Finite Group Databasebeta

← #363652

Subgroups — #363652

Order 512.

1470 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1923C212yesmaximal
2C42×C22⋊C4C211yesmaximal, characteristic
3#11255C212yesmaximal, complemented
4#11042C212yesmaximal
5#10876C212yesmaximal, complemented
6#10832C211yesmaximal, characteristic, complemented
7#10832C211yesmaximal, characteristic, complemented
8#10795C211yesmaximal, characteristic, complemented
9#10712C211yesmaximal, characteristic
10#10712C211yesmaximal, characteristic
11C4×#336C211yesmaximal, characteristic, complemented
12C23×C42C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×C4⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C4×C8C2212yes—
18C2×C4×C22⋊C4C411yescharacteristic
19C2×C4×C22⋊C4C411yescharacteristic
20C2×C4×C22⋊C4C411yescharacteristic
21C2×C4×C22⋊C4C411yescharacteristic
22C2×C4×C22⋊C4C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24C2×C43C2211yescharacteristic
25C2×C43C2211yescharacteristic
26#1923—28no—
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C4×#95C411yescharacteristic
39C4×#95C411yescharacteristic
40C4×#95C411yescharacteristic
41C4×#95C411yescharacteristic
42C4×#95C2211yescharacteristic
43C4×#95C2211yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C22×C22⋊C4C2×C411yescharacteristic
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4C2×C411yescharacteristic
55C22×C42—22no—
56C22×C42—22no—
57C22×C42—42no—
58C22×C42—42no—
59C22×C42C2×C411yescharacteristic
60C22×C42C2×C411yescharacteristic
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42C2×C411yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42C2×C411yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C2×C4⋊C8—24no—
76C2×C4⋊C8—24no—
77C2×C4⋊C8—24no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—42no—
80C2×C4⋊C8—42no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—42no—
86C2×C22⋊C8—42no—
87C2×C4×C8—24no—
88C2×C4×C8—24no—
89C2×C4×C8—42no—
90C4×C22⋊C4—24no—
91C4×C22⋊C4—24no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—24no—
95C4×C22⋊C4C818yescomplemented
96C4×C22⋊C4D414yes—
97C4×C22⋊C4Q814yes—
98C2×#95C2×C411yescharacteristic
99C2×#95C2×C411yescharacteristic
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105C2×#95C2×C411yescharacteristic
106C2×#95C2×C411yescharacteristic
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C411yescharacteristic
109C2×#95C2×C411yescharacteristic
110C43—42no—
111C43—42no—
112#336—24no—
113#336—24no—
114#336—24no—
115#336—24no—
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120#336—42no—
121#336—42no—
122#336—42no—
123C25C22×C411yescharacteristic
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—41no—
140C23×C4—41no—
141C23×C4—41no—
142C23×C4—41no—
143C23×C4—41no—
144C23×C4—41no—
145C23×C4C2×C814yes—
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×Q812yes—
161C23×C4C2411yescharacteristic
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8#4114yes—
189C22×C8#4114yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4C4⋊C414yes—
198C2×C22⋊C4C4⋊C414yes—
199C2×C22⋊C4C2×C814yes—
200C2×C22⋊C4C2×C814yes—
201C2×C22⋊C4M1614yes—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—48no—
230C2×C42—48no—
231C2×C42—41no—
232C2×C42—41no—
233C2×C42—41no—
234C2×C42—41no—
235C2×C42—41no—
236C2×C42—41no—
237C2×C42—41no—
238C2×C42—41no—
239C2×C42—41no—
240C2×C42—41no—
241C2×C42—41no—
242C2×C42—41no—
243C2×C42—41no—
244C2×C42—41no—
245C2×C42—41no—
246C2×C42—41no—
247C2×C42C2×C814yes—
248C2×C42C2×C814yes—
249C2×C42M1614yes—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×Q812yes—
253C2×C42C2×Q812yes—
254C2×C42#4112yes—
255C2×C42#4112yes—
256C2×C42#4112yes—
257C2×C42#4112yes—
258C4⋊C8—48no—
259C4⋊C8—48no—
260C22⋊C8—48no—
261C22⋊C8—48no—
262C4×C8—48no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95C4⋊C414yes—
271#95C4⋊C414yes—
272#95C2×C814yes—
273#95C2×C814yes—
274#95M1614yes—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—21no—
287C24C2×C4⋊C412yes—
288C24C22×C812yes—
289C24C2×M1612yes—
290C24C23×C411yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—28no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
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—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—28no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—44no—
387C22×C4C2×C4⋊C412yes—
388C22×C4C2×C4⋊C412yes—
389C22×C4C2×C4⋊C412yes—
390C22×C4C2×C4⋊C412yes—
391C22×C4C2×C4⋊C412yes—
392C22×C4#11712yes—
393C22×C4#11712yes—
394C22×C4#11712yes—
395C22×C4#11712yes—
396C22×C4C22×C812yes—
397C22×C4C22×C812yes—
398C22×C4C22×C812yes—
399C22×C4C22×C812yes—
400C22×C4C22×C812yes—
401C22×C4C22×C812yes—
402C22×C4C2×M1612yes—
403C22×C4C2×M1612yes—
404C22×C4C2×M1612yes—
405C22×C4C2×M1612yes—
406C22×C4C2×M1612yes—
407C22×C4C2×M1612yes—
408C22×C4C2×M1612yes—
409C22×C4C2×M1612yes—
410C22×C4#13112yes—
411C22×C4#13112yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×Q811yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4C2×#4112yes—
417C22×C4C2×#4112yes—
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4H(5,𝔽2)12yes—
421C22×C4H(5,𝔽2)12yes—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C22⋊C4—28no—
451C22⋊C4—28no—
452C22⋊C4—28no—
453C22⋊C4C4⋊C8116yescomplemented
454C42—24no—
455C42—24no—
456C42—24no—
457C42—24no—
458C42—24no—
459C42—24no—
460C42—24no—
461C42—24no—
462C42—44no—
463C42—44no—
464C42—44no—
465C42—44no—
466C42—44no—
467C42—44no—
468C42—44no—
469C42—44no—
470C42—44no—
471C42—44no—
472C42—42no—
473C42—42no—
474C42—42no—
475C42—42no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23C2×C4⋊C814yes—
511C23C22×C4⋊C411yescharacteristic
512C23C2×#11711yescharacteristic
513C23C2×#11711yescharacteristic
514C23C4×#4111yescharacteristic
515C23C4×#4111yescharacteristic
516C23#51811yescharacteristic
517C23#51811yescharacteristic
518C23C23×C811yescharacteristic
519C23C22×M1611yescharacteristic
520C23C22×M1611yescharacteristic
521C23C22×M1611yescharacteristic
522C23C2×#13111yescharacteristic
523C23C2×#13111yescharacteristic
524C23#56811yescharacteristic
525C23#56811yescharacteristic
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—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—416no—
584C2×C4—416no—
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—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
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—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4C2×C4⋊C814yes—
618C2×C4C2×C4⋊C814yes—
619C2×C4#42314yes—
620C2×C4#43114yes—
621C2×C4#43114yes—
622C2×C4#43314yes—
623C2×C4#52912yes—
624C2×C4#52912yes—
625C2×C4#53211yescharacteristic
626C2×C4#53211yescharacteristic
627C2×C4#53311yescharacteristic
628C2×C4#53311yescharacteristic
629C2×C4#53812yes—
630C2×C4#53812yes—
631C2×C4#54612yes—
632C2×C4#54612yes—
633C2×C4#55112yes—
634C2×C4#55312yes—
635C2×C4#55312yes—
636C2×C4#55412yes—
637C2×C4#55412yes—
638C2×C4#55912yes—
639C8—44no—
640C8—44no—
641C8—82no—
642C8—82no—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22#228511yescharacteristic
659C22#229811yescharacteristic
660C22#231211yescharacteristic
661C22#234911yescharacteristic
662C22#234911yescharacteristic
663C22#235311yescharacteristic
664C22#235711yescharacteristic
665C22C22×C4⋊C811yescharacteristic
666C22C2×#42311yescharacteristic
667C22#290611yescharacteristic
668C22#290611yescharacteristic
669C22C2×#43111yescharacteristic
670C22C2×#43111yescharacteristic
671C22C2×#43311yescharacteristic
672C22#292011yescharacteristic
673C22#292011yescharacteristic
674C22#292111yescharacteristic
675C22#292511yescharacteristic
676C22#292511yescharacteristic
677C22C8×#4111yescharacteristic
678C22C8×#4111yescharacteristic
679C22#296511yescharacteristic
680C22#296511yescharacteristic
681C22#297011yescharacteristic
682C22#297011yescharacteristic
683C22#297011yescharacteristic
684C22#297111yescharacteristic
685C22#297211yescharacteristic
686C22#297211yescharacteristic
687C22#297311yescharacteristic
688C22#297411yescharacteristic
689C22#297411yescharacteristic
690C22#297411yescharacteristic
691C22#297411yescharacteristic
692C22#297511yescharacteristic
693C4—24no—
694C4—24no—
695C4—24no—
696C4—24no—
697C4—24no—
698C4—24no—
699C4—42no—
700C4—42no—
701C4—42no—
702C4—42no—
703C4—42no—
704C4—42no—
705C4—42no—
706C4#255712yes—
707C4#255912yes—
708C4#258712yes—
709C4#258912yes—
710C2—44no—
711C2#1394811yescharacteristic
712C2#2080811yescharacteristic
713C2#2081211yescharacteristic
714C2#2143011yescharacteristic
715C2#2143511yescharacteristic
716C2#2190211yescharacteristic
717C2#2190411yescharacteristic
718C2#2269611yescharacteristic
719C2#2269611yescharacteristic
720C2#2269811yescharacteristic
721C2#2270011yescharacteristic
722C2#2278511yescharacteristic
723C2#2278611yescharacteristic
724C2#2283511yescharacteristic
725C2#2283711yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C23×C8
socleC24C23×C4
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11255C2semidirect
5. #10876C2semidirect
6. #10832C2semidirect
7. #10832C2semidirect
8. #10795C2semidirect
11. C4×#336C2semidirect
95. C4×C22⋊C4C8semidirect
453. C22⋊C4C4⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×#19232
C42×C22⋊C42
#112552
#110422
#108762
#108322
#107952
#107122
C4×#3362