Finite Group Databasebeta

← #406336

Subgroups — #406336

Order 512.

1496 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#484C211yesmaximal, characteristic
2C2×#1873C211yesmaximal, characteristic
3C2×#1823C211yesmaximal, characteristic
4C2×#1798C211yesmaximal, characteristic
5#14258C211yesmaximal, characteristic
6#13795C211yesmaximal, characteristic
7#11256C212yesmaximal, complemented
8#11232C212yesmaximal, complemented
9#11190C212yesmaximal, complemented
10#11077C212yesmaximal, complemented
11#10731C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#484—24no—
17C2×#484—24no—
18C2×#484—24no—
19C22×#107C2212yes—
20C22×C4⋊C8C2212yes—
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×C22⋊C8C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25C2×#391C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#380C2211yescharacteristic
28#1873—24no—
29#1823—24no—
30#1798—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37#1441—22no—
38#1440—22no—
39C4⋊#95—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2212yes—
44C22⋊#95—22no—
45C23×Q8C2311yescharacteristic
46C24×C4—21no—
47C24×C4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8D414yes—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—24no—
59C2×#117—24no—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—42no—
63C2×#117—42no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4D414yes—
72C22×C4⋊C4D412yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4D412yes—
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85#484—232no—
86C2×#107—24no—
87C2×#107—24no—
88C2×#107—24no—
89C2×C4⋊C8—24no—
90C2×C4⋊C8—24no—
91C2×C4⋊C8—42no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—42no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8D414yes—
104#391—24no—
105#391—42no—
106#380—22no—
107#380—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—42no—
122#336—24no—
123C25C2×D411yescharacteristic
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8C2×D412yes—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—24no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—82no—
148C23×C4—82no—
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4#4112yes—
156C23×C4#4112yes—
157C23×C4#4112yes—
158C23×C4C2411yescharacteristic
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
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×C8C2×D414yes—
173C22⋊Q8—28no—
174C22⋊Q8—28no—
175C22⋊Q8—44no—
176C22⋊Q8—44no—
177#117—48no—
178#117—48no—
179#117—48no—
180#117—48no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4C2×D414yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4#4114yes—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—82no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—22no—
234C2×C42—24no—
235C2×C42—21no—
236C2×C42—22no—
237C2×C42—21no—
238C2×C42—22no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42#4114yes—
242#107—216no—
243C4⋊C8—48no—
244#103—216no—
245#103—48no—
246C22⋊C8—28no—
247C22⋊C8—44no—
248#95—44no—
249#95—42no—
250#95—44no—
251#95—82no—
252#95—82no—
253#95—82no—
254#95—82no—
255#95—82no—
256#95—82no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—21no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24#12112yes—
266C24#12112yes—
267C24#12112yes—
268C24C22×D411yescharacteristic
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—48no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—84no—
352C22×C4—84no—
353C22×C4—88no—
354C22×C4—88no—
355C22×C4#12014yes—
356C22×C4#12112yes—
357C22×C4#12112yes—
358C22×C4#12112yes—
359C22×C4#12314yes—
360C22×C4#12314yes—
361C22×C4#12314yes—
362C22×C4#12414yes—
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4112yes—
370C22×C4C2×#4112yes—
371C22×C4H(5,𝔽2)12yes—
372C22×C42−1+412yes—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—48no—
394C4⋊C4—42no—
395C4⋊C4—41no—
396C4⋊C4—42no—
397C4⋊C4—41no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—48no—
401C4⋊C4—48no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C22⋊C4—48no—
407C22⋊C4—48no—
408C22⋊C4—48no—
409C22⋊C4—84no—
410C42—24no—
411C42—24no—
412C42—48no—
413C42—42no—
414C42—42no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—24no—
423C23—22no—
424C23—22no—
425C23—24no—
426C23—22no—
427C23—24no—
428C23—24no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23#39414yes—
438C23C2×#12111yescharacteristic
439C23C2×#12111yescharacteristic
440C23C2×#12111yescharacteristic
441C23C2×#12311yescharacteristic
442C23#52511yescharacteristic
443C23#53411yescharacteristic
444C23#53611yescharacteristic
445C23#57612yes—
446C23#57712yes—
447C23#57812yes—
448C23#57812yes—
449Q8—44no—
450Q8—44no—
451Q8—82no—
452Q8—82no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—28no—
460C2×C4—28no—
461C2×C4—28no—
462C2×C4—28no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—816no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—81no—
506C2×C4—82no—
507C2×C4—81no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4#39414yes—
515C2×C4C2×#12011yescharacteristic
516C2×C4C2×#12311yescharacteristic
517C2×C4C2×#12311yescharacteristic
518C2×C4C2×#12411yescharacteristic
519C2×C4#52912yes—
520C2×C4#53212yes—
521C2×C4#53712yes—
522C2×C4#53912yes—
523C2×C4#54612yes—
524C2×C4#54612yes—
525C2×C4#54712yes—
526C2×C4#54712yes—
527C2×C4#55112yes—
528C2×C4#55112yes—
529C2×C4#55212yes—
530C2×C4#55212yes—
531C8—44no—
532C8—82no—
533C22—24no—
534C22—24no—
535C22—24no—
536C22—44no—
537C22—44no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—44no—
543C22—44no—
544C22C2×#39411yescharacteristic
545C22#241211yescharacteristic
546C22#241311yescharacteristic
547C22#248611yescharacteristic
548C22#253411yescharacteristic
549C22#261811yescharacteristic
550C22#285411yescharacteristic
551C22#304111yescharacteristic
552C22#304211yescharacteristic
553C22#304412yes—
554C22#304411yescharacteristic
555C22#304411yescharacteristic
556C22#304512yes—
557C22#305912yes—
558C22#306212yes—
559C22#309712yes—
560C22#309812yes—
561C22#312012yes—
562C22#312412yes—
563C22#320112yes—
564C22#320312yes—
565C22#324212yes—
566C22#324312yes—
567C4—24no—
568C4—44no—
569C4—82no—
570C4—84no—
571C4—82no—
572C4—82no—
573C4—84no—
574C4—161no—
575C4—161no—
576C4#241912yes—
577C4#243012yes—
578C4#248812yes—
579C4#249812yes—
580C2—44no—
581C2#1492711yescharacteristic
582C2#2354211yescharacteristic
583C2#2354411yescharacteristic
584C2#2456412yes—
585C2#2461412yes—
586C2#2765612yes—
587C2#2813112yes—
588C2#2976411yescharacteristic
589C2#2976711yescharacteristic
590C2#3311911yescharacteristic
591C2#3312011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
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
7. #11256C2semidirect
8. #11232C2semidirect
9. #11190C2semidirect
10. #11077C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4842
C2×#18732
C2×#18232
C2×#17982
#142582
#137952
#112562
#112322
#111902
#110772
#107312