Finite Group Databasebeta

← #385244

Subgroups — #385244

Order 512.

936 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1953C211yesmaximal, characteristic, complemented
2#21178C211yesmaximal, characteristic, complemented
3#21098C211yesmaximal, characteristic
4#21072C211yesmaximal, characteristic
5C2×#1866C211yesmaximal, characteristic
6C2×#1814C211yesmaximal, characteristic
7#20669C211yesmaximal, characteristic
8#11341C211yesmaximal, characteristic, complemented
9#11326C211yesmaximal, characteristic, complemented
10#11250C211yesmaximal, characteristic, complemented
11#11241C211yesmaximal, characteristic, complemented
12#11114C211yesmaximal, characteristic
13#11080C211yesmaximal, characteristic
14#10909C211yesmaximal, characteristic, complemented
15#10901C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C2×#430C2211yescharacteristic
19C2×#430C2211yescharacteristic
20C22×#108C2211yescharacteristic
21C22×#108C2211yescharacteristic
22C22×#108C2211yescharacteristic
23C2×#427C2211yescharacteristic
24#2905C2211yescharacteristic
25C2×#424C2211yescharacteristic
26C2×#423C2211yescharacteristic
27C2×#421C2211yescharacteristic
28C2×#421C2211yescharacteristic
29C22×#104C2211yescharacteristic
30C2×#416C2211yescharacteristic
31#2879C2211yescharacteristic
32#2879C2211yescharacteristic
33#2873C2211yescharacteristic
34C2×#405C2211yescharacteristic
35#1953—24no—
36#1950—22no—
37#1950—22no—
38#1938—22no—
39#1937—22no—
40#1929—22no—
41#1929—22no—
42#1866C412yescomplemented
43#1866C412yescomplemented
44#1866C412yescomplemented
45#1866C412yescomplemented
46#1814—24no—
47#1811—22no—
48#1810—22no—
49C2×#344C2211yescharacteristic
50C2×#344C2211yescharacteristic
51C2×#344C2211yescharacteristic
52C2×#344C2211yescharacteristic
53#1738C2211yescharacteristic
54C2×#341C2211yescharacteristic
55C2×#341C2211yescharacteristic
56C2×#341C2211yescharacteristic
57#1733C2211yescharacteristic
58#1733C2211yescharacteristic
59C2×#339C2211yescharacteristic
60C2×#339C2211yescharacteristic
61C2×#339C2211yescharacteristic
62#1729C2211yescharacteristic
63#1728C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C22×#41C2311yescharacteristic
66C2×#131—21no—
67C2×#131—21no—
68C2×#131—22no—
69C2×#131—22no—
70C2×#131—41no—
71C2×#131—41no—
72C2×#131C2×C412yes—
73C2×#131C2×C412yes—
74C22×M16—21no—
75C22×M16C2311yescharacteristic
76C22×M16C2311yescharacteristic
77C22×M16C2311yescharacteristic
78C22×M16C2311yescharacteristic
79C22×M16C2311yescharacteristic
80C22×M16C2311yescharacteristic
81C22×M16C2311yescharacteristic
82C22×M16C2311yescharacteristic
83C23×C8—21no—
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C2×#117C2311yescharacteristic
87C2×#117C2311yescharacteristic
88C2×#117C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#430—22no—
91#430—22no—
92#430—22no—
93#430—41no—
94#430—41no—
95#430—41no—
96C2×#108—22no—
97C2×#108—22no—
98C2×#108—22no—
99C2×#108—22no—
100C2×#108—22no—
101C2×#108—22no—
102C2×#108—22no—
103C2×#108—21no—
104C2×#108—21no—
105C2×#108—21no—
106C2×#108—21no—
107C2×#108—41no—
108C2×#108—41no—
109C2×#108C2×C412yes—
110C2×#108C2×C412yes—
111#427—22no—
112#427—41no—
113C2×#107—21no—
114C2×#106—21no—
115#424—21no—
116#424—21no—
117#424—22no—
118#424—41no—
119#424C2×C412yes—
120#424C2×C412yes—
121#423—22no—
122#423—41no—
123#423—41no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8—41no—
127#421—22no—
128#421—22no—
129#421—41no—
130#421—41no—
131C2×#104—22no—
132C2×#104—22no—
133C2×#104—41no—
134C2×#104—41no—
135C2×#104C2×C412yes—
136C2×#104C2×C412yes—
137#416C2×C412yes—
138#416C2×C412yes—
139#416D412yes—
140#416D412yes—
141C2×#103Q811yescharacteristic
142C2×#103Q811yescharacteristic
143C2×#102Q811yescharacteristic
144C2×#102Q811yescharacteristic
145#408—22no—
146#408—22no—
147#408—41no—
148#408—41no—
149#407—22no—
150#407—22no—
151#405—22no—
152#405—22no—
153#405—41no—
154#405—41no—
155C4×M16—22no—
156C2×#97—21no—
157C2×C4×C8—21no—
158#344—22no—
159#344—22no—
160#344—41no—
161#344D414yes—
162#341—22no—
163#341—21no—
164#341—21no—
165#341—21no—
166#341—21no—
167#341D412yes—
168#341D412yes—
169#339—22no—
170#339—22no—
171#339—41no—
172#339—41no—
173#339C2×C412yes—
174#339C2×C412yes—
175#336—21no—
176#336—21no—
177#336—41no—
178#336C2×C411yescharacteristic
179#336C2×C411yescharacteristic
180#336C2×C411yescharacteristic
181#336C2×C411yescharacteristic
182C2×#41—21no—
183C2×#41—21no—
184C2×#41—21no—
185C2×#41—21no—
186C2×#41—21no—
187C2×#41—21no—
188C2×#41—21no—
189C2×#41—21no—
190C2×#41—21no—
191C2×#41—21no—
192C2×#41C22×C412yes—
193C2×#41C2×D412yes—
194C22×Q8C2×Q811yescharacteristic
195C22×D4—21no—
196C22×D4C2×Q811yescharacteristic
197C23×C4—21no—
198C23×C4—21no—
199C23×C4C2411yescharacteristic
200#131—24no—
201#131—24no—
202#131—42no—
203#131—42no—
204#131—42no—
205#131—42no—
206#131—42no—
207#131—42no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—22no—
216C2×M16—22no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—21no—
232C2×M16—21no—
233C2×M16—21no—
234C2×M16—21no—
235C2×M16—21no—
236C2×M16—21no—
237C2×M16—21no—
238C2×M16—21no—
239C2×M16—21no—
240C2×M16—21no—
241C2×M16—21no—
242C2×M16—21no—
243C2×M16—21no—
244C2×M16—21no—
245C2×M16—21no—
246C2×M16—21no—
247C2×M16—21no—
248C2×M16—21no—
249C2×M16—21no—
250C2×M16—21no—
251C2×M16—42no—
252C2×M16—42no—
253C2×M16C22×C412yes—
254C2×M16C22×C412yes—
255C2×M16C2×D412yes—
256C2×M16C2×D412yes—
257C2×M16C2×D412yes—
258C2×M16C2×D412yes—
259C2×M16#4112yes—
260C22×C8—21no—
261C22×C8—21no—
262C22×C8—21no—
263C22×C8—21no—
264C22×C8—21no—
265C22×C8—21no—
266C22×C8—21no—
267C22×C8—21no—
268C22×C8—21no—
269C22×C8—21no—
270C22×C8—21no—
271C22×C8—21no—
272C22×C8—21no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8—21no—
276C22×C8—21no—
277C22×C8—21no—
278C22×C8—42no—
279C22×C8—42no—
280C22×C8C22×C411yescharacteristic
281C22×C8C22×C411yescharacteristic
282C22×C8C22×C411yescharacteristic
283C22×C8C22×C411yescharacteristic
284C22×C8C2×D411yescharacteristic
285C22×C8C2×D411yescharacteristic
286C22×C8C2×D411yescharacteristic
287C22×C8C2×D411yescharacteristic
288C22×C8C2×Q811yescharacteristic
289C22×C8C2×Q811yescharacteristic
290C22×C8#4111yescharacteristic
291C22×C8#4111yescharacteristic
292#117—21no—
293#117—21no—
294#117—21no—
295#117—21no—
296#117—21no—
297#117—21no—
298#117—21no—
299#117—21no—
300#117—21no—
301#117C22×C412yes—
302#117C2×D412yes—
303#117C2×D412yes—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4C2×Q811yescharacteristic
307C2×C4⋊C4C2×Q811yescharacteristic
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42C22×C411yescharacteristic
320C2×C42C22×C411yescharacteristic
321#108—24no—
322#108—24no—
323#108—42no—
324#108—42no—
325#108—42no—
326#108—42no—
327#108—44no—
328#108—44no—
329#108—44no—
330#107—42no—
331#106—42no—
332C4⋊C8—22no—
333C4⋊C8—22no—
334C4⋊C8—22no—
335C4⋊C8—22no—
336C4⋊C8—44no—
337C4⋊C8—44no—
338#104—42no—
339#104—42no—
340#104—44no—
341#104—44no—
342#103C4⋊C412yes—
343#103C4⋊C412yes—
344#103C4⋊C412yes—
345#103C4⋊C412yes—
346#102C4⋊C412yes—
347#102C4⋊C412yes—
348#102C4⋊C412yes—
349#102C4⋊C412yes—
350C22⋊C8—44no—
351C22⋊C8—44no—
352#97—42no—
353#97—42no—
354C4×C8—42no—
355C4×C8—42no—
356C24—21no—
357C242−1+411yescharacteristic
358#41—22no—
359#41—22no—
360#41—22no—
361#41—22no—
362#41—22no—
363#41—22no—
364#41—22no—
365#41—22no—
366#41—41no—
367#41—41no—
368#41—41no—
369#41—41no—
370#41—41no—
371#41—41no—
372#41—41no—
373#41—41no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8C2×C4⋊C412yes—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4C2×C4⋊C412yes—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—42no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—81no—
409C22×C4C23×C411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×Q811yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4H(5,𝔽2)11yescharacteristic
417C22×C4H(5,𝔽2)11yescharacteristic
418C22×C4H(5,𝔽2)11yescharacteristic
419C22×C4H(5,𝔽2)11yescharacteristic
420C22×C42−1+411yescharacteristic
421C22×C42−1+411yescharacteristic
422C22×C42−1+411yescharacteristic
423M16—22no—
424M16—22no—
425M16—22no—
426M16—22no—
427M16—22no—
428M16—22no—
429M16—22no—
430M16—22no—
431M16—22no—
432M16—22no—
433M16—22no—
434M16—44no—
435M16—44no—
436M16—41no—
437M16—41no—
438M16—41no—
439M16—41no—
440M16—41no—
441M16—41no—
442M16—41no—
443M16—41no—
444M16—41no—
445M16—41no—
446M16—41no—
447M16—41no—
448M16—41no—
449M16—41no—
450M16—41no—
451M16—41no—
452M16—41no—
453M16—41no—
454M16—41no—
455M16—41no—
456M16C4×D414yes—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—22no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—22no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—22no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—44no—
489C2×C8—44no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—41no—
497C2×C8—41no—
498C2×C8—41no—
499C2×C8—41no—
500C2×C8—41no—
501C2×C8—41no—
502C2×C8—41no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—41no—
506C2×C8—41no—
507C2×C8C2×C4⋊C411yescharacteristic
508C2×C8C2×C4⋊C411yescharacteristic
509C2×C8C2×C4⋊C411yescharacteristic
510C2×C8C2×C4⋊C411yescharacteristic
511C2×C8C4×D411yescharacteristic
512C2×C8C4×D411yescharacteristic
513C2×C8C4×D411yescharacteristic
514C2×C8C4×D411yescharacteristic
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—22no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—41no—
521C4⋊C4C2×C4⋊C412yes—
522C4⋊C4C2×C4⋊C412yes—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—41no—
527C22⋊C4—41no—
528C22⋊C4—41no—
529C42—22no—
530C42—22no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—22no—
536C42—41no—
537C42—41no—
538C42—41no—
539C42—41no—
540C42—41no—
541C23—22no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—81no—
547C23—81no—
548C23#52011yescharacteristic
549C23#53611yescharacteristic
550C23#54111yescharacteristic
551C23#54711yescharacteristic
552C23D4×Q811yescharacteristic
553C23#56211yescharacteristic
554C23#56311yescharacteristic
555Q8—22no—
556Q8—22no—
557Q8—41no—
558Q8—41no—
559D4—22no—
560D4—22no—
561D4—42no—
562D4—42no—
563D4—42no—
564D4—42no—
565D4—41no—
566D4—41no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4C22×C4⋊C411yescharacteristic
593C2×C4C2×C4×D411yescharacteristic
594C2×C4C2×C4×D411yescharacteristic
595C2×C4#51811yescharacteristic
596C2×C4#51911yescharacteristic
597C2×C4#52011yescharacteristic
598C2×C4#53611yescharacteristic
599C2×C4#53711yescharacteristic
600C2×C4#53711yescharacteristic
601C2×C4#53711yescharacteristic
602C2×C4#53711yescharacteristic
603C2×C4#54111yescharacteristic
604C2×C4#54211yescharacteristic
605C2×C4#54211yescharacteristic
606C2×C4#54411yescharacteristic
607C2×C4#54411yescharacteristic
608C2×C4D4211yescharacteristic
609C2×C4D4211yescharacteristic
610C2×C4D4211yescharacteristic
611C2×C4#54611yescharacteristic
612C2×C4#54611yescharacteristic
613C2×C4#54711yescharacteristic
614C2×C4D4×Q811yescharacteristic
615C2×C4#55111yescharacteristic
616C2×C4#55411yescharacteristic
617C2×C4#55411yescharacteristic
618C2×C4#56011yescharacteristic
619C2×C4#56211yescharacteristic
620C8—21no—
621C8—21no—
622C8—21no—
623C8—21no—
624C8—21no—
625C8—21no—
626C8—21no—
627C8—21no—
628C8—24no—
629C8—42no—
630C8—42no—
631C8—42no—
632C8—42no—
633C8—81no—
634C8—81no—
635C8—81no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—82no—
650C22—82no—
651C22#232311yescharacteristic
652C22D4×C4⋊C411yescharacteristic
653C22#237611yescharacteristic
654C22#267211yescharacteristic
655C22#267711yescharacteristic
656C22#270411yescharacteristic
657C22#272211yescharacteristic
658C4—42no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—81no—
663C4—82no—
664C4—161no—
665C4#231611yescharacteristic
666C4#232311yescharacteristic
667C4D4×C4⋊C411yescharacteristic
668C4#235811yescharacteristic
669C4#267211yescharacteristic
670C4#267511yescharacteristic
671C4#270411yescharacteristic
672C4#274211yescharacteristic
673C2—21no—
674C2—21no—
675C2—21no—
676C2—21no—
677C2—21no—
678C2—21no—
679C2—82no—
680C2#1440211yescharacteristic
681C2#2269411yescharacteristic
682C2#2269511yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22D4×C4⋊C4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1953C2semidirect
2. #21178C2semidirect
8. #11341C2semidirect
9. #11326C2semidirect
10. #11250C2semidirect
11. #11241C2semidirect
14. #10909C2semidirect
15. #10901C2semidirect
42. #1866C4semidirect
43. #1866C4semidirect
44. #1866C4semidirect
45. #1866C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19532
#211782
#210982
#210722
C2×#18662
C2×#18142
#206692
#113412
#113262
#112502
#112412
#111142
#110802
#109092
#109012