Finite Group Databasebeta

← #216168

Subgroups — #216168

Order 512.

1976 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2297C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic, complemented
3C2×#1797C211yesmaximal, characteristic, complemented
4#9878C212yesmaximal, complemented
5#9649C212yesmaximal, complemented
6#9079C214yesmaximal, complemented
7#8557C214yesmaximal, complemented
8C23×#41C411yescharacteristic
9C22×#117C411yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#431—21no—
12C22×C4⋊C8—21no—
13C22×C22⋊C8C2212yes—
14#2297—21no—
15#2297—21no—
16#2297—21no—
17#2297—21no—
18#2297—21no—
19#2297—21no—
20#2297—21no—
21#2297—21no—
22C2×#391C411yescharacteristic
23C2×#391C2211yescharacteristic
24C2×#390C411yescharacteristic
25C2×#390C2211yescharacteristic
26C2×#386C412yes—
27C2×#386C2212yes—
28C2×C4×C22⋊C4C412yes—
29C2×C4×C22⋊C4C2212yes—
30#1817—21no—
31#1797—21no—
32#1797—21no—
33#1797—21no—
34#1797C2212yes—
35#1797C2212yes—
36#1797C2212yes—
37#1779C2212yes—
38C4×C22⋊C8—21no—
39C4×C22⋊C8C2212yes—
40C2×#336—21no—
41C2×#323C2218yes—
42#1280C2218yes—
43C22×#41—21no—
44C22×#41—21no—
45C22×#41—21no—
46C22×#41—21no—
47C22×#41—24no—
48C22×#41—44no—
49C22×#41—44no—
50C22×#41C818yescomplemented
51C23×Q8C2×C411yescharacteristic
52C23×D4C2×C412yes—
53C23×D4C2×C411yescharacteristic
54C24×C4C2×C411yescharacteristic
55C24×C4C2×C412yes—
56C24×C4C2×C411yescharacteristic
57C24×C4C2311yescharacteristic
58C23×C8—22no—
59C2×#117—21no—
60C2×#117—21no—
61C2×#117—21no—
62C2×#117—21no—
63C2×#117—44no—
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C412yes—
67C22×C22⋊C4C2×C412yes—
68C22×C22⋊C4C2×C412yes—
69C22×C22⋊C4C2×C412yes—
70C22×C22⋊C4C2312yes—
71C22×C42—44no—
72C22×C42C2×C412yes—
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85#431—44no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—44no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—44no—
94C2×C22⋊C8C2314yes—
95C2×C22⋊C8C2314yes—
96C2×C4×C8—22no—
97#391—21no—
98#391—21no—
99#391—21no—
100#391—21no—
101#391—21no—
102#391—21no—
103#391—21no—
104#391—21no—
105#390—21no—
106#390—21no—
107#390—21no—
108#390—21no—
109#390—21no—
110#390—21no—
111#390—21no—
112#390—21no—
113#386—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#386—22no—
118#386—22no—
119#386—22no—
120#386—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C2×#95C2×C412yes—
130C2×#95C2×C412yes—
131C2×#95C2×C412yes—
132C2×#95C2312yes—
133#336—22no—
134#336—22no—
135#336—22no—
136#336—22no—
137#323—48no—
138C25C22×C411yescharacteristic
139C25C22×C412yes—
140C2×#41—28no—
141C2×#41—28no—
142C2×#41—28no—
143C2×#41—28no—
144C2×#41—432no—
145C2×#41—44no—
146C2×#41—44no—
147C2×#41—432no—
148C2×#41—44no—
149C2×#41—44no—
150C2×#41—44no—
151C2×#41—44no—
152C2×#41—44no—
153C2×#41—44no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—22no—
157C22×Q8—42no—
158C22×Q8—42no—
159C22×Q8C2×C814yescomplemented
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—24no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—22no—
166C22×D4—42no—
167C22×D4—42no—
168C22×D4—44no—
169C22×D4—44no—
170C22×D4C2×C818yes—
171C22×D4C2×C814yescomplemented
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22⋊C411yescharacteristic
190C23×C4C22⋊C411yescharacteristic
191C23×C4C22⋊C411yescharacteristic
192C23×C4C22⋊C411yescharacteristic
193C23×C4C22⋊C411yescharacteristic
194C23×C4C2×C814yes—
195C23×C4C2×C818yescomplemented
196C23×C4M1614yes—
197C23×C4C22×C411yescharacteristic
198C23×C4C22×C411yescharacteristic
199C23×C4C22×C411yescharacteristic
200C23×C4C22×C411yescharacteristic
201C23×C4C22×C412yes—
202C23×C4C22×C412yes—
203C23×C4C22×C411yescharacteristic
204C23×C4C22×C412yes—
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2411yescharacteristic
210C22×C8—24no—
211C22×C8—24no—
212C22×C8—44no—
213C22×C8—44no—
214C22×C8—44no—
215#117—44no—
216#117—44no—
217#117—44no—
218#117—44no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—44no—
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C4⋊C8—44no—
273C4⋊C8—44no—
274C4⋊C8—44no—
275C22⋊C8—24no—
276C22⋊C8—28no—
277C22⋊C8—24no—
278C22⋊C8—44no—
279C22⋊C8—44no—
280C22⋊C8—44no—
281C4×C8—44no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—24no—
297C24—24no—
298C24—41no—
299C24—41no—
300C24—42no—
301C24—42no—
302C24—42no—
303C24—42no—
304C24C2×C22⋊C411yescharacteristic
305C24C2×C22⋊C411yescharacteristic
306C24C22×C812yes—
307C24C22×C814yes—
308C24C2×M1612yes—
309C24C23×C411yescharacteristic
310#41—432no—
311#41—432no—
312#41—432no—
313#41—432no—
314C2×Q8—24no—
315C2×Q8—24no—
316C2×Q8—48no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—48no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×D4—28no—
323C2×D4—28no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—48no—
329C2×D4—48no—
330C2×D4—416no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—416no—
336C2×D4—42no—
337C2×D4—42no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—216no—
344C22×C4—216no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
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—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—416no—
400C22×C4—416no—
401C22×C4—416no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4C22⋊C814yes—
411C22×C4C22⋊C814yes—
412C22×C4C22⋊C814yes—
413C22×C4C22⋊C814yes—
414C22×C4#9914yes—
415C22×C4#10014yes—
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C22×C812yes—
427C22×C4C22×C812yes—
428C22×C4C22×C814yes—
429C22×C4C2×M1612yes—
430C22×C4C2×M1612yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4H(5,𝔽2)12yes—
434C22×C42−1+412yes—
435C2×C8—44no—
436C2×C8—44no—
437C2×C8—44no—
438C2×C8—44no—
439C2×C8—44no—
440C2×C8—44no—
441C2×C8—88no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—41no—
466C42—41no—
467C42—41no—
468C42—41no—
469C42—41no—
470C42—41no—
471C42—41no—
472C42—41no—
473C42—44no—
474C42—44no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—28no—
495C23—42no—
496C23—42no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—84no—
513C23—84no—
514C23—84no—
515C23—84no—
516C23C2×C22⋊C812yes—
517C23C2×C22⋊C812yes—
518C23C2×#9912yes—
519C23C2×#10012yes—
520C23C22×C22⋊C411yescharacteristic
521C23#52012yes—
522C23C23×C811yescharacteristic
523C23C22×M1611yescharacteristic
524C23#56812yes—
525Q8—48no—
526Q8—48no—
527D4—48no—
528D4—416no—
529D4—416no—
530D4—48no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—21no—
553C2×C4—24no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—416no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—416no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4#323116yescomplemented
593C2×C4C2×C22⋊C812yes—
594C2×C4C2×C22⋊C812yes—
595C2×C4C2×C22⋊C812yes—
596C2×C4C2×C22⋊C812yes—
597C2×C4C2×#9912yes—
598C2×C4C2×#9912yes—
599C2×C4#41012yes—
600C2×C4C2×#10012yes—
601C2×C4C2×#10012yes—
602C2×C4#41312yes—
603C2×C4#53511yescharacteristic
604C2×C4#53611yescharacteristic
605C2×C4#53712yes—
606C8—84no—
607C8—84no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—48no—
626C22—48no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—88no—
635C22—88no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22C2×#32314yes—
641C22#231111yescharacteristic
642C22C22×C22⋊C811yescharacteristic
643C22#287911yescharacteristic
644C22C22×#9911yescharacteristic
645C22C2×#41011yescharacteristic
646C22C22×#10011yescharacteristic
647C22C2×#41311yescharacteristic
648C22#298812yes—
649C22#298912yes—
650C4—21no—
651C4—22no—
652C4—21no—
653C4—22no—
654C4—22no—
655C4—22no—
656C4—22no—
657C4—41no—
658C4—44no—
659C4—41no—
660C4—41no—
661C4—41no—
662C4—82no—
663C4—82no—
664C4C2×#32314yes—
665C4C2×#32314yes—
666C2—22no—
667C2—22no—
668C2—24no—
669C2—42no—
670C2—84no—
671C2C22×#32311yescharacteristic
672C2#1734911yescharacteristic
673C2#1740011yescharacteristic
674C2#1740311yescharacteristic
675C2#2184711yescharacteristic
676C2#2187411yescharacteristic
677C2#2187811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC23×#41C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20714C2semidirect
3. C2×#1797C2semidirect
4. #9878C2semidirect
5. #9649C2semidirect
6. #9079C2semidirect
7. #8557C2semidirect
50. C22×#41C8semidirect
159. C22×Q8C2×C8semidirect
171. C22×D4C2×C8semidirect
195. C23×C4C2×C8semidirect
592. C2×C4#323semidirect

Maximal subgroups

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

subgroupindex
C2×#22972
#207142
C2×#17972
#98782
#96492
#90792
#85572