Finite Group Databasebeta

← #110074

Subgroups — #110074

Order 512.

1206 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1869C211yesmaximal, characteristic
2C2×C4×#99C211yesmaximal, characteristic
3C2×C8×C22⋊C4C211yesmaximal, characteristic, complemented
4C2×#1459C211yesmaximal, characteristic
5#10714C211yesmaximal, characteristic, complemented
6C2×#1282C211yesmaximal, characteristic, complemented
7C2×#1282C211yesmaximal, characteristic, complemented
8C24×C8C2211yescharacteristic
9C22×C4×D4C2211yescharacteristic
10C22×#99C412yes—
11C22×C22⋊C8C412yescomplemented
12C22×C22⋊C8C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15C22×C4×C8—21no—
16C2×C4×C22⋊C4C2211yescharacteristic
17C2×C4×C22⋊C4C2211yescharacteristic
18#1869—21no—
19#1869—22no—
20#1869—21no—
21C4×#99—22no—
22C4×#99—22no—
23C8×C22⋊C4—22no—
24C8×C22⋊C4—22no—
25C2×#342C412yes—
26C2×#336—21no—
27C2×#336—22no—
28C2×#336C412yescomplemented
29#1459—22no—
30#1459—22no—
31C2×#324C412yescomplemented
32C2×#323C412yescomplemented
33#1282—22no—
34#1282—22no—
35C23×D4C2×C411yescharacteristic
36C24×C4C2×C411yescharacteristic
37C24×C4C2311yescharacteristic
38C23×C8—21no—
39C23×C8—21no—
40C23×C8—21no—
41C23×C8—21no—
42C23×C8—21no—
43C23×C8—21no—
44C23×C8—44no—
45C23×C8C2×C412yes—
46C23×C8C2×C412yes—
47C23×C8D411yescharacteristic
48C23×C8D412yes—
49C23×C8D411yescharacteristic
50C2×C4×D4—21no—
51C2×C4×D4—21no—
52C2×C4×D4—21no—
53C2×C4×D4—21no—
54C2×C4×D4—21no—
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C2×C4×D4—21no—
60C2×C4×D4—21no—
61C2×C4×D4—21no—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C412yes—
66C22×C22⋊C4C2×C412yes—
67C22×C42—21no—
68C22×C42—21no—
69C22×C42C2×C411yescharacteristic
70C2×#99—24no—
71C2×#99—24no—
72C2×#99C818yescomplemented
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—44no—
90C2×C22⋊C8C2×C412yes—
91C2×C22⋊C8C2×C412yes—
92C2×C22⋊C8D412yes—
93C2×C22⋊C8D412yes—
94C2×C4×C8—24no—
95C2×C4×C8—44no—
96C4×C22⋊C4—21no—
97C4×C22⋊C4—21no—
98C4×C22⋊C4—21no—
99C4×C22⋊C4—21no—
100C4×C22⋊C4—21no—
101C4×C22⋊C4—21no—
102C4×C22⋊C4D412yes—
103C4×C22⋊C4D412yes—
104C2×#95—21no—
105C2×#95—21no—
106#342—24no—
107#336—24no—
108#336—22no—
109#336—22no—
110#336—44no—
111#324—24no—
112#323—24no—
113C25C4211yescharacteristic
114C25C22×C411yescharacteristic
115C22×D4—21no—
116C22×D4—21no—
117C22×D4—21no—
118C22×D4—21no—
119C22×D4—21no—
120C22×D4—21no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4C2×C814yes—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4C4211yescharacteristic
138C23×C4C4211yescharacteristic
139C23×C4C4211yescharacteristic
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4#4111yescharacteristic
149C23×C4#4111yescharacteristic
150C23×C4#4111yescharacteristic
151C23×C4#4111yescharacteristic
152C23×C4#4111yescharacteristic
153C23×C4#4111yescharacteristic
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—24no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—24no—
167C22×C8—24no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—42no—
173C22×C8—42no—
174C22×C8—42no—
175C22×C8—44no—
176C22×C8—44no—
177C22×C8—44no—
178C22×C8—44no—
179C22×C8—42no—
180C22×C8—42no—
181C22×C8—42no—
182C22×C8—42no—
183C22×C8—41no—
184C22×C8—41no—
185C22×C8—42no—
186C22×C8—42no—
187C22×C8C22⋊C412yes—
188C22×C8C22⋊C412yes—
189C4×D4—41no—
190C4×D4—41no—
191C4×D4—41no—
192C4×D4—41no—
193C4×D4—41no—
194C4×D4—41no—
195C4×D4—41no—
196C4×D4—41no—
197C4×D4—41no—
198C4×D4—41no—
199C4×D4—41no—
200C4×D4—41no—
201C4×D4—41no—
202C4×D4—41no—
203C4×D4—41no—
204C4×D4—41no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4C22⋊C414yes—
232C2×C22⋊C4C2×C814yes—
233C2×C22⋊C4C2×C814yes—
234C2×C22⋊C4M1614yes—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—41no—
248C2×C42—41no—
249#99—216no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253C22⋊C8—22no—
254C22⋊C8—22no—
255C22⋊C8—44no—
256C22⋊C8—41no—
257C22⋊C8—41no—
258C22⋊C8—41no—
259C22⋊C8—41no—
260C22⋊C8—41no—
261C22⋊C8—41no—
262C22⋊C8—44no—
263C22⋊C8C22⋊C414yes—
264C4×C8—48no—
265#95—22no—
266#95—22no—
267#95—41no—
268#95—41no—
269C24—21no—
270C24—21no—
271C24—21no—
272C24—21no—
273C24—21no—
274C24—21no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24C4×C812yescomplemented
280C24C2×C4211yescharacteristic
281C24C2×C22⋊C411yescharacteristic
282C24#11711yescharacteristic
283C24C4×D411yescharacteristic
284C24C4×D411yescharacteristic
285C24C4×D411yescharacteristic
286C24C4×D411yescharacteristic
287C24C22×C812yes—
288C24C2×M1612yes—
289C24#13112yes—
290C24#13112yes—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—24no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—24no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—81no—
360C22×C4—81no—
361C22×C4—81no—
362C22×C4—81no—
363C22×C4—82no—
364C22×C4—82no—
365C22×C4C4×C812yescomplemented
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4#11711yescharacteristic
369C22×C4#11711yescharacteristic
370C22×C4C4×D411yescharacteristic
371C22×C4C4×D411yescharacteristic
372C22×C4C4×D411yescharacteristic
373C22×C4C4×D411yescharacteristic
374C22×C4C4×D411yescharacteristic
375C22×C4C4×D411yescharacteristic
376C22×C4C4×D411yescharacteristic
377C22×C4C4×D411yescharacteristic
378C22×C4#12011yescharacteristic
379C22×C4#12111yescharacteristic
380C22×C4#12111yescharacteristic
381C22×C4#12111yescharacteristic
382C22×C4#12111yescharacteristic
383C22×C4#12111yescharacteristic
384C22×C4#12111yescharacteristic
385C22×C4#12311yescharacteristic
386C22×C4#12311yescharacteristic
387C22×C4#12311yescharacteristic
388C22×C4#12411yescharacteristic
389C22×C4#12411yescharacteristic
390C22×C4#12411yescharacteristic
391C22×C4#12611yescharacteristic
392C22×C4#12611yescharacteristic
393C22×C4#12711yescharacteristic
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—24no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—44no—
403C2×C8—44no—
404C2×C8—44no—
405C2×C8—44no—
406C2×C8—42no—
407C2×C8—44no—
408C2×C8—42no—
409C2×C8—44no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—44no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—81no—
418C2×C8—81no—
419C2×C8—82no—
420C2×C8—82no—
421C2×C8—82no—
422C2×C8#9914yescomplemented
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4C22⋊C818yescomplemented
443C42—42no—
444C42—42no—
445C42—44no—
446C42—41no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—81no—
451C42—81no—
452C42—82no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—42no—
481C23—42no—
482C23—81no—
483C23—81no—
484C23—81no—
485C23—81no—
486C23—82no—
487C23—82no—
488C23C4×C22⋊C411yescharacteristic
489C23C4×C22⋊C411yescharacteristic
490C23C4×C22⋊C411yescharacteristic
491C23#38611yescharacteristic
492C23#38811yescharacteristic
493C23#38811yescharacteristic
494C23#39011yescharacteristic
495C23C2×C4×C811yescharacteristic
496C23C4×M1611yescharacteristic
497C23#40511yescharacteristic
498C23#40511yescharacteristic
499C23C2×C22⋊C812yes—
500C23#40812yes—
501C23#43112yes—
502C23#43312yes—
503C23C8×D412yes—
504C23C8×D412yes—
505C23C8×D412yes—
506C23C8×D412yes—
507C23#43512yes—
508C23#43512yes—
509C23#43612yes—
510C23#43612yes—
511D4—44no—
512D4—44no—
513D4—44no—
514D4—44no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—28no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—48no—
555C2×C4—48no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—48no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—84no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—82no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4#38611yescharacteristic
579C2×C4#38611yescharacteristic
580C2×C4#38811yescharacteristic
581C2×C4#38811yescharacteristic
582C2×C4#38811yescharacteristic
583C2×C4#38811yescharacteristic
584C2×C4#39011yescharacteristic
585C2×C4#39011yescharacteristic
586C2×C4C2×#9912yes—
587C2×C4#41012yes—
588C8—24no—
589C8—44no—
590C8—42no—
591C8—82no—
592C8—82no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—24no—
606C22—24no—
607C22—22no—
608C22—22no—
609C22—24no—
610C22—24no—
611C22—22no—
612C22—22no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—44no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—81no—
628C22—81no—
629C22—82no—
630C22#143711yescharacteristic
631C22C4×C22⋊C811yescharacteristic
632C22#175011yescharacteristic
633C22C8×C22⋊C411yescharacteristic
634C22C8×C22⋊C411yescharacteristic
635C22#175311yescharacteristic
636C22#175311yescharacteristic
637C22C4×#9912yes—
638C22#175712yes—
639C22#186912yes—
640C22#192312yes—
641C22#192312yes—
642C22#195412yes—
643C4—21no—
644C4—21no—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—48no—
650C4—42no—
651C4—84no—
652C4—81no—
653C4—81no—
654C4#189811yescharacteristic
655C4#189911yescharacteristic
656C4#196412yes—
657C2—22no—
658C2—22no—
659C2—21no—
660C2—21no—
661C2—21no—
662C2—21no—
663C2—22no—
664C2—22no—
665C2—82no—
666C2C8×#9912yes—
667C2#816112yes—
668C2#1092111yescharacteristic
669C2#1096011yescharacteristic
670C2#1096111yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC23C2×C4×C8
Frattini, cosocleC24×C4C23
socleC23C4×C22⋊C4
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C8 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×C8×C22⋊C4C2semidirect
5. #10714C2semidirect
6. C2×#1282C2semidirect
7. C2×#1282C2semidirect
11. C22×C22⋊C8C4semidirect
28. C2×#336C4semidirect
31. C2×#324C4semidirect
32. C2×#323C4semidirect
72. C2×#99C8semidirect
279. C24C4×C8semidirect
365. C22×C4C4×C8semidirect
422. C2×C8#99semidirect
442. C22⋊C4C22⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×#18692
C2×C4×#992
C2×C8×C22⋊C42
C2×#14592
#107142
C2×#12822