Finite Group Databasebeta

← #228136

Subgroups — #228136

Order 512.

862 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23567C211yesmaximal, characteristic
2#23088C211yesmaximal, characteristic, complemented
3#22402C211yesmaximal, characteristic, complemented
4C2×#1589C211yesmaximal, characteristic, complemented
5C2×#1496C211yesmaximal, characteristic, complemented
6#9659C211yesmaximal, characteristic, complemented
7#9554C211yesmaximal, characteristic, complemented
8#9541C211yesmaximal, characteristic, complemented
9#9074C211yesmaximal, characteristic, complemented
10#8951C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic, complemented
12#8849C211yesmaximal, characteristic, complemented
13#8834C211yesmaximal, characteristic, complemented
14#8200C211yesmaximal, characteristic, complemented
15#8148C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C2×#436C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22C2×#432C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C2×#408—21no—
25C4×#121C2211yescharacteristic
26#2057C411yescharacteristic
27#2041C411yescharacteristic
28#2033C411yescharacteristic
29#2033C411yescharacteristic
30#2023C411yescharacteristic
31#2011C411yescharacteristic
32#1999C411yescharacteristic
33#1999C411yescharacteristic
34#1954—21no—
35#1954C2211yescharacteristic
36#1954C2211yescharacteristic
37#1954C2211yescharacteristic
38#1939C2211yescharacteristic
39#1924—21no—
40#1870—21no—
41#1870—21no—
42#1869—21no—
43#1797C2211yescharacteristic
44#1780—21no—
45C8×C4⋊C4C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1753—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51#1589—24no—
52C2×#335C2211yescharacteristic
53C2×#334C2211yescharacteristic
54#1496—24no—
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊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×#41—21no—
70C22×D8C2×C411yescharacteristic
71C22×D8C2×C411yescharacteristic
72C22×M16—21no—
73C22×M16—21no—
74C23×C8—21no—
75#532—22no—
76#532—22no—
77#532—41no—
78#532—41no—
79C2×C4⋊Q8C2×C411yescharacteristic
80C2×#127C2×C411yescharacteristic
81C2×#124—21no—
82C2×#121—21no—
83C2×#121C2×C411yescharacteristic
84C2×#121C2×C411yescharacteristic
85C2×#121C2×C411yescharacteristic
86C2×C4×D4—21no—
87C2×C4×D4—21no—
88C2×C4×D4C2311yescharacteristic
89C2×C4×D4C2311yescharacteristic
90C2×C4×D4C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C4×D8—22no—
93C4×D8—22no—
94C4×D8—41no—
95C4×D8—41no—
96#436—22no—
97#436—22no—
98#436—41no—
99#436—41no—
100C8×D4—22no—
101C8×D4—41no—
102#432—22no—
103#432—41no—
104#431—22no—
105#431—41no—
106C2×#107C2×C411yescharacteristic
107C2×#107C2×C411yescharacteristic
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×C4⋊C8C2311yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×#102C2×C411yescharacteristic
121C2×#102C2×C411yescharacteristic
122#408—44no—
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⋊C8—21no—
130C2×C22⋊C8—21no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8—21no—
133C2×C22⋊C8—21no—
134C2×C22⋊C8—21no—
135C2×C22⋊C8C2×C411yescharacteristic
136C2×C22⋊C8C2×C411yescharacteristic
137C2×C22⋊C8D411yescharacteristic
138C2×C22⋊C8D411yescharacteristic
139C2×#97—21no—
140C2×C4×C8—21no—
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147#390C2×C411yescharacteristic
148#389C2×C411yescharacteristic
149#388—21no—
150#386—21no—
151#380C2×C411yescharacteristic
152C4×C4⋊C4C2311yescharacteristic
153C4×C22⋊C4—21no—
154#340C2×C411yescharacteristic
155#340C2×C411yescharacteristic
156#340C2×C411yescharacteristic
157#340C2×C411yescharacteristic
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—
167D4⋊C8—22no—
168D4⋊C8—22no—
169D4⋊C8—22no—
170D4⋊C8—22no—
171D4⋊C8—41no—
172D4⋊C8D414yescomplemented
173#322—41no—
174C82—22no—
175C2×#41—42no—
176C2×#41—42no—
177C2×#41—42no—
178C2×#41—42no—
179C2×#41—81no—
180C2×#41—81no—
181C22×Q8—21no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4C22×C411yescharacteristic
188C22×D4C22×C411yescharacteristic
189C22×D4C22×C411yescharacteristic
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4C22×C411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4#4111yescharacteristic
196C2×D8—22no—
197C2×D8—22no—
198C2×D8—22no—
199C2×D8—41no—
200C2×D8—41no—
201C2×D8—41no—
202C2×M16—42no—
203C2×M16—42no—
204C2×M16—42no—
205C2×M16—42no—
206C2×M16—81no—
207C2×M16—81no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—42no—
215C22×C8—42no—
216C22×C8—81no—
217C22×C8C22×C411yescharacteristic
218C22×C8C22×C411yescharacteristic
219C22×C8C22×C411yescharacteristic
220C22×C8C22×C411yescharacteristic
221C22×C8C2×D411yescharacteristic
222C22×C8C2×D411yescharacteristic
223C22×C8C2×D411yescharacteristic
224C22×C8C2×D411yescharacteristic
225C4⋊Q8—22no—
226C4⋊Q8—41no—
227#127—22no—
228#127—41no—
229#124—42no—
230#124—42no—
231#121—22no—
232#121—22no—
233#121—22no—
234#121—42no—
235#121—42no—
236#121—41no—
237#121—41no—
238#121—41no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—42no—
250C4×D4—42no—
251C4×D4—81no—
252C4×D4—81no—
253C4×D4C2×D412yes—
254C4×D4#4112yes—
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×C22⋊C4—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42C22×C411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42#4111yescharacteristic
279C2×C42C2411yescharacteristic
280#107—22no—
281#107—41no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—42no—
290C4⋊C8—81no—
291C4⋊C8C2×D412yes—
292#102—22no—
293#102—22no—
294#102—22no—
295#102—22no—
296#102—22no—
297#102—22no—
298#102—41no—
299#102—41no—
300C22⋊C8—22no—
301C22⋊C8—22no—
302C22⋊C8—22no—
303C22⋊C8—41no—
304C22⋊C8—42no—
305C22⋊C8—42no—
306C22⋊C8—42no—
307C22⋊C8—42no—
308C22⋊C8—42no—
309C22⋊C8—42no—
310C22⋊C8—42no—
311C22⋊C8—42no—
312C22⋊C8—42no—
313C22⋊C8—81no—
314C22⋊C8—81no—
315C22⋊C8—81no—
316#97—42no—
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—21no—
328C4×C8—81no—
329C4×C8C2×D412yes—
330#95—21no—
331#95—21no—
332#95—21no—
333C24—21no—
334C24—21no—
335C24—21no—
336C24C4×D411yescharacteristic
337#41—84no—
338#41—84no—
339#41—84no—
340#41—84no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—81no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
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—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—81no—
372C2×D4—81no—
373C2×D4—81no—
374C2×D4—81no—
375C2×D4—81no—
376C2×D4C4×D412yes—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4C4×D411yescharacteristic
417C22×C4C4×D411yescharacteristic
418C22×C4C4×D411yescharacteristic
419C22×C4C23×C411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C2×#4111yescharacteristic
422D8—44no—
423D8—44no—
424M16—84no—
425M16—84no—
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—21no—
435C2×C8—22no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—22no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—84no—
466C2×C8#13112yes—
467C2×C8#13112yes—
468C2×C8AGL(1,ℤ/8)12yes—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—42no—
478C4⋊C4—81no—
479C4⋊C4C4×D412yes—
480C4⋊C4#13112yes—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—81no—
491C22⋊C4—81no—
492C22⋊C4—81no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—41no—
500C42—41no—
501C42C22×D411yescharacteristic
502C42C2×#4111yescharacteristic
503C42H(5,𝔽2)11yescharacteristic
504C42H(5,𝔽2)11yescharacteristic
505C23—22no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23C2×C4×D411yescharacteristic
534Q8—84no—
535D4—22no—
536D4—22no—
537D4—41no—
538D4—41no—
539D4—41no—
540D4—41no—
541D4—41no—
542D4—41no—
543D4—41no—
544D4—41no—
545D4—41no—
546D4—41no—
547D4—84no—
548D4—84no—
549D4—84no—
550D4—84no—
551D4—84no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
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—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4C2×C4×D411yescharacteristic
601C2×C4#51811yescharacteristic
602C2×C4D4211yescharacteristic
603C2×C4#54611yescharacteristic
604C2×C4#54611yescharacteristic
605C2×C4#55111yescharacteristic
606C2×C4C2×#13111yescharacteristic
607C2×C4C2×#13111yescharacteristic
608C2×C4C2×#13111yescharacteristic
609C2×C4#56811yescharacteristic
610C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
611C2×C4#57511yescharacteristic
612C2×C4#57611yescharacteristic
613C2×C4#57611yescharacteristic
614C8—22no—
615C8—22no—
616C8—22no—
617C8—42no—
618C8—41no—
619C8—41no—
620C8—42no—
621C8—41no—
622C8—41no—
623C8—41no—
624C8—81no—
625C8—81no—
626C8—81no—
627C8—81no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—42no—
632C22—41no—
633C22—41no—
634C22—42no—
635C22—41no—
636C22—42no—
637C22—41no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22#235811yescharacteristic
657C22#293111yescharacteristic
658C22#293311yescharacteristic
659C22C4×AGL(1,ℤ/8)11yescharacteristic
660C22#294811yescharacteristic
661C22#295511yescharacteristic
662C22#295711yescharacteristic
663C4—21no—
664C4—21no—
665C4—41no—
666C4—41no—
667C4—42no—
668C4—41no—
669C4—42no—
670C4—81no—
671C4—81no—
672C4—81no—
673C4#293111yescharacteristic
674C4#293311yescharacteristic
675C4#297611yescharacteristic
676C4#297911yescharacteristic
677C4#326411yescharacteristic
678C4#326811yescharacteristic
679C4#332211yescharacteristic
680C4#332511yescharacteristic
681C2—81no—
682C2—82no—
683C2—161no—
684C2—161no—
685C2#1796611yescharacteristic
686C2#1797011yescharacteristic
687C2#1801111yescharacteristic
688C2#1802611yescharacteristic
689C2#2285311yescharacteristic
690C2#2288011yescharacteristic
691C2#2288911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radical#23567C2
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. #23088C2semidirect
3. #22402C2semidirect
4. C2×#1589C2semidirect
5. C2×#1496C2semidirect
6. #9659C2semidirect
7. #9554C2semidirect
8. #9541C2semidirect
9. #9074C2semidirect
10. #8951C2semidirect
11. #8947C2semidirect
12. #8849C2semidirect
13. #8834C2semidirect
14. #8200C2semidirect
15. #8148C2semidirect
172. D4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
#235672
#230882
#224022
C2×#15892
C2×#14962
#96592
#95542
#95412
#90742
#89512
#89472
#88492
#88342
#82002
#81482