Finite Group Databasebeta

← #227660

Subgroups — #227660

Order 512.

895 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23567C211yesmaximal, characteristic
2#23083C211yesmaximal, characteristic, complemented
3#22402C211yesmaximal, characteristic, complemented
4C2×#1580C211yesmaximal, characteristic, complemented
5C2×#1484C211yesmaximal, characteristic, complemented
6#9663C211yesmaximal, characteristic, complemented
7#9556C211yesmaximal, characteristic, complemented
8#9542C211yesmaximal, characteristic, complemented
9#9078C211yesmaximal, characteristic, complemented
10#8951C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic, complemented
12#8843C211yesmaximal, characteristic, complemented
13#8831C211yesmaximal, characteristic, complemented
14#8210C211yesmaximal, characteristic, complemented
15#8147C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×#408—21no—
24C2×C4×M16C2211yescharacteristic
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#1923—21no—
40#1870—21no—
41#1869—21no—
42#1869—21no—
43#1797C2211yescharacteristic
44#1780—21no—
45C8×C4⋊C4C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C8×C22⋊C4—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51#1580—24no—
52C2×#335C2211yescharacteristic
53C2×#334C2211yescharacteristic
54#1484—24no—
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
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—
75C23×C8—21no—
76#532—22no—
77#532—22no—
78#532—41no—
79#532—41no—
80C2×C4⋊Q8C2×C411yescharacteristic
81C2×#127C2×C411yescharacteristic
82C2×#124—21no—
83C2×#121—21no—
84C2×#121C2×C411yescharacteristic
85C2×#121C2×C411yescharacteristic
86C2×#121C2×C411yescharacteristic
87C2×C4×D4—21no—
88C2×C4×D4—21no—
89C2×C4×D4C2311yescharacteristic
90C2×C4×D4C2311yescharacteristic
91C2×C4×D4C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C4×D8—22no—
94C4×D8—22no—
95C4×D8—41no—
96C4×D8—41no—
97#436—22no—
98#436—41no—
99C8×D4—22no—
100C8×D4—22no—
101C8×D4—41no—
102C8×D4—41no—
103#431—22no—
104#431—41no—
105C2×#107C2×C411yescharacteristic
106C2×#107C2×C411yescharacteristic
107C2×C4⋊C8—21no—
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×#102C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120#408—44no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
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⋊C8C2×C411yescharacteristic
133C2×C22⋊C8C2×C411yescharacteristic
134C2×C22⋊C8D411yescharacteristic
135C2×C22⋊C8D411yescharacteristic
136C4×M16—22no—
137C4×M16—41no—
138C2×#97—21no—
139C2×C4×C8—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—41no—
166#334—22no—
167D4⋊C8—22no—
168D4⋊C8—22no—
169D4⋊C8—22no—
170D4⋊C8—22no—
171D4⋊C8—41no—
172D4⋊C8D414yescomplemented
173#322—22no—
174#322—41no—
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—22no—
200C2×D8—41no—
201C2×D8—41no—
202C2×M16—42no—
203C2×M16—42no—
204C2×M16—42no—
205C2×M16—42no—
206C2×M16—42no—
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—42no—
217C22×C8—42no—
218C22×C8—81no—
219C22×C8—81no—
220C22×C8C22×C411yescharacteristic
221C22×C8C22×C411yescharacteristic
222C22×C8C22×C411yescharacteristic
223C22×C8C22×C411yescharacteristic
224C22×C8C2×D411yescharacteristic
225C22×C8C2×D411yescharacteristic
226C22×C8C2×D411yescharacteristic
227C22×C8C2×D411yescharacteristic
228C4⋊Q8—22no—
229C4⋊Q8—41no—
230#127—22no—
231#127—41no—
232#124—42no—
233#124—42no—
234#121—22no—
235#121—22no—
236#121—22no—
237#121—22no—
238#121—42no—
239#121—42no—
240#121—41no—
241#121—41no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—42no—
253C4×D4—42no—
254C4×D4—81no—
255C4×D4—81no—
256C4×D4C2×D412yes—
257C4×D4#4112yes—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C411yescharacteristic
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×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42C22×C411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42#4111yescharacteristic
282C2×C42C2411yescharacteristic
283#107—22no—
284#107—41no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—81no—
293C4⋊C8C2×D412yes—
294#102—22no—
295#102—22no—
296#102—22no—
297#102—22no—
298#102—22no—
299#102—41no—
300#102—41no—
301#102—41no—
302C22⋊C8—22no—
303C22⋊C8—22no—
304C22⋊C8—41no—
305C22⋊C8—41no—
306C22⋊C8—42no—
307C22⋊C8—42no—
308C22⋊C8—42no—
309C22⋊C8—42no—
310C22⋊C8—42no—
311C22⋊C8—42no—
312C22⋊C8—42no—
313C22⋊C8—42no—
314C22⋊C8—81no—
315C22⋊C8—81no—
316C22⋊C8—81no—
317#97—42no—
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—21no—
329C4×C8—42no—
330C4×C8—81no—
331C4×C8C2×D412yes—
332#95—21no—
333#95—21no—
334#95—21no—
335C24—21no—
336C24—21no—
337C24—21no—
338C24C4×D411yescharacteristic
339#41—84no—
340#41—84no—
341#41—84no—
342#41—84no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—81no—
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—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
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×D4—81no—
377C2×D4C4×D412yes—
378C2×D4#13112yes—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
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—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
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×C4—41no—
417C22×C4—41no—
418C22×C4C4×D411yescharacteristic
419C22×C4C4×D411yescharacteristic
420C22×C4C4×D411yescharacteristic
421C22×C4C23×C411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C2×#4111yescharacteristic
424D8—28no—
425D8—44no—
426M16—48no—
427M16—84no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—22no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—22no—
439C2×C8—21no—
440C2×C8—22no—
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—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
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—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—84no—
469C2×C8—84no—
470C2×C8#13112yes—
471C2×C8AGL(1,ℤ/8)12yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—81no—
482C4⋊C4C4×D412yes—
483C4⋊C4#13112yes—
484C22⋊C4—24no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—41no—
490C22⋊C4—41no—
491C22⋊C4—41no—
492C22⋊C4—41no—
493C22⋊C4—42no—
494C22⋊C4—81no—
495C22⋊C4—81no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—41no—
503C42—41no—
504C42C22×D411yescharacteristic
505C42C2×#4111yescharacteristic
506C42H(5,𝔽2)11yescharacteristic
507C42H(5,𝔽2)11yescharacteristic
508C23—22no—
509C23—21no—
510C23—22no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
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—
533C23—41no—
534C23—41no—
535C23—41no—
536C23C2×C4×D411yescharacteristic
537Q8—84no—
538D4—22no—
539D4—22no—
540D4—22no—
541D4—22no—
542D4—41no—
543D4—41no—
544D4—41no—
545D4—41no—
546D4—41no—
547D4—41no—
548D4—41no—
549D4—41no—
550D4—84no—
551D4—84no—
552D4—84no—
553D4—84no—
554D4—84no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4#44412yes—
603C2×C4C2×C4×D411yescharacteristic
604C2×C4#51811yescharacteristic
605C2×C4D4211yescharacteristic
606C2×C4#54611yescharacteristic
607C2×C4#54611yescharacteristic
608C2×C4#55111yescharacteristic
609C2×C4C2×#13111yescharacteristic
610C2×C4C2×#13111yescharacteristic
611C2×C4C2×#13111yescharacteristic
612C2×C4#56811yescharacteristic
613C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
614C2×C4#57511yescharacteristic
615C2×C4#57611yescharacteristic
616C2×C4#57611yescharacteristic
617C8—22no—
618C8—22no—
619C8—42no—
620C8—42no—
621C8—41no—
622C8—41no—
623C8—41no—
624C8—41no—
625C8—42no—
626C8—41no—
627C8—41no—
628C8—81no—
629C8—81no—
630C8—81no—
631C22—41no—
632C22—42no—
633C22—41no—
634C22—42no—
635C22—42no—
636C22—41no—
637C22—42no—
638C22—41no—
639C22—42no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—42no—
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—81no—
657C22—81no—
658C22—81no—
659C22#235811yescharacteristic
660C22#292911yescharacteristic
661C22#293511yescharacteristic
662C22C4×AGL(1,ℤ/8)11yescharacteristic
663C22#294811yescharacteristic
664C22C2×#44411yescharacteristic
665C22#295611yescharacteristic
666C4—21no—
667C4—21no—
668C4—22no—
669C4—42no—
670C4—41no—
671C4—41no—
672C4—42no—
673C4—81no—
674C4—81no—
675C4—81no—
676C4#292911yescharacteristic
677C4#293511yescharacteristic
678C4#297611yescharacteristic
679C4#297911yescharacteristic
680C4#327311yescharacteristic
681C4#327411yescharacteristic
682C4#331811yescharacteristic
683C4#331911yescharacteristic
684C2—81no—
685C2—82no—
686C2—82no—
687C2—161no—
688C2#1791711yescharacteristic
689C2#1792211yescharacteristic
690C2#1807811yescharacteristic
691C2#1809111yescharacteristic
692C2#2284311yescharacteristic
693C2#2286811yescharacteristic
694C2#2288611yescharacteristic
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. #23083C2semidirect
3. #22402C2semidirect
4. C2×#1580C2semidirect
5. C2×#1484C2semidirect
6. #9663C2semidirect
7. #9556C2semidirect
8. #9542C2semidirect
9. #9078C2semidirect
10. #8951C2semidirect
11. #8947C2semidirect
12. #8843C2semidirect
13. #8831C2semidirect
14. #8210C2semidirect
15. #8147C2semidirect
172. D4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
#235672
#230832
#224022
C2×#15802
C2×#14842
#96632
#95562
#95422
#90782
#89512
#89472
#88432
#88312
#82102
#81472