Finite Group Databasebeta

← #10818131

Subgroups — #10818131

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×#5177C211yesmaximal, characteristic
2#83843C211yesmaximal, characteristic, complemented
3#79146C211yesmaximal, characteristic, complemented
4#78774C211yesmaximal, characteristic, complemented
5#78726C211yesmaximal, characteristic
6C3×#1973C211yesmaximal, characteristic, complemented
7#70625C211yesmaximal, characteristic, complemented
8#70263C211yesmaximal, characteristic, complemented
9#70195C211yesmaximal, characteristic, complemented
10#69839C211yesmaximal, characteristic, complemented
11#69829C211yesmaximal, characteristic, complemented
12#69693C211yesmaximal, characteristic, complemented
13#69672C211yesmaximal, characteristic, complemented
14#69663C211yesmaximal, characteristic
15#69642C211yesmaximal, characteristic, complemented
16#23049—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18#5812C2211yescharacteristic
19C2×#942C2211yescharacteristic
20#5798C2211yescharacteristic
21C2×#932C2211yescharacteristic
22C2×#901C2211yescharacteristic
23#5504C2211yescharacteristic
24#5504C2211yescharacteristic
25C2×C3×#107C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#97C2211yescharacteristic
29C3×#391C2211yescharacteristic
30C3×#390C2211yescharacteristic
31C2×#836C2211yescharacteristic
32#5225C2211yescharacteristic
33C2×#833C2211yescharacteristic
34#5215C2211yescharacteristic
35#5177C418yescomplemented
36#5164—21no—
37#5164—21no—
38Dic3×SD16—21no—
39Dic3×SD16—21no—
40#5113C2211yescharacteristic
41C2×#819C2211yescharacteristic
42#5106C2211yescharacteristic
43#5101C2211yescharacteristic
44#5101C2211yescharacteristic
45#5101C2211yescharacteristic
46#4964C2211yescharacteristic
47#4964C2211yescharacteristic
48#4964C2211yescharacteristic
49#4964C2211yescharacteristic
50C2×#805C2211yescharacteristic
51#4881—21no—
52#4881—21no—
53#4880—21no—
54D3×#107—21no—
55#4817—21no—
56#4790—21no—
57#4789—21no—
58#4788—21no—
59#4785—21no—
60#4758—21no—
61#4757—21no—
62#4756—21no—
63#4552C2211yescharacteristic
64#4530C2211yescharacteristic
65#4530C2211yescharacteristic
66#4523C2211yescharacteristic
67#4523C2211yescharacteristic
68#4523C2211yescharacteristic
69C2×#471—31no—
70#2949—31no—
71#2947—31no—
72#2946—31no—
73#2913—31no—
74#1973D311yescharacteristic, complemented
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
91C22×C3×C8C2311yescharacteristic
92C22×C3×C8C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—41no—
96C2×C3×C42C2311yescharacteristic
97C2×#217—41no—
98Q8×Dic3—21no—
99Q8×Dic3—21no—
100Q8×Dic3—21no—
101Q8×Dic3—21no—
102C2×#216—21no—
103#942—22no—
104#942C2×C414yescomplemented
105C2×#215—21no—
106D4×Dic3—21no—
107D4×Dic3—21no—
108D4×Dic3—21no—
109D4×Dic3—21no—
110C2×#214—21no—
111#932—22no—
112#932C2×C414yes—
113C2×#213—21no—
114C4×#67—41no—
115C4×#67—41no—
116#926—41no—
117#926C2311yescharacteristic
118#926C2311yescharacteristic
119#926C2311yescharacteristic
120C2×#208C2311yescharacteristic
121C2×#208C2311yescharacteristic
122C2×#208C2311yescharacteristic
123C2×#208C2311yescharacteristic
124#901—22no—
125#901C2×C414yescomplemented
126C2×ΓL(1,𝔽25)—21no—
127C2×D3×C8—21no—
128#893—21no—
129#893—21no—
130#892—21no—
131#892—21no—
132D3×C4⋊C4—21no—
133D3×C4⋊C4—21no—
134#887—21no—
135#887—21no—
136C3×#107—21no—
137C3×#107—21no—
138C3×#103—21no—
139C3×#103—21no—
140C3×#102—21no—
141C3×#102—21no—
142C3×#97—22no—
143C3×#95—41no—
144#836C2×C414yescomplemented
145#835—21no—
146#835—21no—
147#833C2×C414yescomplemented
148#832—21no—
149#832—21no—
150#821D411yescharacteristic
151#821D411yescharacteristic
152#819C2×C414yescomplemented
153#818D411yescharacteristic
154#818D411yescharacteristic
155#817D411yescharacteristic
156#817D411yescharacteristic
157#816D411yescharacteristic
158#816D411yescharacteristic
159#815—21no—
160#815—21no—
161#815—21no—
162C8×Dic3—21no—
163C8×Dic3—21no—
164C8×Dic3—21no—
165#808—21no—
166#808—21no—
167#808—21no—
168#808—21no—
169#807—21no—
170#807—21no—
171#807—21no—
172#807—21no—
173#805—22no—
174C2×#137—31no—
175C2×AGL(1,ℤ/8)—31no—
176C22×SD16C2×D311yescharacteristic
177C2×#131—31no—
178#520—31no—
179#520—31no—
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—61no—
205C2×#107C2×D311yescharacteristic
206C2×#104—31no—
207#419—31no—
208#418—31no—
209#416—31no—
210C2×#103C2×D311yescharacteristic
211C2×#102C2×D311yescharacteristic
212C2×#101—31no—
213C2×#100—31no—
214#405—31no—
215#405—31no—
216#405—31no—
217C2×#97C2×D311yescharacteristic
218#391C2×D311yescharacteristic
219#390C2×D311yescharacteristic
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—21no—
241C22×C3×C4—21no—
242C22×C3×C4—21no—
243C22×C3×C4C2411yescharacteristic
244C2×#67—21no—
245C22×Dic3—41no—
246C22×Dic3—41no—
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—
258C2×C3×C8—21no—
259C2×C3×C8—21no—
260C2×C3×C8C22×C412yes—
261C2×C3×C8C2×D411yescharacteristic
262C2×C3×C8C2×D411yescharacteristic
263C2×C3×C8C2×D411yescharacteristic
264C2×C3×C8C2×D411yescharacteristic
265C2×C3×C8#4112yes—
266C3×C4⋊C4—21no—
267C3×C4⋊C4—21no—
268C3×C4⋊C4—21no—
269C3×C4⋊C4—21no—
270C3×C22⋊C4—82no—
271C3×C42—21no—
272C3×C42—21no—
273#217—21no—
274#217—21no—
275#217—21no—
276#217—41no—
277#217—41no—
278#216—24no—
279#215—24no—
280#214—24no—
281#213—24no—
282(C2×D3)⋊C4—41no—
283(C2×D3)⋊C4—41no—
284C4⋊Dic3—41no—
285C4⋊Dic3—41no—
286C4⋊Dic3C2×D411yescharacteristic
287C4⋊Dic3C2×D411yescharacteristic
288C4⋊Dic3C2×D411yescharacteristic
289C4⋊Dic3C2×D411yescharacteristic
290C4⋊Dic3C2×D411yescharacteristic
291C4⋊Dic3C2×D411yescharacteristic
292Dic3⋊C4—41no—
293Dic3⋊C4—41no—
294C4×Dic3—21no—
295C4×Dic3—21no—
296C4×Dic3—21no—
297C4×Dic3—41no—
298C4×Dic3—41no—
299C4×Dic3—41no—
300C4×Dic3—41no—
301#208—21no—
302#208—21no—
303#208—21no—
304#208—21no—
305#208—21no—
306#208C22×C412yes—
307#208C22×C412yes—
308#208C22×C412yes—
309C2×C3⋊C8—21no—
310C2×C3⋊C8—21no—
311C2×C3⋊C8—21no—
312C2×C3⋊C8—21no—
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—21no—
324C2×SD16—21no—
325C2×SD16—22no—
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—31no—
336C2×M16—61no—
337C22×C8—61no—
338C22×C8C22×D311yescharacteristic
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—61no—
361C2×C4⋊C4—61no—
362C2×C4⋊C4C22×D311yescharacteristic
363C2×C4⋊C4C22×D311yescharacteristic
364C2×C22⋊C4—41no—
365C2×C22⋊C4—121no—
366C2×C42C22×D311yescharacteristic
367#108—34no—
368#107—21no—
369#107—21no—
370#107—31no—
371#107—31no—
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—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—22no—
398#97—61no—
399#97—61no—
400#97—61no—
401C4×C8—61no—
402C4×C8—61no—
403C4×C8—61no—
404#95—41no—
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—41no—
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—81no—
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
436#67—22no—
437#67—22no—
438C2×Dic3—21no—
439C2×Dic3—21no—
440C2×Dic3—21no—
441C2×Dic3—21no—
442C2×Dic3—21no—
443C2×Dic3—21no—
444C2×Dic3—21no—
445C2×Dic3—161no—
446C2×Dic3—161no—
447D12—21no—
448D12C4×D412yes—
449C4×D3—22no—
450C4×D3—22no—
451C4×D3—161no—
452C4×D3—161no—
453Dic6—21no—
454Dic6C4×D412yes—
455C3×C8—21no—
456C3×C8—21no—
457C3×C8C4×D412yes—
458C3×C8C4×D412yes—
459C3⋊C8—22no—
460C3⋊C8—22no—
461C3⋊C8—22no—
462C3⋊C8—41no—
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—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—61no—
488C22×C4—121no—
489C22×C4—121no—
490C22×C4—121no—
491C22×C4—121no—
492C22×C4C23×D311yescharacteristic
493Q16—64no—
494SD16—24no—
495SD16—24no—
496SD16—64no—
497SD16—64no—
498D8—64no—
499M16—32no—
500M16—32no—
501M16—32no—
502M16—61no—
503M16—61no—
504M16—61no—
505M16—64no—
506M16—61no—
507M16—61no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—61no—
511C2×C8—61no—
512C2×C8—61no—
513C2×C8—64no—
514C2×C8—61no—
515C2×C8C2×C4×D312yes—
516C2×C8D3×D411yescharacteristic
517C2×C8D3×D411yescharacteristic
518C2×C8D3×D411yescharacteristic
519C2×C8D3×D411yescharacteristic
520C2×C8#23712yes—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—31no—
526C4⋊C4—31no—
527C4⋊C4—31no—
528C4⋊C4—31no—
529C4⋊C4—31no—
530C4⋊C4—31no—
531C4⋊C4—121no—
532C4⋊C4—121no—
533C4⋊C4—121no—
534C4⋊C4—121no—
535C22⋊C4—61no—
536C22⋊C4—61no—
537C22⋊C4—61no—
538C22⋊C4—82no—
539C22⋊C4—121no—
540C22⋊C4—121no—
541C22⋊C4—121no—
542C22⋊C4—121no—
543C42—21no—
544C42—21no—
545C42—61no—
546C42—61no—
547C42—61no—
548C42—121no—
549C42—121no—
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—81no—
567C3×C4—81no—
568C3×C4—81no—
569C3×C4C2×C4×D411yescharacteristic
570C3×C4C2×C4×D411yescharacteristic
571C3×C4#54711yescharacteristic
572C3×C4#54811yescharacteristic
573Dic3—22no—
574Dic3—41no—
575Dic3—41no—
576Dic3—41no—
577Dic3—41no—
578Dic3—41no—
579Dic3—41no—
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—22no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—61no—
610C2×C4—62no—
611C2×C4—62no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—61no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—481no—
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—21no—
631C8—21no—
632C8—62no—
633C8—62no—
634C8—62no—
635C8—121no—
636C8#67⋊C412yes—
637C8#67⋊C412yes—
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#561411yescharacteristic
657C22#571711yescharacteristic
658C4—42no—
659C4—62no—
660C4—81no—
661C4—81no—
662C4—81no—
663C4—121no—
664C4—121no—
665C4—121no—
666C4—121no—
667C4—121no—
668C4—121no—
669C4C2×#67⋊C411yescharacteristic
670C4C4×D3×D411yescharacteristic
671C4#561111yescharacteristic
672C4#567711yescharacteristic
673C3#2304911yescharacteristic, complemented, Sylow
674C2—21no—
675C2—21no—
676C2—82no—
677C2—122no—
678C2#7643311yescharacteristic
679C2#8186011yescharacteristic
680C2#8188511yescharacteristic
subgroupG/H
centerC22#5717
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1973C2
socleC22×C3D42
prime radicalC2×#932C22
cosocleC22×C3×C4C24
hypercenter#1973D3
hypocenterC3#23049
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1973

Complements

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

NH ≅ G/Ntype
2. #83843C2semidirect
3. #79146C2semidirect
4. #78774C2semidirect
6. C3×#1973C2semidirect
7. #70625C2semidirect
8. #70263C2semidirect
9. #70195C2semidirect
10. #69839C2semidirect
11. #69829C2semidirect
12. #69693C2semidirect
13. #69672C2semidirect
15. #69642C2semidirect
35. #5177C4semidirect
74. #1973D3semidirect
90. C2×C3×SD16C2×C4semidirect
104. #942C2×C4semidirect
125. #901C2×C4semidirect
144. #836C2×C4semidirect
147. #833C2×C4semidirect
152. #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
C2×#51772
#838432
#791462
#787742
#787262
C3×#19732
#706252
#702632
#701952
#698392
#698292
#696932
#696722
#696632
#696422
#230493