Finite Group Databasebeta

← #440741

Subgroups — #440741

Order 512.

1430 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×#2503C211yesmaximal, characteristic
2C2×#2024C211yesmaximal, characteristic, complemented
3C2×#1993C211yesmaximal, characteristic, complemented
4#15125C211yesmaximal, characteristic
5#11372C211yesmaximal, characteristic, complemented
6#11268C212yesmaximal, complemented
7#11190C212yesmaximal, complemented
8#10970C212yesmaximal
9#10928C211yesmaximal, characteristic, complemented
10#10922C211yesmaximal, characteristic
11#10730C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×C22⋊C8C2212yes—
18C22×#97C2211yescharacteristic
19#2503—24no—
20#2503—24no—
21C2×#400—21no—
22C2×#400—21no—
23C2×#393—21no—
24C2×#391C2211yescharacteristic
25C2×#391C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×#381—21no—
29C2×#381—21no—
30C2×#380C2212yes—
31C2×C4×C22⋊C4C2211yescharacteristic
32#2024—24no—
33#1993—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42#1446—21no—
43#1442—21no—
44#1439—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C23×Q8—21no—
48C23×Q8C2311yescharacteristic
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—42no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—42no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—42no—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4D411yescharacteristic
76C22×C22⋊C4D411yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#103—42no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—42no—
88C2×#97—22no—
89C2×#97—42no—
90#400—28no—
91#400—28no—
92#393—44no—
93#391—22no—
94#391—22no—
95#391—22no—
96#391D414yes—
97#382—22no—
98#382—22no—
99#382—22no—
100#382D414yes—
101#381—28no—
102#381—28no—
103#380—24no—
104#380—42no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95D411yescharacteristic
121C2×#95D411yescharacteristic
122#340—24no—
123#340—24no—
124#340—24no—
125#340—42no—
126#340—42no—
127#340—42no—
128#336—24no—
129#336—22no—
130C25C2×D411yescharacteristic
131C22×Q8—21no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—44no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8—42no—
142C22×Q8—82no—
143C22×Q8C2×D412yes—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4#4114yes—
170C23×C4C2411yescharacteristic
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8#4114yes—
185C4⋊Q8—28no—
186C4⋊Q8—48no—
187C4⋊Q8—44no—
188C22⋊Q8—48no—
189C22⋊Q8—48no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—48no—
213C2×C4⋊C4—48no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4#4114yes—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—82no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244#103—48no—
245#103—48no—
246C22⋊C8—48no—
247#97—44no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—22no—
257#95—22no—
258#95—44no—
259#95—42no—
260#95—42no—
261#95—44no—
262#95—82no—
263C24—21no—
264C24—21no—
265C24—22no—
266C24—22no—
267C24—21no—
268C24—21no—
269C24—22no—
270C24—22no—
271C24#12412yes—
272C24AGL(1,ℤ/8)12yes—
273C24#13712yes—
274C24C22×D411yescharacteristic
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—48no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—84no—
293C2×Q8—84no—
294C2×Q8—88no—
295C2×Q8—88no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—48no—
340C22×C4—48no—
341C22×C4—48no—
342C22×C4—42no—
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—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—84no—
376C22×C4—84no—
377C22×C4—84no—
378C22×C4—84no—
379C22×C4#12012yes—
380C22×C4#12012yes—
381C22×C4#12412yes—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C2×#4112yes—
385C22×C4C2×#4112yes—
386C22×C4C2×#4112yes—
387C22×C4H(5,𝔽2)12yes—
388C22×C4H(5,𝔽2)12yes—
389C22×C42−1+412yes—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—48no—
410C4⋊C4—48no—
411C4⋊C4—48no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C22⋊C4—44no—
422C22⋊C4—44no—
423C22⋊C4—48no—
424C22⋊C4—88no—
425C42—22no—
426C42—22no—
427C42—24no—
428C42—41no—
429C42—41no—
430C42—42no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—24no—
439C23—22no—
440C23—22no—
441C23—24no—
442C23—24no—
443C23—42no—
444C23—42no—
445C23—41no—
446C23—41no—
447C23—42no—
448C23—42no—
449C23—41no—
450C23—41no—
451C23—42no—
452C23—42no—
453C23#48914yes—
454C23C2×#12011yescharacteristic
455C23C2×#12411yescharacteristic
456C23#53511yescharacteristic
457C23#54612yes—
458C23#54712yes—
459C23C2×AGL(1,ℤ/8)11yescharacteristic
460C23C2×#13711yescharacteristic
461C23#57511yescharacteristic
462C23#57511yescharacteristic
463C23#57712yes—
464C23#57812yes—
465Q8—44no—
466Q8—44no—
467Q8—816no—
468Q8—82no—
469Q8—82no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—816no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—82no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—82no—
525C2×C4—82no—
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#52511yescharacteristic
539C2×C4#52912yes—
540C2×C4#53211yescharacteristic
541C2×C4#53511yescharacteristic
542C2×C4#53611yescharacteristic
543C2×C4#53812yes—
544C2×C4#53912yes—
545C2×C4#54212yes—
546C2×C4#54612yes—
547C2×C4#54612yes—
548C2×C4#54612yes—
549C2×C4#54812yes—
550C2×C4#55112yes—
551C2×C4#55212yes—
552C2×C4#55912yes—
553C2×C4#56212yes—
554C8—44no—
555C8—82no—
556C22—24no—
557C22—28no—
558C22—42no—
559C22—42no—
560C22—44no—
561C22—44no—
562C22—44no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22#240411yescharacteristic
568C22#243211yescharacteristic
569C22#243511yescharacteristic
570C22#248812yes—
571C22#271012yes—
572C22#300611yescharacteristic
573C22#300711yescharacteristic
574C22#301011yescharacteristic
575C22#301111yescharacteristic
576C22C2×#48911yescharacteristic
577C22#313712yes—
578C22#314011yescharacteristic
579C22#321311yescharacteristic
580C22#321511yescharacteristic
581C22#322512yes—
582C22#331012yes—
583C22#331212yes—
584C22#332312yes—
585C22#332712yes—
586C22#334312yes—
587C22#334412yes—
588C22#334812yes—
589C22#334912yes—
590C4—42no—
591C4—42no—
592C4—42no—
593C4—82no—
594C4—82no—
595C4—82no—
596C4—82no—
597C4—81no—
598C4—81no—
599C4—84no—
600C4—162no—
601C4#245812yes—
602C4#255712yes—
603C4#268712yes—
604C4#269612yes—
605C2—44no—
606C2#1566811yescharacteristic
607C2#2398211yescharacteristic
608C2#2398311yescharacteristic
609C2#2571511yescharacteristic
610C2#2571611yescharacteristic
611C2#2586411yescharacteristic
612C2#2586611yescharacteristic
613C2#2770512yes—
614C2#2771312yes—
615C2#3206412yes—
616C2#3206812yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2024C2semidirect
3. C2×#1993C2semidirect
5. #11372C2semidirect
6. #11268C2semidirect
7. #11190C2semidirect
9. #10928C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25032
C2×#20242
C2×#19932
#151252
#113722
#112682
#111902
#109702
#109282
#109222
#107302