Finite Group Databasebeta

← #357347

Subgroups — #357347

Order 512.

1212 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1924C211yesmaximal, characteristic
2C2×#1851C211yesmaximal, characteristic
3C4×#388C211yesmaximal, characteristic
4#10922C211yesmaximal, characteristic
5#10922C211yesmaximal, characteristic
6#10877C211yesmaximal, characteristic, complemented
7#10877C211yesmaximal, characteristic, complemented
8#10877C211yesmaximal, characteristic, complemented
9#10875C211yesmaximal, characteristic, complemented
10#10835C211yesmaximal, characteristic, complemented
11#10832C211yesmaximal, characteristic, complemented
12#10798C211yesmaximal, characteristic, complemented
13#10795C211yesmaximal, characteristic, complemented
14#10715C211yesmaximal, characteristic
15#10715C211yesmaximal, characteristic
16C23×M16C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C22×#97C2211yescharacteristic
25C22×#97C2211yescharacteristic
26C2×#388C412yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31C2×C43C2211yescharacteristic
32#1924C418yescomplemented
33#1851C418yescomplemented
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41C2×#336C2211yescharacteristic
42C2×#336C2211yescharacteristic
43C2×#336C2211yescharacteristic
44C2×#336C2211yescharacteristic
45C2×#336C2211yescharacteristic
46C2×#336C2211yescharacteristic
47C2×#336C2211yescharacteristic
48C2×#336C2211yescharacteristic
49C2×#336C2211yescharacteristic
50C2×#336C2211yescharacteristic
51C2×#336C2211yescharacteristic
52#1437C412yes—
53#1436C412yes—
54#1434C412yes—
55C4×#95C2211yescharacteristic
56C4×#95C2211yescharacteristic
57C24×C4C2311yescharacteristic
58C22×M16—41no—
59C22×M16—41no—
60C22×M16—41no—
61C22×M16—41no—
62C22×M16—41no—
63C22×M16—41no—
64C22×M16C2×C414yes—
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C22×C4⋊C4C2×C412yes—
74C22×C22⋊C4C2×C411yescharacteristic
75C22×C22⋊C4C2×C412yes—
76C22×C22⋊C4C2×C412yes—
77C22×C22⋊C4C2×C411yescharacteristic
78C22×C42C2×C412yes—
79C22×C42C2×C412yes—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×C4⋊C8—41no—
87C2×C4⋊C8—41no—
88C2×C4⋊C8—41no—
89C2×C4⋊C8—41no—
90C2×C4⋊C8—41no—
91C2×C4⋊C8—41no—
92C2×C4⋊C8C2×C414yes—
93C2×C4⋊C8C2×C414yes—
94C2×C4⋊C8C2×C414yes—
95C2×C22⋊C8—41no—
96C2×C22⋊C8—41no—
97C2×C22⋊C8—41no—
98C2×C22⋊C8—41no—
99C2×C22⋊C8—41no—
100C2×C22⋊C8—41no—
101C2×C22⋊C8C2×C414yes—
102C2×C22⋊C8C2×C414yes—
103C2×C22⋊C8C2×C414yes—
104C2×#97—41no—
105C2×#97—41no—
106C2×#97—41no—
107C2×#97—41no—
108C2×#97—41no—
109C2×#97—41no—
110C2×#97—41no—
111C2×#97—41no—
112C2×#97C2×C414yes—
113#388—28no—
114C4×C4⋊C4—41no—
115C4×C4⋊C4—41no—
116C4×C22⋊C4—41no—
117C4×C22⋊C4—41no—
118C4×C22⋊C4—41no—
119C4×C22⋊C4—41no—
120C4×C22⋊C4C2×C414yes—
121C4×C22⋊C4C2×C414yes—
122C2×#95C2×C412yes—
123C2×#95C2×C412yes—
124C2×#95C2×C412yes—
125C2×#95C2×C412yes—
126C2×#95C2×C412yes—
127C2×#95C2×C412yes—
128C2×#95C2×C412yes—
129C2×#95C2×C411yescharacteristic
130C2×#95C2×C411yescharacteristic
131C43—41no—
132C43—41no—
133#336—41no—
134#336—41no—
135#336—41no—
136#336—41no—
137#336—41no—
138#336—41no—
139#336—41no—
140#336—41no—
141#336—41no—
142#336—41no—
143#336—41no—
144#336—41no—
145#336—41no—
146#336—41no—
147#336C2×C414yes—
148#336C2×C414yes—
149#336C2×C414yes—
150#336C2×C414yes—
151C25C22×C411yescharacteristic
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4C22×C412yes—
165C23×C4C22×C412yes—
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C412yes—
171C23×C4C22×C412yes—
172C23×C4C22×C412yes—
173C23×C4C22×C412yes—
174C23×C4C22×C411yescharacteristic
175C23×C4C2411yescharacteristic
176C2×M16—44no—
177C2×M16—44no—
178C2×M16—44no—
179C2×M16—44no—
180C2×M16—44no—
181C2×M16—44no—
182C2×M16—44no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
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—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8C22×C412yes—
232C22×C8C22×C412yes—
233C22×C8C22×C412yes—
234C22×C8C22×C412yes—
235C22×C8C22×C412yes—
236C22×C8C22×C412yes—
237C22×C8C22×C412yes—
238C22×C8C22×C412yes—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4C4214yes—
251C2×C22⋊C4C4214yes—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
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×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42M1618yescomplemented
290C2×C42C22×C412yes—
291C2×C42C22×C412yes—
292C2×C42C22×C412yes—
293C2×C42C22×C412yes—
294C4⋊C8—44no—
295C4⋊C8—44no—
296C4⋊C8—44no—
297C22⋊C8—44no—
298C22⋊C8—44no—
299C22⋊C8—44no—
300#97—44no—
301#97—44no—
302#97—44no—
303#95—24no—
304#95—24no—
305#95—42no—
306#95—42no—
307#95—42no—
308#95—42no—
309#95—42no—
310#95C4214yes—
311#95C4214yes—
312C24—21no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24C2×C4212yes—
323C24C23×C411yescharacteristic
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4C2×C4212yes—
400C22×C4C2×C4212yes—
401C22×C4C2×C4212yes—
402C22×C4C2×C4212yes—
403C22×C4C2×C4212yes—
404C22×C4C2×M1614yes—
405C22×C4C2×M1614yes—
406C22×C4C2×M1614yes—
407C22×C4C23×C411yescharacteristic
408C22×C4C23×C411yescharacteristic
409C22×C4H(5,𝔽2)11yescharacteristic
410C22×C4H(5,𝔽2)11yescharacteristic
411C22×C4H(5,𝔽2)11yescharacteristic
412C22×C4H(5,𝔽2)11yescharacteristic
413C22×C4H(5,𝔽2)11yescharacteristic
414C22×C4H(5,𝔽2)11yescharacteristic
415C22×C4H(5,𝔽2)11yescharacteristic
416C22×C4H(5,𝔽2)11yescharacteristic
417C22×C42−1+411yescharacteristic
418C22×C42−1+411yescharacteristic
419C22×C42−1+411yescharacteristic
420C22×C42−1+411yescharacteristic
421M16—416no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
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—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—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—
478C4⋊C4—48no—
479C22⋊C4—28no—
480C22⋊C4—44no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—48no—
492C42—48no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—41no—
498C42—41no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—41no—
506C42—41no—
507C42—41no—
508C42—41no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23C22×C4211yescharacteristic
533C23#51811yescharacteristic
534C23#51811yescharacteristic
535C23#52011yescharacteristic
536C23#52011yescharacteristic
537C23#52011yescharacteristic
538C23#52011yescharacteristic
539C23C22×M1612yes—
540C23#56812yes—
541C23#56812yes—
542C23#56812yes—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4C4×M1614yes—
606C2×C4#51811yescharacteristic
607C2×C4#51811yescharacteristic
608C2×C4#51811yescharacteristic
609C2×C4#51811yescharacteristic
610C2×C4#51811yescharacteristic
611C2×C4#51911yescharacteristic
612C2×C4#52011yescharacteristic
613C2×C4#52011yescharacteristic
614C2×C4#52011yescharacteristic
615C2×C4#52011yescharacteristic
616C2×C4#52011yescharacteristic
617C2×C4#52011yescharacteristic
618C2×C4#56011yescharacteristic
619C2×C4#56011yescharacteristic
620C2×C4#56011yescharacteristic
621C2×C4#56011yescharacteristic
622C2×C4H(3,𝔽4)11yescharacteristic
623C2×C4#56211yescharacteristic
624C2×C4#56211yescharacteristic
625C2×C4#56211yescharacteristic
626C2×C4#56211yescharacteristic
627C2×C4#56211yescharacteristic
628C2×C4#56211yescharacteristic
629C2×C4#56311yescharacteristic
630C2×C4#56311yescharacteristic
631C2×C4#56311yescharacteristic
632C2×C4#56311yescharacteristic
633C2×C4#56311yescharacteristic
634C8—42no—
635C8—42no—
636C8—42no—
637C8—42no—
638C8—42no—
639C8—42no—
640C8—42no—
641C8—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22#227411yescharacteristic
654C22#227511yescharacteristic
655C22#227511yescharacteristic
656C22#237611yescharacteristic
657C22#237711yescharacteristic
658C22#237911yescharacteristic
659C22#238311yescharacteristic
660C22C2×C4×M1611yescharacteristic
661C22#287311yescharacteristic
662C22#287511yescharacteristic
663C22#287511yescharacteristic
664C22#297312yes—
665C22#297512yes—
666C22#297812yes—
667C22#298012yes—
668C22#298912yes—
669C22#298912yes—
670C22#298912yes—
671C22#298912yes—
672C22#299512yes—
673C22#299512yes—
674C22#299512yes—
675C22#299512yes—
676C4—24no—
677C4—42no—
678C4—44no—
679C4—44no—
680C4—42no—
681C4—42no—
682C4—42no—
683C4—82no—
684C4—82no—
685C4#237511yescharacteristic
686C4#237611yescharacteristic
687C4#237611yescharacteristic
688C4#237711yescharacteristic
689C4#237811yescharacteristic
690C4#237811yescharacteristic
691C4#237911yescharacteristic
692C4#238311yescharacteristic
693C2—82no—
694C2#1383111yescharacteristic
695C2#2054911yescharacteristic
696C2#2055411yescharacteristic
697C2#2055611yescharacteristic
698C2#2056311yescharacteristic
699C2#2056411yescharacteristic
700C2#2056811yescharacteristic
701C2#2332412yes—
702C2#2333412yes—
703C2#2335312yes—
704C2#2337812yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C22×C42
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
6. #10877C2semidirect
7. #10877C2semidirect
8. #10877C2semidirect
9. #10875C2semidirect
10. #10835C2semidirect
11. #10832C2semidirect
12. #10798C2semidirect
13. #10795C2semidirect
32. #1924C4semidirect
33. #1851C4semidirect
289. C2×C42M16semidirect

Maximal subgroups

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

subgroupindex
C2×#19242
C2×#18512
C4×#3882
#109222
#108772
#108752
#108352
#108322
#107982
#107952
#107152