Finite Group Databasebeta

← #380236

Subgroups — #380236

Order 512.

2158 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2311C211yesmaximal, characteristic
2C2×#1854C211yesmaximal, characteristic, complemented
3C2×#1853C211yesmaximal, characteristic, complemented
4C2×#1780C211yesmaximal, characteristic
5#14337C211yesmaximal, characteristic
6#11096C212yesmaximal, complemented
7#11095C212yesmaximal, complemented
8#11095C212yesmaximal, complemented
9#11095C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic, complemented
11#10730C211yesmaximal, characteristic, complemented
12C23×#41C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#106C2212yes—
17C22×#106C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2212yes—
21C22×C22⋊C8C2212yes—
22#2311—24no—
23#2311—24no—
24C2×#391—21no—
25C2×#390—21no—
26C2×#386—22no—
27C2×#380C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C2×#380C2212yes—
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33#1854—24no—
34#1853—24no—
35#1780C418yescomplemented
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43#1446—21no—
44#1438—22no—
45#1438—21no—
46C22×#41—22no—
47C22×#41—22no—
48C22×#41—22no—
49C22×#41—22no—
50C22×#41—42no—
51C22×#41—42no—
52C22×#41—42no—
53C22×#41—42no—
54C22×#41C2×C414yes—
55C22×#41Q814yes—
56C23×Q8D411yescharacteristic
57C23×D4D412yes—
58C23×D4D411yescharacteristic
59C24×C4—21no—
60C24×C4C2312yes—
61C24×C4C2311yescharacteristic
62C23×C8C2312yes—
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C2×#117—22no—
66C2×#117—42no—
67C2×#117—42no—
68C2×#117D414yes—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—42no—
72C22×C4⋊C4—42no—
73C22×C4⋊C4D414yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C42—21no—
88C2×#107—22no—
89C2×#107—42no—
90C2×#106—24no—
91C2×#106—22no—
92C2×#106—42no—
93C2×#106—42no—
94C2×#106—42no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—42no—
100C2×C22⋊C8—42no—
101C2×C22⋊C8—42no—
102C2×C22⋊C8—42no—
103C2×C22⋊C8C2×C414yes—
104C2×C22⋊C8C2×C414yes—
105C2×C22⋊C8C2×C418yescomplemented
106C2×C22⋊C8C2×C418yescomplemented
107#391—28no—
108#390—28no—
109#386—216no—
110#380—24no—
111#380—24no—
112#380—24no—
113#380—24no—
114#380—22no—
115#380—22no—
116#380—22no—
117#380—22no—
118#380—42no—
119#380—42no—
120#380—42no—
121#380—42no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—21no—
127C2×#95—21no—
128#340—24no—
129#340—22no—
130#340—22no—
131#340—42no—
132#340—42no—
133#340—42no—
134#340—42no—
135C25C2×D412yes—
136C25C2×D411yescharacteristic
137C2×#41—28no—
138C2×#41—28no—
139C2×#41—28no—
140C2×#41—28no—
141C2×#41—48no—
142C2×#41—48no—
143C2×#41—48no—
144C2×#41—48no—
145C2×#41—48no—
146C2×#41—44no—
147C2×#41—44no—
148C2×#41—44no—
149C2×#41—44no—
150C2×#41—48no—
151C2×#41—48no—
152C2×#41—48no—
153C2×#41—48no—
154C2×#41—48no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—42no—
158C22×Q8—42no—
159C22×Q8C4⋊C414yes—
160C22×D4—24no—
161C22×D4—24no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—42no—
165C22×D4—42no—
166C22×D4—42no—
167C22×D4—42no—
168C22×D4—42no—
169C22×D4—42no—
170C22×D4C4⋊C418yes—
171C22×D4C4⋊C414yes—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—22no—
195C23×C4—44no—
196C23×C4—81no—
197C23×C4—82no—
198C23×C4—81no—
199C23×C4C22×C414yes—
200C23×C4C22×C412yes—
201C23×C4C2×D412yes—
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D412yes—
204C23×C4C2×D412yes—
205C23×C4C2×Q814yes—
206C23×C4C2×Q812yes—
207C23×C4#4112yes—
208C23×C4#4112yes—
209C23×C4C2411yescharacteristic
210C22×C8—22no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—22no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8C22×C414yes—
225C22×C8C22×C412yes—
226C22×C8C22×C412yes—
227#117—44no—
228#117—44no—
229#117—48no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—48no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—82no—
272C2×C22⋊C4—82no—
273C2×C22⋊C4—82no—
274C2×C22⋊C4—82no—
275C2×C22⋊C4—81no—
276C2×C22⋊C4—81no—
277C2×C22⋊C4—82no—
278C2×C22⋊C4—82no—
279C2×C22⋊C4—82no—
280C2×C22⋊C4—82no—
281C2×C22⋊C4—82no—
282C2×C42—24no—
283C2×C42—44no—
284#107—44no—
285#106—48no—
286#106—44no—
287C22⋊C8—28no—
288C22⋊C8—28no—
289C22⋊C8—48no—
290C22⋊C8—48no—
291#95—28no—
292#95—28no—
293#95—44no—
294#95—44no—
295#95—42no—
296#95—42no—
297C24—22no—
298C24—22no—
299C24—21no—
300C24—21no—
301C24—22no—
302C24—21no—
303C24—22no—
304C24—21no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24C2×C4⋊C414yes—
320C24C2×C4⋊C412yes—
321C24#12012yes—
322C24#12312yes—
323C24C22×D411yescharacteristic
324#41—432no—
325#41—416no—
326#41—416no—
327#41—432no—
328#41—416no—
329#41—416no—
330C2×Q8—28no—
331C2×Q8—48no—
332C2×Q8—44no—
333C2×Q8—44no—
334C2×Q8—48no—
335C2×D4—216no—
336C2×D4—28no—
337C2×D4—48no—
338C2×D4—48no—
339C2×D4—48no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—48no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—48no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—41no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—41no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4—88no—
435C22×C4—88no—
436C22×C4C2×C4⋊C414yes—
437C22×C4C2×C4⋊C412yes—
438C22×C4C4×D414yes—
439C22×C4C4×D414yes—
440C22×C4#12012yes—
441C22×C4C22⋊Q814yes—
442C22×C4C22⋊Q814yes—
443C22×C4#12312yes—
444C22×C4C23×C411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×Q811yescharacteristic
448C22×C4C2×#4111yescharacteristic
449C22×C4C2×#4111yescharacteristic
450C22×C4H(5,𝔽2)12yes—
451C22×C4H(5,𝔽2)12yes—
452C22×C4H(5,𝔽2)12yes—
453C22×C4H(5,𝔽2)12yes—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—42no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—42no—
466C2×C8—41no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—42no—
470C2×C8—42no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—48no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C22⋊C4—48no—
493C22⋊C4—88no—
494C22⋊C4—88no—
495C22⋊C4—84no—
496C22⋊C4—84no—
497C22⋊C4—88no—
498C22⋊C4—88no—
499C42—48no—
500C23—24no—
501C23—22no—
502C23—22no—
503C23—24no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—24no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23#38514yes—
558C23C22×C4⋊C411yescharacteristic
559C23C2×#12011yescharacteristic
560C23C2×#12311yescharacteristic
561C23#53412yes—
562C23#53412yes—
563C23#57611yescharacteristic
564C23#57711yescharacteristic
565C23#57711yescharacteristic
566C23#57712yes—
567C23#57712yes—
568C23#57811yescharacteristic
569Q8—416no—
570D4—416no—
571D4—432no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—28no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—816no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4—81no—
629C2×C4—81no—
630C2×C4—81no—
631C2×C4—81no—
632C2×C4#38514yes—
633C2×C4C2×C4×D411yescharacteristic
634C2×C4C2×C4×D411yescharacteristic
635C2×C4#51812yes—
636C2×C4#51812yes—
637C2×C4C2×C22⋊Q811yescharacteristic
638C2×C4C2×C22⋊Q811yescharacteristic
639C2×C4#53412yes—
640C2×C4#53411yescharacteristic
641C2×C4#53411yescharacteristic
642C2×C4#53511yescharacteristic
643C2×C4#53511yescharacteristic
644C2×C4#53512yes—
645C2×C4#53812yes—
646C2×C4#53812yes—
647C2×C4#53812yes—
648C2×C4#53812yes—
649C2×C4#54312yes—
650C2×C4#54312yes—
651C8—44no—
652C8—42no—
653C8—42no—
654C22—24no—
655C22—24no—
656C22—24no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22C2×#38511yescharacteristic
681C22#231512yes—
682C22#261311yescharacteristic
683C22#261311yescharacteristic
684C22#262212yes—
685C22#291212yes—
686C22#291211yescharacteristic
687C22#291311yescharacteristic
688C22#300911yescharacteristic
689C22#301011yescharacteristic
690C22#301211yescharacteristic
691C22#301311yescharacteristic
692C22#309611yescharacteristic
693C22#309711yescharacteristic
694C22#309912yes—
695C22#319812yes—
696C22#319812yes—
697C22#320012yes—
698C22#320012yes—
699C22#320112yes—
700C22#320112yes—
701C22#320412yes—
702C22#320412yes—
703C4—24no—
704C4—44no—
705C4—84no—
706C4—82no—
707C4—82no—
708C4—161no—
709C4—161no—
710C4—162no—
711C4#232112yes—
712C4#232111yescharacteristic
713C4#232111yescharacteristic
714C4#262612yes—
715C4#262611yescharacteristic
716C4#262611yescharacteristic
717C2—44no—
718C2—84no—
719C2#1426211yescharacteristic
720C2#2084811yescharacteristic
721C2#2084911yescharacteristic
722C2#2194512yes—
723C2#2194612yes—
724C2#2960311yescharacteristic
725C2#2960411yescharacteristic
726C2#2960611yescharacteristic
727C2#2960911yescharacteristic
728C2#2989912yes—
729C2#2990012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×#41 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1854C2semidirect
3. C2×#1853C2semidirect
6. #11096C2semidirect
7. #11095C2semidirect
8. #11095C2semidirect
9. #11095C2semidirect
10. #10731C2semidirect
11. #10730C2semidirect
35. #1780C4semidirect
105. C2×C22⋊C8C2×C4semidirect
106. C2×C22⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23112
C2×#18542
C2×#18532
C2×#17802
#143372
#110962
#110952
#107312
#107302