Finite Group Databasebeta

← #217358

Subgroups — #217358

Order 512.

791 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21853C211yesmaximal, characteristic
2#21847C211yesmaximal, characteristic
3C4×#408C211yesmaximal, characteristic
4#8834C211yesmaximal, characteristic, complemented
5#8834C211yesmaximal, characteristic, complemented
6#8833C211yesmaximal, characteristic, complemented
7#8832C211yesmaximal, characteristic, complemented
8#8831C211yesmaximal, characteristic, complemented
9#8831C211yesmaximal, characteristic, complemented
10#8830C211yesmaximal, characteristic, complemented
11#8829C211yesmaximal, characteristic, complemented
12#8597C211yesmaximal, characteristic, complemented
13#8597C211yesmaximal, characteristic, complemented
14#8596C211yesmaximal, characteristic, complemented
15#8595C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C2×#408C411yescharacteristic
22C2×#408C411yescharacteristic
23C2×C4×M16C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25#1985—21no—
26#1984—21no—
27#1955—21no—
28#1954—21no—
29#1870—21no—
30#1870—21no—
31#1869—21no—
32#1869—21no—
33#1797C2211yescharacteristic
34#1797C2211yescharacteristic
35#1797C2211yescharacteristic
36#1797C2211yescharacteristic
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39#1797C2211yescharacteristic
40#1797C2211yescharacteristic
41#1750C411yescharacteristic
42#1750C411yescharacteristic
43#1750C411yescharacteristic
44#1750C411yescharacteristic
45C4×C22⋊C8C2211yescharacteristic
46C4×C22⋊C8C2211yescharacteristic
47C4×C22⋊C8C2211yescharacteristic
48C4×C22⋊C8C2211yescharacteristic
49#1729C411yescharacteristic
50#1729C411yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×#335C2211yescharacteristic
54C2×#335C2211yescharacteristic
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#334C2211yescharacteristic
59#1277C2211yescharacteristic
60#1277C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×#41C2×C411yescharacteristic
68C22×M16—21no—
69C22×M16C2×C411yescharacteristic
70C22×M16C2×C411yescharacteristic
71C23×C8—21no—
72C23×C8C2×C411yescharacteristic
73C23×C8C2×C411yescharacteristic
74C4×#41—41no—
75C4×#41—41no—
76C4×#41—41no—
77C4×#41—41no—
78C2×C4×Q8D411yescharacteristic
79C2×C4×Q8Q811yescharacteristic
80C2×C4×D4D411yescharacteristic
81C2×C4×D4D411yescharacteristic
82C2×C4×D4D411yescharacteristic
83C2×C4×D4Q811yescharacteristic
84C2×C4×D4Q811yescharacteristic
85C2×C4×D4Q811yescharacteristic
86C2×#117C2×C411yescharacteristic
87C2×#117C2×C411yescharacteristic
88C2×#117C2×C411yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#432—41no—
93#432—41no—
94#431—41no—
95#431—41no—
96#423—41no—
97#423—41no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—41no—
101C2×C4⋊C8—41no—
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110#408—24no—
111#408—24no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8C2×C411yescharacteristic
123C2×C22⋊C8C2×C411yescharacteristic
124C2×C22⋊C8C2×C411yescharacteristic
125C2×C22⋊C8C2×C411yescharacteristic
126C2×C22⋊C8C2×C411yescharacteristic
127C2×C22⋊C8C2×C411yescharacteristic
128C2×C22⋊C8C2×C411yescharacteristic
129C2×C22⋊C8C2×C411yescharacteristic
130C4×M16—41no—
131C4×M16—41no—
132C2×#97—21no—
133C2×#97C2×C411yescharacteristic
134C2×#97C2×C411yescharacteristic
135C2×C4×C8—21no—
136C2×C4×C8—41no—
137C2×C4×C8—41no—
138C2×C4×C8C2×C411yescharacteristic
139C2×C4×C8C2×C411yescharacteristic
140C2×C4×C8C2311yescharacteristic
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336C2×C411yescharacteristic
153#336C2×C411yescharacteristic
154#336C2×C411yescharacteristic
155#336C2×C411yescharacteristic
156#336C2×C411yescharacteristic
157#336C2×C411yescharacteristic
158#336C2×C411yescharacteristic
159#336C2×C411yescharacteristic
160#335—41no—
161#335—41no—
162#335—41no—
163#335—41no—
164#334—41no—
165#334—41no—
166#334—41no—
167#334—41no—
168C2×#41—22no—
169C2×#41—22no—
170C2×#41—22no—
171C2×#41—22no—
172C2×#41—41no—
173C2×#41—41no—
174C22×Q8C4⋊C411yescharacteristic
175C22×D4C4⋊C411yescharacteristic
176C22×D4C4⋊C411yescharacteristic
177C22×D4C4⋊C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×Q811yescharacteristic
185C23×C4C2×Q811yescharacteristic
186C23×C4C2×Q811yescharacteristic
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—41no—
192C2×M16—41no—
193C2×M16—42no—
194C2×M16—81no—
195C2×M16—81no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—41no—
209C22×C8—41no—
210C22×C8—42no—
211C22×C8—81no—
212C22×C8—81no—
213C22×C8C22×C411yescharacteristic
214C22×C8C22×C411yescharacteristic
215C22×C8C22×C411yescharacteristic
216C22×C8C22×C411yescharacteristic
217C22×C8C22×C411yescharacteristic
218C22×C8C22×C411yescharacteristic
219C22×C8C22×C411yescharacteristic
220C22×C8C22×C411yescharacteristic
221C4×Q8—41no—
222C4×Q8—41no—
223C4×Q8—41no—
224C4×Q8—41no—
225C4×D4—41no—
226C4×D4—41no—
227C4×D4—41no—
228C4×D4—41no—
229C4×D4—41no—
230C4×D4—41no—
231C4×D4—41no—
232C4×D4—41no—
233C4×D4—41no—
234C4×D4—41no—
235C4×D4—41no—
236C4×D4—41no—
237#117—41no—
238#117—41no—
239#117—41no—
240#117—41no—
241#117—41no—
242#117—41no—
243C2×C4⋊C4C4⋊C411yescharacteristic
244C2×C4⋊C4C4⋊C411yescharacteristic
245C2×C4⋊C4C4⋊C411yescharacteristic
246C2×C4⋊C4C4⋊C411yescharacteristic
247C2×C4⋊C4C4⋊C411yescharacteristic
248C2×C4⋊C4C4⋊C411yescharacteristic
249C2×C22⋊C4C4⋊C411yescharacteristic
250C2×C22⋊C4C4⋊C411yescharacteristic
251C2×C22⋊C4C4⋊C411yescharacteristic
252C2×C22⋊C4C4⋊C411yescharacteristic
253C2×C22⋊C4C4⋊C411yescharacteristic
254C2×C22⋊C4C4⋊C411yescharacteristic
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42C22×C411yescharacteristic
268C2×C42C22×C411yescharacteristic
269C2×C42C22×C411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×Q811yescharacteristic
274C2×C42C2×Q811yescharacteristic
275C2×C42C2×Q811yescharacteristic
276C2×C42C2411yescharacteristic
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
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—
295C22⋊C8—22no—
296C22⋊C8—22no—
297C22⋊C8—22no—
298C22⋊C8—22no—
299C22⋊C8—22no—
300C22⋊C8—22no—
301C22⋊C8—42no—
302C22⋊C8—42no—
303C22⋊C8—81no—
304C22⋊C8—81no—
305C22⋊C8—81no—
306C22⋊C8—81no—
307C22⋊C8—81no—
308C22⋊C8—81no—
309C22⋊C8—81no—
310C22⋊C8—81no—
311C22⋊C8M1614yescomplemented
312C22⋊C8M1614yescomplemented
313#97—41no—
314#97—41no—
315#97—42no—
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—41no—
325C4×C8—41no—
326C4×C8—42no—
327C24C2×C4⋊C411yescharacteristic
328C24C2×C4⋊C411yescharacteristic
329C24C2×C4⋊C411yescharacteristic
330#41—44no—
331#41—44no—
332#41—44no—
333#41—44no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—41no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4C2×C4⋊C411yescharacteristic
391C22×C4C2×C4⋊C411yescharacteristic
392C22×C4C2×C4⋊C411yescharacteristic
393C22×C4C2×C4⋊C411yescharacteristic
394C22×C4C2×C4⋊C411yescharacteristic
395C22×C4C2×C4⋊C411yescharacteristic
396C22×C4C2×C4⋊C411yescharacteristic
397C22×C4C2×C4⋊C411yescharacteristic
398C22×C4C2×C4⋊C411yescharacteristic
399C22×C4C2×M1612yes—
400C22×C4C2×M1612yes—
401C22×C4#13112yes—
402C22×C4#13112yes—
403C22×C4C23×C411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×Q811yescharacteristic
406M16—44no—
407M16—44no—
408M16—84no—
409C2×C8—21no—
410C2×C8—21no—
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—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—44no—
430C2×C8—44no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
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—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—84no—
456C2×C8C2×M1612yes—
457C2×C8C2×M1612yes—
458C2×C8C2×M1612yes—
459C2×C8C2×M1612yes—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42H(5,𝔽2)11yescharacteristic
491C42H(5,𝔽2)11yescharacteristic
492C42H(5,𝔽2)11yescharacteristic
493C42H(5,𝔽2)11yescharacteristic
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23#42312yes—
514C23#42412yes—
515C23C22×C4⋊C411yescharacteristic
516Q8—44no—
517D4—44no—
518D4—44no—
519D4—44no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
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—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4#42312yes—
578C2×C4#42412yes—
579C2×C4#51811yescharacteristic
580C2×C4#51811yescharacteristic
581C2×C4#53411yescharacteristic
582C2×C4#53411yescharacteristic
583C2×C4#54311yescharacteristic
584C2×C4#54311yescharacteristic
585C2×C4C22×M1611yescharacteristic
586C2×C4C22×M1611yescharacteristic
587C2×C4C2×#13111yescharacteristic
588C2×C4C2×#13111yescharacteristic
589C2×C4#57611yescharacteristic
590C2×C4#57711yescharacteristic
591C2×C4#57711yescharacteristic
592C2×C4#57811yescharacteristic
593C8—22no—
594C8—22no—
595C8—22no—
596C8—22no—
597C8—41no—
598C8—41no—
599C8—41no—
600C8—41no—
601C8—81no—
602C8—81no—
603C8—81no—
604C8—81no—
605C8—81no—
606C8—81no—
607C8—81no—
608C8—81no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22#231511yescharacteristic
631C22C2×#42311yescharacteristic
632C22C2×#42411yescharacteristic
633C22#291211yescharacteristic
634C22#291311yescharacteristic
635C22#291611yescharacteristic
636C22#291611yescharacteristic
637C4—22no—
638C4—22no—
639C4—21no—
640C4—21no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—41no—
646C4—41no—
647C4—41no—
648C4#297411yescharacteristic
649C4#297411yescharacteristic
650C4#297411yescharacteristic
651C4#297411yescharacteristic
652C4#319811yescharacteristic
653C4#319811yescharacteristic
654C4#320111yescharacteristic
655C4#320111yescharacteristic
656C2—42no—
657C2—42no—
658C2—81no—
659C2#1779811yescharacteristic
660C2#1779911yescharacteristic
661C2#1781111yescharacteristic
662C2#1781211yescharacteristic
663C2#2193211yescharacteristic
664C2#2194511yescharacteristic
665C2#2195011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C4⋊C4
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #8834C2semidirect
5. #8834C2semidirect
6. #8833C2semidirect
7. #8832C2semidirect
8. #8831C2semidirect
9. #8831C2semidirect
10. #8830C2semidirect
11. #8829C2semidirect
12. #8597C2semidirect
13. #8597C2semidirect
14. #8596C2semidirect
15. #8595C2semidirect
311. C22⋊C8M16semidirect
312. C22⋊C8M16semidirect

Maximal subgroups

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

subgroupindex
#218532
#218472
C4×#4082
#88342
#88332
#88322
#88312
#88302
#88292
#85972
#85962
#85952