Finite Group Databasebeta

← #6396756

Subgroups — #6396756

Order 512.

1580 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60854C211yesmaximal, characteristic
2#60269C211yesmaximal, characteristic
3#60251C211yesmaximal, characteristic, complemented
4#23082C214yesmaximal, complemented
5#22984C214yesmaximal, complemented
6#22951C214yesmaximal, complemented
7#22949C214yesmaximal, complemented
8#22259C212yesmaximal, complemented
9#22237C212yesmaximal, complemented
10#21994C211yesmaximal, characteristic, complemented
11#21736C212yesmaximal, complemented
12#21662C212yesmaximal, complemented
13#21418C211yesmaximal, characteristic
14#21329C211yesmaximal, characteristic, complemented
15#20467C211yesmaximal, characteristic, complemented
16#3444C2211yescharacteristic
17C2×#533C2211yescharacteristic
18#3240C414yescomplemented
19#3239C412yescomplemented
20#3238C412yescomplemented
21#3237C412yescomplemented
22#3236C412yescomplemented
23#3235C412yescomplemented
24#3234C412yescomplemented
25#3162C2212yes—
26#3135C2212yes—
27#3091C2212yes—
28#3091C2212yes—
29#3076C2214yes—
30#2995—22no—
31#2992—22no—
32#2990—22no—
33#2989—22no—
34#2987—22no—
35#2985—22no—
36#2984—22no—
37#2983—22no—
38#2962C2212yes—
39#2962C2212yes—
40#2961C2212yes—
41#2961C2212yes—
42#2961C2212yes—
43#2933C2214yes—
44#2932C2212yes—
45#2925C2212yes—
46#2924C2214yes—
47#2920C2211yescharacteristic
48C2×#432C2211yescharacteristic
49C2×#432C2211yescharacteristic
50#2906C2212yes—
51C2×#423C2211yescharacteristic
52C4×#117C2211yescharacteristic
53#1995C2214yes—
54#1994C2212yes—
55#1994C2212yes—
56#1994C2211yescharacteristic
57#1994C2211yescharacteristic
58#1962C2211yescharacteristic
59#1962C2211yescharacteristic
60#1950C2214yes—
61#1946C2214yes—
62#1945C2214yes—
63#1938C2214yes—
64#1937C2214yes—
65#1929C2214yes—
66#1929C2214yes—
67#1922C2214yes—
68#1922C2214yes—
69#1922C2214yes—
70#1922C2214yes—
71#1897C2214yes—
72#1866C2214yes—
73#1866C2214yes—
74#1855C2212yes—
75#1855C2212yes—
76#1849C2212yes—
77#1838C2212yes—
78#1773C2214yes—
79#1772C2214yes—
80#1768C2212yes—
81#1768C2212yes—
82#1765C2214yes—
83#1764C2214yes—
84#1759C2214yes—
85#1750C2212yes—
86#1750C2212yes—
87#1745C2212yes—
88#1733C2214yes—
89C4×#97C2212yes—
90C2×#131C2314yes—
91C22×M16C2312yes—
92#533—21no—
93#533—21no—
94#533—21no—
95#533—21no—
96#533C2×C411yescharacteristic
97#533C2×C411yescharacteristic
98#533C2×C412yes—
99#533C2×C412yes—
100#533C2312yes—
101#533C2312yes—
102#533C2311yescharacteristic
103#532C2×C411yescharacteristic
104#532C2×C411yescharacteristic
105#532C2311yescharacteristic
106C2×C4⋊Q8C2×C411yescharacteristic
107C2×C4⋊Q8C2×C411yescharacteristic
108#529—21no—
109#529—21no—
110#529—21no—
111#529—21no—
112#529C2×C412yes—
113#529C2×C412yes—
114C2×#125C2×C412yes—
115#525—21no—
116#525—21no—
117#525C2312yes—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—21no—
120C4×#41—21no—
121C4×#41—21no—
122C4×#41C2312yes—
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×#117C2312yes—
126C2×#117C2312yes—
127C22×C42C2311yescharacteristic
128#501C2×C414yes—
129#501C2×C414yes—
130#501C2×C414yes—
131#501C2×C414yes—
132#491C2×C414yes—
133#490C2×C414yes—
134#489C2×C414yes—
135#489C2×C414yes—
136#485C2×C414yes—
137#485C2×C414yes—
138#484C2×C412yes—
139#484C2×C412yes—
140#483C2×C412yes—
141#483C2×C412yes—
142#482C2×C414yes—
143#482C2×C414yes—
144#481C2×C412yes—
145#481C2×C412yes—
146#480C2×C412yes—
147#480C2×C412yes—
148#479C2×C414yes—
149#479C2×C414yes—
150#478C2×C414yes—
151#478C2×C414yes—
152#477C2×C414yes—
153#476C2×C414yes—
154#475C2×C414yes—
155#474C2×C414yes—
156#446—24no—
157#446—24no—
158#446—22no—
159#446—22no—
160#446—24no—
161#446—24no—
162#446—24no—
163#446—24no—
164#446—24no—
165#446—24no—
166C8×Q8—22no—
167C8×Q8—22no—
168#436—24no—
169#435—24no—
170#435—24no—
171#435—24no—
172#435—24no—
173C8×D4—24no—
174#433—24no—
175#433—24no—
176#433—24no—
177#432—21no—
178#432—21no—
179#432—22no—
180#432—24no—
181#432C2×C412yes—
182#432C2×C412yes—
183#432C2×C412yes—
184#432C2×C412yes—
185#432D412yes—
186#432D412yes—
187#432D412yes—
188#432D412yes—
189#431—21no—
190#431—21no—
191#431—22no—
192#431—24no—
193C2×#108C2314yes—
194C2×#108C2314yes—
195#427C2314yes—
196#427C2314yes—
197#424—24no—
198#424C2314yes—
199#424C2314yes—
200#424C2314yes—
201#423—21no—
202#423—21no—
203#423—22no—
204#423C2311yescharacteristic
205#423C2312yes—
206#423C2311yescharacteristic
207#423C2312yes—
208C2×C4⋊C8—21no—
209C2×C4⋊C8—21no—
210C2×C4⋊C8—22no—
211C2×C4⋊C8C2312yes—
212C2×C4⋊C8C2311yescharacteristic
213C2×C4⋊C8C2311yescharacteristic
214C2×#104C2314yes—
215C2×#104C2314yes—
216#416C2314yes—
217#416C2314yes—
218#408C2314yes—
219#407C2312yes—
220C2×C22⋊C8C2311yescharacteristic
221C2×C22⋊C8C2311yescharacteristic
222C2×C22⋊C8C2312yes—
223#405C2314yes—
224#405C2314yes—
225#405C2314yes—
226C4×M16C2312yes—
227C2×#97C2312yes—
228C2×#97C2312yes—
229C2×#97C2312yes—
230C2×#97C2312yes—
231C2×C4×C8C2312yes—
232C2×C4×C8C2312yes—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—21no—
235C4×C22⋊C4—21no—
236C4×C22⋊C4—21no—
237#376C2312yes—
238C43C2311yescharacteristic
239C43C2311yescharacteristic
240#341C2314yes—
241#341C2314yes—
242#339C2314yes—
243#339C2314yes—
244#339C2314yes—
245#339C2314yes—
246#339C2314yes—
247#339C2314yes—
248#336C2312yes—
249#336C2312yes—
250#336C2312yes—
251#336C2312yes—
252C2×#41—21no—
253C2×#41—21no—
254C2×#41C2412yes—
255C22×Q8—21no—
256C22×Q8—21no—
257C23×C4#4111yescharacteristic
258C23×C4#4111yescharacteristic
259C23×C4C2411yescharacteristic
260#131—44no—
261#131—44no—
262C2×M16—22no—
263C2×M16—24no—
264C2×M16—44no—
265C2×M16C2414yes—
266C2×M16C2412yes—
267C2×M16C2412yes—
268C22×C8—22no—
269C22×C8—24no—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C4⋊Q8—21no—
275C4⋊Q8—21no—
276C4⋊Q8—21no—
277C4⋊Q8—21no—
278C4⋊Q8C22×C411yescharacteristic
279C4⋊Q8C22×C412yes—
280C4⋊Q8C22×C412yes—
281C4⋊Q8C22×C412yes—
282C4⋊Q8C22×C412yes—
283C4⋊Q8C22×C411yescharacteristic
284C4⋊Q8C22×C411yescharacteristic
285#127C22×C411yescharacteristic
286#126—22no—
287#126—22no—
288#125—22no—
289#125—22no—
290#125—21no—
291#125—21no—
292#125C22×C414yes—
293#125C22×C414yes—
294#125C22×C412yes—
295#124—21no—
296#124—21no—
297#124C22×C412yes—
298#123—22no—
299#123—22no—
300#123—22no—
301C22⋊Q8—22no—
302C22⋊Q8—22no—
303C22⋊Q8—22no—
304C22⋊Q8—22no—
305C22⋊Q8—21no—
306C22⋊Q8—21no—
307C22⋊Q8—22no—
308C22⋊Q8—22no—
309C22⋊Q8—22no—
310C22⋊Q8—22no—
311C22⋊Q8—21no—
312C22⋊Q8—21no—
313C22⋊Q8C22×C412yes—
314C22⋊Q8C22×C412yes—
315#121—21no—
316#121—21no—
317#121—21no—
318#121—21no—
319#121C22×C412yes—
320#121C22×C412yes—
321#120—22no—
322C4×Q8—22no—
323C4×Q8—22no—
324C4×Q8—22no—
325C4×Q8—22no—
326C4×Q8—21no—
327C4×Q8—21no—
328C4×Q8—22no—
329C4×Q8—22no—
330C4×Q8—22no—
331C4×Q8—22no—
332C4×Q8—21no—
333C4×Q8—21no—
334C4×Q8C22×C412yes—
335C4×Q8C22×C412yes—
336C4×D4—22no—
337C4×D4—22no—
338C4×D4—22no—
339C4×D4—22no—
340C4×D4—22no—
341C4×D4—22no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4—21no—
345C4×D4—21no—
346C4×D4C22×C412yes—
347C4×D4C22×C412yes—
348#117—21no—
349#117—21no—
350#117—22no—
351#117—22no—
352#117—22no—
353#117—22no—
354#117—22no—
355#117—24no—
356#117—42no—
357#117C22×C414yes—
358#117C2414yes—
359#117C2412yes—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4C22×C412yes—
366C2×C4⋊C4C22×C412yes—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—21no—
372C2×C42—21no—
373C2×C42—21no—
374C2×C42—21no—
375C2×C42C22×C411yescharacteristic
376C2×C42C22×C411yescharacteristic
377C2×C42C22×C411yescharacteristic
378C2×C42C22×C411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42#4111yescharacteristic
384C2×C42#4111yescharacteristic
385C2×C42C2412yes—
386C2×C42C2412yes—
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389#108—44no—
390#108—44no—
391#107C22×C414yes—
392#107C22×C414yes—
393#107C22×C414yes—
394#107C22×C414yes—
395#106C22×C414yes—
396#106C22×C414yes—
397#106C22×C414yes—
398#106C22×C414yes—
399C4⋊C8—24no—
400C4⋊C8—24no—
401C4⋊C8—24no—
402C4⋊C8—24no—
403C4⋊C8—24no—
404C4⋊C8—24no—
405C4⋊C8—22no—
406C4⋊C8—22no—
407C4⋊C8—22no—
408C4⋊C8—22no—
409C4⋊C8—21no—
410C4⋊C8—21no—
411C4⋊C8—21no—
412C4⋊C8—21no—
413C4⋊C8C22×C414yes—
414C4⋊C8C22×C414yes—
415C4⋊C8C2×D412yes—
416C4⋊C8C2×D412yes—
417C4⋊C8C2×D412yes—
418C4⋊C8C2×D412yes—
419#104—28no—
420#104—44no—
421#103C22×C414yes—
422#103C22×C414yes—
423#103C22×C414yes—
424#103C22×C414yes—
425#102C22×C414yes—
426#102C22×C414yes—
427#102C22×C414yes—
428#102C22×C414yes—
429C22⋊C8—22no—
430C22⋊C8—22no—
431C22⋊C8—24no—
432C22⋊C8—24no—
433C22⋊C8C22×C412yes—
434C22⋊C8C22×C412yes—
435C22⋊C8C22×C412yes—
436C22⋊C8C22×C412yes—
437C22⋊C8C2×D414yes—
438C22⋊C8C2×D414yes—
439#97—24no—
440#97—24no—
441#97—24no—
442#97—22no—
443#97—24no—
444#97—42no—
445#97C22×C414yes—
446#97C22×C414yes—
447#97C2×D414yes—
448#97C2×D414yes—
449C4×C8—24no—
450C4×C8—24no—
451C4×C8—24no—
452C4×C8—22no—
453C4×C8—24no—
454C4×C8—42no—
455#95—22no—
456#95—22no—
457C24C2×#4111yescharacteristic
458#41—24no—
459#41—24no—
460#41—82no—
461#41—82no—
462C2×Q8—22no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—22no—
466C2×Q8—22no—
467C2×Q8—22no—
468C2×Q8—82no—
469C2×Q8—82no—
470C2×Q8C23×C412yes—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—22no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4C23×C412yes—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—81no—
487C22×C4—81no—
488C22×C4C4×D411yescharacteristic
489C22×C4C4×D411yescharacteristic
490C22×C4C4×D411yescharacteristic
491C22×C4C4×D411yescharacteristic
492C22×C4C4×D411yescharacteristic
493C22×C4C4×D411yescharacteristic
494C22×C4C4×D411yescharacteristic
495C22×C4C4×D411yescharacteristic
496C22×C4C23×C411yescharacteristic
497C22×C4C23×C411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C2×#4111yescharacteristic
501C22×C4C2×#4111yescharacteristic
502C22×C4C2×#4111yescharacteristic
503C22×C4C2×#4111yescharacteristic
504C22×C4C2×#4111yescharacteristic
505C22×C4C2511yescharacteristic
506M16—24no—
507M16—24no—
508M16—24no—
509M16—24no—
510M16—84no—
511M16—82no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—24no—
523C2×C8—84no—
524C2×C8—82no—
525C2×C8C23×C414yes—
526C2×C8C23×C414yes—
527C2×C8C22×D414yes—
528C2×C8C22×D414yes—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—44no—
550C4⋊C4C23×C414yes—
551C4⋊C4C23×C414yes—
552C4⋊C4C23×C412yes—
553C4⋊C4C23×C412yes—
554C22⋊C4—24no—
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—24no—
558C22⋊C4—22no—
559C22⋊C4—22no—
560C22⋊C4—22no—
561C22⋊C4—22no—
562C22⋊C4—22no—
563C22⋊C4—22no—
564C22⋊C4—22no—
565C22⋊C4—44no—
566C42—22no—
567C42—22no—
568C42—24no—
569C42—22no—
570C42—24no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—81no—
575C42—81no—
576C42—81no—
577C42—81no—
578C42C4×D411yescharacteristic
579C42C4×D411yescharacteristic
580C42C4×D411yescharacteristic
581C42C4×D411yescharacteristic
582C42C4×D411yescharacteristic
583C42C4×D411yescharacteristic
584C42C4×D411yescharacteristic
585C42C4×D411yescharacteristic
586C42C23×C411yescharacteristic
587C42C23×C411yescharacteristic
588C42C23×C411yescharacteristic
589C42C23×C411yescharacteristic
590C42C22×D411yescharacteristic
591C42C22×D411yescharacteristic
592C42C22×D411yescharacteristic
593C42C22×D411yescharacteristic
594C422−1+414yes—
595C23—22no—
596C23—22no—
597C23—41no—
598C23—41no—
599C23C2×C4×D411yescharacteristic
600C23C2×C4×D411yescharacteristic
601C23C22×#4111yescharacteristic
602Q8—24no—
603Q8—44no—
604Q8—82no—
605Q8—82no—
606Q8—82no—
607Q8—81no—
608Q8—81no—
609D4—24no—
610D4—44no—
611D4—44no—
612D4—81no—
613D4—81no—
614D4—82no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—82no—
650C2×C4—82no—
651C2×C4C2×C4×D411yescharacteristic
652C2×C4C2×C4×D411yescharacteristic
653C2×C4C2×C4×D411yescharacteristic
654C2×C4C2×C4×D411yescharacteristic
655C2×C4C2×C4×D411yescharacteristic
656C2×C4C2×C4×D411yescharacteristic
657C2×C4C2×C4×D411yescharacteristic
658C2×C4C2×C4×D411yescharacteristic
659C2×C4C2×C4×D411yescharacteristic
660C2×C4C2×C4×D411yescharacteristic
661C2×C4C24×C411yescharacteristic
662C2×C4C23×D411yescharacteristic
663C2×C4C2×2−1+412yes—
664C2×C4#58512yes—
665C8—24no—
666C8—24no—
667C8—24no—
668C8—24no—
669C8—42no—
670C8—44no—
671C8—42no—
672C22—21no—
673C22—21no—
674C22—21no—
675C22—21no—
676C22—21no—
677C22—21no—
678C22—21no—
679C22—21no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—82no—
684C22C22×C4×D411yescharacteristic
685C22#349512yes—
686C4—21no—
687C4—21no—
688C4—21no—
689C4—21no—
690C4—21no—
691C4—21no—
692C4—21no—
693C4—21no—
694C4—21no—
695C4—21no—
696C4—44no—
697C4—42no—
698C4—44no—
699C4—42no—
700C4—44no—
701C4—42no—
702C4—82no—
703C4C4×2−1+412yes—
704C4#349911yescharacteristic
705C4#350011yescharacteristic
706C2—22no—
707C2—41no—
708C2—41no—
709C2—82no—
710C2#3427611yescharacteristic
711C2#6018211yescharacteristic
712C2#6018211yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC22C22×C4×D4
prime radical#3444C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#432 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #60251C2semidirect
4. #23082C2semidirect
5. #22984C2semidirect
6. #22951C2semidirect
7. #22949C2semidirect
8. #22259C2semidirect
9. #22237C2semidirect
10. #21994C2semidirect
11. #21736C2semidirect
12. #21662C2semidirect
14. #21329C2semidirect
15. #20467C2semidirect
18. #3240C4semidirect
19. #3239C4semidirect
20. #3238C4semidirect
21. #3237C4semidirect
22. #3236C4semidirect
23. #3235C4semidirect
24. #3234C4semidirect

Maximal subgroups

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

subgroupindex
#608542
#602692
#602512
#230822
#229842
#229512
#229492
#222592
#222372
#219942
#217362
#216622
#214182
#213292
#204672