Finite Group Databasebeta

← #228206

Subgroups — #228206

Order 512.

790 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22983C211yesmaximal, characteristic, complemented
2D4×#104C211yesmaximal, characteristic
3#18517C211yesmaximal, characteristic
4#18000C211yesmaximal, characteristic, complemented
5#17507C211yesmaximal, characteristic
6#9851C211yesmaximal, characteristic, complemented
7#9592C211yesmaximal, characteristic, complemented
8#9581C211yesmaximal, characteristic, complemented
9#9231C211yesmaximal, characteristic, complemented
10#9207C211yesmaximal, characteristic, complemented
11#9054C211yesmaximal, characteristic, complemented
12#8906C211yesmaximal, characteristic, complemented
13#8902C211yesmaximal, characteristic, complemented
14#8192C211yesmaximal, characteristic, complemented
15#8172C211yesmaximal, characteristic, complemented
16D4×#41C2211yescharacteristic
17#3167C2211yescharacteristic
18#3101C2211yescharacteristic
19#3076C2211yescharacteristic
20C8×#41C2211yescharacteristic
21#2962C2211yescharacteristic, complemented
22D4×M16C2211yescharacteristic
23#2906C2211yescharacteristic
24C22×#104—21no—
25C42×D4C2211yescharacteristic
26#1994C2211yescharacteristic, complemented
27#1980C2211yescharacteristic
28#1962C2211yescharacteristic
29#1950C2211yescharacteristic, complemented
30#1946—21no—
31#1922—21no—
32#1896—21no—
33#1871—21no—
34#1866—21no—
35#1806C2211yescharacteristic
36#1789—21no—
37#1767C2211yescharacteristic, complemented
38#1765—21no—
39C4×#104C2211yescharacteristic
40C4×#97C2211yescharacteristic, complemented
41#1654C411yescharacteristic, complemented
42#1651C411yescharacteristic, complemented
43#1642C411yescharacteristic, complemented
44#1639C411yescharacteristic, complemented
45#1637C411yescharacteristic, complemented
46#1635C411yescharacteristic, complemented
47#1631C411yescharacteristic, complemented
48#1629C411yescharacteristic, complemented
49#1592—21no—
50#1588—21no—
51#1585—21no—
52SD16⋊C8—21no—
53#1571C2211yescharacteristic, complemented
54#1567C2211yescharacteristic, complemented
55#1540C2211yescharacteristic
56#1503C2211yescharacteristic, complemented
57#1493C2211yescharacteristic, complemented
58#1492—21no—
59#1492C2211yescharacteristic
60#1486—21no—
61#1486C2211yescharacteristic
62#1480—21no—
63#1480C2211yescharacteristic
64#1476C2211yescharacteristic
65C2×D4⋊C8—21no—
66C2×D4⋊C8C2211yescharacteristic
67C2×D4⋊C8C2211yescharacteristic
68#1451C2211yescharacteristic, complemented
69#1303C2211yescharacteristic, complemented
70#1301C2211yescharacteristic, complemented
71#1279C2211yescharacteristic, complemented
72#1279C2211yescharacteristic, complemented
73#1278C2211yescharacteristic, complemented
74#1276C2211yescharacteristic, complemented
75C22×#41—21no—
76C2×#131—21no—
77C22×M16—21no—
78#550C2×C411yescharacteristic
79D4×Q8C2×C411yescharacteristic
80#548—21no—
81#547—21no—
82#547C2×C411yescharacteristic
83#546—21no—
84#546—41no—
85D42—21no—
86D42C2×C411yescharacteristic
87#533C2311yescharacteristic
88#532—21no—
89#532C2311yescharacteristic
90#525—21no—
91#525—41no—
92C4×#41C2311yescharacteristic
93C2×C4×D4—21no—
94C2×C4×D4—21no—
95C2×C4×D4C2311yescharacteristic
96C22×C42—21no—
97#498C2×C411yescharacteristic, complemented
98#492C2×C411yescharacteristic, complemented
99#487C2×C411yescharacteristic, complemented
100#486C2×C411yescharacteristic, complemented
101#483—21no—
102#479—21no—
103#478—21no—
104#477C2×C411yescharacteristic, complemented
105#476C2×C411yescharacteristic
106#476C2×C411yescharacteristic
107#476C2×C411yescharacteristic, complemented
108#475C2×C411yescharacteristic, complemented
109#474C2×C411yescharacteristic, complemented
110#465—41no—
111#461C2×C411yescharacteristic
112#461C2×C411yescharacteristic
113#450—21no—
114#446—21no—
115#446—41no—
116C8×Q8—21no—
117C8×Q8—21no—
118C4×SD16—21no—
119C4×SD16—21no—
120C4×SD16—41no—
121#436—21no—
122#435—41no—
123C8×D4—21no—
124C8×D4—21no—
125C8×D4—41no—
126#431—21no—
127#431C2311yescharacteristic
128#431C2311yescharacteristic
129C2×#108—21no—
130#427—21no—
131#424—21no—
132#424—21no—
133#424—41no—
134#423—21no—
135#423C2311yescharacteristic
136#423C2311yescharacteristic
137C2×C4⋊C8—21no—
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2311yescharacteristic
140C2×#104—21no—
141C2×#104—21no—
142C2×#104—21no—
143C2×#104—21no—
144C2×#104—21no—
145C2×#104—44no—
146C2×#104D411yescharacteristic
147C2×#104D411yescharacteristic
148#416—21no—
149#407—21no—
150#405—21no—
151C4×M16C2311yescharacteristic
152C2×#97—21no—
153C2×#97—41no—
154C2×C4×C8—21no—
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157C4×C4⋊C4—21no—
158C4×C4⋊C4C2311yescharacteristic
159C4×C22⋊C4—21no—
160C4×C22⋊C4—41no—
161C43C2311yescharacteristic
162#341—21no—
163#339—21no—
164#339—21no—
165#339—21no—
166#339D411yescharacteristic
167#339D411yescharacteristic
168#336—21no—
169#335—21no—
170#335—21no—
171#335—41no—
172#332C2×C411yescharacteristic, complemented
173#332C2×C411yescharacteristic, complemented
174#331C2×C411yescharacteristic, complemented
175#331C2×C411yescharacteristic, complemented
176#328—41no—
177Q8⋊C8—21no—
178Q8⋊C8—21no—
179Q8⋊C8C2×C411yescharacteristic, complemented
180Q8⋊C8C2×C411yescharacteristic, complemented
181D4⋊C8—21no—
182D4⋊C8—21no—
183D4⋊C8—21no—
184D4⋊C8—21no—
185D4⋊C8—21no—
186D4⋊C8—21no—
187D4⋊C8C2×C411yescharacteristic
188D4⋊C8C2×C411yescharacteristic
189D4⋊C8C2×C411yescharacteristic
190D4⋊C8C2×C411yescharacteristic
191D4⋊C8C2×C411yescharacteristic, complemented
192D4⋊C8C2×C411yescharacteristic, complemented
193D4⋊C8D411yescharacteristic
194D4⋊C8D411yescharacteristic
195D4⋊C8D411yescharacteristic
196D4⋊C8D411yescharacteristic
197#322—21no—
198#322—21no—
199C2×#41—21no—
200C2×#41—21no—
201C2×#41—21no—
202C2×#41—41no—
203C2×#41—41no—
204C2×#41—42no—
205C2×#41—42no—
206C2×#41—81no—
207C2×#41—81no—
208C2×#41C2×D411yescharacteristic
209C22×Q8—21no—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4—41no—
213C22×D4#4111yescharacteristic
214C23×C4—21no—
215C23×C4—41no—
216C23×C4—41no—
217C2×SD16—21no—
218C2×SD16—21no—
219C2×SD16—41no—
220#131—42no—
221#131—81no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—41no—
225C2×M16—41no—
226C2×M16—42no—
227C2×M16—81no—
228C2×M16C2×D411yescharacteristic
229C2×M16C2×D411yescharacteristic
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—41no—
235C22×C8—41no—
236C22×C8—41no—
237C4⋊Q8—21no—
238C4⋊Q8C22×C411yescharacteristic
239C4⋊Q8C22×C411yescharacteristic
240C4⋊Q8C22×C411yescharacteristic
241#127—21no—
242#127C22×C411yescharacteristic
243#125—21no—
244#124—21no—
245#124—41no—
246#123—41no—
247#123—41no—
248#123—41no—
249C22⋊Q8—21no—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252C22⋊Q8—21no—
253C22⋊Q8—41no—
254C22⋊Q8—41no—
255#121—21no—
256#121—21no—
257#121—21no—
258#121—21no—
259#121—21no—
260#121—21no—
261#121—41no—
262#121—41no—
263#121—41no—
264#120—41no—
265#120—41no—
266#120—41no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8C22×C411yescharacteristic
270C4×Q8#4111yescharacteristic
271C4×D4—21no—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—41no—
280C4×D4—41no—
281C4×D4—41no—
282C4×D4—42no—
283C4×D4—42no—
284C4×D4C22×C411yescharacteristic
285C4×D4C22×C411yescharacteristic
286C4×D4C22×C411yescharacteristic
287C4×D4C2×D411yescharacteristic
288C4×D4C2×D411yescharacteristic
289C4×D4#4111yescharacteristic
290#117—21no—
291#117—21no—
292#117C2×D411yescharacteristic
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4#4111yescharacteristic
296C2×C22⋊C4—21no—
297C2×C22⋊C4—41no—
298C2×C22⋊C4—41no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—42no—
304C2×C42—41no—
305C2×C42—81no—
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2411yescharacteristic
309#108—81no—
310#107—21no—
311#107—21no—
312#106—21no—
313#106—21no—
314#106—21no—
315#106—21no—
316#106—41no—
317C4⋊C8—21no—
318C4⋊C8—21no—
319C4⋊C8—21no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8—21no—
323C4⋊C8—21no—
324C4⋊C8—41no—
325C4⋊C8—41no—
326C4⋊C8—41no—
327C4⋊C8—41no—
328C4⋊C8—41no—
329C4⋊C8C22×C411yescharacteristic
330C4⋊C8C22×C411yescharacteristic
331C4⋊C8C22×C411yescharacteristic
332C4⋊C8C22×C411yescharacteristic
333C4⋊C8C2×D411yescharacteristic
334C4⋊C8C2×D411yescharacteristic
335#104—22no—
336#104—22no—
337#104—42no—
338#104—42no—
339#104—42no—
340#104—81no—
341#104—81no—
342#103—21no—
343#103—21no—
344#103—21no—
345#103—21no—
346#103—41no—
347#102—21no—
348#102—21no—
349#102—21no—
350#102—21no—
351#102—41no—
352C22⋊C8—21no—
353C22⋊C8—21no—
354C22⋊C8—41no—
355C22⋊C8—41no—
356C22⋊C8—41no—
357#97—21no—
358#97—41no—
359#97—44no—
360#97—81no—
361C4×C8—21no—
362C4×C8—21no—
363C4×C8—21no—
364C4×C8—21no—
365C4×C8—21no—
366C4×C8—41no—
367C4×C8—41no—
368C4×C8C22×C411yescharacteristic
369C4×C8C22×C411yescharacteristic
370C4×C8C2×D411yescharacteristic
371C4×C8C2×D411yescharacteristic
372#95—21no—
373#95—41no—
374C24—21no—
375C24—41no—
376#41—21no—
377#41—21no—
378#41—42no—
379#41—42no—
380#41—41no—
381#41—41no—
382#41—41no—
383#41—41no—
384#41—82no—
385#41—82no—
386#41—81no—
387#41—82no—
388#41—82no—
389#41—81no—
390#41AGL(1,ℤ/8)12yes—
391#41#13712yes—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—41no—
396C2×Q8—42no—
397C2×Q8—81no—
398C2×Q8C4×D411yescharacteristic
399C2×D4—21no—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—81no—
415C2×D4—81no—
416C2×D4—82no—
417C2×D4—82no—
418C2×D4C4×D411yescharacteristic
419C2×D4C4×D411yescharacteristic
420C2×D4C4×D411yescharacteristic
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—44no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4C22×D411yescharacteristic
446C22×C4C2×#4111yescharacteristic
447C22×C4H(5,𝔽2)11yescharacteristic
448SD16—42no—
449SD16—82no—
450SD16—81no—
451M16—21no—
452M16—21no—
453M16—21no—
454M16—21no—
455M16—41no—
456M16—41no—
457M16—41no—
458M16—41no—
459M16—82no—
460M16—82no—
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—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—82no—
483C2×C8—82no—
484C2×C8—82no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4C4×D411yescharacteristic
501C4⋊C4C4×D411yescharacteristic
502C4⋊C4C4×D411yescharacteristic
503C4⋊C4C4×D411yescharacteristic
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—41no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—44no—
516C22⋊C4—41no—
517C22⋊C4—41no—
518C22⋊C4—41no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—42no—
527C42—41no—
528C42—41no—
529C42—41no—
530C42—41no—
531C42—41no—
532C42—82no—
533C42C23×C411yescharacteristic
534C42C22×D411yescharacteristic
535C42C2×#4111yescharacteristic
536C42H(5,𝔽2)11yescharacteristic
537C23—21no—
538C23—21no—
539C23—21no—
540C23—42no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—81no—
545C23—82no—
546C23—82no—
547C23—161no—
548C23#54611yescharacteristic
549Q8—21no—
550Q8—41no—
551Q8—41no—
552Q8—41no—
553Q8—41no—
554Q8—82no—
555Q8—81no—
556Q8#44012yes—
557D4—22no—
558D4—22no—
559D4—21no—
560D4—21no—
561D4—21no—
562D4—21no—
563D4—42no—
564D4—41no—
565D4—41no—
566D4—41no—
567D4—41no—
568D4—82no—
569D4—81no—
570D4—82no—
571D4—82no—
572D4—82no—
573D4—81no—
574D4—81no—
575D4—161no—
576D4#44012yes—
577D4#44412yes—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—81no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—88no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4C2×C4×D411yescharacteristic
620C2×C4C2×C4×D411yescharacteristic
621C2×C4#51811yescharacteristic
622C2×C4D4211yescharacteristic
623C2×C4#54611yescharacteristic
624C2×C4#55111yescharacteristic
625C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
626C2×C4C2×#13711yescharacteristic
627C2×C4#57611yescharacteristic
628C2×C4#57711yescharacteristic
629C8—22no—
630C8—22no—
631C8—41no—
632C8—41no—
633C8—41no—
634C8—41no—
635C8—41no—
636C8—41no—
637C8—41no—
638C8—41no—
639C8—41no—
640C8—41no—
641C8—81no—
642C8—81no—
643C8—81no—
644C22—21no—
645C22—21no—
646C22—22no—
647C22—22no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—41no—
655C22—82no—
656C22—82no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—164no—
661C22#235811yescharacteristic
662C22#331011yescharacteristic
663C22#331411yescharacteristic
664C4—22no—
665C4—21no—
666C4—21no—
667C4—21no—
668C4—42no—
669C4—41no—
670C4—41no—
671C4—41no—
672C4—41no—
673C4—81no—
674C4—82no—
675C4—81no—
676C4—81no—
677C4—81no—
678C4C2×#44011yescharacteristic
679C4#294611yescharacteristic
680C4C2×#44411yescharacteristic
681C4#295611yescharacteristic
682C4#326411yescharacteristic
683C4#326611yescharacteristic
684C2—21no—
685C2—21no—
686C2—42no—
687C2—42no—
688C2—81no—
689C2—161no—
690C2#2285411yescharacteristic
691C2#2288611yescharacteristic
692C2#2289911yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2358
prime radicalD4×#41C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22983C2semidirect
4. #18000C2semidirect
6. #9851C2semidirect
7. #9592C2semidirect
8. #9581C2semidirect
9. #9231C2semidirect
10. #9207C2semidirect
11. #9054C2semidirect
12. #8906C2semidirect
13. #8902C2semidirect
14. #8192C2semidirect
15. #8172C2semidirect
21. #2962C22semidirect
26. #1994C22semidirect
29. #1950C22semidirect
37. #1767C22semidirect
40. C4×#97C22semidirect
41. #1654C4semidirect
42. #1651C4semidirect
43. #1642C4semidirect
44. #1639C4semidirect
45. #1637C4semidirect
46. #1635C4semidirect
47. #1631C4semidirect
48. #1629C4semidirect
53. #1571C22semidirect
54. #1567C22semidirect
56. #1503C22semidirect
57. #1493C22semidirect
68. #1451C22semidirect
69. #1303C22semidirect
70. #1301C22semidirect
71. #1279C22semidirect
72. #1279C22semidirect
73. #1278C22semidirect
74. #1276C22semidirect
97. #498C2×C4semidirect
98. #492C2×C4semidirect
99. #487C2×C4semidirect
100. #486C2×C4semidirect
104. #477C2×C4semidirect
107. #476C2×C4semidirect
108. #475C2×C4semidirect
109. #474C2×C4semidirect
172. #332C2×C4semidirect
173. #332C2×C4semidirect
174. #331C2×C4semidirect
175. #331C2×C4semidirect
179. Q8⋊C8C2×C4semidirect
180. Q8⋊C8C2×C4semidirect
191. D4⋊C8C2×C4semidirect
192. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#229832
D4×#1042
#185172
#180002
#175072
#98512
#95922
#95812
#92312
#92072
#90542
#89062
#89022
#81922
#81722