Finite Group Databasebeta

← #227705

Subgroups — #227705

Order 512.

826 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23087C211yesmaximal, characteristic
2Q8×C22⋊C8C211yesmaximal, characteristic, complemented
3C4×#448C211yesmaximal, characteristic
4C2×#1588C211yesmaximal, characteristic, complemented
5C2×#1479C211yesmaximal, characteristic, complemented
6#9661C211yesmaximal, characteristic, complemented
7#9559C211yesmaximal, characteristic, complemented
8#9548C211yesmaximal, characteristic, complemented
9#9074C211yesmaximal, characteristic
10#8949C211yesmaximal, characteristic, complemented
11#8948C211yesmaximal, characteristic, complemented
12#8846C211yesmaximal, characteristic, complemented
13#8831C211yesmaximal, characteristic, complemented
14#8203C211yesmaximal, characteristic, complemented
15#8149C211yesmaximal, characteristic, complemented
16C22×C4×Q8C2211yescharacteristic
17C2×#448C411yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20C2×C4×SD16C2211yescharacteristic
21C2×C4×SD16C2211yescharacteristic
22C2×#436C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C4×#121C2211yescharacteristic
26#1984—21no—
27#1984—21no—
28#1984C2211yescharacteristic
29#1984C2211yescharacteristic
30#1955—21no—
31#1954C2211yescharacteristic
32#1939C2211yescharacteristic
33#1923—21no—
34#1880C411yescharacteristic
35#1880C411yescharacteristic
36#1874C411yescharacteristic
37#1870—21no—
38#1865C411yescharacteristic
39#1865C411yescharacteristic
40#1853C411yescharacteristic
41#1797—21no—
42#1797C2211yescharacteristic
43#1787C411yescharacteristic
44#1770C2211yescharacteristic
45C4×#103C2211yescharacteristic
46C4×#103C2211yescharacteristic
47#1753—21no—
48C4×C22⋊C8—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51#1588—24no—
52C2×#335C2211yescharacteristic
53C2×#334C2211yescharacteristic
54#1479—24no—
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C23×Q8C2×C411yescharacteristic
70C22×SD16C2×C411yescharacteristic
71C22×SD16C2×C411yescharacteristic
72C22×M16—21no—
73C2×#121C2×C411yescharacteristic
74C2×C4×Q8—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8—22no—
77C2×C4×Q8—22no—
78C2×C4×Q8—41no—
79C2×C4×Q8—41no—
80C2×C4×Q8C2×C411yescharacteristic
81C2×C4×Q8C2×C411yescharacteristic
82C2×C4×Q8C2311yescharacteristic
83C2×C4×Q8C2311yescharacteristic
84C2×C4×D4—21no—
85C2×C4×D4C2311yescharacteristic
86C22×C4⋊C4—21no—
87C22×C4⋊C4C2×C411yescharacteristic
88C22×C42—21no—
89C22×C42C2311yescharacteristic
90#448—24no—
91C8×Q8—22no—
92C8×Q8—22no—
93C8×Q8—41no—
94C8×Q8—41no—
95C4×SD16—22no—
96C4×SD16—22no—
97C4×SD16—41no—
98C4×SD16—41no—
99#436—22no—
100#436—41no—
101#431—22no—
102#431—22no—
103#431—41no—
104#431—41no—
105C2×#106C2×C411yescharacteristic
106C2×#106C2×C411yescharacteristic
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×C4⋊C8C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115C2×#103C2×C411yescharacteristic
116C2×#103C2×C411yescharacteristic
117C2×#103C2×C411yescharacteristic
118C2×#103C2×C411yescharacteristic
119C2×#103C2×C411yescharacteristic
120C2×#103C2×C411yescharacteristic
121C2×#102C2×C411yescharacteristic
122C2×#102C2×C411yescharacteristic
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—21no—
129C2×C22⋊C8C2×C411yescharacteristic
130C2×C22⋊C8C2×C411yescharacteristic
131C2×C22⋊C8D411yescharacteristic
132C2×C22⋊C8D411yescharacteristic
133C2×#97—21no—
134C2×C4×C8—21no—
135C2×C4×C8—21no—
136C2×C4×C8—21no—
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140C2×C4×C8C2311yescharacteristic
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2311yescharacteristic
143#390C2×C411yescharacteristic
144#389C2×C411yescharacteristic
145#388—21no—
146#386—21no—
147#380C2×C411yescharacteristic
148C4×C4⋊C4C2311yescharacteristic
149C4×C22⋊C4—21no—
150#340C2×C411yescharacteristic
151#340C2×C411yescharacteristic
152#340C2×C411yescharacteristic
153#340C2×C411yescharacteristic
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#336—21no—
160#336—21no—
161#336C2×C411yescharacteristic
162#336C2×C411yescharacteristic
163#336D411yescharacteristic
164#336D411yescharacteristic
165#335—22no—
166#334—41no—
167Q8⋊C8—22no—
168Q8⋊C8—22no—
169Q8⋊C8—22no—
170Q8⋊C8D414yescomplemented
171D4⋊C8—22no—
172D4⋊C8—41no—
173#322—22no—
174#322—41no—
175C22×Q8—21no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8—41no—
180C22×Q8—41no—
181C22×Q8M1614yescomplemented
182C22×Q8C22×C411yescharacteristic
183C22×Q8C22×C411yescharacteristic
184C22×D4—21no—
185C22×D4C22×C411yescharacteristic
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4C22×C411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4#4111yescharacteristic
192C2×SD16—22no—
193C2×SD16—22no—
194C2×SD16—22no—
195C2×SD16—22no—
196C2×SD16—41no—
197C2×SD16—41no—
198C2×M16—42no—
199C2×M16—42no—
200C2×M16—81no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8C22×C411yescharacteristic
208C22×C8C22×C411yescharacteristic
209C22×C8C22×C411yescharacteristic
210C22×C8C22×C411yescharacteristic
211C22×C8C2×D411yescharacteristic
212C22×C8C2×D411yescharacteristic
213C22×C8C2×D411yescharacteristic
214C22×C8C2×D411yescharacteristic
215#121—22no—
216#121—22no—
217C4×Q8—21no—
218C4×Q8—21no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8—42no—
226C4×Q8—42no—
227C4×Q8—41no—
228C4×Q8—41no—
229C4×Q8—42no—
230C4×Q8—42no—
231C4×Q8C2×D412yes—
232C4×Q8#4112yes—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—81no—
238C4×D4—81no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C4⋊C4C22×C411yescharacteristic
253C2×C4⋊C4C22×C411yescharacteristic
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42C22×C411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42#4111yescharacteristic
273C2×C42C2411yescharacteristic
274#106—22no—
275#106—41no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—42no—
284C4⋊C8—42no—
285C4⋊C8—42no—
286C4⋊C8—81no—
287C4⋊C8C2×D412yes—
288#103—22no—
289#103—22no—
290#103—22no—
291#103—22no—
292#103—41no—
293#103—41no—
294#102—22no—
295#102—41no—
296C22⋊C8—22no—
297C22⋊C8—22no—
298C22⋊C8—22no—
299C22⋊C8—41no—
300C22⋊C8—42no—
301C22⋊C8—42no—
302C22⋊C8—42no—
303C22⋊C8—42no—
304C22⋊C8—42no—
305C22⋊C8—81no—
306#97—81no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—42no—
319C4×C8—42no—
320C4×C8—42no—
321C4×C8C2×D412yes—
322#95—21no—
323#95—21no—
324#95—21no—
325C24—21no—
326C24C4×D411yescharacteristic
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—24no—
332C2×Q8—28no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—44no—
340C2×Q8—44no—
341C2×Q8—44no—
342C2×Q8C4×D412yes—
343C2×Q8C2×M1612yes—
344C2×Q8C2×M1612yes—
345C2×Q8#13112yes—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—42no—
353C2×D4—81no—
354C2×D4—81no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—81no—
388C22×C4C4×D411yescharacteristic
389C22×C4C4×D411yescharacteristic
390C22×C4C4×D411yescharacteristic
391C22×C4C2×M1612yes—
392C22×C4#13112yes—
393C22×C4C23×C411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C2×#4111yescharacteristic
396SD16—44no—
397SD16—44no—
398M16—84no—
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—22no—
408C2×C8—21no—
409C2×C8—22no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8#13712yes—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—81no—
458C4⋊C4C4×D412yes—
459C22⋊C4—42no—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C22⋊C4—42no—
463C22⋊C4—81no—
464C22⋊C4—81no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—22no—
469C42—22no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—42no—
474C42—42no—
475C42—41no—
476C42—41no—
477C42—41no—
478C42—41no—
479C42C22×D411yescharacteristic
480C42C2×#4111yescharacteristic
481C42H(5,𝔽2)11yescharacteristic
482C42H(5,𝔽2)11yescharacteristic
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23#43512yes—
497C23C2×C4×D411yescharacteristic
498Q8—22no—
499Q8—22no—
500Q8—22no—
501Q8—22no—
502Q8—22no—
503Q8—48no—
504Q8—41no—
505Q8—41no—
506Q8#43514yes—
507D4—41no—
508D4—41no—
509D4—41no—
510D4—41no—
511D4—84no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—48no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
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—41no—
551C2×C4—41no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—84no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4#43512yes—
560C2×C4C2×C4×D411yescharacteristic
561C2×C4#51811yescharacteristic
562C2×C4D4211yescharacteristic
563C2×C4#54611yescharacteristic
564C2×C4#54611yescharacteristic
565C2×C4#55111yescharacteristic
566C2×C4C22×M1611yescharacteristic
567C2×C4C2×#13111yescharacteristic
568C2×C4C2×#13111yescharacteristic
569C2×C4#56811yescharacteristic
570C2×C4C2×#13711yescharacteristic
571C2×C4#57511yescharacteristic
572C2×C4#57711yescharacteristic
573C2×C4#57711yescharacteristic
574C8—22no—
575C8—42no—
576C8—41no—
577C8—41no—
578C8—41no—
579C8—41no—
580C8—41no—
581C8—42no—
582C8—41no—
583C8—41no—
584C8—81no—
585C8—81no—
586C8—81no—
587C8—81no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22#235811yescharacteristic
603C22C2×#43511yescharacteristic
604C22#293311yescharacteristic
605C22C4×#13711yescharacteristic
606C22#294911yescharacteristic
607C22#295511yescharacteristic
608C22#295711yescharacteristic
609C4—22no—
610C4—24no—
611C4—21no—
612C4—21no—
613C4—42no—
614C4—41no—
615C4—41no—
616C4—42no—
617C4—81no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4C2×#43511yescharacteristic
622C4#293311yescharacteristic
623C4#297311yescharacteristic
624C4#297711yescharacteristic
625C4#327511yescharacteristic
626C4#327811yescharacteristic
627C4#332011yescharacteristic
628C4#332111yescharacteristic
629C2—42no—
630C2—161no—
631C2#1787011yescharacteristic
632C2#1787811yescharacteristic
633C2#1801011yescharacteristic
634C2#1803011yescharacteristic
635C2#2284411yescharacteristic
636C2#2286511yescharacteristic
637C2#2288911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×C4
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
2. Q8×C22⋊C8C2semidirect
4. C2×#1588C2semidirect
5. C2×#1479C2semidirect
6. #9661C2semidirect
7. #9559C2semidirect
8. #9548C2semidirect
10. #8949C2semidirect
11. #8948C2semidirect
12. #8846C2semidirect
13. #8831C2semidirect
14. #8203C2semidirect
15. #8149C2semidirect
170. Q8⋊C8D4semidirect
181. C22×Q8M16semidirect

Maximal subgroups

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

subgroupindex
#230872
Q8×C22⋊C82
C4×#4482
C2×#15882
C2×#14792
#96612
#95592
#95482
#90742
#89492
#89482
#88462
#88312
#82032
#81492