Finite Group Databasebeta

← #361539

Subgroups — #361539

Order 512.

1594 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1839C211yesmaximal, characteristic, complemented
2C22×#336C211yesmaximal, characteristic
3C42×C22⋊C4C211yesmaximal, characteristic
4#11298C212yesmaximal, complemented
5#11298C212yesmaximal, complemented
6#10795C211yesmaximal, characteristic, complemented
7#10795C211yesmaximal, characteristic, complemented
8#10795C211yesmaximal, characteristic, complemented
9#10714C212yesmaximal
10#10714C212yesmaximal
11#10712C211yesmaximal, characteristic
12C24×C8C2212yes—
13C23×C42C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15C2×C4×C22⋊C4C411yescharacteristic
16C2×C4×C22⋊C4C411yescharacteristic
17C2×C4×C22⋊C4C411yescharacteristic
18C2×C4×C22⋊C4C411yescharacteristic
19C2×C4×C22⋊C4C2211yescharacteristic
20C2×C4×C22⋊C4C2211yescharacteristic
21C2×C43C2211yescharacteristic
22C2×C43C2211yescharacteristic
23#1839—24no—
24C2×#336—24no—
25C2×#336C2212yes—
26C2×#336C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C4×#95C411yescharacteristic
40C4×#95C411yescharacteristic
41C4×#95C411yescharacteristic
42C4×#95C411yescharacteristic
43C4×#95C2211yescharacteristic
44C4×#95C2211yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8—24no—
49C23×C8—24no—
50C23×C8—24no—
51C23×C8—24no—
52C23×C8—24no—
53C23×C8—42no—
54C23×C8—42no—
55C23×C8C2312yes—
56C23×C8C2312yes—
57C23×C8C2312yes—
58C23×C8C2312yes—
59C22×C22⋊C4C2×C411yescharacteristic
60C22×C22⋊C4C2×C411yescharacteristic
61C22×C22⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C42—24no—
64C22×C42—22no—
65C22×C42—22no—
66C22×C42—42no—
67C22×C42C2×C411yescharacteristic
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×C22⋊C8—24no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86C4×C22⋊C4—24no—
87C4×C22⋊C4—22no—
88C4×C22⋊C4—24no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—24no—
91C4×C22⋊C4—24no—
92C4×C22⋊C4C818yescomplemented
93C2×#95C2×C411yescharacteristic
94C2×#95C2×C411yescharacteristic
95C2×#95C2×C411yescharacteristic
96C2×#95C2×C411yescharacteristic
97C2×#95C2×C411yescharacteristic
98C2×#95C2×C411yescharacteristic
99C2×#95C2×C411yescharacteristic
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105C43—22no—
106C43—22no—
107C43—42no—
108#336—24no—
109#336—24no—
110#336—24no—
111#336—24no—
112#336—24no—
113#336—22no—
114#336—22no—
115#336—22no—
116#336—42no—
117#336—42no—
118#336—42no—
119#336—42no—
120#336—42no—
121#336D414yes—
122C25C22×C411yescharacteristic
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—41no—
138C23×C4—41no—
139C23×C4—41no—
140C23×C4—41no—
141C23×C4C2×C814yes—
142C23×C4M1614yes—
143C23×C4C22×C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
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×C4#4112yes—
157C23×C4#4114yes—
158C23×C4#4114yes—
159C23×C4#4112yes—
160C23×C4C2411yescharacteristic
161C22×C8—216no—
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—48no—
189C22×C8—48no—
190C22×C8—48no—
191C22×C8—48no—
192C22×C8—48no—
193C22×C8—48no—
194C22×C8C2×D414yes—
195C22×C8#4114yes—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4C2×C814yes—
205C2×C22⋊C4C2×C814yes—
206C2×C22⋊C4M1614yes—
207C2×C22⋊C4M1614yes—
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—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—48no—
251C2×C42—48no—
252C2×C42—41no—
253C2×C42—41no—
254C2×C42—41no—
255C2×C42—41no—
256C2×C42—41no—
257C2×C42—41no—
258C2×C42—41no—
259C2×C42—41no—
260C2×C42—41no—
261C2×C42—41no—
262C2×C42C2×C814yescomplemented
263C2×C42C2×C814yescomplemented
264C2×C42C2×D412yes—
265C2×C42#4112yes—
266C22⋊C8—48no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95C2×C814yescomplemented
278#95C2×C814yescomplemented
279C24—21no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—21no—
287C24#11712yes—
288C24#11712yes—
289C24#11712yes—
290C24C22×C812yes—
291C24C2×M1612yes—
292C24C2×M1612yes—
293C24C2×M1612yes—
294C24C23×C411yescharacteristic
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—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
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—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—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4#11712yes—
396C22×C4#11712yes—
397C22×C4#11712yes—
398C22×C4C4×D412yes—
399C22×C4C4×D412yes—
400C22×C4#12618yes—
401C22×C4#12618yes—
402C22×C4C22×C812yes—
403C22×C4C22×C812yes—
404C22×C4C22×C812yes—
405C22×C4C22×C812yes—
406C22×C4C22×C812yes—
407C22×C4C22×C812yes—
408C22×C4C2×M1612yes—
409C22×C4C2×M1612yes—
410C22×C4C2×M1612yes—
411C22×C4C2×M1612yes—
412C22×C4C2×M1612yes—
413C22×C4C2×M1612yes—
414C22×C4#13112yes—
415C22×C4#13112yes—
416C22×C4C22×D411yescharacteristic
417C22×C4C2×#4112yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4112yes—
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4H(5,𝔽2)11yescharacteristic
424C22×C4H(5,𝔽2)12yes—
425C22×C42−1+411yescharacteristic
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—432no—
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—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C22⋊C4—28no—
456C22⋊C4—28no—
457C22⋊C4—28no—
458C22⋊C4—28no—
459C42—24no—
460C42—24no—
461C42—28no—
462C42—44no—
463C42—44no—
464C42—44no—
465C42—44no—
466C42—44no—
467C42—44no—
468C42—44no—
469C42—44no—
470C42—42no—
471C42—42no—
472C42—42no—
473C42—42no—
474C42—41no—
475C42—42no—
476C42—41no—
477C42—42no—
478C42—42no—
479C42—41no—
480C42—41no—
481C42—42no—
482C42—44no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23#38314yes—
512C23#43114yes—
513C23#43114yes—
514C23#43114yes—
515C23#43214yes—
516C23#43214yes—
517C23#43214yes—
518C23C2×#11711yescharacteristic
519C23C2×#11711yescharacteristic
520C23C2×#11711yescharacteristic
521C23C2×C4×D411yescharacteristic
522C23C4×#4111yescharacteristic
523C23#51811yescharacteristic
524C23#52011yescharacteristic
525C23C23×C811yescharacteristic
526C23C22×M1611yescharacteristic
527C23C22×M1611yescharacteristic
528C23C22×M1611yescharacteristic
529C23C2×#13111yescharacteristic
530C23C2×#13111yescharacteristic
531C23#56811yescharacteristic
532C23#56811yescharacteristic
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—28no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—28no—
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—22no—
584C2×C4—22no—
585C2×C4—416no—
586C2×C4—416no—
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—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4#38314yes—
628C2×C4C8×D414yes—
629C2×C4#43514yes—
630C2×C4#52512yes—
631C2×C4C2×#12612yes—
632C2×C4C2×#12612yes—
633C2×C4#52912yes—
634C2×C4#53812yes—
635C2×C4#53912yes—
636C2×C4#54611yescharacteristic
637C2×C4#54612yes—
638C2×C4#54811yescharacteristic
639C2×C4#55112yes—
640C2×C4#55112yes—
641C2×C4#55112yes—
642C2×C4#55212yes—
643C2×C4#55211yescharacteristic
644C2×C4#55312yes—
645C2×C4#55311yescharacteristic
646C8—44no—
647C8—44no—
648C8—82no—
649C8—82no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—24no—
661C22—24no—
662C22—24no—
663C22—24no—
664C22#183918yescomplemented
665C22C2×#38311yescharacteristic
666C22#229011yescharacteristic
667C22#231811yescharacteristic
668C22#234211yescharacteristic
669C22#234311yescharacteristic
670C22#235211yescharacteristic
671C22#235311yescharacteristic
672C22C2×#43111yescharacteristic
673C22C2×#43111yescharacteristic
674C22C2×#43111yescharacteristic
675C22C2×#43211yescharacteristic
676C22C2×#43211yescharacteristic
677C22C2×#43211yescharacteristic
678C22#292211yescharacteristic
679C22#292311yescharacteristic
680C22#292411yescharacteristic
681C22#292411yescharacteristic
682C22#292511yescharacteristic
683C22#292511yescharacteristic
684C22C2×C8×D411yescharacteristic
685C22C2×#43511yescharacteristic
686C22D4×M1611yescharacteristic
687C22D4×M1611yescharacteristic
688C22C8×#4111yescharacteristic
689C22#296511yescharacteristic
690C22#297011yescharacteristic
691C22#297011yescharacteristic
692C22#297211yescharacteristic
693C22#297311yescharacteristic
694C22#297411yescharacteristic
695C22#297411yescharacteristic
696C22#298811yescharacteristic
697C22#298911yescharacteristic
698C22#299011yescharacteristic
699C22#299011yescharacteristic
700C4—24no—
701C4—24no—
702C4—24no—
703C4—42no—
704C4—42no—
705C4—42no—
706C4—42no—
707C4—42no—
708C4—42no—
709C4—42no—
710C4—42no—
711C4—42no—
712C4—42no—
713C4#248012yes—
714C4#248112yes—
715C4#249312yes—
716C4#251012yes—
717C2—28no—
718C2#1390111yescharacteristic
719C2C2×#183911yescharacteristic
720C2#2077411yescharacteristic
721C2#2110111yescharacteristic
722C2#2110711yescharacteristic
723C2#2199611yescharacteristic
724C2#2200011yescharacteristic
725C2#2255211yescharacteristic
726C2#2255411yescharacteristic
727C2#2258911yescharacteristic
728C2#2259111yescharacteristic
729C2#2274811yescharacteristic
730C2#2275011yescharacteristic
731C2#2278511yescharacteristic
732C2#2278711yescharacteristic
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
1. C2×#1839C2semidirect
4. #11298C2semidirect
5. #11298C2semidirect
6. #10795C2semidirect
7. #10795C2semidirect
8. #10795C2semidirect
92. C4×C22⋊C4C8semidirect
262. C2×C42C2×C8semidirect
263. C2×C42C2×C8semidirect
277. #95C2×C8semidirect
278. #95C2×C8semidirect
664. C22#1839semidirect

Maximal subgroups

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

subgroupindex
C2×#18392
C22×#3362
C42×C22⋊C42
#112982
#107952
#107142
#107122