Finite Group Databasebeta

← #227982

Subgroups — #227982

Order 512.

1015 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22990C211yesmaximal, characteristic, complemented
2D4×C22⋊C8C211yesmaximal, characteristic, complemented
3C4×#447C211yesmaximal, characteristic
4C2×#1581C211yesmaximal, characteristic, complemented
5C2×#1486C211yesmaximal, characteristic, complemented
6#9657C211yesmaximal, characteristic, complemented
7#9558C211yesmaximal, characteristic, complemented
8#9541C211yesmaximal, characteristic, complemented
9#9076C211yesmaximal, characteristic, complemented
10#8951C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic, complemented
12#8843C211yesmaximal, characteristic, complemented
13#8832C211yesmaximal, characteristic, complemented
14C8×#102C211yesmaximal, characteristic, complemented
15#8148C211yesmaximal, characteristic, complemented
16C22×C4×D4C2211yescharacteristic
17C2×#447C411yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×#436—21no—
21C2×#436C2211yescharacteristic, complemented
22C2×C8×D4C2211yescharacteristic
23C2×C8×D4C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#423C2211yescharacteristic
26C22×C22⋊C8—21no—
27C4×#121C2211yescharacteristic
28#1954C2211yescharacteristic
29#1954C2211yescharacteristic
30#1954C2211yescharacteristic, complemented
31#1939C2211yescharacteristic, complemented
32#1923—21no—
33#1879C411yescharacteristic
34#1879C411yescharacteristic
35#1874C411yescharacteristic
36#1870—21no—
37#1869—21no—
38#1869—21no—
39#1862C411yescharacteristic
40#1862C411yescharacteristic
41#1852C411yescharacteristic
42#1851—21no—
43#1797C2211yescharacteristic
44#1786C411yescharacteristic
45#1779—21no—
46C8×C4⋊C4C2211yescharacteristic, complemented
47C4×#102C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51#1581—24no—
52C2×#335C2211yescharacteristic, complemented
53C2×#335C2211yescharacteristic, complemented
54#1486—24no—
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic, complemented
60C2×D4⋊C8C2211yescharacteristic, complemented
61C2×#322C2211yescharacteristic, complemented
62C2×C82C2211yescharacteristic, complemented
63#1279C2211yescharacteristic, complemented
64#1279C2211yescharacteristic, complemented
65#1277C2211yescharacteristic, complemented
66#1277C2211yescharacteristic, complemented
67#1273C2211yescharacteristic, complemented
68#1273C2211yescharacteristic, complemented
69C23×D4C2×C411yescharacteristic
70C24×C4—21no—
71C22×D8C2×C411yescharacteristic
72C22×D8C2×C411yescharacteristic
73C22×M16—21no—
74C22×M16—21no—
75C23×C8—21no—
76C23×C8—21no—
77C2×#121C2×C411yescharacteristic
78C2×C4×D4—21no—
79C2×C4×D4—22no—
80C2×C4×D4—22no—
81C2×C4×D4—21no—
82C2×C4×D4—21no—
83C2×C4×D4—41no—
84C2×C4×D4—41no—
85C2×C4×D4C2×C411yescharacteristic
86C2×C4×D4C2×C411yescharacteristic
87C2×C4×D4C2311yescharacteristic
88C2×C4×D4C2311yescharacteristic
89C2×C4×D4C2311yescharacteristic
90C22×C4⋊C4C2×C411yescharacteristic
91C22×C22⋊C4—21no—
92C22×C42C2311yescharacteristic
93#447—24no—
94C4×D8—22no—
95C4×D8—22no—
96C4×D8—41no—
97C4×D8—41no—
98#436—22no—
99#436—41no—
100#436—44no—
101C8×D4—22no—
102C8×D4—22no—
103C8×D4—41no—
104C8×D4—41no—
105#431—22no—
106#431—41no—
107C2×#107C2×C411yescharacteristic
108C2×#107C2×C411yescharacteristic
109#423—22no—
110#423—41no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8C2311yescharacteristic
115C2×C4⋊C8C2311yescharacteristic
116C2×C4⋊C8C2311yescharacteristic
117C2×C4⋊C8C2311yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×#102C2×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⋊C8—21no—
134C2×C22⋊C8—21no—
135C2×C22⋊C8—21no—
136C2×C22⋊C8—21no—
137C2×C22⋊C8—44no—
138C2×C22⋊C8C2×C411yescharacteristic
139C2×C22⋊C8C2×C411yescharacteristic
140C2×C22⋊C8D411yescharacteristic
141C2×C22⋊C8D411yescharacteristic
142C2×C4×C8—21no—
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149#390C2×C411yescharacteristic
150#389C2×C411yescharacteristic
151#388—21no—
152#386—21no—
153#380C2×C411yescharacteristic
154C4×C4⋊C4C2311yescharacteristic
155C4×C22⋊C4—21no—
156#340C2×C411yescharacteristic
157#340C2×C411yescharacteristic
158#340C2×C411yescharacteristic
159#340C2×C411yescharacteristic
160#336—21no—
161#336—21no—
162#336—21no—
163#336C2×C411yescharacteristic
164#336C2×C411yescharacteristic
165#336D411yescharacteristic
166#336D411yescharacteristic
167#335—22no—
168#335—41no—
169D4⋊C8—22no—
170D4⋊C8—22no—
171D4⋊C8—22no—
172D4⋊C8—22no—
173D4⋊C8—41no—
174D4⋊C8D414yescomplemented
175#322—22no—
176C82—41no—
177C25—21no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4—41no—
185C22×D4—41no—
186C22×D4C22×C411yescharacteristic
187C22×D4C22×C411yescharacteristic
188C22×D4C22×C411yescharacteristic
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—81no—
200C23×C4—81no—
201C23×C4C22×C411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4#4111yescharacteristic
204C2×D8—22no—
205C2×D8—22no—
206C2×D8—22no—
207C2×D8—22no—
208C2×D8—22no—
209C2×D8—41no—
210C2×M16—42no—
211C2×M16—42no—
212C2×M16—42no—
213C2×M16—42no—
214C2×M16—81no—
215C2×M16—81no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—42no—
223C22×C8—42no—
224C22×C8—42no—
225C22×C8—42no—
226C22×C8—81no—
227C22×C8—81no—
228C22×C8C22×C411yescharacteristic
229C22×C8C22×C411yescharacteristic
230C22×C8C22×C411yescharacteristic
231C22×C8C22×C411yescharacteristic
232C22×C8C2×D411yescharacteristic
233C22×C8C2×D411yescharacteristic
234C22×C8C2×D411yescharacteristic
235C22×C8C2×D411yescharacteristic
236#121—22no—
237#121—41no—
238C4×D4—22no—
239C4×D4—22no—
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—21no—
250C4×D4—42no—
251C4×D4—42no—
252C4×D4—41no—
253C4×D4—41no—
254C4×D4—42no—
255C4×D4—42no—
256C4×D4—42no—
257C4×D4—42no—
258C4×D4C2×D412yes—
259C4×D4#4112yes—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C4⋊C4C22×C411yescharacteristic
266C2×C4⋊C4C22×C411yescharacteristic
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C4⋊C4C22×C411yescharacteristic
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×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42C22×C411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42#4111yescharacteristic
288C2×C42C2411yescharacteristic
289#107—22no—
290#107—41no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—42no—
299C4⋊C8—42no—
300C4⋊C8—42no—
301C4⋊C8C2×D412yes—
302#102—22no—
303#102—22no—
304#102—22no—
305#102—22no—
306#102—22no—
307#102—41no—
308#102—41no—
309#102M1614yescomplemented
310C22⋊C8—22no—
311C22⋊C8—22no—
312C22⋊C8—41no—
313C22⋊C8—42no—
314C22⋊C8—42no—
315C22⋊C8—42no—
316C22⋊C8—42no—
317C22⋊C8—42no—
318C22⋊C8—42no—
319C22⋊C8—42no—
320C22⋊C8—42no—
321C22⋊C8—81no—
322C22⋊C8—81no—
323C22⋊C8—81no—
324C22⋊C8D814yescomplemented
325C4×C8—21no—
326C4×C8—21no—
327C4×C8—21no—
328C4×C8—21no—
329C4×C8—21no—
330C4×C8—21no—
331C4×C8—21no—
332C4×C8—21no—
333C4×C8—21no—
334C4×C8—21no—
335C4×C8—21no—
336C4×C8—42no—
337C4×C8C2×D412yes—
338#95—21no—
339#95—21no—
340#95—21no—
341C24—21no—
342C24—24no—
343C24—21no—
344C24—21no—
345C24—42no—
346C24—42no—
347C24—42no—
348C24—42no—
349C24—81no—
350C24—81no—
351C24C4×D411yescharacteristic
352C2×D4—24no—
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—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—42no—
375C2×D4—81no—
376C2×D4C4×D412yes—
377C2×D4C2×M1612yes—
378C2×D4#13112yes—
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—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—42no—
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×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4—84no—
426C22×C4—84no—
427C22×C4—84no—
428C22×C4—84no—
429C22×C4—81no—
430C22×C4C4×D411yescharacteristic
431C22×C4C4×D411yescharacteristic
432C22×C4C4×D411yescharacteristic
433C22×C4C2×D812yes—
434C22×C4#13512yes—
435C22×C4C23×C411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C2×#4111yescharacteristic
438D8—44no—
439D8—44no—
440M16—84no—
441M16—84no—
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—22no—
453C2×C8—21no—
454C2×C8—22no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—84no—
482C2×C8—84no—
483C2×C8C2×M1612yes—
484C2×C8C2×D812yes—
485C2×C8C2×D812yes—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4C4×D412yes—
497C4⋊C4C2×M1612yes—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—41no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—42no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—41no—
517C42—41no—
518C42C22×D411yescharacteristic
519C42C2×#4111yescharacteristic
520C42H(5,𝔽2)11yescharacteristic
521C42H(5,𝔽2)11yescharacteristic
522C23—22no—
523C23—21no—
524C23—21no—
525C23—22no—
526C23—21no—
527C23—22no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—48no—
533C23—48no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—84no—
552C23—84no—
553C23—84no—
554C23—84no—
555C23—84no—
556C23C4×D812yes—
557C23C2×C4×D411yescharacteristic
558D4—22no—
559D4—22no—
560D4—22no—
561D4—22no—
562D4—22no—
563D4—48no—
564D4—41no—
565D4—41no—
566D4—41no—
567D4—41no—
568D4—41no—
569D4—41no—
570D4—84no—
571D4#43614yescomplemented
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4C4×D812yes—
623C2×C4C2×C4×D411yescharacteristic
624C2×C4#51811yescharacteristic
625C2×C4D4211yescharacteristic
626C2×C4#54611yescharacteristic
627C2×C4#54611yescharacteristic
628C2×C4#55111yescharacteristic
629C2×C4C22×M1611yescharacteristic
630C2×C4C2×#13111yescharacteristic
631C2×C4C2×#13111yescharacteristic
632C2×C4#56811yescharacteristic
633C2×C4C22×D811yescharacteristic
634C2×C4C2×#13511yescharacteristic
635C2×C4#57611yescharacteristic
636C2×C4#57711yescharacteristic
637C8—22no—
638C8—22no—
639C8—22no—
640C8—42no—
641C8—41no—
642C8—41no—
643C8—41no—
644C8—42no—
645C8—41no—
646C8—41no—
647C8—81no—
648C8—81no—
649C8—81no—
650C8—81no—
651C22—24no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—41no—
656C22—42no—
657C22—41no—
658C22—42no—
659C22—42no—
660C22—41no—
661C22—42no—
662C22—42no—
663C22—41no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—816no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22#235811yescharacteristic
681C22#292911yescharacteristic
682C22D4×M1611yescharacteristic
683C22C2×C4×D811yescharacteristic
684C22#294611yescharacteristic
685C22#295511yescharacteristic
686C22#295711yescharacteristic
687C4—22no—
688C4—21no—
689C4—21no—
690C4—41no—
691C4—42no—
692C4—41no—
693C4—42no—
694C4—81no—
695C4—81no—
696C4—81no—
697C4C2×#43611yescharacteristic
698C4#293311yescharacteristic
699C4#297411yescharacteristic
700C4#297611yescharacteristic
701C4D4×D811yescharacteristic
702C4#328211yescharacteristic
703C4#329411yescharacteristic
704C4#329711yescharacteristic
705C2—42no—
706C2—44no—
707C2—82no—
708C2—161no—
709C2#1790011yescharacteristic
710C2#1791811yescharacteristic
711C2M16×D811yescharacteristic
712C2#1805211yescharacteristic
713C2#2284811yescharacteristic
714C2#2287011yescharacteristic
715C2#2288711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#447C4
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
1. #22990C2semidirect
2. D4×C22⋊C8C2semidirect
4. C2×#1581C2semidirect
5. C2×#1486C2semidirect
6. #9657C2semidirect
7. #9558C2semidirect
8. #9541C2semidirect
9. #9076C2semidirect
10. #8951C2semidirect
11. #8947C2semidirect
12. #8843C2semidirect
13. #8832C2semidirect
14. C8×#102C2semidirect
15. #8148C2semidirect
21. C2×#436C22semidirect
30. #1954C22semidirect
31. #1939C22semidirect
46. C8×C4⋊C4C22semidirect
52. C2×#335C22semidirect
53. C2×#335C22semidirect
59. C2×D4⋊C8C22semidirect
60. C2×D4⋊C8C22semidirect
61. C2×#322C22semidirect
62. C2×C82C22semidirect
63. #1279C22semidirect
64. #1279C22semidirect
65. #1277C22semidirect
66. #1277C22semidirect
67. #1273C22semidirect
68. #1273C22semidirect
174. D4⋊C8D4semidirect
309. #102M16semidirect
324. C22⋊C8D8semidirect
571. D4#436semidirect

Maximal subgroups

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

subgroupindex
#229902
D4×C22⋊C82
C4×#4472
C2×#15812
C2×#14862
#96572
#95582
#95412
#90762
#89512
#89472
#88432
#88322
C8×#1022
#81482