Finite Group Databasebeta

← #227642

Subgroups — #227642

Order 512.

933 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22985C211yesmaximal, characteristic, complemented
2#22469C211yesmaximal, characteristic, complemented
3C4×#449C211yesmaximal, characteristic
4C2×#1580C211yesmaximal, characteristic, complemented
5C2×#1477C211yesmaximal, characteristic, complemented
6#9655C211yesmaximal, characteristic, complemented
7#9555C211yesmaximal, characteristic, complemented
8#9543C211yesmaximal, characteristic, complemented
9#9074C211yesmaximal, characteristic, complemented
10#8947C211yesmaximal, characteristic, complemented
11#8945C211yesmaximal, characteristic, complemented
12#8845C211yesmaximal, characteristic, complemented
13#8830C211yesmaximal, characteristic, complemented
14C8×#102C211yesmaximal, characteristic, complemented
15#8148C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17C2×#449C411yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×C8×D4—21no—
21C2×C8×D4C2211yescharacteristic
22C2×C8×D4C2211yescharacteristic
23C2×C8×D4C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C4×#121C2211yescharacteristic
27#1984C2211yescharacteristic
28#1984C2211yescharacteristic
29#1954C2211yescharacteristic
30#1939C2211yescharacteristic
31#1923—21no—
32#1880C411yescharacteristic
33#1879C411yescharacteristic
34#1875C411yescharacteristic
35#1869—21no—
36#1869—21no—
37#1869—21no—
38#1864C411yescharacteristic
39#1863C411yescharacteristic
40#1854C411yescharacteristic
41#1850—21no—
42#1817—21no—
43#1797C2211yescharacteristic
44#1788C411yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46C4×#103C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×C22⋊C8C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50C2×#336—21no—
51#1580—24no—
52C2×#334C2211yescharacteristic
53C2×#334C2211yescharacteristic
54#1477—24no—
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×C82C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×#41C2×C411yescharacteristic
70C22×SD16C2×C411yescharacteristic
71C22×D8C2×C411yescharacteristic
72C23×C8—21no—
73C23×C8—21no—
74C23×C8—21no—
75C23×C8—21no—
76C2×#121C2×C411yescharacteristic
77C4×#41—22no—
78C4×#41—22no—
79C4×#41—41no—
80C4×#41—41no—
81C2×C4×Q8C2×C411yescharacteristic
82C2×C4×Q8C2311yescharacteristic
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×C4×D4—21no—
86C2×C4×D4C2×C411yescharacteristic
87C2×C4×D4C2311yescharacteristic
88C2×C4×D4C2311yescharacteristic
89C2×#117—21no—
90C2×#117C2×C411yescharacteristic
91C22×C42—21no—
92C22×C42C2311yescharacteristic
93#449C818yescomplemented
94C4×SD16—41no—
95C4×SD16—41no—
96C4×D8—22no—
97C4×D8—22no—
98C8×D4—22no—
99C8×D4—22no—
100C8×D4—22no—
101C8×D4—41no—
102C8×D4—41no—
103C8×D4—41no—
104C8×D4—44no—
105#431—22no—
106#431—41no—
107C2×#107C2×C411yescharacteristic
108C2×#106C2×C411yescharacteristic
109C2×C4⋊C8—22no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—41no—
114C2×C4⋊C8C2311yescharacteristic
115C2×C4⋊C8C2311yescharacteristic
116C2×C4⋊C8C2311yescharacteristic
117C2×C4⋊C8C2311yescharacteristic
118C2×#103C2×C411yescharacteristic
119C2×#103C2×C411yescharacteristic
120C2×#103C2×C411yescharacteristic
121C2×#102C2×C411yescharacteristic
122C2×#102C2×C411yescharacteristic
123C2×#102C2×C411yescharacteristic
124C2×#102C2×C411yescharacteristic
125C2×#102C2×C411yescharacteristic
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—21no—
129C2×C22⋊C8—21no—
130C2×C22⋊C8—21no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8—21no—
133C2×C22⋊C8C2×C411yescharacteristic
134C2×C22⋊C8C2×C411yescharacteristic
135C2×C22⋊C8D411yescharacteristic
136C2×C22⋊C8D411yescharacteristic
137C2×C4×C8—21no—
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140C2×C4×C8C2311yescharacteristic
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144#390C2×C411yescharacteristic
145#389C2×C411yescharacteristic
146#388—21no—
147#386—21no—
148#380C2×C411yescharacteristic
149C4×C4⋊C4C2311yescharacteristic
150C4×C22⋊C4—21no—
151#340C2×C411yescharacteristic
152#340C2×C411yescharacteristic
153#340C2×C411yescharacteristic
154#340C2×C411yescharacteristic
155#336—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#336—21no—
160#336—21no—
161#336—21no—
162#336C2×C411yescharacteristic
163#336C2×C411yescharacteristic
164#336D411yescharacteristic
165#336D411yescharacteristic
166#334—22no—
167#334—41no—
168Q8⋊C8—22no—
169Q8⋊C8—22no—
170D4⋊C8—22no—
171D4⋊C8—22no—
172D4⋊C8—41no—
173D4⋊C8D414yescomplemented
174#322—22no—
175C82—41no—
176C2×#41—22no—
177C2×#41—22no—
178C2×#41—22no—
179C2×#41—41no—
180C2×#41—41no—
181C2×#41C2×C814yescomplemented
182C22×Q8C22×C411yescharacteristic
183C22×D4—21no—
184C22×D4—21no—
185C22×D4C22×C411yescharacteristic
186C22×D4C22×C411yescharacteristic
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4C22×C411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4#4111yescharacteristic
194C2×SD16—41no—
195C2×SD16—41no—
196C2×SD16C2×C814yescomplemented
197C2×D8—22no—
198C2×D8—22no—
199C2×D8C2×C814yescomplemented
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—42no—
207C22×C8—42no—
208C22×C8—42no—
209C22×C8—42no—
210C22×C8—42no—
211C22×C8—42no—
212C22×C8—42no—
213C22×C8—42no—
214C22×C8—81no—
215C22×C8—81no—
216C22×C8—81no—
217C22×C8—81no—
218C22×C8C22×C411yescharacteristic
219C22×C8C22×C411yescharacteristic
220C22×C8C22×C411yescharacteristic
221C22×C8C22×C411yescharacteristic
222C22×C8C2×D411yescharacteristic
223C22×C8C2×D411yescharacteristic
224C22×C8C2×D411yescharacteristic
225C22×C8C2×D411yescharacteristic
226#121—22no—
227#121C2×C814yescomplemented
228C4×Q8—22no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8—21no—
232C4×Q8—21no—
233C4×Q8—41no—
234C4×D4—22no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—41no—
242C4×D4—42no—
243C4×D4—42no—
244C4×D4—42no—
245C4×D4—42no—
246C4×D4—42no—
247C4×D4—42no—
248C4×D4C2×D412yes—
249C4×D4#4112yes—
250#117—22no—
251#117—42no—
252#117—42no—
253#117—41no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C4⋊C4C22×C411yescharacteristic
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—42no—
279C2×C42—42no—
280C2×C42C22×C411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42#4111yescharacteristic
283C2×C42C2411yescharacteristic
284#107—22no—
285#106—41no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—42no—
294C4⋊C8—42no—
295C4⋊C8—42no—
296C4⋊C8C2×D412yes—
297#103—41no—
298#103—41no—
299#103C2×C814yescomplemented
300#102—22no—
301#102—22no—
302#102—22no—
303#102—41no—
304#102C2×C814yescomplemented
305C22⋊C8—22no—
306C22⋊C8—22no—
307C22⋊C8—41no—
308C22⋊C8—42no—
309C22⋊C8—42no—
310C22⋊C8—42no—
311C22⋊C8—42no—
312C22⋊C8—42no—
313C22⋊C8—81no—
314C22⋊C8—81no—
315C22⋊C8C2×C814yescomplemented
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—21no—
324C4×C8—21no—
325C4×C8—21no—
326C4×C8—21no—
327C4×C8—42no—
328C4×C8C2×D412yes—
329#95—21no—
330#95—21no—
331#95—21no—
332C24—21no—
333C24—21no—
334C24C4×D411yescharacteristic
335#41—28no—
336#41—44no—
337#41—44no—
338#41—44no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8C22×C812yes—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4C4×D412yes—
360C2×D4C22×C812yes—
361C2×D4C22×C812yes—
362C2×D4#13112yes—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—81no—
401C22×C4C4×D411yescharacteristic
402C22×C4C4×D411yescharacteristic
403C22×C4C4×D411yescharacteristic
404C22×C4C22×C812yes—
405C22×C4#13112yes—
406C22×C4C23×C411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C2×#4111yescharacteristic
409SD16—44no—
410D8—28no—
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—22no—
422C2×C8—21no—
423C2×C8—22no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—84no—
451C2×C8—84no—
452C2×C8—84no—
453C2×C8—84no—
454C2×C8C22×C812yes—
455C2×C8C22×C812yes—
456C2×C8AGL(1,ℤ/8)12yes—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4C4×D412yes—
470C4⋊C4C22×C812yes—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—24no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—81no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—42no—
488C42—42no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42—41no—
493C42C22×D411yescharacteristic
494C42C2×#4111yescharacteristic
495C42H(5,𝔽2)11yescharacteristic
496C42H(5,𝔽2)11yescharacteristic
497C23—22no—
498C23—21no—
499C23—22no—
500C23—21no—
501C23—22no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—22no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23C8×D412yes—
518C23C2×C4×D411yescharacteristic
519Q8—22no—
520Q8—22no—
521Q8—41no—
522Q8—41no—
523D4—22no—
524D4—22no—
525D4—22no—
526D4—22no—
527D4—22no—
528D4—48no—
529D4—48no—
530D4—41no—
531D4—41no—
532D4C8×D414yescomplemented
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—48no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—84no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4C8×D412yes—
582C2×C4C2×C4×D411yescharacteristic
583C2×C4#51811yescharacteristic
584C2×C4D4211yescharacteristic
585C2×C4#54611yescharacteristic
586C2×C4#54611yescharacteristic
587C2×C4#55111yescharacteristic
588C2×C4C23×C811yescharacteristic
589C2×C4C2×#13111yescharacteristic
590C2×C4C2×#13111yescharacteristic
591C2×C4#56811yescharacteristic
592C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
593C2×C4#57511yescharacteristic
594C2×C4#57711yescharacteristic
595C2×C4#57711yescharacteristic
596C8—22no—
597C8—22no—
598C8—22no—
599C8—42no—
600C8—41no—
601C8—41no—
602C8—41no—
603C8—42no—
604C8—41no—
605C8—41no—
606C8—81no—
607C8—81no—
608C8—81no—
609C8—81no—
610C22—22no—
611C22—24no—
612C22—22no—
613C22—22no—
614C22—41no—
615C22—42no—
616C22—41no—
617C22—42no—
618C22—41no—
619C22—42no—
620C22—41no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22#235811yescharacteristic
632C22C2×C8×D411yescharacteristic
633C22#293311yescharacteristic
634C22C4×AGL(1,ℤ/8)11yescharacteristic
635C22#295211yescharacteristic
636C22#295511yescharacteristic
637C22#295711yescharacteristic
638C4—22no—
639C4—21no—
640C4—21no—
641C4—42no—
642C4—41no—
643C4—42no—
644C4—41no—
645C4—42no—
646C4—81no—
647C4—81no—
648C4—81no—
649C4C2×C8×D411yescharacteristic
650C4#293311yescharacteristic
651C4#297211yescharacteristic
652C4#297711yescharacteristic
653C4#327211yescharacteristic
654C4#327511yescharacteristic
655C4#330611yescharacteristic
656C4#330711yescharacteristic
657C2—42no—
658C2—44no—
659C2—82no—
660C2C8×AGL(1,ℤ/8)11yescharacteristic
661C2#1785811yescharacteristic
662C2#1800711yescharacteristic
663C2#1802911yescharacteristic
664C2#2284211yescharacteristic
665C2#2286611yescharacteristic
666C2#2288711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#449C4
LCSG ▷ C2×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. #22985C2semidirect
2. #22469C2semidirect
4. C2×#1580C2semidirect
5. C2×#1477C2semidirect
6. #9655C2semidirect
7. #9555C2semidirect
8. #9543C2semidirect
9. #9074C2semidirect
10. #8947C2semidirect
11. #8945C2semidirect
12. #8845C2semidirect
13. #8830C2semidirect
14. C8×#102C2semidirect
15. #8148C2semidirect
93. #449C8semidirect
173. D4⋊C8D4semidirect
181. C2×#41C2×C8semidirect
196. C2×SD16C2×C8semidirect
199. C2×D8C2×C8semidirect
227. #121C2×C8semidirect
299. #103C2×C8semidirect
304. #102C2×C8semidirect
315. C22⋊C8C2×C8semidirect
532. D4C8×D4semidirect

Maximal subgroups

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

subgroupindex
#229852
#224692
C4×#4492
C2×#15802
C2×#14772
#96552
#95552
#95432
#90742
#89472
#89452
#88452
#88302
C8×#1022
#81482