Finite Group Databasebeta

← #385169

Subgroups — #385169

Order 512.

1309 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21190C211yesmaximal, characteristic
2#20898C211yesmaximal, characteristic
3C2×#1869C211yesmaximal, characteristic
4C2×#1855C212yesmaximal, complemented
5C2×#1810C212yesmaximal, complemented
6#11326C212yesmaximal, complemented
7#11097C212yesmaximal, complemented
8#11097C212yesmaximal, complemented
9#10911C212yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C24×C8C2211yescharacteristic
12C22×C4×D4C2211yescharacteristic
13C22×#108C2212yes—
14C22×#108C2212yes—
15C22×#108C2212yes—
16C2×#423C2211yescharacteristic
17C2×#407C2212yes—
18C2×#407C2212yes—
19C2×#407C2212yes—
20C2×#407C2211yescharacteristic
21C22×C22⋊C8C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24#1955—21no—
25#1955—21no—
26#1954—21no—
27#1954—21no—
28#1870—21no—
29#1870—21no—
30#1869—21no—
31#1869—21no—
32#1869C412yes—
33#1869C412yes—
34#1869C412yes—
35#1869C412yes—
36#1855—28no—
37#1810—28no—
38C2×#341C2212yes—
39C2×#341C2212yes—
40C2×#341C2212yes—
41C2×#341C2212yes—
42C2×#341C2212yes—
43#1728C2212yes—
44C2×#336C2212yes—
45C23×D4Q811yescharacteristic
46C24×C4C2311yescharacteristic
47C24×C4C2311yescharacteristic
48C22×M16—21no—
49C22×M16—24no—
50C22×M16—42no—
51C22×M16—41no—
52C22×M16—41no—
53C22×M16C2312yes—
54C22×M16C2312yes—
55C22×M16C2312yes—
56C22×M16C2312yes—
57C23×C8—24no—
58C23×C8—24no—
59C23×C8—21no—
60C23×C8—41no—
61C23×C8—41no—
62C23×C8C2×C412yes—
63C23×C8C2×C412yes—
64C23×C8C2312yes—
65C23×C8C2312yes—
66C2×C4×D4—41no—
67C2×C4×D4—41no—
68C2×C4×D4—41no—
69C2×C4×D4—41no—
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C2×C4×D4D411yescharacteristic
77C2×C4×D4D411yescharacteristic
78C22×C4⋊C4Q811yescharacteristic
79C22×C22⋊C4Q811yescharacteristic
80C22×C22⋊C4Q811yescharacteristic
81C22×C42C2311yescharacteristic
82C2×#108—24no—
83C2×#108—24no—
84C2×#108—24no—
85C2×#108—24no—
86C2×#108—24no—
87C2×#108—42no—
88C2×#108—42no—
89C2×#108—42no—
90C2×#108—42no—
91#423—41no—
92#423—41no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95#407—24no—
96#407—24no—
97#407—24no—
98#407—42no—
99#407—42no—
100#407—42no—
101#407—41no—
102#407—41no—
103C2×C22⋊C8—24no—
104C2×C22⋊C8—24no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—41no—
114C2×C22⋊C8—41no—
115C2×C22⋊C8C2×C414yes—
116C2×C22⋊C8C2×C414yes—
117C2×C22⋊C8C2×C412yes—
118C2×C22⋊C8C2×C412yes—
119C4×M16—41no—
120C4×M16—41no—
121C2×#97—21no—
122C2×C4×C8—21no—
123#341—24no—
124#341—24no—
125#341—42no—
126#341—42no—
127#341D418yescomplemented
128#336—22no—
129#336—22no—
130#336C2×C414yes—
131#336C2×C414yes—
132C25C2×Q811yescharacteristic
133C25C2×Q811yescharacteristic
134C22×D4—24no—
135C22×D4—41no—
136C22×D4—41no—
137C22×D4—42no—
138C22×D4C4⋊C411yescharacteristic
139C22×D4C4⋊C411yescharacteristic
140C22×D4C4⋊C411yescharacteristic
141C22×D4C4⋊C411yescharacteristic
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×Q811yescharacteristic
167C23×C4C2×Q811yescharacteristic
168C23×C4C2×Q811yescharacteristic
169C23×C4C2×Q811yescharacteristic
170C23×C4C2411yescharacteristic
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—44no—
194C2×M16—44no—
195C2×M16—48no—
196C2×M16—42no—
197C2×M16—44no—
198C2×M16C2×D414yes—
199C2×M16C2×D414yes—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—28no—
213C22×C8—28no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—42no—
217C22×C8—42no—
218C22×C8—44no—
219C22×C8—48no—
220C22×C8—44no—
221C22×C8—44no—
222C22×C8—42no—
223C22×C8C22×C414yes—
224C22×C8C22×C414yes—
225C22×C8C2×D414yes—
226C22×C8#4114yes—
227C4×D4—41no—
228C4×D4—41no—
229C4×D4—41no—
230C4×D4—41no—
231C4×D4—41no—
232C4×D4—41no—
233C4×D4—41no—
234C4×D4—41no—
235C4×D4—41no—
236C4×D4—41no—
237C4×D4—41no—
238C4×D4—41no—
239C4×D4—41no—
240C4×D4—41no—
241C4×D4—41no—
242C4×D4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4C4⋊C411yescharacteristic
246C2×C4⋊C4C4⋊C411yescharacteristic
247C2×C4⋊C4C4⋊C411yescharacteristic
248C2×C4⋊C4C4⋊C411yescharacteristic
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4C4⋊C411yescharacteristic
254C2×C22⋊C4C4⋊C411yescharacteristic
255C2×C22⋊C4C4⋊C411yescharacteristic
256C2×C22⋊C4C4⋊C411yescharacteristic
257C2×C22⋊C4C4⋊C411yescharacteristic
258C2×C22⋊C4C4⋊C411yescharacteristic
259C2×C22⋊C4C4⋊C411yescharacteristic
260C2×C22⋊C4C4⋊C411yescharacteristic
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42C22×C411yescharacteristic
266C2×C42C22×C411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269#108—48no—
270#108—48no—
271#108—48no—
272C4⋊C8—42no—
273C4⋊C8—42no—
274C22⋊C8—28no—
275C22⋊C8—28no—
276C22⋊C8—42no—
277C22⋊C8—42no—
278C22⋊C8—44no—
279C22⋊C8—44no—
280C22⋊C8—44no—
281C22⋊C8—44no—
282C22⋊C8—44no—
283C22⋊C8—44no—
284C22⋊C8—42no—
285C22⋊C8—42no—
286#97—42no—
287C4×C8—42no—
288C24—21no—
289C24—21no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—21no—
296C24—21no—
297C24—22no—
298C24—22no—
299C24—42no—
300C24—41no—
301C24—42no—
302C24C2×C4⋊C411yescharacteristic
303C24C2×C4⋊C411yescharacteristic
304C24C2×C4⋊C411yescharacteristic
305C24C2×C4⋊C411yescharacteristic
306C24C22×Q811yescharacteristic
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—48no—
319C2×D4—48no—
320C2×D4#10814yescomplemented
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4C2×C4⋊C411yescharacteristic
389C22×C4C2×C4⋊C411yescharacteristic
390C22×C4C2×C4⋊C411yescharacteristic
391C22×C4C2×C4⋊C411yescharacteristic
392C22×C4C2×C4⋊C411yescharacteristic
393C22×C4C2×C4⋊C411yescharacteristic
394C22×C4C2×C4⋊C411yescharacteristic
395C22×C4C2×C4⋊C411yescharacteristic
396C22×C4C23×C411yescharacteristic
397C22×C4C22×D412yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C2×#4112yes—
400C22×C4H(5,𝔽2)11yescharacteristic
401C22×C4H(5,𝔽2)11yescharacteristic
402C22×C4H(5,𝔽2)11yescharacteristic
403C22×C4H(5,𝔽2)11yescharacteristic
404C22×C4H(5,𝔽2)12yes—
405C22×C42−1+412yes—
406M16—24no—
407M16—24no—
408M16—24no—
409M16—24no—
410M16—48no—
411M16—42no—
412M16—42no—
413M16—42no—
414M16—42no—
415M16—42no—
416M16—42no—
417M16—42no—
418M16—42no—
419M16—42no—
420M16—42no—
421M16—42no—
422M16—42no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—48no—
435C2×C8—48no—
436C2×C8—48no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8C4×D418yes—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C23—21no—
473C23—21no—
474C23—22no—
475C23—24no—
476C23—24no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—21no—
490C23—22no—
491C23—21no—
492C23—21no—
493C23—22no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—44no—
502C23—42no—
503C23—42no—
504C23—41no—
505C23—42no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—41no—
510C23—82no—
511C23—82no—
512C23—81no—
513C23—81no—
514C23C2×#10812yes—
515C23C2×#10812yes—
516C23#43012yes—
517C23C22×C4⋊C411yescharacteristic
518C23#54311yescharacteristic
519C23#54311yescharacteristic
520C23D4×Q812yes—
521C23#55412yes—
522D4—44no—
523D4—44no—
524D4—44no—
525D4—44no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4C2×#10812yes—
578C2×C4C2×C4×D412yes—
579C2×C4#51811yescharacteristic
580C2×C4#51811yescharacteristic
581C2×C4#52012yes—
582C2×C4#53411yescharacteristic
583C2×C4#53411yescharacteristic
584C2×C4#53512yes—
585C2×C4#53712yes—
586C2×C4#54012yes—
587C2×C4#54212yes—
588C2×C4D4211yescharacteristic
589C2×C4D4212yes—
590C2×C4#54612yes—
591C2×C4#54712yes—
592C2×C4#55111yescharacteristic
593C2×C4#56011yescharacteristic
594C2×C4#56212yes—
595C2×C4#56311yescharacteristic
596C8—28no—
597C8—44no—
598C8—44no—
599C8—44no—
600C8—82no—
601C8—82no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—44no—
617C22—48no—
618C22—44no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—42no—
630C22—42no—
631C22—41no—
632C22—41no—
633C22—82no—
634C22—84no—
635C22—82no—
636C22#231511yescharacteristic
637C22D4×C4⋊C412yes—
638C22#268311yescharacteristic
639C22#269212yes—
640C22#274811yescharacteristic
641C22C22×#10811yescharacteristic
642C22C2×#43011yescharacteristic
643C22#291612yes—
644C4—22no—
645C4—41no—
646C4—41no—
647C4—41no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4#232412yes—
653C4#235811yescharacteristic
654C4#237611yescharacteristic
655C4#266911yescharacteristic
656C4#267112yes—
657C4#271711yescharacteristic
658C2—21no—
659C2—22no—
660C2—21no—
661C2—42no—
662C2—42no—
663C2—42no—
664C2—81no—
665C2#1440111yescharacteristic
666C2#2194811yescharacteristic
667C2#2195011yescharacteristic
668C2D4×#10812yes—
669C2#2266812yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C4⋊C4
prime radicalC23×D4Q8
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1855C2semidirect
5. C2×#1810C2semidirect
6. #11326C2semidirect
7. #11097C2semidirect
8. #11097C2semidirect
9. #10911C2semidirect
127. #341D4semidirect
320. C2×D4#108semidirect

Maximal subgroups

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

subgroupindex
#211902
#208982
C2×#18692
C2×#18552
C2×#18102
#113262
#110972
#109112