Finite Group Databasebeta

← #394696

Subgroups — #394696

Order 512.

2236 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C22×#389C211yesmaximal, characteristic
3C2×#2043C212yesmaximal, complemented
4C2×#1872C212yesmaximal
5C2×#1872C212yesmaximal
6#11207C212yesmaximal, complemented
7#11067C214yesmaximal, complemented
8#10712C211yesmaximal, characteristic
9C22×C4⋊Q8C2211yescharacteristic
10C22×C22⋊Q8C2212yes—
11C23×C4⋊C4—21no—
12C23×C4⋊C4C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×#103C2214yes—
16C22×#103C2214yes—
17C22×C22⋊C8C2212yes—
18#2857—28no—
19C2×#391C2212yes—
20C2×#389—21no—
21C2×#389—21no—
22C2×#389—21no—
23C2×#389—21no—
24C2×#389—28no—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#384C2211yescharacteristic
28C2×#380—21no—
29C2×#380—21no—
30C22×#95—21no—
31#2043—28no—
32#1872—28no—
33#1872—28no—
34C2×#340C2214yes—
35C2×#340C2214yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C23×Q8C2312yes—
39C24×C4—21no—
40C24×C4—21no—
41C24×C4D411yescharacteristic
42C24×C4D411yescharacteristic
43C24×C4C2311yescharacteristic
44C23×C8C2314yes—
45C2×C4⋊Q8—24no—
46C2×C4⋊Q8—24no—
47C2×C4⋊Q8—42no—
48C2×C22⋊Q8—24no—
49C2×C22⋊Q8—24no—
50C2×C22⋊Q8—24no—
51C2×C22⋊Q8—24no—
52C2×C22⋊Q8—24no—
53C2×C22⋊Q8D418yes—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—28no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—24no—
69C22×C4⋊C4—42no—
70C22×C4⋊C4—44no—
71C22×C4⋊C4—44no—
72C22×C4⋊C4D414yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C22⋊C4—22no—
77C22×C42—24no—
78C22×C42—24no—
79C22×C42—42no—
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C2×#103—28no—
85C2×#103—28no—
86C2×#103—28no—
87C2×#103—28no—
88C2×#103—28no—
89C2×#103—44no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—24no—
93#391—24no—
94#391—24no—
95#389—28no—
96#389—28no—
97#389—28no—
98#389—28no—
99#389—24no—
100#389—24no—
101#389—24no—
102#389D418yes—
103#384—24no—
104#380—28no—
105#380—28no—
106C2×#95—28no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111#340—28no—
112#340—28no—
113#336—24no—
114#336—42no—
115C25C2×D411yescharacteristic
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—24no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—24no—
123C22×Q8—24no—
124C22×Q8—24no—
125C22×Q8C2×D414yes—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—24no—
131C23×C4—24no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—24no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—24no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—41no—
147C23×C4—41no—
148C23×C4—41no—
149C23×C4—41no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4SD1614yes—
161C23×C4Q1614yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D414yes—
168C23×C4C2×D412yes—
169C23×C4#4114yes—
170C23×C4#4112yes—
171C23×C4C2411yescharacteristic
172C22×C8—24no—
173C22×C8—24no—
174C22×C8—24no—
175C22×C8—24no—
176C22×C8—24no—
177C22×C8—24no—
178C22×C8—24no—
179C4⋊Q8—416no—
180C22⋊Q8—216no—
181C22⋊Q8—216no—
182C22⋊Q8—48no—
183C22⋊Q8—48no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—216no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—216no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—416no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—432no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4C2×D414yes—
227C2×C4⋊C4C2×D414yes—
228C2×C4⋊C4C2×D414yes—
229C2×C4⋊C4C2×D414yes—
230C2×C4⋊C4#4114yes—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—416no—
244C2×C42—41no—
245C2×C42—41no—
246C2×C42C2×D414yes—
247#103—416no—
248#103—416no—
249C22⋊C8—48no—
250#95—28no—
251#95—28no—
252#95—28no—
253#95—28no—
254C24—21no—
255C24—21no—
256C24—21no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—21no—
261C24—21no—
262C24#12012yes—
263C24#12112yes—
264C24#12412yes—
265C24C2×SD1612yes—
266C24C2×Q1612yes—
267C24#13712yes—
268C24#13712yes—
269C24C22×D411yescharacteristic
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—28no—
273C2×Q8—28no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—44no—
277C2×Q8—44no—
278C2×Q8—44no—
279C2×Q8—44no—
280C2×Q8—44no—
281C2×Q8—44no—
282C2×Q8—44no—
283C2×Q8—44no—
284C2×Q8—44no—
285C2×Q8—44no—
286C22×C4—21no—
287C22×C4—21no—
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—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—28no—
303C22×C4—28no—
304C22×C4—28no—
305C22×C4—28no—
306C22×C4—216no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—24no—
311C22×C4—21no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—28no—
323C22×C4—28no—
324C22×C4—28no—
325C22×C4—28no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—48no—
340C22×C4—48no—
341C22×C4—48no—
342C22×C4—48no—
343C22×C4—48no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—48no—
349C22×C4—48no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—84no—
371C22×C4—84no—
372C22×C4—84no—
373C22×C4—84no—
374C22×C4#12012yes—
375C22×C4#12012yes—
376C22×C4#12012yes—
377C22×C4#12118yes—
378C22×C4#12112yes—
379C22×C4#12318yes—
380C22×C4#12412yes—
381C22×C4C2×SD1612yes—
382C22×C4C2×SD1612yes—
383C22×C4C2×Q1612yes—
384C22×C4C2×Q1612yes—
385C22×C4AGL(1,ℤ/8)12yes—
386C22×C4#13712yes—
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D412yes—
389C22×C4C22×D412yes—
390C22×C4C2×#4112yes—
391C22×C4C2×#4112yes—
392C22×C4C2×#4111yescharacteristic
393C22×C4H(5,𝔽2)12yes—
394C22×C42−1+412yes—
395C2×C8—44no—
396C2×C8—44no—
397C2×C8—44no—
398C2×C8—44no—
399C2×C8—44no—
400C2×C8—44no—
401C2×C8—44no—
402C4⋊C4—216no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—28no—
409C4⋊C4—48no—
410C4⋊C4—48no—
411C4⋊C4—48no—
412C4⋊C4—416no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—44no—
418C4⋊C4—416no—
419C4⋊C4—416no—
420C4⋊C4#12118yes—
421C22⋊C4—416no—
422C42—28no—
423C42—44no—
424C42—44no—
425C42—44no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
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—22no—
454C23#39414yes—
455C23#44814yes—
456C23#45114yes—
457C23#46114yes—
458C23#46214yes—
459C23#48714yes—
460C23#49114yes—
461C23C2×#12011yescharacteristic
462C23C2×#12011yescharacteristic
463C23C2×#12111yescharacteristic
464C23#52511yescharacteristic
465C23C2×#12411yescharacteristic
466C23#53411yescharacteristic
467C23#53611yescharacteristic
468C23C22×SD1611yescharacteristic
469C23C22×Q1611yescharacteristic
470C23C2×AGL(1,ℤ/8)11yescharacteristic
471C23C2×#13711yescharacteristic
472C23C2×#13711yescharacteristic
473C23C2×#13711yescharacteristic
474C23#57511yescharacteristic
475C23#57511yescharacteristic
476Q8—48no—
477Q8—48no—
478Q8—84no—
479Q8—84no—
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—216no—
499C2×C4—216no—
500C2×C4—28no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—416no—
508C2×C4—416no—
509C2×C4—432no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—44no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—816no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4#39414yes—
547C2×C4#45014yes—
548C2×C4#45114yes—
549C2×C4C2×#12112yes—
550C2×C4C2×#12112yes—
551C2×C4C2×#12312yes—
552C2×C4#52512yes—
553C2×C4#53712yes—
554C2×C4#53912yes—
555C2×C4D4214yes—
556C2×C4#54714yes—
557C2×C4#54814yes—
558C2×C4#55314yes—
559C8—84no—
560C22—24no—
561C22—24no—
562C22—24no—
563C22—24no—
564C22—24no—
565C22—24no—
566C22—24no—
567C22—24no—
568C22—24no—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22#204318yescomplemented
575C22C2×#39411yescharacteristic
576C22#240311yescharacteristic
577C22#240611yescharacteristic
578C22#243611yescharacteristic
579C22#259111yescharacteristic
580C22#261811yescharacteristic
581C22#284711yescharacteristic
582C22C2×#45011yescharacteristic
583C22C2×#44811yescharacteristic
584C22C2×#45111yescharacteristic
585C22C2×#45111yescharacteristic
586C22#300611yescharacteristic
587C22#300711yescharacteristic
588C22#300711yescharacteristic
589C22#300711yescharacteristic
590C22C2×#46111yescharacteristic
591C22C2×#46211yescharacteristic
592C22#303611yescharacteristic
593C22#303711yescharacteristic
594C22#310111yescharacteristic
595C22#310411yescharacteristic
596C22#311011yescharacteristic
597C22#311311yescharacteristic
598C22C2×#48711yescharacteristic
599C22C2×#49111yescharacteristic
600C22#314111yescharacteristic
601C22#314211yescharacteristic
602C22#318711yescharacteristic
603C22#318911yescharacteristic
604C22#319311yescharacteristic
605C22#319511yescharacteristic
606C22#322911yescharacteristic
607C22#323011yescharacteristic
608C22#323611yescharacteristic
609C22#323611yescharacteristic
610C4—24no—
611C4—24no—
612C4—24no—
613C4—42no—
614C4—42no—
615C4—48no—
616C4—84no—
617C4—84no—
618C4—84no—
619C4#241614yes—
620C4#244314yes—
621C2—28no—
622C2#1471911yescharacteristic
623C2C2×#204311yescharacteristic
624C2#2350811yescharacteristic
625C2#2391911yescharacteristic
626C2#2392111yescharacteristic
627C2#2406311yescharacteristic
628C2#2406411yescharacteristic
629C2#2590211yescharacteristic
630C2#2591211yescharacteristic
631C2#2932911yescharacteristic
632C2#2933511yescharacteristic
633C2#2975611yescharacteristic
634C2#2975711yescharacteristic
635C2#3290511yescharacteristic
636C2#3290611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
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
3. C2×#2043C2semidirect
6. #11207C2semidirect
7. #11067C2semidirect
574. C22#2043semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C22×#3892
C2×#20432
C2×#18722
#112072
#110672
#107122