Finite Group Databasebeta

← #404091

Subgroups — #404091

Order 512.

1642 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#432C211yesmaximal, characteristic
2C2×#2056C211yesmaximal, characteristic, complemented
3C2×#2001C211yesmaximal, characteristic
4C2×#1971C211yesmaximal, characteristic, complemented
5#14702C212yesmaximal
6#11231C211yesmaximal, characteristic, complemented
7#11095C211yesmaximal, characteristic
8#11077C212yesmaximal, complemented
9#10970C212yesmaximal
10#10928C212yesmaximal, complemented
11#10821C211yesmaximal, characteristic, complemented
12C23×Q16C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C23×C42C2211yescharacteristic
15C2×#432—24no—
16C2×#432—24no—
17C2×#432—24no—
18C22×#106C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25C22×C22⋊C8C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C22×#97C2211yescharacteristic
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#380C2212yes—
31#2056—24no—
32#2001—24no—
33#1971—24no—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2212yes—
39#1436—22no—
40#1436—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C22⋊#95—22no—
44C22⋊#95—22no—
45C23×Q8C2312yes—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C22×Q16—24no—
50C22×Q16—22no—
51C22×Q16—22no—
52C22×Q16—22no—
53C22×Q16—42no—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C2×C22⋊Q8—24no—
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—24no—
62C2×C22⋊Q8—42no—
63C2×C22⋊Q8—42no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C42—22no—
71C22×C42—22no—
72C22×C42—22no—
73C22×C42—22no—
74C22×C42—22no—
75C22×C42—22no—
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80#432—232no—
81C2×#106—22no—
82C2×#106—42no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
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—42no—
95C2×#103—42no—
96C2×#103—42no—
97C2×#103—42no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8D414yes—
104C2×#97—22no—
105C2×#97—22no—
106C2×#97—22no—
107C2×#97—22no—
108C2×#97—22no—
109C2×#97D414yes—
110#391—24no—
111#391—24no—
112#391—24no—
113#391—24no—
114#380—42no—
115#380—42no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125#340—22no—
126#340—22no—
127#340—24no—
128#340—42no—
129#340—42no—
130C25C2×D411yescharacteristic
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8#4114yes—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4C2×D411yescharacteristic
173C23×C4#4112yes—
174C23×C4C2411yescharacteristic
175C2×Q16—28no—
176C2×Q16—48no—
177C2×Q16—48no—
178C2×Q16—44no—
179C2×Q16—44no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8C2×D412yes—
195C22×C8C2×D412yes—
196C22×C8C2×D412yes—
197C22×C8C2×D412yes—
198C22×C8C2×D412yes—
199C22×C8C2×D412yes—
200C22⋊Q8—48no—
201C22⋊Q8—48no—
202C22⋊Q8—48no—
203C22⋊Q8—48no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4#4114yes—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C42—28no—
232C2×C42—28no—
233C2×C42—28no—
234C2×C42—28no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42#4112yes—
252C2×C42#4112yes—
253#106—44no—
254C4⋊C8—28no—
255C4⋊C8—28no—
256#103—48no—
257#103—48no—
258#103—48no—
259C22⋊C8—28no—
260C22⋊C8—28no—
261#97—28no—
262#97—28no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—44no—
272C24—21no—
273C24—21no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24#12112yes—
280C24#13714yes—
281C24C22×D411yescharacteristic
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—28no—
314C22×C4—28no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—48no—
342C22×C4—48no—
343C22×C4—48no—
344C22×C4—48no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—44no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
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#12112yes—
386C22×C4#12112yes—
387C22×C4#12112yes—
388C22×C4#12112yes—
389C22×C4#12112yes—
390C22×C4#13714yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4112yes—
398C22×C4C2×#4112yes—
399C22×C4H(5,𝔽2)12yes—
400C22×C4H(5,𝔽2)12yes—
401Q16—416no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8#12014yes—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—416no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C22⋊C4—416no—
436C22⋊C4—88no—
437C42—28no—
438C42—28no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C23—22no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—22no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—42no—
458C23—42no—
459C23—41no—
460C23—41no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—41no—
465C23—41no—
466C23—42no—
467C23#47014yes—
468C23C2×#12111yescharacteristic
469C23C2×#12111yescharacteristic
470C23C2×#12111yescharacteristic
471C23#52512yes—
472C23#53512yes—
473C23C2×#13712yes—
474C23C2×#13712yes—
475C23#57512yes—
476C23#57512yes—
477Q8—44no—
478Q8—44no—
479Q8—44no—
480Q8—44no—
481Q8—82no—
482Q8—82no—
483Q8—82no—
484Q8—82no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—416no—
498C2×C4—416no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4#47014yes—
546C2×C4C2×#12011yescharacteristic
547C2×C4#52511yescharacteristic
548C2×C4#52511yescharacteristic
549C2×C4#52511yescharacteristic
550C2×C4#52511yescharacteristic
551C2×C4#52912yes—
552C2×C4#53211yescharacteristic
553C2×C4#53511yescharacteristic
554C2×C4#53511yescharacteristic
555C2×C4#54012yes—
556C2×C4#54612yes—
557C2×C4#54612yes—
558C2×C4#54612yes—
559C2×C4#55012yes—
560C2×C4#55112yes—
561C2×C4#55112yes—
562C2×C4#55312yes—
563C2×C4#55512yes—
564C8—24no—
565C8—42no—
566C8—42no—
567C8—42no—
568C22—24no—
569C22—28no—
570C22—42no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—42no—
576C22—44no—
577C22#239211yescharacteristic
578C22#241112yes—
579C22#241412yes—
580C22#261911yescharacteristic
581C22#284711yescharacteristic
582C22C2×#47011yescharacteristic
583C22C2×#47011yescharacteristic
584C22#305411yescharacteristic
585C22#305411yescharacteristic
586C22#305612yes—
587C22#305612yes—
588C22#305612yes—
589C22#305612yes—
590C22#311112yes—
591C22#311312yes—
592C22#311612yes—
593C22#311712yes—
594C22#321512yes—
595C22#321612yes—
596C22#321812yes—
597C22#321912yes—
598C4—24no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—84no—
604C4—84no—
605C4—162no—
606C4—162no—
607C4#240411yescharacteristic
608C4#243111yescharacteristic
609C4#247112yes—
610C4#247612yes—
611C4#259811yescharacteristic
612C4#261011yescharacteristic
613C2—44no—
614C2#1488611yescharacteristic
615C2#2370912yes—
616C2#2450812yes—
617C2#2451112yes—
618C2#2465712yes—
619C2#2466012yes—
620C2#2981512yes—
621C2#3293711yescharacteristic
622C2#3293711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2056C2semidirect
4. C2×#1971C2semidirect
6. #11231C2semidirect
8. #11077C2semidirect
10. #10928C2semidirect
11. #10821C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4322
C2×#20562
C2×#20012
C2×#19712
#147022
#112312
#110952
#110772
#109702
#109282
#108212