Finite Group Databasebeta

← #362628

Subgroups — #362628

Order 512.

2058 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2361C211yesmaximal, characteristic
2C2×#1964C212yesmaximal
3C2×#1799C212yesmaximal, complemented
4#13907C212yesmaximal, complemented
5#11043C214yesmaximal, complemented
6#10846C211yesmaximal, characteristic, complemented
7#10761C212yesmaximal, complemented
8C4×#342C211yesmaximal, characteristic, complemented
9C22×#121C2211yescharacteristic
10C22×C4×D4C411yescharacteristic, complemented
11C22×C4×D4C2212yes—
12C23×C4⋊C4C2211yescharacteristic
13C22×#99C2214yes—
14C22×#99C2214yes—
15#2361—22no—
16#2361—22no—
17#2361—24no—
18C2×#390C411yescharacteristic, complemented
19C2×#389—22no—
20C2×#389C411yescharacteristic, complemented
21C2×#388C412yescomplemented
22C2×#386C2212yes—
23C2×#385C412yescomplemented
24C2×#380C2212yes—
25C2×#380C2214yes—
26C2×C4×C4⋊C4C411yescharacteristic, complemented
27C2×C4×C22⋊C4C2212yes—
28C2×C4×C22⋊C4C2211yescharacteristic
29#1964—28no—
30#1799—28no—
31C2×#342C2211yescharacteristic
32C2×#342C2212yes—
33C2×#342C2212yes—
34C2×#342C2212yes—
35C2×#342C2211yescharacteristic
36#1441—24no—
37C4⋊#95—22no—
38C22⋊#95C2214yes—
39C23×D4C2×C411yescharacteristic
40C23×D4C2312yes—
41C24×C4C2×C412yes—
42C24×C4C2312yes—
43C24×C4C2311yescharacteristic
44C2×#121—22no—
45C2×#121—22no—
46C2×#121—24no—
47C2×#121D414yes—
48C2×#121Q814yes—
49C2×C4×D4—24no—
50C2×C4×D4—24no—
51C2×C4×D4—22no—
52C2×C4×D4—21no—
53C2×C4×D4—21no—
54C2×C4×D4—24no—
55C2×C4×D4—24no—
56C2×C4×D4—44no—
57C2×C4×D4—44no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—24no—
60C22×C4⋊C4—44no—
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4D412yes—
67C22×C4⋊C4D412yes—
68C22×C4⋊C4D414yes—
69C22×C22⋊C4C2×C412yes—
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C412yes—
72C22×C22⋊C4C2×C412yes—
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2311yescharacteristic
75C22×C22⋊C4C2312yes—
76C22×C22⋊C4C2312yes—
77C22×C22⋊C4C2311yescharacteristic
78C22×C42—22no—
79C22×C42C2×C412yes—
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C2×#99—28no—
83C2×#99—28no—
84C2×#99—28no—
85C2×#99—28no—
86C2×#99—44no—
87C2×#99—44no—
88#390—22no—
89#390—22no—
90#389—22no—
91#389—22no—
92#389—48no—
93#388—24no—
94#388—24no—
95#386—24no—
96#386—24no—
97#385—24no—
98#385—24no—
99#380—24no—
100#380—24no—
101#380—28no—
102#380—44no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—21no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—24no—
109C4×C22⋊C4—24no—
110C2×#95—24no—
111C2×#95—24no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—24no—
115C2×#95C2×C411yescharacteristic
116C2×#95C2×C411yescharacteristic
117C2×#95C2×C412yes—
118C2×#95C2×C412yes—
119#342—24no—
120#342—22no—
121#342—24no—
122#342—42no—
123#342D414yescomplemented
124C25C22×C412yes—
125C25C2411yescharacteristic
126C22×D4—24no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—24no—
130C22×D4—24no—
131C22×D4—24no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—42no—
138C22×D4—42no—
139C22×D4C4⋊C414yescomplemented
140C22×D4C2×D414yes—
141C22×D4C2×Q814yes—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—24no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—44no—
154C23×C4—42no—
155C23×C4—44no—
156C23×C4—82no—
157C23×C4—82no—
158C23×C4C22×C412yes—
159C23×C4C22×C412yes—
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C412yes—
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C2×D414yes—
168C23×C4C2×D412yes—
169C23×C4C2×D414yes—
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4C2×Q812yes—
173C23×C4#4114yes—
174#121—28no—
175#121—28no—
176#121—28no—
177#121—28no—
178C4×D4—48no—
179C4×D4—48no—
180C4×D4—48no—
181C4×D4—48no—
182C4×D4—48no—
183C4×D4—44no—
184C4×D4—44no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—48no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—48no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—48no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—84no—
203C2×C4⋊C4C22⋊C411yescharacteristic
204C2×C4⋊C4C22⋊C412yes—
205C2×C4⋊C4C22⋊C414yes—
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22⋊C414yes—
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22⋊C411yescharacteristic
210C2×C4⋊C4C4⋊C414yescomplemented
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—82no—
243C2×C22⋊C4C4⋊C418yescomplemented
244C2×C22⋊C4C2×D414yes—
245C2×C22⋊C4C2×D412yes—
246C2×C22⋊C4#4114yes—
247C2×C22⋊C4#4112yes—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—24no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—44no—
259C2×C42—44no—
260C2×C42—44no—
261#99—416no—
262#99—416no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—24no—
268#95—48no—
269#95—48no—
270#95—44no—
271#95—44no—
272#95—84no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—41no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24—42no—
287C24C2×C4⋊C414yes—
288C24C23×C411yescharacteristic
289C24C22×D412yes—
290C24C22×D411yescharacteristic
291C24C22×D411yescharacteristic
292C24C22×Q811yescharacteristic
293C24C2×#4112yes—
294C24C2×#4111yescharacteristic
295C24C2×#4112yes—
296C24H(5,𝔽2)12yes—
297C24H(5,𝔽2)11yescharacteristic
298C242−1+411yescharacteristic
299C2×D4—28no—
300C2×D4—28no—
301C2×D4—24no—
302C2×D4—28no—
303C2×D4—24no—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—28no—
307C2×D4—24no—
308C2×D4—24no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—48no—
312C2×D4—48no—
313C2×D4—44no—
314C2×D4—44no—
315C2×D4—44no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—42no—
322C2×D4—42no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—28no—
346C22×C4—24no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—48no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—44no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—44no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—82no—
400C22×C4—82no—
401C22×C4—84no—
402C22×C4—88no—
403C22×C4—84no—
404C22×C4—88no—
405C22×C4—88no—
406C22×C4C2×C22⋊C412yes—
407C22×C4C2×C22⋊C414yes—
408C22×C4C2×C22⋊C412yes—
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C412yes—
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C4⋊C414yes—
413C22×C4C2×C4⋊C412yes—
414C22×C4C2×C4⋊C412yes—
415C22×C4#11712yes—
416C22×C4#11712yes—
417C22×C4C4×D412yes—
418C22×C4C4×D412yes—
419C22×C4#12114yes—
420C22×C4#12114yes—
421C22×C4C22⋊Q814yes—
422C22×C4C22⋊Q814yes—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—28no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—42no—
430C4⋊C4—44no—
431C4⋊C4—48no—
432C4⋊C4—88no—
433C4⋊C4—816no—
434C4⋊C4#9914yescomplemented
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—216no—
438C22⋊C4—24no—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—48no—
443C22⋊C4—44no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—84no—
456C42—44no—
457C42—44no—
458C42—42no—
459C42—42no—
460C42—42no—
461C42—42no—
462C42—88no—
463C23—22no—
464C23—21no—
465C23—22no—
466C23—21no—
467C23—22no—
468C23—21no—
469C23—22no—
470C23—21no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—41no—
475C23—41no—
476C23—42no—
477C23—41no—
478C23—41no—
479C23—42no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—42no—
485C23—41no—
486C23—42no—
487C23—41no—
488C23—41no—
489C23—42no—
490C23—41no—
491C23—41no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—82no—
505C23—82no—
506C23—82no—
507C23—82no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23—82no—
512C23C22×C22⋊C411yescharacteristic
513C23C22×C4⋊C411yescharacteristic
514C23C2×#11711yescharacteristic
515C23C2×C4×D411yescharacteristic
516C23C4×#4111yescharacteristic
517C23#51811yescharacteristic
518C23#52011yescharacteristic
519C23C2×#12112yes—
520C23C2×C22⋊Q812yes—
521C23#52912yes—
522C23#53212yes—
523C23#54012yes—
524C23#54212yes—
525C23D4212yes—
526C23D4211yescharacteristic
527C23#54612yes—
528C23#54612yes—
529C23#54711yescharacteristic
530C23#54812yes—
531C23D4×Q811yescharacteristic
532C23#55012yes—
533C23#55411yescharacteristic
534C23#55412yes—
535D4—48no—
536D4—48no—
537D4—48no—
538D4—48no—
539D4—48no—
540D4—48no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—48no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—88no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—88no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4#38014yes—
620C2×C4#38012yes—
621C2×C4#38012yes—
622C2×C4C2×#9914yes—
623C2×C4C2×#9912yes—
624C2×C4#41012yes—
625C22—22no—
626C22—22no—
627C22—21no—
628C22—21no—
629C22—22no—
630C22—22no—
631C22—21no—
632C22—21no—
633C22—22no—
634C22—21no—
635C22—21no—
636C22—22no—
637C22—22no—
638C22—21no—
639C22—21no—
640C22—22no—
641C22—22no—
642C22—21no—
643C22—21no—
644C22—22no—
645C22—22no—
646C22—21no—
647C22—21no—
648C22—22no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—41no—
654C22—42no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—42no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—84no—
663C22—84no—
664C22—84no—
665C22—84no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22C2×#38011yescharacteristic
671C22#229311yescharacteristic
672C22#231911yescharacteristic
673C22D4×C22⋊C411yescharacteristic
674C22#234111yescharacteristic
675C22D4×C4⋊C411yescharacteristic
676C22#234911yescharacteristic
677C22#242712yes—
678C22#242812yes—
679C22#245212yes—
680C22#245412yes—
681C22C22×#9911yescharacteristic
682C22C2×#41011yescharacteristic
683C22#288311yescharacteristic
684C22#288411yescharacteristic
685C4—24no—
686C4—24no—
687C4—22no—
688C4—44no—
689C4—44no—
690C4—42no—
691C4—42no—
692C4—42no—
693C4—84no—
694C4—84no—
695C4—81no—
696C4—81no—
697C4—164no—
698C4#179914yescomplemented
699C2—22no—
700C2—22no—
701C2—21no—
702C2—21no—
703C2—21no—
704C2—21no—
705C2—22no—
706C2—22no—
707C2—84no—
708C2#1392011yescharacteristic
709C2C2×#179911yescharacteristic
710C2#2063411yescharacteristic
711C2D4×#9911yescharacteristic
712C2#2240611yescharacteristic
713C2#2253411yescharacteristic
714C2#2253511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1799C2semidirect
4. #13907C2semidirect
5. #11043C2semidirect
6. #10846C2semidirect
7. #10761C2semidirect
8. C4×#342C2semidirect
10. C22×C4×D4C4semidirect
18. C2×#390C4semidirect
20. C2×#389C4semidirect
21. C2×#388C4semidirect
23. C2×#385C4semidirect
26. C2×C4×C4⋊C4C4semidirect
123. #342D4semidirect
139. C22×D4C4⋊C4semidirect
210. C2×C4⋊C4C4⋊C4semidirect
243. C2×C22⋊C4C4⋊C4semidirect
434. C4⋊C4#99semidirect
698. C4#1799semidirect

Maximal subgroups

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

subgroupindex
C2×#23612
C2×#19642
C2×#17992
#139072
#110432
#108462
#107612
C4×#3422