Finite Group Databasebeta

← #396576

Subgroups — #396576

Order 512.

1666 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2075C211yesmaximal, characteristic, complemented
2C2×#1869C211yesmaximal, characteristic
3C2×#1808C211yesmaximal, characteristic, complemented
4C22×#340C211yesmaximal, characteristic, complemented
5#13872C212yesmaximal
6#11379C211yesmaximal, characteristic, complemented
7#11315C212yesmaximal, complemented
8#11256C211yesmaximal, characteristic, complemented
9#11199C212yesmaximal, complemented
10#11096C211yesmaximal, characteristic, complemented
11#10730C212yesmaximal, complemented
12C24×C8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C23×C4⋊C4C2212yes—
15C22×#107C2211yescharacteristic
16C22×#107C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C2×#386—22no—
21C2×#380C2212yes—
22C2×#380C2212yes—
23C2×#380C2212yes—
24#2075—24no—
25#1869—24no—
26#1808—24no—
27C2×#340—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1441—22no—
39#1439—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C22⋊#95—22no—
43C23×D4D411yescharacteristic
44C24×C4—22no—
45C24×C4C2311yescharacteristic
46C24×C4C2311yescharacteristic
47C23×C8—24no—
48C23×C8—22no—
49C23×C8—22no—
50C23×C8—42no—
51C23×C8D414yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—24no—
59C2×C4×D4—24no—
60C22×C4⋊C4—24no—
61C22×C4⋊C4—24no—
62C22×C4⋊C4—24no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—42no—
66C22×C4⋊C4—42no—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4D411yescharacteristic
76C22×C22⋊C4D411yescharacteristic
77C22×C42C2311yescharacteristic
78C2×#107—22no—
79C2×#107—22no—
80C2×#107—22no—
81C2×#107—22no—
82C2×#107—42no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—42no—
90#386—48no—
91#380—24no—
92#380—24no—
93#380—24no—
94#380—24no—
95#380—42no—
96#380—42no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104#340—24no—
105#340—22no—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—42no—
110#340—42no—
111#340D414yescomplemented
112#336—22no—
113#336—22no—
114C25C2×D411yescharacteristic
115C25C2×D411yescharacteristic
116C22×D4—22no—
117C22×D4—24no—
118C22×D4—22no—
119C22×D4—24no—
120C22×D4D814yes—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—44no—
142C23×C4—44no—
143C23×C4—44no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4—82no—
147C23×C4—82no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4#4112yes—
154C23×C4#4114yes—
155C23×C4#4114yes—
156C23×C4C2411yescharacteristic
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—28no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—28no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—44no—
176C22×C8—48no—
177C22×C8—48no—
178C22×C8C2×D412yes—
179C22×C8C2×D412yes—
180C22×C8C2×D412yes—
181C22×C8#4112yes—
182C4×D4—44no—
183C4×D4—44no—
184C4×D4—48no—
185C2×C4⋊C4—216no—
186C2×C4⋊C4—22no—
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—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—48no—
211C2×C4⋊C4—48no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—48no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4C2×D414yes—
217C2×C4⋊C4#4114yes—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—82no—
230C2×C22⋊C4—82no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—21no—
235#107—44no—
236#107—44no—
237C22⋊C8—44no—
238C22⋊C8—44no—
239C22⋊C8—44no—
240#95—44no—
241#95—44no—
242#95—44no—
243#95—44no—
244#95—44no—
245#95—82no—
246#95—82no—
247C24—21no—
248C24—22no—
249C24—21no—
250C24—21no—
251C24—22no—
252C24—21no—
253C24—21no—
254C24—22no—
255C24—21no—
256C24—22no—
257C24—21no—
258C24—21no—
259C24—22no—
260C24—21no—
261C24—22no—
262C24—21no—
263C24#12112yes—
264C24#12314yes—
265C24C2×D812yes—
266C24C2×D812yes—
267C24#13512yes—
268C24#13714yes—
269C24C22×D411yescharacteristic
270C2×D4—28no—
271C2×D4—216no—
272C2×D4—44no—
273C2×D4—44no—
274C2×D4—48no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—416no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—88no—
355C22×C4—88no—
356C22×C4—88no—
357C22×C4—88no—
358C22×C4—88no—
359C22×C4—88no—
360C22×C4#12112yes—
361C22×C4#12314yes—
362C22×C4#12314yes—
363C22×C4#12414yes—
364C22×C4#12618yes—
365C22×C4C2×D812yes—
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4112yes—
370C22×C4C2×#4112yes—
371C22×C4C2×#4112yes—
372C22×C4C2×#4111yescharacteristic
373C22×C4H(5,𝔽2)12yes—
374C22×C4H(5,𝔽2)11yescharacteristic
375C22×C42−1+411yescharacteristic
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—416no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—41no—
387C2×C8—42no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—42no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8#12114yes—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—416no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—44no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—88no—
418C22⋊C4—88no—
419C22⋊C4—88no—
420C42—41no—
421C42—41no—
422C42—42no—
423C23—22no—
424C23—22no—
425C23—24no—
426C23—24no—
427C23—24no—
428C23—22no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—22no—
433C23—22no—
434C23—24no—
435C23—24no—
436C23—22no—
437C23—24no—
438C23—24no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—24no—
443C23—22no—
444C23—24no—
445C23—22no—
446C23—41no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—41no—
452C23—41no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—41no—
457C23—42no—
458C23—41no—
459C23—41no—
460C23—42no—
461C23—42no—
462C23—41no—
463C23—41no—
464C23#39714yes—
465C23#46614yes—
466C23#47014yes—
467C23#48018yes—
468C23#48518yes—
469C23C2×#12111yescharacteristic
470C23C2×#12312yes—
471C23D4211yescharacteristic
472C23#54612yes—
473C23#54711yescharacteristic
474C23C22×D811yescharacteristic
475C23C2×#13511yescharacteristic
476C23C2×#13712yes—
477C23#57611yescharacteristic
478C23#57712yes—
479C23#57811yescharacteristic
480D4—416no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—44no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—832no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—82no—
525C2×C4—81no—
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#39714yes—
542C2×C4C2×#12111yescharacteristic
543C2×C4C2×#12311yescharacteristic
544C2×C4#52512yes—
545C2×C4C2×#12411yescharacteristic
546C2×C4C2×#12612yes—
547C2×C4#52912yes—
548C2×C4#53211yescharacteristic
549C2×C4#53411yescharacteristic
550C2×C4#53711yescharacteristic
551C2×C4#53812yes—
552C2×C4#53912yes—
553C2×C4#54612yes—
554C2×C4#54811yescharacteristic
555C2×C4#55112yes—
556C2×C4#55112yes—
557C2×C4#55212yes—
558C2×C4#55312yes—
559C2×C4#55311yescharacteristic
560C8—24no—
561C8—42no—
562C8—81no—
563C8—81no—
564C22—28no—
565C22—28no—
566C22—24no—
567C22—28no—
568C22—28no—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—44no—
575C22—42no—
576C22—44no—
577C22—42no—
578C22—42no—
579C22—82no—
580C22—81no—
581C22—82no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22#207518yescomplemented
586C22C2×#39711yescharacteristic
587C22#241611yescharacteristic
588C22#241812yes—
589C22#241912yes—
590C22#254311yescharacteristic
591C22C2×#46611yescharacteristic
592C22C2×#47011yescharacteristic
593C22#306011yescharacteristic
594C22#306111yescharacteristic
595C22C2×#48012yes—
596C22C2×#48512yes—
597C22#309612yes—
598C22#309712yes—
599C22#326612yes—
600C22D4×D811yescharacteristic
601C22#328511yescharacteristic
602C22#329412yes—
603C22#330112yes—
604C22#331212yes—
605C22#331412yes—
606C22#333411yescharacteristic
607C22#333911yescharacteristic
608C22#334412yes—
609C4—24no—
610C4—42no—
611C4—84no—
612C4—84no—
613C4—81no—
614C4—81no—
615C4—162no—
616C4—162no—
617C4#241211yescharacteristic
618C4#248112yes—
619C4#249811yescharacteristic
620C4#251012yes—
621C4#263911yescharacteristic
622C4#285011yescharacteristic
623C2—28no—
624C2—82no—
625C2#1473511yescharacteristic
626C2C2×#207511yescharacteristic
627C2#2364111yescharacteristic
628C2#2478011yescharacteristic
629C2#2478311yescharacteristic
630C2#2486112yes—
631C2#2487512yes—
632C2#2493312yes—
633C2#2494712yes—
634C2#2855111yescharacteristic
635C2#2856311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2075C2semidirect
3. C2×#1808C2semidirect
4. C22×#340C2semidirect
6. #11379C2semidirect
7. #11315C2semidirect
8. #11256C2semidirect
9. #11199C2semidirect
10. #11096C2semidirect
11. #10730C2semidirect
111. #340D4semidirect
585. C22#2075semidirect

Maximal subgroups

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

subgroupindex
C2×#20752
C2×#18692
C2×#18082
C22×#3402
#138722
#113792
#113152
#112562
#111992
#110962
#107302