Finite Group Databasebeta

← #374705

Subgroups — #374705

Order 512.

1562 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1954C211yesmaximal, characteristic
2C2×#1954C211yesmaximal, characteristic
3C2×#1939C211yesmaximal, characteristic, complemented
4C2×#1923C212yesmaximal, complemented
5C2×C4×C4⋊C8C211yesmaximal, characteristic, complemented
6C4×#390C211yesmaximal, characteristic
7#11147C212yesmaximal, complemented
8#11042C212yesmaximal, complemented
9#10921C212yesmaximal, complemented
10#10834C211yesmaximal, characteristic, complemented
11C4×#336C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2212yes—
20C22×C4×C8C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C22×C4×C8C2212yes—
23C2×#390C411yescharacteristic
24C2×#390C411yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C22⋊C4C2212yes—
27C2×C4×C22⋊C4C2212yes—
28C2×C43C2211yescharacteristic
29#1954—24no—
30#1954—24no—
31#1939—24no—
32#1923—28no—
33C4×C4⋊C8—24no—
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42#1437C412yes—
43#1437C412yes—
44C4⋊#95C411yescharacteristic
45C4⋊#95C411yescharacteristic
46C23×D4C2×C411yescharacteristic
47C24×C4C2312yes—
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C412yes—
64C22×C22⋊C4C2×C412yes—
65C22×C22⋊C4C2×C412yes—
66C22×C22⋊C4C2×C412yes—
67C22×C22⋊C4C2×C412yes—
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—24no—
81C2×C4⋊C8—24no—
82C2×C4⋊C8—42no—
83C2×C4⋊C8D414yes—
84C2×C4⋊C8D414yes—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—42no—
94C2×C22⋊C8—42no—
95C2×C22⋊C8—42no—
96C2×C22⋊C8—42no—
97C2×C4×C8—24no—
98C2×C4×C8—24no—
99C2×C4×C8—22no—
100C2×C4×C8—22no—
101C2×C4×C8—22no—
102C2×C4×C8—22no—
103C2×C4×C8—22no—
104C2×C4×C8—22no—
105C2×C4×C8—42no—
106#390—24no—
107#390C818yescomplemented
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C22⋊C4—24no—
111C4×C22⋊C4—24no—
112C4×C22⋊C4—42no—
113C4×C22⋊C4—42no—
114C2×#95C2×C412yes—
115C2×#95C2×C412yes—
116C2×#95C2×C412yes—
117C2×#95C2×C412yes—
118C2×#95C2×C412yes—
119C43—22no—
120C43—22no—
121#336—24no—
122#336—24no—
123#336—24no—
124#336—22no—
125#336—22no—
126#336—22no—
127#336—42no—
128#336D414yescomplemented
129C25C22×C412yes—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4C2×C814yes—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4C22×C412yes—
152C23×C4C22×C412yes—
153C23×C4C22×C412yes—
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C2411yescharacteristic
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—21no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—22no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8C2×D412yes—
198C22×C8C2×D414yes—
199C22×C8C2×D414yes—
200C22×C8C2×D412yes—
201C22×C8#4112yes—
202C22×C8#4112yes—
203C4×D4—44no—
204C4×D4—44no—
205C4×D4—44no—
206C4×D4—44no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4C2×C814yes—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4C2×C818yes—
230C2×C22⋊C4C2×C818yes—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—21no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—41no—
256C2×C42—41no—
257C2×C42C2×C814yes—
258C2×C42M1614yes—
259C2×C42C2×D412yes—
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262C2×C42#4112yes—
263C2×C42#4112yes—
264C2×C42#4112yes—
265C2×C42#4112yes—
266C2×C42#4112yes—
267C4⋊C8—28no—
268C4⋊C8—28no—
269C4⋊C8—48no—
270C22⋊C8—48no—
271C22⋊C8—48no—
272C22⋊C8—48no—
273C22⋊C8—48no—
274C4×C8—48no—
275C4×C8—44no—
276C4×C8—44no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—42no—
281#95—42no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24C22×C814yes—
297C24C23×C411yescharacteristic
298C2×D4—28no—
299C2×D4—28no—
300C2×D4—28no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—44no—
304C2×D4—44no—
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—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4#11712yes—
378C22×C4#11712yes—
379C22×C4C4×D412yes—
380C22×C4C4×D412yes—
381C22×C4C4×D412yes—
382C22×C4C4×D412yes—
383C22×C4C4×D412yes—
384C22×C4C4×D412yes—
385C22×C4C22×C812yes—
386C22×C4C22×C814yes—
387C22×C4C22×C812yes—
388C22×C4C22×C812yes—
389C22×C4C2×M1612yes—
390C22×C4C2×M1612yes—
391C22×C4C2×M1612yes—
392C22×C4#13112yes—
393C22×C4#13112yes—
394C22×C4#13112yes—
395C22×C4#13112yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4H(5,𝔽2)11yescharacteristic
407C22×C4H(5,𝔽2)11yescharacteristic
408C22×C4H(5,𝔽2)11yescharacteristic
409C22×C42−1+411yescharacteristic
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—24no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—24no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—41no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—42no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—42no—
446C2×C8—41no—
447C2×C8—42no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—42no—
452C4⋊C4—28no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C22⋊C4—216no—
456C22⋊C4—48no—
457C22⋊C4—48no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—28no—
465C42—28no—
466C42—44no—
467C42—44no—
468C42—42no—
469C42—41no—
470C42—41no—
471C42—42no—
472C42—42no—
473C42—41no—
474C42—41no—
475C42—42no—
476C42#12114yes—
477C42#12314yes—
478C42#12414yes—
479C42#12614yes—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23C2×#11711yescharacteristic
516C23C2×C4×D411yescharacteristic
517C23C2×C4×D411yescharacteristic
518C23C2×C4×D411yescharacteristic
519C23C4×#4111yescharacteristic
520C23#51811yescharacteristic
521C23#52011yescharacteristic
522C23C23×C811yescharacteristic
523C23C22×M1611yescharacteristic
524C23C2×#13111yescharacteristic
525C23C2×#13111yescharacteristic
526C23C2×#13111yescharacteristic
527C23C2×#13111yescharacteristic
528C23#56811yescharacteristic
529C23#56811yescharacteristic
530D4—416no—
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—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
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—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4#38814yes—
593C2×C4#38814yes—
594C2×C4#43114yes—
595C2×C4#43314yes—
596C2×C4C8×D414yes—
597C2×C4C8×D414yes—
598C2×C4C8×D414yes—
599C2×C4#43514yes—
600C2×C4#43614yes—
601C2×C4#43614yes—
602C2×C4C2×#12111yescharacteristic
603C2×C4C2×#12311yescharacteristic
604C2×C4C2×#12411yescharacteristic
605C2×C4C2×#12611yescharacteristic
606C2×C4#52911yescharacteristic
607C2×C4#52911yescharacteristic
608C2×C4#53211yescharacteristic
609C2×C4#53211yescharacteristic
610C2×C4#54011yescharacteristic
611C2×C4#54011yescharacteristic
612C2×C4#54211yescharacteristic
613C2×C4#54211yescharacteristic
614C2×C4D4211yescharacteristic
615C2×C4D4211yescharacteristic
616C2×C4#54611yescharacteristic
617C2×C4#54611yescharacteristic
618C2×C4#54711yescharacteristic
619C2×C4#54811yescharacteristic
620C2×C4#54811yescharacteristic
621C2×C4#55011yescharacteristic
622C2×C4#55011yescharacteristic
623C2×C4#55011yescharacteristic
624C2×C4#55311yescharacteristic
625C2×C4#55311yescharacteristic
626C2×C4#55511yescharacteristic
627C2×C4#55611yescharacteristic
628C2×C4#55911yescharacteristic
629C2×C4#55911yescharacteristic
630C8—42no—
631C8—44no—
632C8—42no—
633C8—44no—
634C8—42no—
635C8—42no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22C2×#38811yescharacteristic
652C22#229311yescharacteristic
653C22#231911yescharacteristic
654C22#236111yescharacteristic
655C22#236311yescharacteristic
656C22#236611yescharacteristic
657C22#236711yescharacteristic
658C22C2×#43111yescharacteristic
659C22C2×#43311yescharacteristic
660C22#292211yescharacteristic
661C22#292311yescharacteristic
662C22C2×C8×D411yescharacteristic
663C22C2×C8×D411yescharacteristic
664C22C2×C8×D411yescharacteristic
665C22C2×#43511yescharacteristic
666C22C2×#43611yescharacteristic
667C22C2×#43611yescharacteristic
668C22#293311yescharacteristic
669C22#293311yescharacteristic
670C22#293511yescharacteristic
671C22#293511yescharacteristic
672C22#293511yescharacteristic
673C22#293511yescharacteristic
674C22C8×#4111yescharacteristic
675C22#296511yescharacteristic
676C22#296911yescharacteristic
677C22#296911yescharacteristic
678C22#297211yescharacteristic
679C22#297511yescharacteristic
680C22#297911yescharacteristic
681C22#297911yescharacteristic
682C22#298811yescharacteristic
683C22#298911yescharacteristic
684C22#299411yescharacteristic
685C22#299411yescharacteristic
686C4—24no—
687C4—24no—
688C4—24no—
689C4—42no—
690C4—42no—
691C4—44no—
692C4—82no—
693C4—82no—
694C4#192318yescomplemented
695C4#242611yescharacteristic
696C4#242811yescharacteristic
697C4#244511yescharacteristic
698C4#244811yescharacteristic
699C4#249011yescharacteristic
700C4#249111yescharacteristic
701C4#251211yescharacteristic
702C4#251311yescharacteristic
703C2—84no—
704C2#1414111yescharacteristic
705C2C2×#192311yescharacteristic
706C2#2104811yescharacteristic
707C2#2126411yescharacteristic
708C2#2126811yescharacteristic
709C2#2200811yescharacteristic
710C2#2200911yescharacteristic
711C2#2298511yescharacteristic
712C2#2298911yescharacteristic
713C2#2308311yescharacteristic
714C2#2308711yescharacteristic
715C2#2317311yescharacteristic
716C2#2317611yescharacteristic
717C2#2320411yescharacteristic
718C2#2320811yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C23×C8
socleC24C23×C4
prime radicalC23×D4C2×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. C2×#1939C2semidirect
4. C2×#1923C2semidirect
5. C2×C4×C4⋊C8C2semidirect
7. #11147C2semidirect
8. #11042C2semidirect
9. #10921C2semidirect
10. #10834C2semidirect
11. C4×#336C2semidirect
107. #390C8semidirect
128. #336D4semidirect
694. C4#1923semidirect

Maximal subgroups

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

subgroupindex
C2×#19542
C2×#19392
C2×#19232
C2×C4×C4⋊C82
C4×#3902
#111472
#110422
#109212
#108342
C4×#3362