Finite Group Databasebeta

← #396495

Subgroups — #396495

Order 512.

2184 conjugacy classes of subgroups.

The subgroup lattice has 635 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2311C211yesmaximal, characteristic
2C2×#2095C211yesmaximal, characteristic, complemented
3C2×#2093C211yesmaximal, characteristic, complemented
4C2×#1852C212yesmaximal, complemented
5C2×#1788C212yesmaximal
6#14340C211yesmaximal, characteristic, complemented
7#11124C212yesmaximal, complemented
8#11123C212yesmaximal, complemented
9#11052C211yesmaximal, characteristic, complemented
10#11051C211yesmaximal, characteristic, complemented
11#10713C211yesmaximal, characteristic, complemented
12C23×#41C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2212yes—
16C22×#107C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#102C2212yes—
20C22×#102C2212yes—
21C22×C22⋊C8C2212yes—
22#2311—24no—
23#2311—24no—
24C2×#391C2211yescharacteristic
25C2×#391C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#386—22no—
29C2×#384C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×#380—22no—
32C2×#380C2212yes—
33#2095—24no—
34#2093—24no—
35#1852—28no—
36#1788—28no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42#1441—22no—
43#1439—22no—
44C4⋊#95C2212yes—
45C22×#41—22no—
46C22×#41—22no—
47C22×#41—22no—
48C22×#41—22no—
49C22×#41—22no—
50C22×#41—24no—
51C22×#41—24no—
52C22×#41—42no—
53C22×#41—42no—
54C22×#41Q814yes—
55C23×Q8C2311yescharacteristic
56C23×D4—21no—
57C23×D4C2311yescharacteristic
58C24×C4—21no—
59C24×C4—21no—
60C24×C4C2311yescharacteristic
61C23×C8C2312yes—
62C23×C8C2312yes—
63C2×#117—24no—
64C2×#117—22no—
65C2×#117—22no—
66C2×#117—42no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—42no—
73C22×C4⋊C4D412yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4D412yes—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C2×#107—24no—
85C2×#107—24no—
86C2×#107—24no—
87C2×#107—24no—
88C2×#107—42no—
89C2×#107—42no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—42no—
95C2×#103—42no—
96C2×#102—24no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—42no—
101C2×#102—42no—
102C2×C22⋊C8—24no—
103C2×C22⋊C8—24no—
104C2×C22⋊C8D418yes—
105#391—22no—
106#391—22no—
107#391—22no—
108#391D414yes—
109#390—22no—
110#390—22no—
111#390—22no—
112#390D414yes—
113#386—216no—
114#384—22no—
115#384—22no—
116#382—42no—
117#380—216no—
118#380—24no—
119#380—24no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—24no—
125#340—24no—
126#340—24no—
127#336—22no—
128#336—22no—
129#336—42no—
130C25—21no—
131C25C2×D411yescharacteristic
132C2×#41—28no—
133C2×#41—28no—
134C2×#41—28no—
135C2×#41—28no—
136C2×#41—216no—
137C2×#41—48no—
138C2×#41—48no—
139C2×#41—48no—
140C2×#41—48no—
141C2×#41—48no—
142C2×#41—48no—
143C2×#41—48no—
144C2×#41—48no—
145C2×#41—48no—
146C2×#41—48no—
147C2×#41—48no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8C2×D412yes—
156C22×Q8C2×Q812yes—
157C22×Q8#4112yes—
158C22×D4—24no—
159C22×D4—24no—
160C22×D4—24no—
161C22×D4—22no—
162C22×D4—21no—
163C22×D4—22no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—42no—
169C22×D4—42no—
170C22×D4—42no—
171C22×D4—42no—
172C22×D4C2×D412yes—
173C22×D4C2×Q812yes—
174C22×D4#4112yes—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—21no—
184C23×C4—24no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—84no—
197C23×C4C2×D412yes—
198C23×C4C2×D412yes—
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D414yes—
201C23×C4C2×Q812yes—
202C23×C4#4114yes—
203C23×C4#4112yes—
204C23×C4C2411yescharacteristic
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8C2×D414yes—
218C22×C8C2×D414yes—
219#117—48no—
220#117—48no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—48no—
243C2×C4⋊C4—48no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—84no—
249C2×C4⋊C4C2×D412yes—
250C2×C4⋊C4C2×D412yes—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—84no—
258C2×C22⋊C4—84no—
259C2×C22⋊C4—84no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268#107—48no—
269#107—48no—
270#103—48no—
271#103—48no—
272#102—48no—
273#102—48no—
274C22⋊C8—216no—
275#95—28no—
276#95—28no—
277#95—44no—
278#95—44no—
279#95—84no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—24no—
285C24—24no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—41no—
290C24—41no—
291C24—41no—
292C24—41no—
293C24—44no—
294C24—44no—
295C24C22⋊Q812yes—
296C24#12312yes—
297C24#12712yes—
298C24C22×D411yescharacteristic
299#41—432no—
300#41—432no—
301#41—432no—
302#41—432no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—24no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8C22⋊Q814yes—
319C2×D4—216no—
320C2×D4—24no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—48no—
330C2×D4—48no—
331C2×D4—48no—
332C2×D4—48no—
333C2×D4—48no—
334C2×D4—48no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4C22⋊Q814yes—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—216no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—48no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—48no—
385C22×C4—48no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—88no—
403C22×C4—88no—
404C22×C4—88no—
405C22×C4#12014yes—
406C22×C4C22⋊Q818yes—
407C22×C4C22⋊Q812yes—
408C22×C4#12318yes—
409C22×C4#12312yes—
410C22×C4#12712yes—
411C22×C4C22×D412yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×Q811yescharacteristic
415C22×C4C2×#4112yes—
416C22×C4C2×#4111yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4H(5,𝔽2)12yes—
420C22×C4H(5,𝔽2)12yes—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—88no—
450C22⋊C4—48no—
451C22⋊C4—88no—
452C22⋊C4—88no—
453C22⋊C4—88no—
454C42—24no—
455C42—24no—
456C42—42no—
457C42—42no—
458C23—28no—
459C23—28no—
460C23—22no—
461C23—22no—
462C23—24no—
463C23—24no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—24no—
469C23—44no—
470C23—44no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—48no—
476C23—48no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—44no—
483C23—44no—
484C23—42no—
485C23—84no—
486C23—82no—
487C23—82no—
488C23—84no—
489C23—82no—
490C23—82no—
491C23#39914yes—
492C23C2×#12011yescharacteristic
493C23C2×C22⋊Q811yescharacteristic
494C23C2×#12311yescharacteristic
495C23#52511yescharacteristic
496C23C2×#12711yescharacteristic
497C23#53411yescharacteristic
498C23#53511yescharacteristic
499C23#57614yes—
500C23#57814yes—
501Q8—28no—
502Q8—44no—
503Q8—44no—
504Q8—44no—
505D4—28no—
506D4—432no—
507D4—44no—
508D4—44no—
509D4—44no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—432no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—816no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—84no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—84no—
556C2×C4—82no—
557C2×C4#39914yes—
558C2×C4C2×C22⋊Q811yescharacteristic
559C2×C4C2×C22⋊Q811yescharacteristic
560C2×C4C2×C22⋊Q812yes—
561C2×C4C2×#12312yes—
562C2×C4#52511yescharacteristic
563C2×C4#52511yescharacteristic
564C2×C4#53812yes—
565C2×C4#54312yes—
566C2×C4D4212yes—
567C2×C4D4212yes—
568C2×C4#54612yes—
569C2×C4#54612yes—
570C2×C4#54612yes—
571C2×C4#54612yes—
572C2×C4#55112yes—
573C2×C4#55112yes—
574C8—44no—
575C8—44no—
576C22—24no—
577C22—28no—
578C22—44no—
579C22—42no—
580C22—42no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—48no—
587C22—84no—
588C22—82no—
589C22—82no—
590C22—84no—
591C22—84no—
592C22—84no—
593C22—84no—
594C22C2×#39911yescharacteristic
595C22#240511yescharacteristic
596C22#243311yescharacteristic
597C22#248211yescharacteristic
598C22#253911yescharacteristic
599C22#262611yescharacteristic
600C22#284811yescharacteristic
601C22#300812yes—
602C22#301112yes—
603C22#308314yes—
604C22#309814yes—
605C22#312414yes—
606C22#315512yes—
607C22#315612yes—
608C22#320314yes—
609C22#322012yes—
610C22#322312yes—
611C4—24no—
612C4—44no—
613C4—44no—
614C4—82no—
615C4—82no—
616C4—82no—
617C4—162no—
618C4—162no—
619C4#241812yes—
620C4#241812yes—
621C4#243712yes—
622C4#243712yes—
623C2—44no—
624C2—84no—
625C2#1473411yescharacteristic
626C2#2373712yes—
627C2#2401112yes—
628C2#2573411yescharacteristic
629C2#2573511yescharacteristic
630C2#2759512yes—
631C2#2832712yes—
632C2#2996412yes—
633C2#3296312yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2095C2semidirect
3. C2×#2093C2semidirect
4. C2×#1852C2semidirect
6. #14340C2semidirect
7. #11124C2semidirect
8. #11123C2semidirect
9. #11052C2semidirect
10. #11051C2semidirect
11. #10713C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23112
C2×#20952
C2×#20932
C2×#18522
C2×#17882
#143402
#111242
#111232
#110522
#110512
#107132