Finite Group Databasebeta

← #367008

Subgroups — #367008

Order 512.

1802 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2C2×#1981C211yesmaximal, characteristic, complemented
3C2×#1959C212yesmaximal
4C2×#1956C211yesmaximal, characteristic, complemented
5C2×#1802C211yesmaximal, characteristic
6C2×#1770C211yesmaximal, characteristic, complemented
7C2×C4×#102C211yesmaximal, characteristic
8#13910C211yesmaximal, characteristic, complemented
9#11050C212yesmaximal, complemented
10#11009C212yesmaximal
11#10928C212yesmaximal, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124C2212yes—
14C22×C4×D4C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#102C2211yescharacteristic
21C22×#102C2212yes—
22C22×#102C2211yescharacteristic
23C22×#97C2212yes—
24C22×C4×C8C2211yescharacteristic
25#2547—24no—
26#2547C418yescomplemented
27C2×#397—22no—
28C2×#391C2212yes—
29C2×#389—22no—
30C2×#384C2211yescharacteristic
31C2×#384C2212yes—
32C2×#380—22no—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1981—24no—
36#1959—28no—
37#1956—24no—
38#1802—24no—
39#1770—24no—
40C4×#102—24no—
41C2×#340C2211yescharacteristic
42C2×#340C2212yes—
43C2×#340C2211yescharacteristic
44C2×#336C2212yes—
45#1444—22no—
46C4⋊#95C2212yes—
47C23×Q8C2312yes—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C23×C8C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8C2×C414yes—
59C2×#124—24no—
60C2×#124—24no—
61C2×#124—24no—
62C2×#124—42no—
63C2×#124—42no—
64C2×#124C2×C418yes—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4C2×C414yes—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4D412yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—21no—
82C22×C22⋊C4—24no—
83C22×C42—21no—
84C22×C42C2312yes—
85C22×C42C2311yescharacteristic
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8D414yes—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—42no—
98C2×#103—42no—
99C2×#103—42no—
100C2×#103—42no—
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—22no—
105C2×#102—22no—
106C2×#102—24no—
107C2×#102—24no—
108C2×#102—42no—
109C2×#102D414yes—
110C2×#97—24no—
111C2×#97—24no—
112C2×#97—42no—
113C2×C4×C8—22no—
114C2×C4×C8—22no—
115C2×C4×C8D414yes—
116#397—216no—
117#391—24no—
118#391C2×C418yes—
119#389—48no—
120#384—24no—
121#384—22no—
122#384—22no—
123#384—42no—
124#380—216no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4C2×C414yes—
128C4×C4⋊C4D414yes—
129C2×#95—24no—
130C2×#95—24no—
131C2×#95—21no—
132C2×#95D412yes—
133#340—24no—
134#340—22no—
135#340—22no—
136#336—24no—
137C25—21no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8C22×C414yes—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4C22×C412yes—
164C22×D4C2×D412yes—
165C22×D4#4112yes—
166C23×C4—22no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—24no—
171C23×C4—22no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—44no—
175C23×C4—44no—
176C23×C4C2×D414yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2411yescharacteristic
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8C2×D412yes—
197C22×C8C2×D414yes—
198C22×C8C2×D412yes—
199C22×C8C2×D412yes—
200C22×C8#4112yes—
201C4⋊Q8—28no—
202C4⋊Q8—28no—
203C4⋊Q8—44no—
204C4⋊Q8—44no—
205#124—216no—
206#124—48no—
207#124—48no—
208#124—48no—
209C4×D4—28no—
210C4×D4—28no—
211C4×D4—44no—
212C4×D4—44no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
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—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—48no—
243C2×C4⋊C4—48no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—82no—
248C2×C4⋊C4—84no—
249C2×C4⋊C4C22⋊C418yes—
250C2×C4⋊C4C22×C412yes—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4#4112yes—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—216no—
255C2×C22⋊C4—48no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—42no—
264C2×C42—44no—
265C2×C42C22×C414yes—
266C2×C42C22×C412yes—
267C2×C42C2×D414yes—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42#4112yes—
272C2×C42#4112yes—
273C4⋊C8—28no—
274#103—48no—
275#103—48no—
276#103—48no—
277#103—48no—
278#102—28no—
279#102—28no—
280#102—48no—
281#97—48no—
282C4×C8—28no—
283#95—216no—
284#95—48no—
285#95—42no—
286#95C22⋊C418yes—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—41no—
291C24—41no—
292C24—41no—
293C24—41no—
294C24—42no—
295C24—42no—
296C24—42no—
297C24—42no—
298C24C22×D411yescharacteristic
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4C4×D414yes—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—41no—
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—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—48no—
386C22×C4—416no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—88no—
393C22×C4—84no—
394C22×C4—84no—
395C22×C4C2×C22⋊C418yes—
396C22×C4C2×C22⋊C412yes—
397C22×C4C2×C22⋊C412yes—
398C22×C4#12412yes—
399C22×C4#12412yes—
400C22×C4#12712yes—
401C22×C4#12712yes—
402C22×C4AGL(1,ℤ/8)14yes—
403C22×C4#13714yes—
404C22×C4C23×C411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4H(5,𝔽2)11yescharacteristic
415C22×C4H(5,𝔽2)11yescharacteristic
416C22×C4H(5,𝔽2)11yescharacteristic
417C22×C42−1+411yescharacteristic
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—24no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—42no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8#12114yes—
439C4⋊C4—216no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—84no—
459C4⋊C4—816no—
460C4⋊C4—84no—
461C22⋊C4—432no—
462C22⋊C4—48no—
463C42—24no—
464C42—24no—
465C42—24no—
466C42—24no—
467C42—84no—
468C42C4×D414yes—
469C42C4×D414yes—
470C42#12114yes—
471C42#12114yes—
472C23—24no—
473C23—44no—
474C23—44no—
475C23—42no—
476C23—44no—
477C23—44no—
478C23—42no—
479C23—42no—
480C23—44no—
481C23—44no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—82no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23—82no—
490C23—82no—
491C23—81no—
492C23—81no—
493C23—81no—
494C23—81no—
495C23C22×C22⋊C411yescharacteristic
496C23C2×#12411yescharacteristic
497C23C2×#12711yescharacteristic
498C23D4211yescharacteristic
499C23#54611yescharacteristic
500C23#54811yescharacteristic
501C23#55011yescharacteristic
502C23C2×AGL(1,ℤ/8)12yes—
503C23C2×#13712yes—
504C23#57712yes—
505C23#57812yes—
506Q8—44no—
507Q8—44no—
508Q8—44no—
509Q8—44no—
510Q8—82no—
511Q8—82no—
512Q8—82no—
513Q8—82no—
514D4—24no—
515D4—24no—
516D4—42no—
517D4—42no—
518D4—42no—
519D4—42no—
520D4—42no—
521D4—42no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—816no—
556C2×C4—84no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—84no—
560C2×C4—82no—
561C2×C4#39014yes—
562C2×C4#39014yes—
563C2×C4#41718yes—
564C2×C4#48918yes—
565C2×C4#49614yes—
566C2×C4#49714yes—
567C2×C4C2×C4×D411yescharacteristic
568C2×C4C2×C4×D411yescharacteristic
569C2×C4C2×C4×D411yescharacteristic
570C2×C4C4×#4111yescharacteristic
571C2×C4#51811yescharacteristic
572C2×C4#52011yescharacteristic
573C2×C4C2×#12111yescharacteristic
574C2×C4C2×#12111yescharacteristic
575C2×C4C2×#12111yescharacteristic
576C2×C4#52511yescharacteristic
577C2×C4#52911yescharacteristic
578C2×C4#53211yescharacteristic
579C2×C4#53511yescharacteristic
580C2×C4#53711yescharacteristic
581C2×C4#54011yescharacteristic
582C2×C4#54211yescharacteristic
583C2×C4D4211yescharacteristic
584C2×C4D4211yescharacteristic
585C2×C4#54611yescharacteristic
586C2×C4#54611yescharacteristic
587C2×C4#54711yescharacteristic
588C2×C4#54811yescharacteristic
589C2×C4#54811yescharacteristic
590C2×C4#55011yescharacteristic
591C2×C4#55011yescharacteristic
592C2×C4#55311yescharacteristic
593C2×C4#55311yescharacteristic
594C2×C4#55511yescharacteristic
595C8—24no—
596C8—42no—
597C8—44no—
598C22—48no—
599C22—44no—
600C22—82no—
601C22—84no—
602C22—82no—
603C22—82no—
604C22—84no—
605C22—84no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22C2×#39011yescharacteristic
610C22D4×C22⋊C411yescharacteristic
611C22#234111yescharacteristic
612C22#249011yescharacteristic
613C22#249111yescharacteristic
614C22#254011yescharacteristic
615C22#254111yescharacteristic
616C22C2×#41712yes—
617C22#289812yes—
618C22C2×#48912yes—
619C22#313712yes—
620C22C2×#49611yescharacteristic
621C22C2×#49711yescharacteristic
622C22#315411yescharacteristic
623C22#315611yescharacteristic
624C22#327212yes—
625C22#327612yes—
626C22#333012yes—
627C22#333112yes—
628C22#336712yes—
629C22#336912yes—
630C22#339712yes—
631C22#339812yes—
632C4—24no—
633C4—48no—
634C4—82no—
635C4—84no—
636C4—82no—
637C4—82no—
638C4—162no—
639C4#195918yescomplemented
640C4C4×#12111yescharacteristic
641C4#232411yescharacteristic
642C4#236111yescharacteristic
643C4#236311yescharacteristic
644C4#241411yescharacteristic
645C4#241611yescharacteristic
646C4#242611yescharacteristic
647C4#242811yescharacteristic
648C2—84no—
649C2#1399811yescharacteristic
650C2C2×#195911yescharacteristic
651C2#2120511yescharacteristic
652C2#2243112yes—
653C2#2254412yes—
654C2#2775612yes—
655C2#2777612yes—
656C2#2841411yescharacteristic
657C2#2841911yescharacteristic
658C2#2847111yescharacteristic
659C2#2847211yescharacteristic
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×#1981C2semidirect
4. C2×#1956C2semidirect
6. C2×#1770C2semidirect
8. #13910C2semidirect
9. #11050C2semidirect
11. #10928C2semidirect
26. #2547C4semidirect
639. C4#1959semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#19812
C2×#19592
C2×#19562
C2×#18022
C2×#17702
C2×C4×#1022
#139102
#110502
#110092
#109282