Finite Group Databasebeta

← #392942

Subgroups — #392942

Order 512.

1474 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#131C211yesmaximal, characteristic
2C2×#1950C211yesmaximal, characteristic, complemented
3#21143C211yesmaximal, characteristic, complemented
4C2×#1866C212yesmaximal
5C2×#1806C211yesmaximal, characteristic
6#20645C211yesmaximal, characteristic
7#10910C212yesmaximal, complemented
8#10887C212yesmaximal, complemented
9#10851C211yesmaximal, characteristic, complemented
10#10834C211yesmaximal, characteristic, complemented
11#10791C212yesmaximal
12C22×#131C2211yescharacteristic
13C22×#131C2211yescharacteristic
14C2×C4×#41C2211yescharacteristic
15C22×#108C2212yes—
16C2×#424C2212yes—
17C2×#423C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×#104C2212yes—
20C2×#416C2212yes—
21C4×#131—24no—
22C4×#131—24no—
23C2×#405—21no—
24C2×#405—21no—
25C2×#405C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C2×C4×M16—21no—
28C2×C4×M16C2211yescharacteristic
29C22×C4×C8—21no—
30C22×C4×C8C2211yescharacteristic
31#2285C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1950—24no—
34#1939—21no—
35#1939—21no—
36#1939—21no—
37#1939—21no—
38#1866—28no—
39#1806—24no—
40#1805C412yescomplemented
41#1804C412yescomplemented
42#1803C411yescharacteristic, complemented
43#1803C411yescharacteristic, complemented
44#1802C411yescharacteristic, complemented
45#1802C411yescharacteristic, complemented
46C2×#341C2212yes—
47#1735C2212yes—
48C2×#339C2212yes—
49C2×#339C2212yes—
50C2×#339C2212yes—
51#1729C2212yes—
52C2×#336C2212yes—
53C2×#336C2212yes—
54C22×#41C2311yescharacteristic
55C2×#131—22no—
56C2×#131—22no—
57C2×#131—22no—
58C2×#131—22no—
59C2×#131—22no—
60C2×#131—22no—
61C2×#131—22no—
62C2×#131—22no—
63C2×#131—22no—
64C2×#131—22no—
65C2×#131D414yes—
66C2×#131Q814yes—
67C22×M16—21no—
68C22×M16—21no—
69C22×M16C2312yes—
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C23×C8—21no—
73C23×C8—21no—
74C23×C8C2312yes—
75C23×C8C2311yescharacteristic
76C23×C8C2311yescharacteristic
77C4×#41—22no—
78C4×#41—22no—
79C4×#41D414yes—
80C4×#41Q814yes—
81C2×C4×Q8C2×C411yescharacteristic
82C2×C4×Q8C2×C411yescharacteristic
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×C4×D4C2×C411yescharacteristic
86C2×C4×D4C2×C411yescharacteristic
87C2×#117—21no—
88C2×#117C2312yes—
89C2×#117C2311yescharacteristic
90C22×C4⋊C4—21no—
91C22×C42—21no—
92C22×C42C2312yes—
93C22×C42C2311yescharacteristic
94C2×#108—24no—
95C2×#108—24no—
96C2×#108—24no—
97#424—24no—
98#424—24no—
99#423—22no—
100#423—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8C2×C411yescharacteristic
108C2×C4⋊C8C2×C411yescharacteristic
109C2×C4⋊C8C2×C412yes—
110C2×#104—24no—
111C2×#104—24no—
112C2×#104—24no—
113#416—24no—
114#416—24no—
115C2×#103C2×C412yes—
116C2×#103C2×C412yes—
117C2×#102C2×C412yes—
118C2×#102C2×C412yes—
119#405—28no—
120#405—28no—
121#405—22no—
122#405—22no—
123#405—22no—
124#405—22no—
125C4×M16—28no—
126C4×M16—22no—
127C4×M16D414yes—
128C2×#97—21no—
129C2×#97—21no—
130C2×#97—21no—
131C2×C4×C8—28no—
132C2×C4×C8—21no—
133C2×C4×C8—21no—
134C2×C4×C8—21no—
135C2×C4×C8—22no—
136C2×C4×C8—22no—
137C2×C4×C8D411yescharacteristic
138C2×C4×C8D411yescharacteristic
139C2×C4×C8D411yescharacteristic
140C2×C4×C8D411yescharacteristic
141#384C2×C411yescharacteristic
142#384C2×C411yescharacteristic
143#382C2×C412yes—
144#380—21no—
145#380—21no—
146C4×C4⋊C4—21no—
147C4×C4⋊C4—21no—
148C4×C4⋊C4—22no—
149C4×C4⋊C4—22no—
150C2×#95—21no—
151#341—24no—
152#340C2×C412yes—
153#340C2×C412yes—
154#340C2×C412yes—
155#340C2×C412yes—
156#339—24no—
157#339—24no—
158#339—24no—
159#336—24no—
160#336—22no—
161#336—22no—
162#336—24no—
163C2×#41—21no—
164C2×#41—21no—
165C2×#41—21no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41—21no—
169C2×#41C2×D412yes—
170C2×#41C2×D412yes—
171C2×#41C2×Q812yes—
172C2×#41C2×Q812yes—
173C2×#41#4112yes—
174C2×#41#4112yes—
175C22×Q8C22×C411yescharacteristic
176C22×D4—21no—
177C22×D4C22×C411yescharacteristic
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—22no—
182C23×C4#4111yescharacteristic
183C23×C4#4111yescharacteristic
184C23×C4C2411yescharacteristic
185#131—28no—
186#131—28no—
187#131—28no—
188#131—28no—
189#131—28no—
190#131—28no—
191#131—28no—
192#131—28no—
193C2×M16—24no—
194C2×M16—24no—
195C2×M16—24no—
196C2×M16—24no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—24no—
204C2×M16—24no—
205C2×M16—22no—
206C2×M16—22no—
207C2×M16—22no—
208C2×M16—22no—
209C2×M16—22no—
210C2×M16—22no—
211C2×M16C2×D412yes—
212C2×M16C2×D412yes—
213C2×M16C2×D412yes—
214C2×M16C2×Q812yes—
215C2×M16#4112yes—
216C2×M16#4112yes—
217C22×C8—24no—
218C22×C8—24no—
219C22×C8—24no—
220C22×C8—24no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—24no—
232C22×C8—24no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8C22×C412yes—
241C22×C8C22×C412yes—
242C22×C8C2×D412yes—
243C22×C8C2×D411yescharacteristic
244C22×C8C2×D411yescharacteristic
245C22×C8C2×D411yescharacteristic
246C22×C8C2×D411yescharacteristic
247C22×C8C2×Q812yes—
248C22×C8#4112yes—
249C22×C8#4112yes—
250C4×Q8—22no—
251C4×Q8—22no—
252C4×Q8C4⋊C414yes—
253C4×Q8C4⋊C414yes—
254C4×D4—24no—
255C4×D4—24no—
256C4×D4—24no—
257C4×D4—24no—
258C4×D4—22no—
259C4×D4—22no—
260C4×D4C4⋊C414yes—
261C4×D4C4⋊C414yes—
262#117—21no—
263#117—21no—
264#117—24no—
265#117—24no—
266#117—22no—
267#117—22no—
268#117—22no—
269#117—22no—
270#117C2×D412yes—
271#117C2×Q812yes—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4C22×C412yes—
282C2×C4⋊C4C22×C412yes—
283C2×C4⋊C4C22×C411yescharacteristic
284C2×C4⋊C4C22×C411yescharacteristic
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—22no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42C22×C411yescharacteristic
302C2×C42C22×C411yescharacteristic
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×Q812yes—
308C2×C42#4112yes—
309#108—48no—
310C4⋊C8—22no—
311C4⋊C8—22no—
312C4⋊C8—24no—
313C4⋊C8—44no—
314C4⋊C8—44no—
315C4⋊C8—42no—
316C4⋊C8—42no—
317#104—48no—
318#103—24no—
319#103—24no—
320#102—24no—
321#102—24no—
322#97—24no—
323#97—24no—
324#97—24no—
325C4×C8—24no—
326C4×C8—24no—
327C4×C8—22no—
328C4×C8—24no—
329C4×C8—22no—
330C4×C8—22no—
331C4×C8—22no—
332#95—22no—
333#95—22no—
334#95—42no—
335C24—21no—
336C24C2×#4111yescharacteristic
337#41—22no—
338#41—22no—
339#41—22no—
340#41—22no—
341#41—22no—
342#41—22no—
343#41—22no—
344#41—22no—
345#41—22no—
346#41—22no—
347#41—22no—
348#41—22no—
349#41#12114yes—
350#41C22⋊Q814yes—
351#41#12514yes—
352#41C4⋊Q814yes—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8C2×C4⋊C412yes—
357C2×Q8C4×D412yes—
358C2×Q8C4×Q812yes—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4C2×C4⋊C412yes—
369C2×D4C4×D412yes—
370C2×D4C4×Q812yes—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4C4×D411yescharacteristic
400C22×C4C4×D411yescharacteristic
401C22×C4C4×D411yescharacteristic
402C22×C4C4×D411yescharacteristic
403C22×C4C22⋊Q812yes—
404C22×C4C22⋊Q812yes—
405C22×C4#12312yes—
406C22×C4#12312yes—
407C22×C4C23×C411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×Q811yescharacteristic
413C22×C4C22×Q811yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4C2×#4111yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4111yescharacteristic
421M16—216no—
422M16—216no—
423M16—22no—
424M16—22no—
425M16—22no—
426M16—22no—
427M16—22no—
428M16—22no—
429M16—42no—
430M16—42no—
431M16—42no—
432M16—42no—
433M16#12114yes—
434M16C22⋊Q814yes—
435C2×C8—216no—
436C2×C8—216no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8#12112yes—
473C2×C8#12112yes—
474C2×C8C22⋊Q812yes—
475C2×C8C22⋊Q812yes—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—48no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4C2×C4⋊C412yes—
490C4⋊C4C2×C4⋊C412yes—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C42—22no—
498C42—22no—
499C42—24no—
500C42—24no—
501C42—42no—
502C42—42no—
503C42—42no—
504C42—42no—
505C42—42no—
506C42—42no—
507C42C2×C4⋊C412yes—
508C42C2×C4⋊C412yes—
509C42C4×D412yes—
510C42C4×D412yes—
511C42#12012yes—
512C42#12012yes—
513C42C22⋊Q812yes—
514C42C22⋊Q812yes—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23C2×C4×D411yescharacteristic
527C23C2×C22⋊Q811yescharacteristic
528C23C2×#12311yescharacteristic
529C23#52911yescharacteristic
530C23#52911yescharacteristic
531C23#53211yescharacteristic
532C23#53311yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—22no—
536Q8#38914yes—
537D4—24no—
538D4—24no—
539D4—24no—
540D4—24no—
541D4—22no—
542D4—22no—
543D4—22no—
544D4#38914yes—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—48no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—48no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4#38512yes—
584C2×C4#38512yes—
585C2×C4#38512yes—
586C2×C4#38512yes—
587C2×C4#44312yes—
588C2×C4#44412yes—
589C2×C4C22×C4⋊C411yescharacteristic
590C2×C4C2×C4×D411yescharacteristic
591C2×C4C2×C4×D411yescharacteristic
592C2×C4C2×C4×Q811yescharacteristic
593C2×C4C4×#4111yescharacteristic
594C2×C4C4×#4111yescharacteristic
595C2×C4C2×#12011yescharacteristic
596C2×C4C2×#12111yescharacteristic
597C2×C4C2×#12111yescharacteristic
598C2×C4C2×#12111yescharacteristic
599C2×C4C2×C22⋊Q811yescharacteristic
600C2×C4C2×C22⋊Q811yescharacteristic
601C2×C4C2×C22⋊Q811yescharacteristic
602C2×C4C2×C22⋊Q811yescharacteristic
603C2×C4#52511yescharacteristic
604C2×C4#52511yescharacteristic
605C2×C4#52511yescharacteristic
606C2×C4#52511yescharacteristic
607C2×C4#52511yescharacteristic
608C2×C4#52511yescharacteristic
609C2×C4C2×#12511yescharacteristic
610C2×C4#52911yescharacteristic
611C2×C4#52911yescharacteristic
612C2×C4C2×C4⋊Q811yescharacteristic
613C2×C4#53211yescharacteristic
614C2×C4#53311yescharacteristic
615C2×C4#53311yescharacteristic
616C2×C4#53311yescharacteristic
617C8—22no—
618C8—24no—
619C8—22no—
620C8—22no—
621C8—24no—
622C8—22no—
623C8—42no—
624C8—42no—
625C8—82no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—24no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22C2×#38511yescharacteristic
646C22C4×#12111yescharacteristic
647C22C4×C22⋊Q811yescharacteristic
648C22#285111yescharacteristic
649C22#285411yescharacteristic
650C22#285511yescharacteristic
651C22#285711yescharacteristic
652C22C2×#44311yescharacteristic
653C22C2×#44411yescharacteristic
654C22#295511yescharacteristic
655C22#295511yescharacteristic
656C4—24no—
657C4—22no—
658C4—22no—
659C4—22no—
660C4—42no—
661C4—42no—
662C4—84no—
663C4—82no—
664C4—82no—
665C4#186614yescomplemented
666C4C2×#38911yescharacteristic
667C4#229611yescharacteristic
668C4C4×#12111yescharacteristic
669C4C4×C22⋊Q811yescharacteristic
670C4#284711yescharacteristic
671C4#284811yescharacteristic
672C4#285111yescharacteristic
673C4#285711yescharacteristic
674C2—21no—
675C2—21no—
676C2—21no—
677C2—21no—
678C2—44no—
679C2#1469711yescharacteristic
680C2C2×#186611yescharacteristic
681C2#2088411yescharacteristic
682C2#2151711yescharacteristic
683C2#2151811yescharacteristic
684C2#2162711yescharacteristic
685C2#2162811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1950C2semidirect
3. #21143C2semidirect
7. #10910C2semidirect
8. #10887C2semidirect
9. #10851C2semidirect
10. #10834C2semidirect
40. #1805C4semidirect
41. #1804C4semidirect
42. #1803C4semidirect
43. #1803C4semidirect
44. #1802C4semidirect
45. #1802C4semidirect
665. C4#1866semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1312
C2×#19502
#211432
C2×#18662
C2×#18062
#206452
#109102
#108872
#108512
#108342
#107912