Finite Group Databasebeta

← #399159

Subgroups — #399159

Order 512.

1870 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2285C211yesmaximal, characteristic, complemented
2C2×#2094C211yesmaximal, characteristic, complemented
3C2×#2093C211yesmaximal, characteristic, complemented
4C2×#1875C212yesmaximal
5C2×#1854C212yesmaximal, complemented
6#14698C211yesmaximal, characteristic
7#11373C211yesmaximal, characteristic, complemented
8#11371C211yesmaximal, characteristic, complemented
9#11325C212yesmaximal, complemented
10#11051C212yesmaximal, complemented
11#10712C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C22×#117C2212yes—
14C23×C42C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×C22⋊C8C2212yes—
20C2×#390C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#386—22no—
23#2285—24no—
24#2285—24no—
25C2×#384C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#382C2212yes—
28C2×#380—22no—
29C2×#380C2211yescharacteristic
30C2×C4×C4⋊C4—22no—
31#2094—24no—
32#2093—24no—
33#1875—28no—
34#1854—28no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2211yescharacteristic
42#1437—22no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C23×D4—21no—
46C23×D4C2311yescharacteristic
47C24×C4D412yes—
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×#121—22no—
52C2×#121—24no—
53C2×#121—24no—
54C2×#121Q814yes—
55C2×#117—24no—
56C2×#117—24no—
57C2×#117—24no—
58C2×#117—24no—
59C2×#117—42no—
60C2×#117—42no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C22⋊C4—22no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C42—24no—
74C22×C42—22no—
75C22×C42—22no—
76C22×C42—24no—
77C22×C42—22no—
78C22×C42D412yes—
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C2×#107—24no—
82C2×#107—24no—
83C2×#107—42no—
84C2×#106—24no—
85C2×#106—24no—
86C2×#106—42no—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—42no—
92C2×#102—42no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8D418yes—
96#390—22no—
97#390—22no—
98#390—24no—
99#386—48no—
100#384—22no—
101#384—22no—
102#384D414yescomplemented
103#384D414yescomplemented
104#382—24no—
105#382—24no—
106#380—216no—
107#380—24no—
108C4×C4⋊C4—216no—
109C2×#95—24no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113#340—24no—
114#340—24no—
115#340—24no—
116#340—24no—
117#336—22no—
118#336—22no—
119#336—24no—
120C25—21no—
121C25C2×D411yescharacteristic
122C22×D4—24no—
123C22×D4—22no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—21no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—44no—
133C22×D4—44no—
134C22×D4—42no—
135C22×D4—42no—
136C22×D4C2×Q812yes—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D414yes—
159C23×C4C2×Q812yes—
160C23×C4#4114yes—
161C23×C4#4112yes—
162C23×C4C2411yescharacteristic
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8C2×D414yes—
176C22×C8C2×D414yes—
177#121—216no—
178#121—48no—
179#117—48no—
180#117—48no—
181#117—48no—
182#117—48no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
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—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4C2×D414yes—
211C2×C4⋊C4C2×D414yes—
212C2×C4⋊C4C2×Q812yes—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C42—216no—
228C2×C42—216no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—28no—
238C2×C42—44no—
239C2×C42—44no—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42#4114yes—
243#107—48no—
244#106—48no—
245#102—48no—
246#102—48no—
247C22⋊C8—216no—
248#95—216no—
249#95—44no—
250#95—42no—
251#95—44no—
252C24—21no—
253C24—21no—
254C24—21no—
255C24—21no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—41no—
262C24—41no—
263C24—41no—
264C24—41no—
265C24—41no—
266C24—41no—
267C24—42no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24C22⋊Q812yes—
272C24#12312yes—
273C24#12712yes—
274C24C22×D411yescharacteristic
275C2×D4—24no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—24no—
279C2×D4—416no—
280C2×D4—48no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—42no—
293C2×D4—88no—
294C2×D4—84no—
295C2×D4—84no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—216no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—48no—
329C22×C4—48no—
330C22×C4—48no—
331C22×C4—48no—
332C22×C4—48no—
333C22×C4—48no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—44no—
341C22×C4—42no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4#12014yes—
357C22×C4C22⋊Q818yes—
358C22×C4C22⋊Q812yes—
359C22×C4#12314yes—
360C22×C4#12318yes—
361C22×C4#12312yes—
362C22×C4#12712yes—
363C22×C4#13514yes—
364C22×C4AGL(1,ℤ/8)14yes—
365C22×C4C22×D412yes—
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×Q811yescharacteristic
369C22×C4C2×#4112yes—
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4112yes—
372C22×C4H(5,𝔽2)12yes—
373C22×C42−1+411yescharacteristic
374C22×C42−1+411yescharacteristic
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
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—416no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—416no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C22⋊C4—416no—
406C22⋊C4—48no—
407C22⋊C4—88no—
408C22⋊C4—84no—
409C22⋊C4—88no—
410C42—216no—
411C42—24no—
412C42—24no—
413C42—24no—
414C42—24no—
415C42—416no—
416C23—24no—
417C23—24no—
418C23—24no—
419C23—24no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—44no—
439C23—44no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23—81no—
448C23—81no—
449C23—81no—
450C23—81no—
451C23—81no—
452C23—81no—
453C23—81no—
454C23—82no—
455C23—82no—
456C23—82no—
457C23—82no—
458C23—82no—
459C23—82no—
460C23—81no—
461C23#39914yes—
462C23C2×#12011yescharacteristic
463C23C2×C22⋊Q811yescharacteristic
464C23C2×#12311yescharacteristic
465C23C2×#12311yescharacteristic
466C23C2×#12711yescharacteristic
467C23#53511yescharacteristic
468C23#53611yescharacteristic
469C23C2×#13512yes—
470C23C2×AGL(1,ℤ/8)12yes—
471C23#57512yes—
472C23#57512yes—
473D4—44no—
474D4—44no—
475D4—816no—
476D4—82no—
477D4—82no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—28no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—432no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—48no—
507C2×C4—48no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—81no—
513C2×C4—82no—
514C2×C4—81no—
515C2×C4—84no—
516C2×C4—84no—
517C2×C4—82no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—81no—
527C2×C4#39914yes—
528C2×C4#44918yes—
529C2×C4C2×C22⋊Q812yes—
530C2×C4C2×#12312yes—
531C2×C4#52512yes—
532C2×C4#52512yes—
533C2×C4#53912yes—
534C2×C4#53912yes—
535C2×C4D4212yes—
536C2×C4D4212yes—
537C2×C4#54712yes—
538C2×C4#54712yes—
539C2×C4#54811yescharacteristic
540C2×C4#54811yescharacteristic
541C2×C4D4×Q811yescharacteristic
542C2×C4D4×Q811yescharacteristic
543C2×C4#55312yes—
544C2×C4#55412yes—
545C8—44no—
546C8—44no—
547C22—24no—
548C22—24no—
549C22—24no—
550C22—42no—
551C22—42no—
552C22—44no—
553C22—42no—
554C22—42no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22—84no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—84no—
563C22—161no—
564C22—161no—
565C22—161no—
566C22—161no—
567C22—162no—
568C22—162no—
569C22C2×#39911yescharacteristic
570C22#240911yescharacteristic
571C22#243311yescharacteristic
572C22#248711yescharacteristic
573C22#250111yescharacteristic
574C22#262311yescharacteristic
575C22#284911yescharacteristic
576C22C2×#44912yes—
577C22#300512yes—
578C22#307612yes—
579C22#307712yes—
580C22#309112yes—
581C22#309112yes—
582C22#309412yes—
583C22#309412yes—
584C22#314711yescharacteristic
585C22#315011yescharacteristic
586C22#315111yescharacteristic
587C22#315311yescharacteristic
588C22#320712yes—
589C22#321912yes—
590C22#323312yes—
591C22#323512yes—
592C4—24no—
593C4—28no—
594C4—44no—
595C4—84no—
596C4—84no—
597C4—82no—
598C4—161no—
599C4—161no—
600C4#244312yes—
601C4#244312yes—
602C4#244912yes—
603C4#244912yes—
604C2—44no—
605C2—162no—
606C2#1477711yescharacteristic
607C2#2373011yescharacteristic
608C2#2373111yescharacteristic
609C2#2430012yes—
610C2#2572011yescharacteristic
611C2#2572711yescharacteristic
612C2#2768312yes—
613C2#2791111yescharacteristic
614C2#2791711yescharacteristic
615C2#2991412yes—
616C2#3298012yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×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
1. C2×#2285C2semidirect
2. C2×#2094C2semidirect
3. C2×#2093C2semidirect
5. C2×#1854C2semidirect
7. #11373C2semidirect
8. #11371C2semidirect
9. #11325C2semidirect
10. #11051C2semidirect
11. #10712C2semidirect
102. #384D4semidirect
103. #384D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22852
C2×#20942
C2×#20932
C2×#18752
C2×#18542
#146982
#113732
#113712
#113252
#110512
#107122