Finite Group Databasebeta

← #468184

Subgroups — #468184

Order 512.

1368 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2859C211yesmaximal, characteristic, complemented
2C2×#2067C212yesmaximal
3#14700C211yesmaximal, characteristic
4#11328C212yesmaximal, complemented
5#11324C211yesmaximal, characteristic, complemented
6#11267C211yesmaximal, characteristic, complemented
7#11227C212yesmaximal, complemented
8#11066C212yesmaximal
9#11049C211yesmaximal, characteristic
10#11049C211yesmaximal, characteristic
11#10832C211yesmaximal, characteristic, complemented
12C22×#125—21no—
13C22×#107C2212yes—
14C22×C4⋊C8C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17#2859—28no—
18C2×#390C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#389—21no—
22C2×#389—21no—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2211yescharacteristic
26C2×#388—21no—
27C2×#388—21no—
28C2×#384C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×#382—21no—
31C2×C43C2211yescharacteristic
32#2067—28no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1437—21no—
42#1437—21no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C23×D4C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×#125—28no—
49C2×#125—44no—
50C2×#125—44no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C2×#107—24no—
77C2×#107—42no—
78C2×#107—42no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—24no—
81C2×C4⋊C8—42no—
82C2×#102—24no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—42no—
86C2×#102—42no—
87C2×#102—42no—
88#390—24no—
89#390—22no—
90#390—42no—
91#390Q814yes—
92#389—28no—
93#389—28no—
94#389—24no—
95#389—24no—
96#389—24no—
97#389—42no—
98#389D418yes—
99#388—44no—
100#388—44no—
101#384—24no—
102#384—24no—
103#382—28no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C43—24no—
112#340—24no—
113#340—24no—
114#340—42no—
115#340—42no—
116#340—42no—
117#336—22no—
118#336—22no—
119#336—42no—
120C25—21no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—21no—
124C22×D4—22no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4C2×Q812yes—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2411yescharacteristic
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8#4114yes—
157#125—432no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4C2×D414yes—
201C2×C4⋊C4C2×D414yes—
202C2×C4⋊C4C2×Q812yes—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—21no—
237C2×C42—22no—
238C2×C42—21no—
239C2×C42—41no—
240C2×C42—41no—
241C2×C42D814yes—
242C2×C42C2×D414yes—
243C2×C42C2×Q812yes—
244C2×C42#4112yes—
245#107—48no—
246C4⋊C8—48no—
247#102—48no—
248#102—48no—
249#95—28no—
250#95—28no—
251#95—42no—
252#95—42no—
253#95—42no—
254#95—42no—
255#95—42no—
256C24—22no—
257C24—41no—
258C24—41no—
259C24—41no—
260C24—41no—
261C24—41no—
262C24—41no—
263C24—41no—
264C24—41no—
265C24—41no—
266C24—41no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24C22×D411yescharacteristic
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4—41no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
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—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
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#12012yes—
354C22×C4#12012yes—
355C22×C4C22⋊Q812yes—
356C22×C4C22⋊Q812yes—
357C22×C4C2×D812yes—
358C22×C4C2×D812yes—
359C22×C4C2×D812yes—
360C22×C4#13512yes—
361C22×C4AGL(1,ℤ/8)12yes—
362C22×C4AGL(1,ℤ/8)12yes—
363C22×C4#13712yes—
364C22×C4#13712yes—
365C22×C4C22×D412yes—
366C22×C4C22×Q811yescharacteristic
367C22×C4C2×#4112yes—
368C22×C4C2×#4111yescharacteristic
369C22×C4H(5,𝔽2)11yescharacteristic
370C22×C4H(5,𝔽2)12yes—
371C22×C4H(5,𝔽2)11yescharacteristic
372C22×C42−1+411yescharacteristic
373C22×C42−1+412yes—
374C22×C42−1+411yescharacteristic
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—48no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—48no—
398C4⋊C4—416no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—416no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—84no—
410C22⋊C4—48no—
411C22⋊C4—48no—
412C22⋊C4—84no—
413C22⋊C4—84no—
414C22⋊C4—84no—
415C22⋊C4—84no—
416C42—28no—
417C42—24no—
418C42—24no—
419C42—24no—
420C42—24no—
421C42—28no—
422C42—44no—
423C42—44no—
424C42—42no—
425C42—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—81no—
434C23—81no—
435C23—81no—
436C23—81no—
437C23—81no—
438C23—81no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—81no—
443C23—81no—
444C23—81no—
445C23—81no—
446C23—81no—
447C23—81no—
448C23—81no—
449C23—81no—
450C23—81no—
451C23—81no—
452C23—81no—
453C23—81no—
454C23—81no—
455C23—81no—
456C23—81no—
457C23—81no—
458C23—81no—
459C23—81no—
460C23—81no—
461C23C2×#12011yescharacteristic
462C23C2×C22⋊Q811yescharacteristic
463C23#53211yescharacteristic
464C23#53511yescharacteristic
465C23#53611yescharacteristic
466C23#53711yescharacteristic
467C23#53711yescharacteristic
468C23C22×D811yescharacteristic
469C23C2×#13511yescharacteristic
470C23C2×AGL(1,ℤ/8)11yescharacteristic
471C23C2×AGL(1,ℤ/8)11yescharacteristic
472C23C2×#13711yescharacteristic
473C23C2×#13711yescharacteristic
474C23#57511yescharacteristic
475C23#57511yescharacteristic
476D4—44no—
477D4—44no—
478D4—82no—
479D4—82no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
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—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—81no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—82no—
534C2×C4—81no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
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#44714yes—
555C2×C4#47414yes—
556C2×C4#47814yes—
557C2×C4#53512yes—
558C2×C4#53712yes—
559C2×C4#53912yes—
560C2×C4#54011yescharacteristic
561C2×C4#54112yes—
562C2×C4#54111yescharacteristic
563C2×C4#54411yescharacteristic
564C2×C4#54411yescharacteristic
565C2×C4#54612yes—
566C2×C4#54712yes—
567C2×C4#54812yes—
568C2×C4D4×Q812yes—
569C2×C4#55312yes—
570C2×C4#55412yes—
571C2×C4#56212yes—
572C2×C4#56312yes—
573C8—44no—
574C8—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—161no—
583C22—161no—
584C22—161no—
585C22—161no—
586C22—161no—
587C22—161no—
588C22—161no—
589C22—161no—
590C22#240911yescharacteristic
591C22#243011yescharacteristic
592C22#256611yescharacteristic
593C22#261511yescharacteristic
594C22#262911yescharacteristic
595C22#266411yescharacteristic
596C22#282011yescharacteristic
597C22C2×#44711yescharacteristic
598C22#300211yescharacteristic
599C22#300411yescharacteristic
600C22#300611yescharacteristic
601C22C2×#47411yescharacteristic
602C22C2×#47811yescharacteristic
603C22#307911yescharacteristic
604C22#308011yescharacteristic
605C22#316911yescharacteristic
606C22#317211yescharacteristic
607C22#318011yescharacteristic
608C22#318211yescharacteristic
609C22#320511yescharacteristic
610C22#321011yescharacteristic
611C22#321411yescharacteristic
612C22#321511yescharacteristic
613C22#322611yescharacteristic
614C22#323211yescharacteristic
615C22#323411yescharacteristic
616C22#323911yescharacteristic
617C22#324511yescharacteristic
618C22#324511yescharacteristic
619C22#324711yescharacteristic
620C22#324811yescharacteristic
621C22#325211yescharacteristic
622C22#325211yescharacteristic
623C22#325311yescharacteristic
624C22#325411yescharacteristic
625C4—24no—
626C4—24no—
627C4—24no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—82no—
632C4—84no—
633C4—84no—
634C4—161no—
635C4—161no—
636C4#270412yes—
637C4#271112yes—
638C4#272012yes—
639C4#272512yes—
640C2—162no—
641C2#1660411yescharacteristic
642C2#2429811yescharacteristic
643C2#2430511yescharacteristic
644C2#2549611yescharacteristic
645C2#2550311yescharacteristic
646C2#2919211yescharacteristic
647C2#2919811yescharacteristic
648C2#2966011yescharacteristic
649C2#2966811yescharacteristic
650C2#2999411yescharacteristic
651C2#2999511yescharacteristic
652C2#3037611yescharacteristic
653C2#3038311yescharacteristic
654C2#3281811yescharacteristic
655C2#3282611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2859C2semidirect
4. #11328C2semidirect
5. #11324C2semidirect
6. #11267C2semidirect
7. #11227C2semidirect
11. #10832C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28592
C2×#20672
#147002
#113282
#113242
#112672
#112272
#110662
#110492
#108322