—21no—
135C4×Dic6—21no—
136C3×#107D411yescharacteristic
137C3×#107D411yescharacteristic
138C3×#106D411yescharacteristic
139C3×#106D411yescharacteristic
140C3×#103—21no—
141C3×#103—21no—
142C3×#102—21no—
143C3×#102—21no—
144C3×#97—21no—
145C3×C4×C8—21no—
146#836C2×C414yescomplemented
147#835—21no—
148#835—21no—
149#833C2×C414yescomplemented
150#832—21no—
151#832—21no—
152C3⋊#95—41no—
153#825—22no—
154#819C2×C414yescomplemented
155#818—21no—
156#818—21no—
157#815—22no—
158#810D411yescharacteristic
159#810D411yescharacteristic
160#809D411yescharacteristic
161#809D411yescharacteristic
162#808—21no—
163#808—21no—
164#808—21no—
165#808—21no—
166#807—21no—
167#807—21no—
168#807—21no—
169#807—21no—
170#803—21no—
171#803—21no—
172C4×C3⋊C8—21no—
173C4×C3⋊C8—21no—
174C2×#137—31no—
175C2×AGL(1,ℤ/8)—31no—
176C22×SD16C2×D311yescharacteristic
177C2×#131—31no—
178#520—31no—
179#520—31no—
180#519C2×D311yescharacteristic
181#518C2×D311yescharacteristic
182#471—38no—
183#442—61no—
184#441—61no—
185#440—21no—
186#440—21no—
187#440—61no—
188#440—61no—
189C4×Q16—61no—
190C4×SD16—21no—
191C4×SD16—21no—
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—61no—
203#427C2×D311yescharacteristic
204C2×#107—31no—
205C2×#107—61no—
206C2×#104—31no—
207#419C2×D311yescharacteristic
208#418C2×D311yescharacteristic
209#416—31no—
210C2×#103—31no—
211C2×#102—31no—
212C2×#101—31no—
213C2×#100—31no—
214#405—31no—
215#405—31no—
216#405C2×D311yescharacteristic
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—
227C2×C3×Q8—21no—
228C2×C3×Q8—21no—
229C2×C3×Q8—21no—
230C2×C3×Q8—21no—
231C2×C3×Q8—21no—
232C2×C3×Q8C22×C412yes—
233C2×C3×D4—21no—
234C2×C3×D4—21no—
235C2×C3×D4—21no—
236C2×C3×D4—21no—
237C2×C3×D4—21no—
238C2×C3×D4C22×C412yes—
239C22×C3×C4—21no—
240C22×C3×C4—41no—
241C22×C3×C4—41no—
242C22×C3×C4C2411yescharacteristic
243C2×#67—21no—
244C22×Dic3—21no—
245C22×Dic3—21no—
246C22×Dic3—21no—
247#235—21no—
248#235—21no—
249#235C22×C412yes—
250#235#4112yes—
251C2×D12C2×D411yescharacteristic
252C2×C4×D3—21no—
253C2×C4×D3—41no—
254C2×C4×D3—41no—
255C2×Dic6C2×D411yescharacteristic
256C3×SD16—24no—
257C3×SD16—24no—
258C3×M16—21no—
259C3×M16—21no—
260C2×C3×C8—21no—
261C2×C3×C8C22×C412yes—
262C2×C3×C8C2×D411yescharacteristic
263C2×C3×C8C2×D411yescharacteristic
264C2×C3×C8#4112yes—
265C3×C4⋊C4—41no—
266C3×C4⋊C4—41no—
267C3×C4⋊C4C2×D411yescharacteristic
268C3×C4⋊C4C2×D411yescharacteristic
269C3×C4⋊C4C2×D411yescharacteristic
270C3×C4⋊C4C2×D411yescharacteristic
271C3×C4⋊C4C2×D411yescharacteristic
272C3×C4⋊C4C2×D411yescharacteristic
273C3×C22⋊C4—21no—
274C3×C22⋊C4—21no—
275C3×C22⋊C4—21no—
276C3×C22⋊C4—41no—
277C3×C22⋊C4—41no—
278C3×C42—21no—
279C3×C42—21no—
280C3×C42—21no—
281C3×C42—41no—
282C3×C42—41no—
283#217—82no—
284#216—24no—
285#215—24no—
286#214—24no—
287#213—24no—
288(C2×D3)⋊C4—41no—
289(C2×D3)⋊C4—41no—
290C4⋊Dic3—21no—
291C4⋊Dic3—21no—
292C4⋊Dic3—21no—
293C4⋊Dic3—21no—
294Dic3⋊C4—41no—
295Dic3⋊C4—41no—
296C4×Dic3—21no—
297C4×Dic3—21no—
298C4×Dic3—41no—
299C4×Dic3—41no—
300#208—21no—
301#208—21no—
302#208—21no—
303#208C22×C412yes—
304#208C22×C412yes—
305#208C22×C412yes—
306C2×C3⋊C8—21no—
307C2×C3⋊C8—21no—
308C2×C3⋊C8—21no—
309C2×C3⋊C8—21no—
310C2×C3⋊C8—21no—
311C2×C3⋊C8C2×D411yescharacteristic
312C2×C3⋊C8C2×D411yescharacteristic
313ΓL(1,𝔽25)—24no—
314D3×C8—24no—
315C2×#41—31no—
316C22×Q8C22×D311yescharacteristic
317C22×D4C22×D311yescharacteristic
318#137—34no—
319#137—62no—
320AGL(1,ℤ/8)—34no—
321AGL(1,ℤ/8)—62no—
322C2×Q16—61no—
323C2×SD16—22no—
324C2×SD16—21no—
325C2×SD16—21no—
326C2×SD16—61no—
327C2×SD16—61no—
328C2×SD16C4×D314yescomplemented
329C2×D8—61no—
330#131—34no—
331#131—62no—
332C2×M16—31no—
333C2×M16—31no—
334C2×M16—31no—
335C2×M16—61no—
336C2×M16C22×D311yescharacteristic
337C22×C8—31no—
338C22×C8—61no—
339C22×C8C22×D311yescharacteristic
340C4×Q8—21no—
341C4×Q8—21no—
342C4×Q8—21no—
343C4×Q8—21no—
344C4×Q8—61no—
345C4×Q8—61no—
346C4×D4—21no—
347C4×D4—21no—
348C4×D4—21no—
349C4×D4—21no—
350C4×D4—61no—
351C4×D4—61no—
352C4×D4—121no—
353C4×D4—121no—
354#117—41no—
355#117—61no—
356#117—61no—
357#117C22×D311yescharacteristic
358#117C22×D311yescharacteristic
359#117C22×D311yescharacteristic
360C2×C4⋊C4—31no—
361C2×C4⋊C4—31no—
362C2×C4⋊C4—61no—
363C2×C4⋊C4—61no—
364C2×C22⋊C4—41no—
365C2×C22⋊C4—121no—
366C2×C42—31no—
367#108—34no—
368#107—61no—
369#107—61no—
370#107—61no—
371#107—61no—
372#107—61no—
373#107—61no—
374#107#6711yescharacteristic
375#107#6711yescharacteristic
376#106—61no—
377#106—61no—
378#106—61no—
379#106—61no—
380#106#6711yescharacteristic
381#106#6711yescharacteristic
382#104—62no—
383#103—21no—
384#103—21no—
385#103—31no—
386#103—31no—
387#103—61no—
388#103—61no—
389#102—21no—
390#102—21no—
391#102—31no—
392#102—31no—
393#102—61no—
394#102—61no—
395#101—34no—
396#100—34no—
397#97—21no—
398#97—61no—
399#97—61no—
400#97—62no—
401C4×C8—21no—
402C4×C8—61no—
403C4×C8—61no—
404#95—121no—
405C23×C3—22no—
406C23×C3—41no—
407C23×C3—41no—
408C23×C3—81no—
409C23×C3—81no—
410C23×C3H(5,𝔽2)11yescharacteristic
411C22×D3—21no—
412C3×Q8—22no—
413C3×Q8—22no—
414C3×Q8—41no—
415C3×Q8—41no—
416C3×D4—22no—
417C3×D4—22no—
418C3×D4—41no—
419C3×D4—41no—
420C2×C3×C4—22no—
421C2×C3×C4—21no—
422C2×C3×C4—21no—
423C2×C3×C4—21no—
424C2×C3×C4—21no—
425C2×C3×C4—21no—
426C2×C3×C4—21no—
427C2×C3×C4—41no—
428C2×C3×C4—41no—
429C2×C3×C4—161no—
430C2×C3×C4—161no—
431C2×C3×C4C23×C411yescharacteristic
432C2×C3×C4C22×D411yescharacteristic
433C2×C3×C4C22×D411yescharacteristic
434C2×C3×C4C2×#4111yescharacteristic
435C2×C3×C4C2×#4111yescharacteristic
436C2×C3×C42−1+411yescharacteristic
437#67—22no—
438#67—22no—
439C2×Dic3—21no—
440C2×Dic3—41no—
441C2×Dic3—41no—
442C2×Dic3—41no—
443C2×Dic3—41no—
444C2×Dic3—41no—
445C2×Dic3—81no—
446C2×Dic3—81no—
447D12—21no—
448D12C4×D412yes—
449C4×D3—22no—
450C4×D3—22no—
451C4×D3—161no—
452C4×D3—161no—
453Dic6—21no—
454Dic6C4×D412yes—
455C3×C8—41no—
456C3×C8C4×D412yes—
457C3×C8C4×D412yes—
458C3⋊C8—22no—
459C3⋊C8—22no—
460C3⋊C8—22no—
461C3⋊C8—21no—
462C3⋊C8—21no—
463C24—21no—
464#41—32no—
465#41—32no—
466#41—61no—
467#41—61no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—31no—
474C2×Q8C2×C4×D312yes—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—31no—
481C2×D4—61no—
482C2×D4C2×C4×D312yes—
483C22×C4—21no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—61no—
487C22×C4—61no—
488C22×C4—61no—
489C22×C4—61no—
490C22×C4—121no—
491C22×C4—121no—
492C22×C4C23×D311yescharacteristic
493Q16—64no—
494SD16—24no—
495SD16—24no—
496SD16—64no—
497SD16—64no—
498D8—64no—
499M16—21no—
500M16—21no—
501M16—32no—
502M16—32no—
503M16—32no—
504M16—61no—
505M16—61no—
506M16—61no—
507M16—64no—
508C2×C8—21no—
509C2×C8—31no—
510C2×C8—31no—
511C2×C8—61no—
512C2×C8—61no—
513C2×C8—61no—
514C2×C8—61no—
515C2×C8—61no—
516C2×C8—64no—
517C2×C8C2×C4×D312yes—
518C2×C8#23512yes—
519C2×C8C2×#6711yescharacteristic
520C2×C8C2×#6711yescharacteristic
521C4⋊C4—41no—
522C4⋊C4—41no—
523C4⋊C4—61no—
524C4⋊C4—61no—
525C4⋊C4—61no—
526C4⋊C4—61no—
527C4⋊C4—121no—
528C4⋊C4—121no—
529C4⋊C4D3×D411yescharacteristic
530C4⋊C4D3×D411yescharacteristic
531C4⋊C4C2×#6711yescharacteristic
532C4⋊C4C2×#6711yescharacteristic
533C4⋊C4C2×#6711yescharacteristic
534C4⋊C4C2×#6711yescharacteristic
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—41no—
539C22⋊C4—41no—
540C22⋊C4—121no—
541C22⋊C4—121no—
542C22⋊C4—242no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—41no—
547C42—41no—
548C42—61no—
549C42—61no—
550C42—121no—
551C42—121no—
552C22×C3—21no—
553C22×C3—21no—
554C22×C3—42no—
555C22×C3—82no—
556C22×C3—81no—
557C22×C3—81no—
558C22×C3—161no—
559C22×C3#52011yescharacteristic
560C22×C3D4211yescharacteristic
561C22×C3#55311yescharacteristic
562C2×D3—22no—
563C2×D3—41no—
564C2×D3—41no—
565C3×C4—42no—
566C3×C4—41no—
567C3×C4—41no—
568C3×C4—41no—
569C3×C4—41no—
570C3×C4—41no—
571C3×C4—41no—
572C3×C4C2×C4×D411yescharacteristic
573C3×C4C2×C4×D411yescharacteristic
574C3×C4#54711yescharacteristic
575C3×C4#54811yescharacteristic
576Dic3—22no—
577Dic3—81no—
578Dic3—81no—
579Dic3—81no—
580C23—22no—
581C23—41no—
582C23—41no—
583C23—61no—
584C23—81no—
585C23—81no—
586C23#100411yescharacteristic
587Q8—22no—
588Q8—22no—
589Q8—32no—
590Q8—41no—
591Q8—41no—
592Q8—61no—
593D4—22no—
594D4—22no—
595D4—32no—
596D4—41no—
597D4—41no—
598D4—62no—
599D4—62no—
600D4—61no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—61no—
611C2×C4—62no—
612C2×C4—62no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—161no—
619C2×C4—161no—
620C2×C4—241no—
621C2×C4—241no—
622C2×C4—481no—
623C2×C4—481no—
624C2×C4C22×C4×D311yescharacteristic
625C2×C4C2×#23511yescharacteristic
626C2×C4C2×D3×D411yescharacteristic
627C2×C4D3×#4111yescharacteristic
628C2×C4#101011yescharacteristic
629C2×C4C22×#6711yescharacteristic
630C8—41no—
631C8—62no—
632C8—62no—
633C8—62no—
634C8—61no—
635C8—61no—
636C8C4×#6712yes—
637C8C4×#6712yes—
638C2×C3—21no—
639C2×C3—21no—
640C2×C3—82no—
641C2×C3#236111yescharacteristic
642C2×C3#328811yescharacteristic
643C2×C3#328911yescharacteristic
644D3—42no—
645C22—21no—
646C22—21no—
647C22—42no—
648C22—62no—
649C22—82no—
650C22—81no—
651C22—81no—
652C22—121no—
653C22—121no—
654C22—161no—
655C22#554611yescharacteristic
656C22#555811yescharacteristic
657C22D4×#6711yescharacteristic
658C4—41no—
659C4—41no—
660C4—41no—
661C4—41no—
662C4—41no—
663C4—41no—
664C4—42no—
665C4—62no—
666C4—241no—
667C4—241no—
668C4—241no—
669C4C4×D3×D411yescharacteristic
670C4#555111yescharacteristic
671C4C2×C4×#6711yescharacteristic
672C4#583111yescharacteristic
673C3#2304911yescharacteristic, complemented, Sylow
674C2—21no—
675C2—21no—
676C2—82no—
677C2—122no—
678C2#8239911yescharacteristic
679C2#8401311yescharacteristic
680C2#8401511yescharacteristic
subgroupG/H
centerC22D4×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2949C2
socleC22×C3D42
prime radicalC2×#932C22
cosocleC22×C3×C4C24
hypercenter#2949D3
hypocenterC3#23049
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2949

Complements

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

NH ≅ G/Ntype
1. C3×#2949C2semidirect
3. #82547C2semidirect
5. #79179C2semidirect
6. #72050C2semidirect
7. #71658C2semidirect
8. #71647C2semidirect
9. #70660C2semidirect
10. #70348C2semidirect
11. #70338C2semidirect
13. #70318C2semidirect
14. #69894C2semidirect
15. #69888C2semidirect
39. #5177C4semidirect
70. #2949D3semidirect
90. C2×C3×SD16C2×C4semidirect
109. #942C2×C4semidirect
123. #901C2×C4semidirect
146. #836C2×C4semidirect
149. #833C2×C4semidirect
154. #819C2×C4semidirect
328. C2×SD16C4×D3semidirect
673. C3#23049semidirect

Maximal subgroups

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

subgroupindex
C3×#29492
C2×#51772
#825472
#825372
#791792
#720502
#716582
#716472
#706602
#703482
#703382
#703282
#703182
#698942
#698882
#230493