Finite Group Databasebeta

← #357101

Subgroups — #357101

Order 512.

1418 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1924C211yesmaximal, characteristic
2C2×#1771C211yesmaximal, characteristic, complemented
3C2×#1752C211yesmaximal, characteristic
4C2×C8×C22⋊C4C211yesmaximal, characteristic
5C4×#388C211yesmaximal, characteristic
6#10922C211yesmaximal, characteristic
7#10921C211yesmaximal, characteristic
8#10876C211yesmaximal, characteristic, complemented
9#10874C211yesmaximal, characteristic, complemented
10#10798C211yesmaximal, characteristic, complemented
11#10798C211yesmaximal, characteristic, complemented
12#10714C211yesmaximal, characteristic
13#10696C211yesmaximal, characteristic, complemented
14#10693C211yesmaximal, characteristic, complemented
15C4×#336C211yesmaximal, characteristic, complemented
16C24×C8C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×#97C2211yescharacteristic
22C22×#97C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C22×#97C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28C2×#388C412yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33C2×C43C2211yescharacteristic
34#1924C418yescomplemented
35#1771—24no—
36#1752C418yescomplemented
37C8×C22⋊C4—24no—
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
52C2×#336C2211yescharacteristic
53C2×#336C2211yescharacteristic
54#1437C412yes—
55#1436C412yes—
56#1434C412yes—
57C4×#95C2211yescharacteristic
58C4×#95C2211yescharacteristic
59C24×C4C2311yescharacteristic
60C23×C8—22no—
61C23×C8—22no—
62C23×C8—22no—
63C23×C8—22no—
64C23×C8—41no—
65C23×C8—41no—
66C23×C8C2×C414yes—
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C22×C4⋊C4C2×C412yes—
76C22×C22⋊C4C2×C412yes—
77C22×C22⋊C4C2×C411yescharacteristic
78C22×C22⋊C4C2×C412yes—
79C22×C22⋊C4C2×C411yescharacteristic
80C22×C42C2×C412yes—
81C22×C42C2×C412yes—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8C2×C414yes—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—41no—
96C2×C22⋊C8—41no—
97C2×C22⋊C8C2×C414yes—
98C2×C22⋊C8C2×C414yes—
99C2×C22⋊C8C2×C414yes—
100C2×#97—22no—
101C2×#97—22no—
102C2×#97—22no—
103C2×#97—22no—
104C2×#97—41no—
105C2×#97—41no—
106C2×#97—41no—
107C2×#97—41no—
108C2×#97—41no—
109C2×#97C2×C414yes—
110C2×#97C2×C414yes—
111C2×#97C2×C414yes—
112C2×C4×C8—22no—
113C2×C4×C8—22no—
114C2×C4×C8—22no—
115C2×C4×C8—22no—
116C2×C4×C8—22no—
117C2×C4×C8—22no—
118C2×C4×C8—41no—
119C2×C4×C8—41no—
120C2×C4×C8—41no—
121#388—28no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—41no—
127C4×C22⋊C4—41no—
128C4×C22⋊C4C2×C414yes—
129C4×C22⋊C4C2×C414yes—
130C2×#95C2×C412yes—
131C2×#95C2×C412yes—
132C2×#95C2×C412yes—
133C2×#95C2×C412yes—
134C2×#95C2×C412yes—
135C2×#95C2×C412yes—
136C2×#95C2×C412yes—
137C2×#95C2×C411yescharacteristic
138C2×#95C2×C411yescharacteristic
139C43—41no—
140C43—41no—
141#336—22no—
142#336—22no—
143#336—22no—
144#336—22no—
145#336—22no—
146#336—22no—
147#336—22no—
148#336—22no—
149#336—22no—
150#336—41no—
151#336—41no—
152#336C2×C414yes—
153#336C2×C414yes—
154#336C2×C414yes—
155#336C2×C414yes—
156#336D414yes—
157C25C22×C411yescharacteristic
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4C22×C412yes—
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C412yes—
177C23×C4C22×C412yes—
178C23×C4C22×C412yes—
179C23×C4C22×C412yes—
180C23×C4C22×C412yes—
181C23×C4C2411yescharacteristic
182C22×C8—21no—
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—28no—
223C22×C8—28no—
224C22×C8—44no—
225C22×C8—44no—
226C22×C8—44no—
227C22×C8—44no—
228C22×C8—44no—
229C22×C8C22×C412yes—
230C22×C8C22×C412yes—
231C22×C8C22×C412yes—
232C22×C8C22×C412yes—
233C22×C8C22×C412yes—
234C22×C8C22×C412yes—
235C22×C8C22×C412yes—
236C22×C8C22×C412yes—
237C22×C8C2×D412yes—
238C22×C8C2×D412yes—
239C22×C8#4112yes—
240C22×C8#4112yes—
241C22×C8#4112yes—
242C22×C8#4112yes—
243C22×C8#4112yes—
244C22×C8#4112yes—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4C4214yes—
256C2×C22⋊C4C4214yes—
257C2×C22⋊C4M1618yes—
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—24no—
271C2×C42—24no—
272C2×C42—24no—
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—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42C22×C412yes—
295C2×C42C22×C412yes—
296C2×C42C22×C412yes—
297C2×C42C22×C412yes—
298C2×C42C2×D412yes—
299C2×C42#4112yes—
300C4⋊C8—28no—
301C22⋊C8—28no—
302C22⋊C8—28no—
303C22⋊C8—44no—
304#97—28no—
305#97—44no—
306#97—44no—
307#97—44no—
308C4×C8—44no—
309C4×C8—44no—
310C4×C8—44no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—42no—
317#95—42no—
318#95C4214yes—
319#95C4214yes—
320C24—21no—
321C24—21no—
322C24—21no—
323C24—21no—
324C24—21no—
325C24—21no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24C2×C4212yes—
330C24C2×M1614yes—
331C24C23×C411yescharacteristic
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
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—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
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×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4C2×C4212yes—
406C22×C4C2×C4212yes—
407C22×C4C2×C4212yes—
408C22×C4C2×C4212yes—
409C22×C4C2×C4212yes—
410C22×C4C4×D414yes—
411C22×C4C2×M1614yes—
412C22×C4C2×M1614yes—
413C22×C4#13114yes—
414C22×C4#13114yes—
415C22×C4C23×C411yescharacteristic
416C22×C4C23×C411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4H(5,𝔽2)11yescharacteristic
426C22×C4H(5,𝔽2)11yescharacteristic
427C22×C4H(5,𝔽2)11yescharacteristic
428C22×C42−1+411yescharacteristic
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—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—416no—
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—41no—
482C2×C8#11714yes—
483C2×C8#11714yes—
484C2×C8C4×D414yes—
485C2×C8C4×D414yes—
486C4⋊C4—48no—
487C22⋊C4—28no—
488C22⋊C4—216no—
489C22⋊C4—44no—
490C22⋊C4—48no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—48no—
504C42—48no—
505C42—41no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42—41no—
510C42—41no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23C4×M1614yes—
540C23C22×C4211yescharacteristic
541C23C2×C4×D411yescharacteristic
542C23C4×#4111yescharacteristic
543C23C4×#4111yescharacteristic
544C23C4×#4111yescharacteristic
545C23#51811yescharacteristic
546C23#52011yescharacteristic
547C23C22×M1612yes—
548C23C2×#13112yes—
549C23C2×#13112yes—
550C23#56812yes—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
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—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
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×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4#43518yes—
614C2×C4C2×#11711yescharacteristic
615C2×C4C2×#11711yescharacteristic
616C2×C4C2×C4×D411yescharacteristic
617C2×C4C2×C4×D411yescharacteristic
618C2×C4C4×#4111yescharacteristic
619C2×C4C4×#4111yescharacteristic
620C2×C4C4×#4111yescharacteristic
621C2×C4C4×#4111yescharacteristic
622C2×C4#51811yescharacteristic
623C2×C4#51811yescharacteristic
624C2×C4#52011yescharacteristic
625C2×C4#52011yescharacteristic
626C2×C4#54611yescharacteristic
627C2×C4#54611yescharacteristic
628C2×C4#54611yescharacteristic
629C2×C4#54811yescharacteristic
630C2×C4#55111yescharacteristic
631C2×C4#55111yescharacteristic
632C2×C4#55211yescharacteristic
633C2×C4#55311yescharacteristic
634C2×C4#55311yescharacteristic
635C2×C4#55311yescharacteristic
636C2×C4#55311yescharacteristic
637C2×C4#55311yescharacteristic
638C2×C4#55611yescharacteristic
639C2×C4#55611yescharacteristic
640C2×C4#55911yescharacteristic
641C2×C4#55911yescharacteristic
642C8—24no—
643C8—24no—
644C8—24no—
645C8—24no—
646C8—42no—
647C8—42no—
648C8—42no—
649C8—42no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—44no—
654C22—42no—
655C22—44no—
656C22—42no—
657C22—44no—
658C22—44no—
659C22—42no—
660C22—42no—
661C22C4×#11711yescharacteristic
662C22C42×D411yescharacteristic
663C22#227511yescharacteristic
664C22#236411yescharacteristic
665C22#236711yescharacteristic
666C22#237311yescharacteristic
667C22#237411yescharacteristic
668C22C2×C4×M1611yescharacteristic
669C22#287311yescharacteristic
670C22C4×#13111yescharacteristic
671C22C4×#13111yescharacteristic
672C22C2×#43512yes—
673C22#293512yes—
674C22#296512yes—
675C22#296612yes—
676C22#296612yes—
677C22#296812yes—
678C22#297012yes—
679C22#297012yes—
680C22#297412yes—
681C22#297612yes—
682C22#299012yes—
683C22#299312yes—
684C4—28no—
685C4—44no—
686C4—42no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—44no—
691C4—42no—
692C4—82no—
693C4#234211yescharacteristic
694C4#234711yescharacteristic
695C4#235211yescharacteristic
696C4#235711yescharacteristic
697C4#236411yescharacteristic
698C4#236711yescharacteristic
699C4#237311yescharacteristic
700C4#237411yescharacteristic
701C2—44no—
702C2#1382211yescharacteristic
703C2#2047411yescharacteristic
704C2#2047911yescharacteristic
705C2C4×#43511yescharacteristic
706C2#2049211yescharacteristic
707C2#2055911yescharacteristic
708C2#2056111yescharacteristic
709C2#2311712yes—
710C2#2320912yes—
711C2#2328212yes—
712C2#2330012yes—
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
2. C2×#1771C2semidirect
8. #10876C2semidirect
9. #10874C2semidirect
10. #10798C2semidirect
11. #10798C2semidirect
13. #10696C2semidirect
14. #10693C2semidirect
15. C4×#336C2semidirect
34. #1924C4semidirect
36. #1752C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19242
C2×#17712
C2×#17522
C2×C8×C22⋊C42
C4×#3882
#109222
#109212
#108762
#108742
#107982
#107142
#106962
#106932
C4×#3362