Finite Group Databasebeta

← #373828

Subgroups — #373828

Order 512.

1922 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C4⋊C4C211yesmaximal, characteristic
2C2×#1947C211yesmaximal, characteristic, complemented
3C2×#1862C212yesmaximal
4C2×#1802C211yesmaximal, characteristic
5C2×C4×#102C211yesmaximal, characteristic
6#14151C211yesmaximal, characteristic, complemented
7#11328C212yesmaximal, complemented
8#11292C212yesmaximal, complemented
9#10841C211yesmaximal, characteristic, complemented
10#10834C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×C4×D4C2212yes—
13C22×C4×D4C2211yescharacteristic
14C23×C4⋊C4—21no—
15C22×#107C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×#102C2212yes—
18C22×#102C2211yescharacteristic
19C22×#102C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21D4×C4⋊C4—24no—
22D4×C4⋊C4—24no—
23C2×#389—21no—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#385—22no—
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×#380—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#1947—24no—
33#1862—28no—
34#1802C418yescomplemented
35C4×#102—24no—
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1441—22no—
45C4⋊#95—21no—
46C4⋊#95C2212yes—
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C24×C4—22no—
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×C4×D4—24no—
54C2×C4×D4—24no—
55C2×C4×D4—24no—
56C2×C4×D4—24no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—42no—
61C2×C4×D4—42no—
62C2×C4×D4C2×C414yes—
63C2×C4×D4D414yes—
64C2×C4×D4Q814yes—
65C22×C4⋊C4—28no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—44no—
73C22×C4⋊C4—44no—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—21no—
82C22×C22⋊C4—22no—
83C22×C42—21no—
84C22×C42C2312yes—
85C22×C42C2311yescharacteristic
86C2×#107—24no—
87C2×#107—24no—
88C2×#107—42no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8C2×C414yescomplemented
92C2×#102—24no—
93C2×#102—22no—
94C2×#102—22no—
95C2×#102—24no—
96C2×#102—22no—
97C2×#102—22no—
98C2×#102C2×C418yes—
99C2×#102D414yes—
100C2×#102Q814yes—
101C2×C4×C8—22no—
102C2×C4×C8—22no—
103C2×C4×C8—22no—
104#389—24no—
105#389—24no—
106#389—24no—
107#389—42no—
108#389—44no—
109#385—216no—
110#384—22no—
111#384—22no—
112#384C2×C414yescomplemented
113#384D414yes—
114#380—28no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95D412yes—
125#340—24no—
126#340—24no—
127#340—22no—
128#340—22no—
129#340—42no—
130#340C2×C418yescomplemented
131#336—24no—
132#336—42no—
133C25—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4C22×C412yes—
142C22×D4C2×D412yes—
143C22×D4C2×D412yes—
144C22×D4C2×Q812yes—
145C22×D4C2×Q812yes—
146C22×D4#4112yes—
147C22×D4#4112yes—
148C23×C4—21no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—28no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4—44no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—44no—
165C23×C4—82no—
166C23×C4—82no—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2411yescharacteristic
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—22no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—21no—
187C22×C8—22no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8C22×C414yes—
191C22×C8C2×D412yes—
192C22×C8C2×Q812yes—
193C4×D4—28no—
194C4×D4—28no—
195C4×D4—28no—
196C4×D4—48no—
197C4×D4—48no—
198C4×D4—48no—
199C4×D4—48no—
200C4×D4C4⋊C4116yescomplemented
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—432no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—81no—
244C2×C4⋊C4—81no—
245C2×C4⋊C4—81no—
246C2×C4⋊C4—81no—
247C2×C4⋊C4—82no—
248C2×C4⋊C4D814yes—
249C2×C4⋊C4C22×C414yes—
250C2×C4⋊C4C22×C412yes—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2×Q812yes—
255C2×C4⋊C4C2×Q812yes—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42C22×C412yes—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×Q812yes—
275C2×C42#4112yes—
276C2×C42#4112yes—
277#107—48no—
278C4⋊C8—28no—
279#102—216no—
280#102—28no—
281#102—28no—
282C4×C8—44no—
283#95—28no—
284#95—24no—
285#95—44no—
286#95—42no—
287#95—44no—
288#95—82no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—41no—
297C24—41no—
298C24—41no—
299C24—41no—
300C24—41no—
301C24—41no—
302C24—41no—
303C24—41no—
304C24C22×D411yescharacteristic
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4C2×C4⋊C414yes—
327C2×D4C4×D414yes—
328C2×D4C4×Q814yes—
329C2×D4#12114yes—
330C2×D4C22⋊Q814yes—
331C2×D4#12514yes—
332C2×D4C4⋊Q814yes—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
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—216no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—24no—
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—416no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—48no—
364C22×C4—48no—
365C22×C4—48no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—416no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—88no—
395C22×C4—88no—
396C22×C4—88no—
397C22×C4—88no—
398C22×C4—88no—
399C22×C4C4×D412yes—
400C22×C4C4×D412yes—
401C22×C4#12012yes—
402C22×C4#12012yes—
403C22×C4C22⋊Q812yes—
404C22×C4C22⋊Q812yes—
405C22×C4C2×D814yes—
406C22×C4C2×D812yes—
407C22×C4#13512yes—
408C22×C4AGL(1,ℤ/8)12yes—
409C22×C4#13712yes—
410C22×C4C23×C411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×Q811yescharacteristic
415C22×C4C22×Q811yescharacteristic
416C22×C4C2×#4111yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4H(5,𝔽2)11yescharacteristic
421C22×C4H(5,𝔽2)11yescharacteristic
422C22×C42−1+411yescharacteristic
423C22×C42−1+411yescharacteristic
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—42no—
443C2×C8—41no—
444C2×C8—41no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—28no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—416no—
458C4⋊C4—42no—
459C4⋊C4—41no—
460C4⋊C4—42no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—42no—
464C4⋊C4—41no—
465C4⋊C4—42no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—44no—
469C4⋊C4—84no—
470C4⋊C4—84no—
471C4⋊C4—88no—
472C4⋊C4C2×C4⋊C414yes—
473C22⋊C4—216no—
474C22⋊C4—416no—
475C42—42no—
476C42—42no—
477C42—42no—
478C42—42no—
479C42—84no—
480C42C2×C4⋊C414yes—
481C42C4×D414yes—
482C42C22⋊Q814yes—
483C42#12314yes—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
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—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23C2×C4×D411yescharacteristic
520C23C2×#12011yescharacteristic
521C23C2×C22⋊Q811yescharacteristic
522C23D4211yescharacteristic
523C23#54611yescharacteristic
524C23#54811yescharacteristic
525C23D4×Q811yescharacteristic
526C23C22×D811yescharacteristic
527C23C2×#13511yescharacteristic
528C23C2×AGL(1,ℤ/8)11yescharacteristic
529C23C2×#13711yescharacteristic
530C23#57611yescharacteristic
531C23#57711yescharacteristic
532C23#57711yescharacteristic
533C23#57811yescharacteristic
534D4—24no—
535D4—24no—
536D4—24no—
537D4—24no—
538D4—24no—
539D4—24no—
540D4—24no—
541D4#38918yescomplemented
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—432no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—832no—
583C2×C4—81no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—81no—
596C2×C4—82no—
597C2×C4—81no—
598C2×C4—82no—
599C2×C4#38514yes—
600C2×C4#38514yes—
601C2×C4C4×D814yes—
602C2×C4#44014yes—
603C2×C4#44714yes—
604C2×C4#45214yes—
605C2×C4#47414yes—
606C2×C4#47814yes—
607C2×C4C22×C4⋊C411yescharacteristic
608C2×C4C2×C4×D411yescharacteristic
609C2×C4C2×C4×D411yescharacteristic
610C2×C4C2×C4×Q811yescharacteristic
611C2×C4#52011yescharacteristic
612C2×C4#52011yescharacteristic
613C2×C4C2×#12111yescharacteristic
614C2×C4C2×C22⋊Q811yescharacteristic
615C2×C4C2×C22⋊Q811yescharacteristic
616C2×C4C2×#12311yescharacteristic
617C2×C4C2×#12511yescharacteristic
618C2×C4C2×C4⋊Q811yescharacteristic
619C2×C4#53511yescharacteristic
620C2×C4#53611yescharacteristic
621C2×C4#54411yescharacteristic
622C2×C4#54411yescharacteristic
623C2×C4D4211yescharacteristic
624C2×C4#54611yescharacteristic
625C2×C4#54711yescharacteristic
626C2×C4#54711yescharacteristic
627C2×C4#54811yescharacteristic
628C2×C4D4×Q811yescharacteristic
629C2×C4#55211yescharacteristic
630C2×C4#55211yescharacteristic
631C2×C4#55311yescharacteristic
632C2×C4#55311yescharacteristic
633C2×C4#55411yescharacteristic
634C2×C4#55411yescharacteristic
635C8—44no—
636C8—42no—
637C8—42no—
638C22—28no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22C2×#38511yescharacteristic
654C22#236011yescharacteristic
655C22#236111yescharacteristic
656C22#240811yescharacteristic
657C22#240911yescharacteristic
658C22#244011yescharacteristic
659C22#244211yescharacteristic
660C22C2×C4×D811yescharacteristic
661C22C2×#44011yescharacteristic
662C22#294611yescharacteristic
663C22#294711yescharacteristic
664C22C2×#44711yescharacteristic
665C22C2×#45211yescharacteristic
666C22#301211yescharacteristic
667C22#301311yescharacteristic
668C22C2×#47411yescharacteristic
669C22C2×#47811yescharacteristic
670C22#308311yescharacteristic
671C22#308411yescharacteristic
672C22#327211yescharacteristic
673C22#327611yescharacteristic
674C22D4×D811yescharacteristic
675C22#328411yescharacteristic
676C22#329411yescharacteristic
677C22#330211yescharacteristic
678C22#332511yescharacteristic
679C22#333111yescharacteristic
680C22#335811yescharacteristic
681C22#336111yescharacteristic
682C22#336711yescharacteristic
683C22#337211yescharacteristic
684C22Q8×D811yescharacteristic
685C22#338111yescharacteristic
686C22#338411yescharacteristic
687C22#338511yescharacteristic
688C4—24no—
689C4—44no—
690C4—44no—
691C4—82no—
692C4—82no—
693C4—84no—
694C4—82no—
695C4—84no—
696C4#186218yescomplemented
697C4C2×#38911yescharacteristic
698C4#231711yescharacteristic
699C4#236011yescharacteristic
700C4#236111yescharacteristic
701C4#244011yescharacteristic
702C4#244211yescharacteristic
703C4#248611yescharacteristic
704C4#248711yescharacteristic
705C2—48no—
706C2#1413011yescharacteristic
707C2C2×#186211yescharacteristic
708C2#2087811yescharacteristic
709C2#2296211yescharacteristic
710C2#2296911yescharacteristic
711C2#2299511yescharacteristic
712C2#2300211yescharacteristic
713C2#2423611yescharacteristic
714C2#2424111yescharacteristic
715C2#2432611yescharacteristic
716C2#2433111yescharacteristic
717C2#2604011yescharacteristic
718C2#2604511yescharacteristic
719C2#2610611yescharacteristic
720C2#2611111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1947C2semidirect
6. #14151C2semidirect
7. #11328C2semidirect
8. #11292C2semidirect
9. #10841C2semidirect
10. #10834C2semidirect
34. #1802C4semidirect
91. C2×C4⋊C8C2×C4semidirect
112. #384C2×C4semidirect
130. #340C2×C4semidirect
200. C4×D4C4⋊C4semidirect
541. D4#389semidirect
696. C4#1862semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C4⋊C42
C2×#19472
C2×#18622
C2×#18022
C2×C4×#1022
#141512
#113282
#112922
#108412
#108342
#107882