Finite Group Databasebeta

← #360391

Subgroups — #360391

Order 512.

1020 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1837C211yesmaximal, characteristic, complemented
2C2×#1806C211yesmaximal, characteristic
3#20629C211yesmaximal, characteristic
4#20431C211yesmaximal, characteristic, complemented
5C4×#416C211yesmaximal, characteristic
6#11317C212yesmaximal, complemented
7#11191C212yesmaximal
8#10934C212yesmaximal
9#10852C211yesmaximal, characteristic, complemented
10#10834C211yesmaximal, characteristic, complemented
11#10809C212yesmaximal, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#104C2212yes—
17C2×#416C2211yescharacteristic
18C2×#416C2211yescharacteristic
19C2×#408C2212yes—
20C2×#405C2212yes—
21C22×C4×C8C2211yescharacteristic
22C4×#117C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1837—24no—
25#1806—24no—
26#1798—22no—
27#1797—21no—
28#1797—21no—
29#1768—22no—
30#1767—21no—
31C4×C4⋊C8—21no—
32C4×#103C411yescharacteristic, complemented
33C4×#103C411yescharacteristic, complemented
34C4×#103C411yescharacteristic, complemented
35C4×#103C411yescharacteristic, complemented
36C4×#102C411yescharacteristic, complemented
37C4×#102C411yescharacteristic, complemented
38C4×#102C411yescharacteristic, complemented
39C4×#102C411yescharacteristic, complemented
40C2×#341C2212yes—
41C2×#341C2212yes—
42#1735C2211yescharacteristic
43#1735C2211yescharacteristic
44C2×#339C2212yes—
45C2×#339C2212yes—
46C2×#339C2212yes—
47#1729C2212yes—
48#1729C2212yes—
49C2×#336C2212yes—
50C2×#336C2212yes—
51C22×#41C2311yescharacteristic
52C22×M16C2312yes—
53C22×M16C2312yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C23×C8C2312yes—
57C4×#41—22no—
58C4×#41—22no—
59C4×#41—22no—
60C4×#41—22no—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4C2×C411yescharacteristic
67C2×#117—21no—
68C2×#117C2311yescharacteristic
69C2×#117C2311yescharacteristic
70C2×#117C2311yescharacteristic
71C22×C4⋊C4—21no—
72C22×C42—21no—
73C22×C42C2312yes—
74C22×C42C2311yescharacteristic
75#423—22no—
76#423—22no—
77#423—22no—
78#423D414yes—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8D411yescharacteristic
86C2×C4⋊C8D411yescharacteristic
87C2×C4⋊C8D412yes—
88C2×#104—24no—
89C2×#104—24no—
90C2×#104—42no—
91#416—22no—
92#416—22no—
93#416—22no—
94#416—22no—
95C2×#103C2×C411yescharacteristic
96C2×#103C2×C411yescharacteristic
97C2×#103C2×C411yescharacteristic
98C2×#103C2×C411yescharacteristic
99C2×#102C2×C411yescharacteristic
100C2×#102C2×C411yescharacteristic
101C2×#102C2×C411yescharacteristic
102C2×#102C2×C411yescharacteristic
103#408—24no—
104#408—42no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107#405—24no—
108#405—42no—
109C2×#97—22no—
110C2×C4×C8—22no—
111C2×C4×C8—22no—
112C2×C4×C8—22no—
113C2×C4×C8C2×C411yescharacteristic
114C2×C4×C8C2×C411yescharacteristic
115C2×C4×C8C2×C411yescharacteristic
116C2×C4×C8C2×C411yescharacteristic
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C22⋊C4—21no—
124C4×C22⋊C4—21no—
125#376—21no—
126C2×#95—21no—
127C43—21no—
128#341—24no—
129#341—24no—
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#339—24no—
139#339—24no—
140#339—42no—
141#336—22no—
142#336—22no—
143#336—22no—
144#336—22no—
145#336—42no—
146#336—42no—
147C2×#41—21no—
148C2×#41—21no—
149C2×#41—21no—
150C2×#41—21no—
151C2×#41—21no—
152C2×#41—21no—
153C2×#41—21no—
154C2×#41—21no—
155C2×#41—21no—
156C2×#41—21no—
157C2×#41#4112yes—
158C2×#41#4112yes—
159C22×Q8C22×C411yescharacteristic
160C22×D4—21no—
161C22×D4C22×C411yescharacteristic
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4#4111yescharacteristic
167C23×C4#4111yescharacteristic
168C23×C4C2411yescharacteristic
169C2×M16—22no—
170C2×M16—22no—
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×M16C2×D414yes—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—22no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—22no—
201C22×C8C22×C411yescharacteristic
202C22×C8C22×C411yescharacteristic
203C22×C8C22×C411yescharacteristic
204C22×C8C22×C411yescharacteristic
205C22×C8C2×D412yes—
206C22×C8C2×D412yes—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×Q8—22no—
210C4×Q8—22no—
211C4×D4—22no—
212C4×D4—22no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—42no—
216C4×D4—42no—
217C4×D4—42no—
218C4×D4—42no—
219#117—21no—
220#117—21no—
221#117—21no—
222#117—21no—
223#117—21no—
224#117—21no—
225#117—21no—
226#117—21no—
227#117—21no—
228#117—21no—
229#117—42no—
230#117—42no—
231#117#4112yes—
232#117#4112yes—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42C22×C411yescharacteristic
267C2×C42C22×C411yescharacteristic
268C2×C42C2×D412yes—
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42#4112yes—
272C2×C42#4112yes—
273C2×C42#4112yes—
274C4⋊C8—24no—
275C4⋊C8—24no—
276C4⋊C8—22no—
277C4⋊C8—22no—
278C4⋊C8—24no—
279C4⋊C8—42no—
280C4⋊C8—42no—
281#104—48no—
282#103—22no—
283#103—22no—
284#103—22no—
285#103—22no—
286#102—22no—
287#102—22no—
288#102—22no—
289#102—22no—
290C22⋊C8—44no—
291C22⋊C8—44no—
292#97—44no—
293C4×C8—22no—
294C4×C8—22no—
295C4×C8—22no—
296C4×C8—22no—
297C4×C8—44no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—42no—
303C24—21no—
304C24C2×#4111yescharacteristic
305#41—22no—
306#41—22no—
307#41—22no—
308#41—22no—
309#41—22no—
310#41—22no—
311#41—22no—
312#41—22no—
313#41—41no—
314#41—41no—
315#41—41no—
316#41—41no—
317#41—41no—
318#41—41no—
319#41—41no—
320#41—41no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8#11712yes—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4#11712yes—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—81no—
368C22×C4—81no—
369C22×C4C4×D411yescharacteristic
370C22×C4C4×D411yescharacteristic
371C22×C4C4×D411yescharacteristic
372C22×C4C4×D411yescharacteristic
373C22×C4#12412yes—
374C22×C4#12412yes—
375C22×C4#12612yes—
376C22×C4#12612yes—
377C22×C4C23×C411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4H(5,𝔽2)11yescharacteristic
388C22×C4H(5,𝔽2)11yescharacteristic
389C22×C42−1+411yescharacteristic
390C22×C42−1+411yescharacteristic
391M16—24no—
392M16—24no—
393M16—42no—
394M16—42no—
395M16—42no—
396M16—42no—
397M16—42no—
398M16—42no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—24no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—42no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—42no—
429C2×C8—41no—
430C2×C8—42no—
431C2×C8—41no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—48no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4#11712yes—
449C4⋊C4#11712yes—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—41no—
453C22⋊C4—41no—
454C22⋊C4—41no—
455C22⋊C4—41no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C42—22no—
459C42—22no—
460C42—41no—
461C42—41no—
462C42—41no—
463C42—41no—
464C42—42no—
465C42—42no—
466C42—42no—
467C42—42no—
468C42—42no—
469C42—42no—
470C42—42no—
471C42—42no—
472C42—81no—
473C42—81no—
474C42#11712yes—
475C42#11712yes—
476C42C4×D412yes—
477C42C4×D412yes—
478C42#12112yes—
479C42#12112yes—
480C42#12312yes—
481C42#12312yes—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—22no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23C2×C4×D411yescharacteristic
494C23C2×#12411yescharacteristic
495C23C2×#12611yescharacteristic
496C23#55211yescharacteristic
497C23#55211yescharacteristic
498C23#55311yescharacteristic
499C23#55311yescharacteristic
500Q8—22no—
501Q8—22no—
502Q8—41no—
503Q8—41no—
504D4—22no—
505D4—22no—
506D4—42no—
507D4—42no—
508D4—42no—
509D4—42no—
510D4—41no—
511D4—41no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—84no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—84no—
556C2×C4#38812yes—
557C2×C4#38812yes—
558C2×C4#38812yes—
559C2×C4#38812yes—
560C2×C4#44312yes—
561C2×C4#44412yes—
562C2×C4C2×#11711yescharacteristic
563C2×C4C2×#11711yescharacteristic
564C2×C4C2×#11711yescharacteristic
565C2×C4C2×C4×D411yescharacteristic
566C2×C4#52011yescharacteristic
567C2×C4#52011yescharacteristic
568C2×C4C2×#12111yescharacteristic
569C2×C4C2×#12311yescharacteristic
570C2×C4#52911yescharacteristic
571C2×C4#52911yescharacteristic
572C2×C4#52911yescharacteristic
573C2×C4#52911yescharacteristic
574C2×C4#54011yescharacteristic
575C2×C4#54111yescharacteristic
576C2×C4#54211yescharacteristic
577C2×C4#54211yescharacteristic
578C2×C4#54611yescharacteristic
579C2×C4#54611yescharacteristic
580C2×C4#54611yescharacteristic
581C2×C4#54711yescharacteristic
582C2×C4#54811yescharacteristic
583C2×C4#54811yescharacteristic
584C2×C4#54811yescharacteristic
585C2×C4#55011yescharacteristic
586C2×C4#55211yescharacteristic
587C2×C4#55311yescharacteristic
588C2×C4#55511yescharacteristic
589C2×C4#55611yescharacteristic
590C8—41no—
591C8—41no—
592C8—41no—
593C8—41no—
594C8—42no—
595C8—42no—
596C8—44no—
597C8—82no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22C2×#38811yescharacteristic
618C22#235211yescharacteristic
619C22#235211yescharacteristic
620C22#248911yescharacteristic
621C22#249011yescharacteristic
622C22#249411yescharacteristic
623C22#249511yescharacteristic
624C22C2×#44311yescharacteristic
625C22C2×#44411yescharacteristic
626C22#295711yescharacteristic
627C22#295711yescharacteristic
628C4—22no—
629C4—22no—
630C4—22no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—84no—
635C4—81no—
636C4—81no—
637C4—81no—
638C4—81no—
639C4#192914yes—
640C4#228311yescharacteristic
641C4#231411yescharacteristic
642C4#235211yescharacteristic
643C4#235411yescharacteristic
644C4#242011yescharacteristic
645C4#242111yescharacteristic
646C4#242611yescharacteristic
647C4#242711yescharacteristic
648C2—21no—
649C2—21no—
650C2—21no—
651C2—21no—
652C2—82no—
653C2#1387511yescharacteristic
654C2C2×#192911yescharacteristic
655C2#2107211yescharacteristic
656C2#2278211yescharacteristic
657C2#2278211yescharacteristic
658C2#2278311yescharacteristic
659C2#2278311yescharacteristic
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
1. C2×#1837C2semidirect
4. #20431C2semidirect
6. #11317C2semidirect
9. #10852C2semidirect
10. #10834C2semidirect
11. #10809C2semidirect
32. C4×#103C4semidirect
33. C4×#103C4semidirect
34. C4×#103C4semidirect
35. C4×#103C4semidirect
36. C4×#102C4semidirect
37. C4×#102C4semidirect
38. C4×#102C4semidirect
39. C4×#102C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18372
C2×#18062
#206292
#204312
C4×#4162
#113172
#111912
#109342
#108522
#108342
#108092