Finite Group Databasebeta

← #374184

Subgroups — #374184

Order 512.

1830 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#484C211yesmaximal, characteristic
2C2×#2344C211yesmaximal, characteristic
3C2×#1894C211yesmaximal, characteristic
4C2×#1736C211yesmaximal, characteristic
5#14138C211yesmaximal, characteristic
6#11219C212yesmaximal, complemented
7#11067C212yesmaximal, complemented
8#10932C212yesmaximal, complemented
9#10907C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic
11#10715C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×#117—21no—
15C22×#117C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#484—24no—
18C2×#484—24no—
19C2×#484C418yescomplemented
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#103C2212yes—
23C22×#103C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×C22⋊C8C2211yescharacteristic
26#2344—24no—
27#2344—24no—
28C2×#391C2212yes—
29C2×#389C2212yes—
30C2×#387—22no—
31C2×#381—21no—
32C2×#380C2211yescharacteristic
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C22⋊C4C2211yescharacteristic
35#1894—24no—
36#1736—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1440—21no—
45#1440—21no—
46#1436—22no—
47C4⋊#95C2212yes—
48C23×Q8C2311yescharacteristic
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C22×M16—22no—
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—24no—
55C22×M16—42no—
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C23×C8C2312yes—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8C2×C414yes—
65C2×#117—28no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—24no—
69C2×#117—48no—
70C2×#117—42no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4C2×C414yes—
78C22×C4⋊C4D414yes—
79C22×C4⋊C4D414yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4D411yescharacteristic
89C22×C22⋊C4D411yescharacteristic
90C22×C42—22no—
91C22×C42—21no—
92C22×C42C2312yes—
93#484—232no—
94C2×#107—22no—
95C2×#107—22no—
96C2×#107—22no—
97C2×#107—22no—
98C2×#107C2×C414yes—
99C2×#107C2×C414yes—
100C2×#103—22no—
101C2×#103—22no—
102C2×#103—22no—
103C2×#103—22no—
104C2×#103—24no—
105C2×#103—24no—
106C2×#103—42no—
107C2×#103C2×C414yes—
108C2×#103C2×C414yes—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8C2×C414yes—
112#391—24no—
113#391—42no—
114#389—24no—
115#389—24no—
116#387—216no—
117#381—28no—
118#380—22no—
119#380D414yes—
120C4×C4⋊C4—24no—
121C4×C4⋊C4—24no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C2×#95—22no—
125C2×#95—21no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95D412yes—
132#340—24no—
133#340—24no—
134#340—24no—
135#340—24no—
136#340—42no—
137#336—24no—
138#336—42no—
139C25C2×D411yescharacteristic
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8C22×C412yes—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4C22×C412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4#4112yes—
173C23×C4#4112yes—
174C23×C4C2411yescharacteristic
175C2×M16—28no—
176C2×M16—48no—
177C2×M16—48no—
178C2×M16—44no—
179C2×M16—44no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8C22×C412yes—
195C22×C8C22×C412yes—
196C22×C8#4114yes—
197C22⋊Q8—28no—
198C22⋊Q8—28no—
199C22⋊Q8—44no—
200C22⋊Q8—44no—
201#117—28no—
202#117—432no—
203#117—48no—
204#117—44no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4C22⋊C4116yes—
239C2×C4⋊C4C22×C412yes—
240C2×C4⋊C4C22×C412yes—
241C2×C4⋊C4C22×C412yes—
242C2×C4⋊C4C2×D414yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4#4114yes—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C42—28no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—44no—
267C2×C42—44no—
268C2×C42—42no—
269C2×C42—44no—
270#107—28no—
271#107—28no—
272#103—28no—
273#103—28no—
274#103—48no—
275C22⋊C8—28no—
276#95—28no—
277#95—28no—
278#95—24no—
279#95—24no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24C4×D412yes—
291C24#12012yes—
292C24#12112yes—
293C24C22×D411yescharacteristic
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—42no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—216no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—48no—
353C22×C4—416no—
354C22×C4—48no—
355C22×C4—416no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
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×C4C2×C22⋊C414yes—
381C22×C4C4×D414yes—
382C22×C4C4×D412yes—
383C22×C4#12012yes—
384C22×C4#12114yes—
385C22×C4#12112yes—
386C22×C4#12314yes—
387C22×C4AGL(1,ℤ/8)14yes—
388C22×C4C23×C411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4112yes—
394C22×C4C2×#4112yes—
395C22×C4C2×#4111yescharacteristic
396C22×C4H(5,𝔽2)12yes—
397C22×C42−1+412yes—
398M16—416no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—42no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—42no—
410C2×C8—41no—
411C2×C8—42no—
412C2×C8—41no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—216no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—416no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—48no—
437C4⋊C4C2×C22⋊C414yes—
438C4⋊C4C2×C22⋊C414yes—
439C22⋊C4—28no—
440C22⋊C4—44no—
441C22⋊C4—48no—
442C22⋊C4—48no—
443C22⋊C4—48no—
444C42—416no—
445C42—48no—
446C42—42no—
447C42—42no—
448C42—42no—
449C42—42no—
450C23—24no—
451C23—22no—
452C23—22no—
453C23—24no—
454C23—24no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23#38614yes—
471C23C2×C4×D411yescharacteristic
472C23C2×#12011yescharacteristic
473C23C2×#12111yescharacteristic
474C23#54612yes—
475C23#54712yes—
476C23C2×AGL(1,ℤ/8)12yes—
477C23#57512yes—
478C23#57812yes—
479C23#57812yes—
480Q8—42no—
481Q8—44no—
482Q8—42no—
483Q8—81no—
484Q8—81no—
485Q8—82no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—28no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—28no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—432no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—82no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4#38614yes—
546C2×C4C22×C22⋊C411yescharacteristic
547C2×C4C2×C4×D411yescharacteristic
548C2×C4C4×#4112yes—
549C2×C4#52012yes—
550C2×C4C2×#12111yescharacteristic
551C2×C4C2×#12311yescharacteristic
552C2×C4#52511yescharacteristic
553C2×C4#52511yescharacteristic
554C2×C4#53212yes—
555C2×C4#53511yescharacteristic
556C2×C4#53611yescharacteristic
557C2×C4#53712yes—
558C2×C4#54612yes—
559C2×C4#54712yes—
560C2×C4#55112yes—
561C2×C4#55112yes—
562C2×C4#55212yes—
563C2×C4#55212yes—
564C8—42no—
565C8—44no—
566C8—42no—
567C22—24no—
568C22—24no—
569C22—24no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22C2×#38611yescharacteristic
577C22#235912yes—
578C22#241111yescharacteristic
579C22#241311yescharacteristic
580C22#243012yes—
581C22C4×AGL(1,ℤ/8)12yes—
582C22#295011yescharacteristic
583C22#295011yescharacteristic
584C22#300311yescharacteristic
585C22#300511yescharacteristic
586C22#301112yes—
587C22#303612yes—
588C22#304412yes—
589C22#330812yes—
590C22#331512yes—
591C22#332712yes—
592C22#333012yes—
593C22#334312yes—
594C22#334912yes—
595C22#335212yes—
596C22#335312yes—
597C4—24no—
598C4—44no—
599C4—44no—
600C4—44no—
601C4—82no—
602C4—84no—
603C4—82no—
604C4—84no—
605C4—82no—
606C4#229711yescharacteristic
607C4#231111yescharacteristic
608C4#235912yes—
609C4#241111yescharacteristic
610C4#241311yescharacteristic
611C4#249812yes—
612C2—44no—
613C2#1413511yescharacteristic
614C2#2092511yescharacteristic
615C2#2093511yescharacteristic
616C2#2292512yes—
617C2#2294712yes—
618C2#2448912yes—
619C2#2458812yes—
620C2#2554712yes—
621C2#2555712yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11219C2semidirect
7. #11067C2semidirect
8. #10932C2semidirect
9. #10907C2semidirect
19. C2×#484C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4842
C2×#23442
C2×#18942
C2×#17362
#141382
#112192
#110672
#109322
#109072
#107312
#107152