Finite Group Databasebeta

← #225060

Subgroups — #225060

Order 512.

773 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22639C211yesmaximal, characteristic, complemented
2#22402C211yesmaximal, characteristic
3#21390C211yesmaximal, characteristic
4#9560C211yesmaximal, characteristic, complemented
5#9554C211yesmaximal, characteristic, complemented
6#9542C211yesmaximal, characteristic, complemented
7#9541C211yesmaximal, characteristic, complemented
8#8849C211yesmaximal, characteristic, complemented
9#8843C211yesmaximal, characteristic, complemented
10#8708C211yesmaximal, characteristic, complemented
11#8600C211yesmaximal, characteristic, complemented
12#8313C211yesmaximal, characteristic
13#8212C211yesmaximal, characteristic, complemented
14C8×#102C211yesmaximal, characteristic, complemented
15#8148C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#436C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#408—21no—
21C4×#117C2211yescharacteristic
22#1987C411yescharacteristic
23#1987C411yescharacteristic
24#1982C411yescharacteristic
25#1982C411yescharacteristic
26#1954C2211yescharacteristic
27#1954C2211yescharacteristic
28#1939C2211yescharacteristic
29#1939C2211yescharacteristic
30#1923—21no—
31#1923—21no—
32#1875C411yescharacteristic
33#1875C411yescharacteristic
34#1875C411yescharacteristic
35#1875C411yescharacteristic
36#1870—21no—
37#1869—21no—
38#1839—21no—
39#1797C2211yescharacteristic
40#1797C2211yescharacteristic
41#1780—21no—
42C8×C4⋊C4C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44C4×C4⋊C8—21no—
45C4×#102C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×#322C2211yescharacteristic
58C2×C82C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×#41—21no—
68C22×M16—21no—
69C23×C8—21no—
70#532—22no—
71#532—22no—
72#532—41no—
73#532—41no—
74C2×C4⋊Q8C2×C411yescharacteristic
75C2×#127C2×C411yescharacteristic
76C2×#124—21no—
77C2×#121—21no—
78C2×#121C2×C411yescharacteristic
79C2×#121C2×C411yescharacteristic
80C2×C4×D4—21no—
81C2×C4×D4C2311yescharacteristic
82C2×C4×D4C2311yescharacteristic
83C2×#117C2×C411yescharacteristic
84C2×#117C2×C411yescharacteristic
85C22×C42C2311yescharacteristic
86#436—22no—
87#436—41no—
88C8×D4—22no—
89C8×D4—41no—
90#431—41no—
91#431—41no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2×C411yescharacteristic
95C2×C4⋊C8C2×C411yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×#102C2×C411yescharacteristic
101C2×#102C2×C411yescharacteristic
102C2×#102C2×C411yescharacteristic
103C2×#102C2×C411yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108#408—44no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8C2×C411yescharacteristic
119C2×C22⋊C8C2×C411yescharacteristic
120C2×C22⋊C8D411yescharacteristic
121C2×C22⋊C8D411yescharacteristic
122C2×C4×C8—21no—
123C2×C4×C8—21no—
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134C4×C22⋊C4—21no—
135C4×C22⋊C4—21no—
136#376—21no—
137C43—21no—
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336D411yescharacteristic
154#336D411yescharacteristic
155#335—41no—
156#334—41no—
157D4⋊C8—22no—
158D4⋊C8—22no—
159D4⋊C8—41no—
160D4⋊C8—41no—
161#322—41no—
162C82—41no—
163C2×#41—42no—
164C2×#41—42no—
165C2×#41—42no—
166C2×#41—42no—
167C2×#41—81no—
168C2×#41—81no—
169C22×Q8—21no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4C22×C411yescharacteristic
175C22×D4C22×C411yescharacteristic
176C23×C4—21no—
177C23×C4—21no—
178C23×C4C22×C411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4#4111yescharacteristic
181C2×M16—42no—
182C2×M16—42no—
183C2×M16—81no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—42no—
191C22×C8—42no—
192C22×C8—81no—
193C22×C8C22×C411yescharacteristic
194C22×C8C22×C411yescharacteristic
195C22×C8C22×C411yescharacteristic
196C22×C8C22×C411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C22×C8C2×D411yescharacteristic
201C4⋊Q8—22no—
202C4⋊Q8—41no—
203#127—22no—
204#127—41no—
205#124—42no—
206#124—42no—
207#121—22no—
208#121—22no—
209#121—42no—
210#121—42no—
211#121—41no—
212#121—41no—
213C4×D4—21no—
214C4×D4—21no—
215C4×D4—21no—
216C4×D4—21no—
217C4×D4—21no—
218C4×D4—21no—
219C4×D4—42no—
220C4×D4—42no—
221C4×D4#4112yes—
222C4×D4#4112yes—
223#117—22no—
224#117—22no—
225#117—22no—
226#117—41no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C4⋊C4C22×C411yescharacteristic
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42C22×C411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42#4111yescharacteristic
253C2×C42C2411yescharacteristic
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—41no—
263C4⋊C8—41no—
264C4⋊C8—42no—
265C4⋊C8—42no—
266#102—22no—
267#102—22no—
268#102—22no—
269#102—22no—
270#102—22no—
271#102—22no—
272#102—41no—
273#102—41no—
274C22⋊C8—24no—
275C22⋊C8—22no—
276C22⋊C8—41no—
277C22⋊C8—41no—
278C22⋊C8—41no—
279C22⋊C8—42no—
280C22⋊C8—42no—
281C22⋊C8—42no—
282C22⋊C8—42no—
283C22⋊C8—42no—
284C22⋊C8—81no—
285C22⋊C8—81no—
286C22⋊C8—81no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—21no—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8—21no—
299C4×C8—81no—
300C4×C8—81no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305C24—21no—
306C24—21no—
307C24C4×D411yescharacteristic
308#41—84no—
309#41—84no—
310#41—84no—
311#41—84no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—81no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—81no—
335C2×D4—81no—
336C2×D4—81no—
337C2×D4—81no—
338C2×D4#11712yes—
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—21no—
348C22×C4—21no—
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—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4C4×D411yescharacteristic
374C22×C4C4×D411yescharacteristic
375C22×C4C4×D411yescharacteristic
376C22×C4C23×C411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C2×#4111yescharacteristic
379M16—84no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—84no—
422C2×C8AGL(1,ℤ/8)12yes—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—42no—
432C4⋊C4#11712yes—
433C4⋊C4#13112yes—
434C4⋊C4#13112yes—
435C4⋊C4#13112yes—
436C22⋊C4—24no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C42—24no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—42no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—42no—
457C42—81no—
458C42C2×#4111yescharacteristic
459C42C2×#4111yescharacteristic
460C42H(5,𝔽2)11yescharacteristic
461C42H(5,𝔽2)11yescharacteristic
462C23—22no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23C2×C4×D411yescharacteristic
484Q8—84no—
485D4—22no—
486D4—22no—
487D4—41no—
488D4—41no—
489D4—41no—
490D4—41no—
491D4—41no—
492D4—41no—
493D4—84no—
494D4—84no—
495D4—84no—
496D4—84no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4#44312yes—
544C2×C4C2×#11711yescharacteristic
545C2×C4#51811yescharacteristic
546C2×C4#54611yescharacteristic
547C2×C4#54611yescharacteristic
548C2×C4#55111yescharacteristic
549C2×C4#55111yescharacteristic
550C2×C4C2×#13111yescharacteristic
551C2×C4C2×#13111yescharacteristic
552C2×C4C2×#13111yescharacteristic
553C2×C4#56811yescharacteristic
554C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
555C2×C4#57511yescharacteristic
556C2×C4#57611yescharacteristic
557C2×C4#57711yescharacteristic
558C8—22no—
559C8—42no—
560C8—42no—
561C8—41no—
562C8—41no—
563C8—41no—
564C8—41no—
565C8—41no—
566C8—41no—
567C8—41no—
568C8—81no—
569C8—81no—
570C8—81no—
571C8—81no—
572C22—41no—
573C22—41no—
574C22—42no—
575C22—41no—
576C22—42no—
577C22—41no—
578C22—42no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22#234211yescharacteristic
594C22#292911yescharacteristic
595C22#293311yescharacteristic
596C22C4×AGL(1,ℤ/8)11yescharacteristic
597C22#294611yescharacteristic
598C22C2×#44311yescharacteristic
599C22#295711yescharacteristic
600C4—21no—
601C4—21no—
602C4—22no—
603C4—42no—
604C4—41no—
605C4—41no—
606C4—42no—
607C4—42no—
608C4—42no—
609C4—81no—
610C4—81no—
611C4#292211yescharacteristic
612C4#292511yescharacteristic
613C4#297911yescharacteristic
614C4#298111yescharacteristic
615C4#331111yescharacteristic
616C4#331811yescharacteristic
617C4#332211yescharacteristic
618C4#332411yescharacteristic
619C2—81no—
620C2—82no—
621C2—161no—
622C2#1791211yescharacteristic
623C2#1792011yescharacteristic
624C2#1801111yescharacteristic
625C2#1801511yescharacteristic
626C2#2255611yescharacteristic
627C2#2258011yescharacteristic
628C2#2258711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22639C2semidirect
4. #9560C2semidirect
5. #9554C2semidirect
6. #9542C2semidirect
7. #9541C2semidirect
8. #8849C2semidirect
9. #8843C2semidirect
10. #8708C2semidirect
11. #8600C2semidirect
13. #8212C2semidirect
14. C8×#102C2semidirect
15. #8148C2semidirect

Maximal subgroups

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

subgroupindex
#226392
#224022
#213902
#95602
#95542
#95422
#95412
#88492
#88432
#87082
#86002
#83132
#82122
C8×#1022
#81482