Finite Group Databasebeta

← #380279

Subgroups — #380279

Order 512.

1566 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2285C211yesmaximal, characteristic
2C2×#1854C212yesmaximal
3C2×#1817C211yesmaximal, characteristic
4#14685C211yesmaximal, characteristic
5#11325C212yesmaximal, complemented
6#11324C211yesmaximal, characteristic, complemented
7#11323C211yesmaximal, characteristic, complemented
8#11268C212yesmaximal, complemented
9#11267C211yesmaximal, characteristic, complemented
10#11267C211yesmaximal, characteristic, complemented
11#10731C212yesmaximal
12C22×#117C2212yes—
13C23×C42C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16C22×C22⋊C8C2212yes—
17#2285—28no—
18C2×#384C2211yescharacteristic
19C2×#384C2211yescharacteristic
20C2×#382C2212yes—
21C2×#381—21no—
22C2×#380—21no—
23C2×#380—21no—
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4—21no—
28#1854—28no—
29#1817C418yescomplemented
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43#1434—21no—
44#1434—21no—
45C22⋊#95—21no—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×#117—24no—
52C2×#117—42no—
53C2×#117—42no—
54C2×#117—42no—
55C2×#117—42no—
56C2×#117D418yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C42—24no—
71C22×C42—24no—
72C22×C42—22no—
73C22×C42C2×C414yes—
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42Q814yes—
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C2×#107—24no—
80C2×#107—42no—
81C2×#107—42no—
82C2×#106—24no—
83C2×#106—42no—
84C2×#106—42no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8C2×C418yes—
88#384—24no—
89#384—24no—
90#382—24no—
91#382—24no—
92#381—44no—
93#380—28no—
94#380—28no—
95#380—24no—
96#380—24no—
97C4×C4⋊C4—28no—
98C4×C4⋊C4—28no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111#340—24no—
112#340—24no—
113#340—42no—
114#340—42no—
115#340—42no—
116#340—42no—
117#336C2×C414yes—
118#336C2×C414yes—
119#336C2×C418yes—
120C25C2×D411yescharacteristic
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—41no—
142C23×C4—41no—
143C23×C4—41no—
144C23×C4—41no—
145C23×C4C4⋊C414yes—
146C23×C4C4⋊C414yes—
147C23×C4C22×C412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D414yes—
154C23×C4C2×Q812yes—
155C23×C4#4114yes—
156C23×C4C2411yescharacteristic
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8C22×C414yes—
170C22×C8C22×C414yes—
171#117—48no—
172#117—48no—
173#117—48no—
174#117—48no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4C2×D414yes—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C42—216no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—216no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—28no—
235C2×C42—44no—
236C2×C42—44no—
237C2×C42C4⋊C414yes—
238C2×C42C4⋊C414yes—
239C2×C42C22×C412yes—
240C2×C42C22×C412yes—
241C2×C42C2×Q812yes—
242C2×C42C2×Q812yes—
243#107—48no—
244#106—48no—
245C22⋊C8—216no—
246#95—28no—
247#95—28no—
248#95—42no—
249#95—42no—
250#95—42no—
251#95—42no—
252#95—42no—
253#95—42no—
254#95—42no—
255#95—42no—
256#95—42no—
257#95—42no—
258C24—21no—
259C24—21no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—21no—
268C24—21no—
269C24—21no—
270C24C2×C4⋊C412yes—
271C24#12012yes—
272C24#12312yes—
273C24C22×D411yescharacteristic
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—28no—
313C22×C4—28no—
314C22×C4—28no—
315C22×C4—28no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4C2×C4⋊C412yes—
369C22×C4C2×C4⋊C412yes—
370C22×C4C2×C4⋊C412yes—
371C22×C4C2×C4⋊C412yes—
372C22×C4C2×C4⋊C412yes—
373C22×C4C4×D418yes—
374C22×C4#12012yes—
375C22×C4C22⋊Q818yes—
376C22×C4#12312yes—
377C22×C4#13512yes—
378C22×C4#13512yes—
379C22×C4AGL(1,ℤ/8)12yes—
380C22×C4#13712yes—
381C22×C4C23×C411yescharacteristic
382C22×C4C22×D412yes—
383C22×C4C22×Q811yescharacteristic
384C22×C4C2×#4112yes—
385C22×C4H(5,𝔽2)12yes—
386C22×C4H(5,𝔽2)11yescharacteristic
387C22×C4H(5,𝔽2)11yescharacteristic
388C22×C42−1+412yes—
389C22×C42−1+411yescharacteristic
390C22×C42−1+411yescharacteristic
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—48no—
410C4⋊C4—48no—
411C4⋊C4—48no—
412C4⋊C4—48no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C22⋊C4—416no—
426C22⋊C4—84no—
427C22⋊C4—84no—
428C42—24no—
429C42—24no—
430C42—24no—
431C42—24no—
432C42—28no—
433C42—28no—
434C42—416no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23#38514yes—
470C23C22×C4⋊C411yescharacteristic
471C23C2×#12011yescharacteristic
472C23C2×#12311yescharacteristic
473C23#53511yescharacteristic
474C23#53511yescharacteristic
475C23#53611yescharacteristic
476C23#53611yescharacteristic
477C23C2×#13511yescharacteristic
478C23C2×#13511yescharacteristic
479C23C2×AGL(1,ℤ/8)11yescharacteristic
480C23C2×#13711yescharacteristic
481C23#57511yescharacteristic
482C23#57511yescharacteristic
483C23#57511yescharacteristic
484C23#57511yescharacteristic
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—28no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—416no—
514C2×C4—416no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
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—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4#38514yes—
566C2×C4#42714yes—
567C2×C4#42814yes—
568C2×C4C2×C4×D412yes—
569C2×C4#52012yes—
570C2×C4#52011yescharacteristic
571C2×C4#52011yescharacteristic
572C2×C4C2×C22⋊Q812yes—
573C2×C4#53412yes—
574C2×C4#53412yes—
575C2×C4#53612yes—
576C2×C4#53612yes—
577C2×C4#53912yes—
578C2×C4#53912yes—
579C2×C4#53912yes—
580C2×C4#53912yes—
581C2×C4#54412yes—
582C2×C4#54411yescharacteristic
583C2×C4#54411yescharacteristic
584C8—44no—
585C8—44no—
586C22—24no—
587C22—24no—
588C22—24no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22C2×#38511yescharacteristic
602C22#231711yescharacteristic
603C22#231711yescharacteristic
604C22#261311yescharacteristic
605C22#261411yescharacteristic
606C22#262311yescharacteristic
607C22#262311yescharacteristic
608C22C2×#42711yescharacteristic
609C22C2×#42811yescharacteristic
610C22#291111yescharacteristic
611C22#291111yescharacteristic
612C22#300211yescharacteristic
613C22#300211yescharacteristic
614C22#300611yescharacteristic
615C22#300711yescharacteristic
616C22#309111yescharacteristic
617C22#309111yescharacteristic
618C22#309411yescharacteristic
619C22#309511yescharacteristic
620C22#320711yescharacteristic
621C22#321011yescharacteristic
622C22#321211yescharacteristic
623C22#321211yescharacteristic
624C22#321411yescharacteristic
625C22#321611yescharacteristic
626C22#321911yescharacteristic
627C22#321911yescharacteristic
628C22#322811yescharacteristic
629C22#323111yescharacteristic
630C22#323311yescharacteristic
631C22#323311yescharacteristic
632C22#323511yescharacteristic
633C22#323711yescharacteristic
634C22#324011yescharacteristic
635C22#324011yescharacteristic
636C4—24no—
637C4—24no—
638C4—24no—
639C4—42no—
640C4—42no—
641C4—84no—
642C4—84no—
643C4—161no—
644C4—161no—
645C4—161no—
646C4—161no—
647C4#232212yes—
648C4#232212yes—
649C4#262712yes—
650C4#262712yes—
651C2—44no—
652C2#1426311yescharacteristic
653C2#2084311yescharacteristic
654C2#2084611yescharacteristic
655C2#2198411yescharacteristic
656C2#2198511yescharacteristic
657C2#2198811yescharacteristic
658C2#2198911yescharacteristic
659C2#2962811yescharacteristic
660C2#2962911yescharacteristic
661C2#2964411yescharacteristic
662C2#2964511yescharacteristic
663C2#2991011yescharacteristic
664C2#2991111yescharacteristic
665C2#2991411yescharacteristic
666C2#2991511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×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
5. #11325C2semidirect
6. #11324C2semidirect
7. #11323C2semidirect
8. #11268C2semidirect
9. #11267C2semidirect
10. #11267C2semidirect
29. #1817C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22852
C2×#18542
C2×#18172
#146852
#113252
#113242
#113232
#112682
#112672
#107312