Finite Group Databasebeta

← #374179

Subgroups — #374179

Order 512.

1810 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#482C211yesmaximal, characteristic
2C2×#2347C211yesmaximal, characteristic, complemented
3C2×#1893C211yesmaximal, characteristic
4C2×#1735C211yesmaximal, characteristic, complemented
5#14136C211yesmaximal, characteristic
6#11220C212yesmaximal, complemented
7#11066C212yesmaximal, complemented
8#10931C212yesmaximal, complemented
9#10907C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic, complemented
11#10714C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C2×#482—24no—
17C2×#482—24no—
18C2×#482C418yescomplemented
19C22×#107C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#102C2212yes—
22C22×#102C2211yescharacteristic
23C22×#102C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25#2347—24no—
26#2347—24no—
27C2×#390C2212yes—
28C2×#389C2212yes—
29C2×#388—22no—
30C2×#383—21no—
31C2×#380C2211yescharacteristic
32C2×C4×C4⋊C4—21no—
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C22⋊C4C2211yescharacteristic
35#1893—24no—
36#1735—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1438—22no—
45#1437—21no—
46#1437—21no—
47C4⋊#95C2212yes—
48C23×D4—21no—
49C23×D4C2311yescharacteristic
50C24×C4C2311yescharacteristic
51C23×C8—22no—
52C23×C8—24no—
53C23×C8—22no—
54C23×C8—22no—
55C23×C8—42no—
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C23×C8C2312yes—
59C2×#121—22no—
60C2×#121—22no—
61C2×#121—22no—
62C2×#121—22no—
63C2×#121—22no—
64C2×#121C2×C414yes—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—24no—
71C2×#117—42no—
72C2×#117C2×C414yes—
73C2×#117D414yes—
74C2×#117D414yes—
75C22×C4⋊C4—22no—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—22no—
86C22×C22⋊C4—22no—
87C22×C22⋊C4D412yes—
88C22×C42—21no—
89C22×C42—22no—
90C22×C42—21no—
91C22×C42C2312yes—
92#482—232no—
93C2×#107—22no—
94C2×#107—22no—
95C2×#107C2×C414yes—
96C2×#106—22no—
97C2×#106—22no—
98C2×#106C2×C414yes—
99C2×#102—22no—
100C2×#102—22no—
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—24no—
104C2×#102—24no—
105C2×#102—42no—
106C2×#102C2×C414yes—
107C2×#102C2×C414yes—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8C2×C414yes—
111#390—24no—
112#390—42no—
113#389—24no—
114#389—24no—
115#388—216no—
116#383—28no—
117#380—22no—
118#380D414yes—
119C4×C4⋊C4—28no—
120C4×C4⋊C4—24no—
121C4×C4⋊C4—24no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95D411yescharacteristic
130C2×#95D411yescharacteristic
131#340—24no—
132#340—24no—
133#340—24no—
134#340—24no—
135#340—42no—
136#336—24no—
137#336—42no—
138C25—21no—
139C25C2×D411yescharacteristic
140C22×D4—24no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—21no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—42no—
151C22×D4—42no—
152C22×D4—82no—
153C22×D4—82no—
154C22×D4C22×C412yes—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—21no—
164C23×C4C22×C412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D412yes—
171C23×C4#4112yes—
172C23×C4#4112yes—
173C23×C4C2411yescharacteristic
174C22×C8—28no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—48no—
190C22×C8—48no—
191C22×C8—44no—
192C22×C8—44no—
193C22×C8C22×C412yes—
194C22×C8C22×C412yes—
195C22×C8#4114yes—
196#121—28no—
197#121—28no—
198#121—44no—
199#121—44no—
200#117—28no—
201#117—28no—
202#117—28no—
203#117—28no—
204#117—48no—
205#117—44no—
206#117C22⋊C4116yescomplemented
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4C22×C412yes—
232C2×C4⋊C4C22×C412yes—
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C2×D414yes—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4#4114yes—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—82no—
257C2×C22⋊C4—82no—
258C2×C22⋊C4—82no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—28no—
269C2×C42—44no—
270C2×C42—42no—
271C2×C42—82no—
272C2×C42—82no—
273#107—28no—
274#106—28no—
275#102—28no—
276#102—28no—
277#102—48no—
278C22⋊C8—28no—
279#95—28no—
280#95—28no—
281#95—24no—
282#95—22no—
283#95—22no—
284#95—82no—
285#95—82no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—41no—
294C24—41no—
295C24—42no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24—42no—
300C24—42no—
301C24C4×D412yes—
302C24#12012yes—
303C24#12112yes—
304C24C22×D411yescharacteristic
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—24no—
308C2×D4—24no—
309C2×D4—48no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—42no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—88no—
325C2×D4—84no—
326C2×D4—84no—
327C2×D4—88no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
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—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—41no—
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—42no—
386C22×C4C2×C22⋊C414yes—
387C22×C4C4×D414yes—
388C22×C4C4×D412yes—
389C22×C4#12012yes—
390C22×C4#12114yes—
391C22×C4#12112yes—
392C22×C4#12314yes—
393C22×C4#13514yes—
394C22×C4C23×C411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4112yes—
400C22×C4C2×#4112yes—
401C22×C4C2×#4111yescharacteristic
402C22×C4H(5,𝔽2)12yes—
403C22×C42−1+412yes—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—416no—
412C2×C8—42no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—42no—
416C2×C8—41no—
417C2×C8—42no—
418C2×C8—41no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—416no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4C2×C22⋊C414yes—
442C4⋊C4C2×C22⋊C414yes—
443C22⋊C4—216no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C22⋊C4—88no—
448C22⋊C4—84no—
449C22⋊C4—88no—
450C42—216no—
451C42—48no—
452C42—42no—
453C42—42no—
454C42—42no—
455C42—42no—
456C42—88no—
457C23—24no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—24no—
462C23—22no—
463C23—24no—
464C23—24no—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—24no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—44no—
478C23—42no—
479C23—42no—
480C23—44no—
481C23—42no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—81no—
487C23—81no—
488C23—82no—
489C23—82no—
490C23—81no—
491C23—81no—
492C23—82no—
493C23—82no—
494C23—82no—
495C23—82no—
496C23—82no—
497C23—82no—
498C23#38614yes—
499C23C2×C4×D411yescharacteristic
500C23C2×#12011yescharacteristic
501C23C2×#12111yescharacteristic
502C23#54612yes—
503C23#54712yes—
504C23C2×#13512yes—
505C23#57512yes—
506C23#57612yes—
507C23#57712yes—
508D4—42no—
509D4—44no—
510D4—42no—
511D4—816no—
512D4—81no—
513D4—81no—
514D4—82no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4#38614yes—
569C2×C4C22×C22⋊C411yescharacteristic
570C2×C4C2×C4×D411yescharacteristic
571C2×C4C4×#4112yes—
572C2×C4#52012yes—
573C2×C4C2×#12111yescharacteristic
574C2×C4C2×#12311yescharacteristic
575C2×C4#52511yescharacteristic
576C2×C4#52511yescharacteristic
577C2×C4#53212yes—
578C2×C4#53511yescharacteristic
579C2×C4#53611yescharacteristic
580C2×C4#53712yes—
581C2×C4#54612yes—
582C2×C4#54712yes—
583C2×C4#55112yes—
584C2×C4#55112yes—
585C2×C4#55212yes—
586C2×C4#55212yes—
587C8—42no—
588C8—44no—
589C8—42no—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—82no—
600C22—84no—
601C22—84no—
602C22—82no—
603C22—82no—
604C22—162no—
605C22—162no—
606C22—162no—
607C22—162no—
608C22C2×#38611yescharacteristic
609C22#235912yes—
610C22#241111yescharacteristic
611C22#241311yescharacteristic
612C22#243012yes—
613C22C4×#13511yescharacteristic
614C22#294311yescharacteristic
615C22#294612yes—
616C22#300212yes—
617C22#300811yescharacteristic
618C22#301011yescharacteristic
619C22#303512yes—
620C22#304512yes—
621C22#329712yes—
622C22#330412yes—
623C22#330912yes—
624C22#331112yes—
625C22#333912yes—
626C22#334012yes—
627C22#334712yes—
628C22#335012yes—
629C4—24no—
630C4—44no—
631C4—44no—
632C4—44no—
633C4—84no—
634C4—84no—
635C4—82no—
636C4—82no—
637C4#229711yescharacteristic
638C4#231111yescharacteristic
639C4#235912yes—
640C4#241111yescharacteristic
641C4#241311yescharacteristic
642C4#249812yes—
643C2—44no—
644C2—162no—
645C2#1413511yescharacteristic
646C2#2092411yescharacteristic
647C2#2093211yescharacteristic
648C2#2292312yes—
649C2#2293312yes—
650C2#2448412yes—
651C2#2458512yes—
652C2#2553412yes—
653C2#2554012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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
2. C2×#2347C2semidirect
4. C2×#1735C2semidirect
6. #11220C2semidirect
7. #11066C2semidirect
8. #10931C2semidirect
9. #10907C2semidirect
10. #10731C2semidirect
11. #10714C2semidirect
18. C2×#482C4semidirect
206. #117C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4822
C2×#23472
C2×#18932
C2×#17352
#141362
#112202
#110662
#109312
#109072
#107312
#107142