Finite Group Databasebeta

← #227678

Subgroups — #227678

Order 512.

917 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23567C211yesmaximal, characteristic
2#22985C211yesmaximal, characteristic, complemented
3#22402C211yesmaximal, characteristic, complemented
4C2×C8×D8C211yesmaximal, characteristic, complemented
5C2×#1480C211yesmaximal, characteristic, complemented
6#9663C211yesmaximal, characteristic, complemented
7#9554C211yesmaximal, characteristic, complemented
8#9541C211yesmaximal, characteristic, complemented
9#9076C211yesmaximal, characteristic, complemented
10#8951C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic, complemented
12#8843C211yesmaximal, characteristic, complemented
13#8829C211yesmaximal, characteristic, complemented
14#8210C211yesmaximal, characteristic, complemented
15#8148C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C2×C8×D4—21no—
21C2×C8×D4C2211yescharacteristic
22C2×C8×D4C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C2×#408—21no—
26C4×#121C2211yescharacteristic
27#2057C411yescharacteristic
28#2041C411yescharacteristic
29#2033C411yescharacteristic
30#2033C411yescharacteristic
31#2023C411yescharacteristic
32#2011C411yescharacteristic
33#1999C411yescharacteristic
34#1999C411yescharacteristic
35#1954C2211yescharacteristic
36#1954C2211yescharacteristic
37#1954C2211yescharacteristic
38#1939C2211yescharacteristic
39#1923—21no—
40#1870—21no—
41#1869—21no—
42#1869—21no—
43#1850—21no—
44#1797C2211yescharacteristic
45#1780—21no—
46C8×C4⋊C4C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C8×D8—24no—
52C2×#335C2211yescharacteristic
53C2×#335C2211yescharacteristic
54#1480—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—
73C23×C8—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—
103C8×D4—44no—
104#431—22no—
105#431—41no—
106C2×#107C2×C411yescharacteristic
107C2×#107C2×C411yescharacteristic
108C2×C4⋊C8—22no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—41no—
113C2×C4⋊C8C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115C2×C4⋊C8C2311yescharacteristic
116C2×C4⋊C8C2311yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×#102C2×C411yescharacteristic
121C2×#102C2×C411yescharacteristic
122C2×#102C2×C411yescharacteristic
123C2×#102C2×C411yescharacteristic
124C2×#102C2×C411yescharacteristic
125#408—44no—
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⋊C8—21no—
136C2×C22⋊C8—21no—
137C2×C22⋊C8C2×C411yescharacteristic
138C2×C22⋊C8C2×C411yescharacteristic
139C2×C22⋊C8D411yescharacteristic
140C2×C22⋊C8D411yescharacteristic
141C2×C4×C8—21no—
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148#390C2×C411yescharacteristic
149#389C2×C411yescharacteristic
150#388—21no—
151#386—21no—
152#380C2×C411yescharacteristic
153C4×C4⋊C4C2311yescharacteristic
154C4×C22⋊C4—21no—
155#340C2×C411yescharacteristic
156#340C2×C411yescharacteristic
157#340C2×C411yescharacteristic
158#340C2×C411yescharacteristic
159#336—21no—
160#336—21no—
161#336—21no—
162#336C2×C411yescharacteristic
163#336C2×C411yescharacteristic
164#336D411yescharacteristic
165#336D411yescharacteristic
166#335—22no—
167#335—41no—
168D4⋊C8—22no—
169D4⋊C8—22no—
170D4⋊C8—22no—
171D4⋊C8—22no—
172D4⋊C8—41no—
173D4⋊C8D414yescomplemented
174#322—41no—
175C82—22no—
176C2×#41—42no—
177C2×#41—42no—
178C2×#41—42no—
179C2×#41—42no—
180C2×#41—81no—
181C2×#41—81no—
182C22×Q8—21no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—21no—
188C22×D4C22×C411yescharacteristic
189C22×D4C22×C411yescharacteristic
190C22×D4C22×C411yescharacteristic
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4C22×C411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4#4111yescharacteristic
197C2×D8—22no—
198C2×D8—22no—
199C2×D8—22no—
200C2×D8—22no—
201C2×D8—41no—
202C2×D8—41no—
203C2×M16—42no—
204C2×M16—42no—
205C2×M16—81no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—42no—
213C22×C8—42no—
214C22×C8—42no—
215C22×C8—42no—
216C22×C8—42no—
217C22×C8—42no—
218C22×C8—81no—
219C22×C8—81no—
220C22×C8—81no—
221C22×C8C22×C411yescharacteristic
222C22×C8C22×C411yescharacteristic
223C22×C8C22×C411yescharacteristic
224C22×C8C22×C411yescharacteristic
225C22×C8C2×D411yescharacteristic
226C22×C8C2×D411yescharacteristic
227C22×C8C2×D411yescharacteristic
228C22×C8C2×D411yescharacteristic
229C4⋊Q8—22no—
230C4⋊Q8—41no—
231#127—22no—
232#127—41no—
233#124—42no—
234#124—42no—
235#121—22no—
236#121—22no—
237#121—22no—
238#121—22no—
239#121—42no—
240#121—42no—
241#121—41no—
242#121—41no—
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—21no—
253C4×D4—42no—
254C4×D4—42no—
255C4×D4—42no—
256C4×D4—42no—
257C4×D4C2×D412yes—
258C4×D4#4112yes—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C4⋊C4C22×C411yescharacteristic
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×C22⋊C4—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42C22×C411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42#4111yescharacteristic
283C2×C42C2411yescharacteristic
284#107—22no—
285#107—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#102—22no—
298#102—22no—
299#102—22no—
300#102—22no—
301#102—22no—
302#102—41no—
303#102—41no—
304#102—41no—
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—42no—
314C22⋊C8—42no—
315C22⋊C8—42no—
316C22⋊C8—81no—
317C22⋊C8—81no—
318C22⋊C8—81no—
319C22⋊C8D814yescomplemented
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—21no—
330C4×C8—21no—
331C4×C8—42no—
332C4×C8C2×D412yes—
333#95—21no—
334#95—21no—
335#95—21no—
336C24—21no—
337C24—21no—
338C24—21no—
339C24C4×D411yescharacteristic
340#41—84no—
341#41—84no—
342#41—84no—
343#41—84no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—81no—
347C2×D4—24no—
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—21no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—81no—
375C2×D4—81no—
376C2×D4—81no—
377C2×D4—81no—
378C2×D4C4×D412yes—
379C2×D4#13112yes—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
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—21no—
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×C4—41no—
417C22×C4C4×D411yescharacteristic
418C22×C4C4×D411yescharacteristic
419C22×C4C4×D411yescharacteristic
420C22×C4C2×D812yes—
421C22×C4#13512yes—
422C22×C4C23×C411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C2×#4111yescharacteristic
425D8—28no—
426D8—44no—
427M16—84no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
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—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—41no—
466C2×C8—41no—
467C2×C8—84no—
468C2×C8—84no—
469C2×C8—84no—
470C2×C8#13112yes—
471C2×C8C2×D812yes—
472C2×C8C2×D812yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4C4×D412yes—
484C4⋊C4#13112yes—
485C22⋊C4—24no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—41no—
492C22⋊C4—41no—
493C22⋊C4—41no—
494C22⋊C4—41no—
495C22⋊C4—42no—
496C22⋊C4—81no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—41no—
504C42—41no—
505C42C22×D411yescharacteristic
506C42C2×#4111yescharacteristic
507C42H(5,𝔽2)11yescharacteristic
508C42H(5,𝔽2)11yescharacteristic
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—22no—
516C23—22no—
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—
536C23C4×D812yes—
537C23C2×C4×D411yescharacteristic
538Q8—84no—
539D4—22no—
540D4—22no—
541D4—22no—
542D4—22no—
543D4—48no—
544D4—41no—
545D4—41no—
546D4—41no—
547D4—41no—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—41no—
552D4—84no—
553D4—84no—
554D4—84no—
555D4—84no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
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—81no—
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—81no—
603C2×C4C4×D812yes—
604C2×C4C2×C4×D411yescharacteristic
605C2×C4#51811yescharacteristic
606C2×C4D4211yescharacteristic
607C2×C4#54611yescharacteristic
608C2×C4#54611yescharacteristic
609C2×C4#55111yescharacteristic
610C2×C4C2×#13111yescharacteristic
611C2×C4C2×#13111yescharacteristic
612C2×C4C2×#13111yescharacteristic
613C2×C4#56811yescharacteristic
614C2×C4C22×D811yescharacteristic
615C2×C4C2×#13511yescharacteristic
616C2×C4#57611yescharacteristic
617C2×C4#57711yescharacteristic
618C8—22no—
619C8—22no—
620C8—22no—
621C8—42no—
622C8—41no—
623C8—41no—
624C8—41no—
625C8—42no—
626C8—41no—
627C8—41no—
628C8—81no—
629C8—81no—
630C8—81no—
631C8—81no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
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—81no—
660C22#235811yescharacteristic
661C22#292911yescharacteristic
662C22#293511yescharacteristic
663C22C2×C4×D811yescharacteristic
664C22#294611yescharacteristic
665C22#295511yescharacteristic
666C22#295711yescharacteristic
667C4—22no—
668C4—21no—
669C4—21no—
670C4—41no—
671C4—42no—
672C4—41no—
673C4—42no—
674C4—81no—
675C4—81no—
676C4—81no—
677C4#292911yescharacteristic
678C4#293511yescharacteristic
679C4#297611yescharacteristic
680C4#297911yescharacteristic
681C4D4×D811yescharacteristic
682C4#328211yescharacteristic
683C4#329411yescharacteristic
684C4#329711yescharacteristic
685C2—42no—
686C2—82no—
687C2—82no—
688C2—161no—
689C2#1790011yescharacteristic
690C2#1792011yescharacteristic
691C2#1806611yescharacteristic
692C2#1808011yescharacteristic
693C2#2284311yescharacteristic
694C2#2287011yescharacteristic
695C2#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. #22985C2semidirect
3. #22402C2semidirect
4. C2×C8×D8C2semidirect
5. C2×#1480C2semidirect
6. #9663C2semidirect
7. #9554C2semidirect
8. #9541C2semidirect
9. #9076C2semidirect
10. #8951C2semidirect
11. #8947C2semidirect
12. #8843C2semidirect
13. #8829C2semidirect
14. #8210C2semidirect
15. #8148C2semidirect
173. D4⋊C8D4semidirect
319. C22⋊C8D8semidirect

Maximal subgroups

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

subgroupindex
#235672
#229852
#224022
C2×C8×D82
C2×#14802
#96632
#95542
#95412
#90762
#89512
#89472
#88432
#88292
#82102
#81482