Finite Group Databasebeta

← #405402

Subgroups — #405402

Order 512.

1482 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#485C211yesmaximal, characteristic
2C2×#1892C211yesmaximal, characteristic
3C2×#1850C211yesmaximal, characteristic
4C2×#1735C211yesmaximal, characteristic
5#14138C211yesmaximal, characteristic
6#13939C211yesmaximal, characteristic
7#11256C212yesmaximal, complemented
8#11231C212yesmaximal, complemented
9#11190C212yesmaximal, complemented
10#11077C212yesmaximal, complemented
11#10731C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C2×#485—24no—
17C2×#485—24no—
18C2×#485—24no—
19C22×#107C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×C4⋊C8C2212yes—
22C22×#103C2211yescharacteristic
23C22×#103C2211yescharacteristic
24C22×#103C2212yes—
25C22×C22⋊C8C2211yescharacteristic
26C2×#391C2212yes—
27C2×#380C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29#1892—24no—
30#1850—24no—
31#1735—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38#1440—21no—
39#1440—21no—
40#1440—21no—
41#1440—21no—
42#1436—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46#1434—22no—
47C23×Q8C2311yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8—22no—
50C23×C8—24no—
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—42no—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C23×C8C2312yes—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8D414yes—
63C2×#117—22no—
64C2×#117—22no—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—24no—
71C2×#117—42no—
72C2×#117D414yes—
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4D411yescharacteristic
84C22×C22⋊C4D411yescharacteristic
85C22×C42—21no—
86C22×C42—21no—
87C22×C42C2312yes—
88#485—232no—
89C2×#107—22no—
90C2×#107—22no—
91C2×#107—22no—
92C2×#106—22no—
93C2×#106—22no—
94C2×#106—22no—
95C2×C4⋊C8—24no—
96C2×C4⋊C8—24no—
97C2×C4⋊C8—42no—
98C2×#103—22no—
99C2×#103—22no—
100C2×#103—22no—
101C2×#103—22no—
102C2×#103—22no—
103C2×#103—22no—
104C2×#103—24no—
105C2×#103—24no—
106C2×#103—42no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8D414yes—
110#391—24no—
111#391—42no—
112#380—22no—
113#380—22no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95D412yes—
126#340—24no—
127#340—24no—
128#340—24no—
129#340—24no—
130#340—42no—
131#336—24no—
132C25C2×D411yescharacteristic
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8C2×D412yes—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4#4112yes—
160C23×C4#4112yes—
161C23×C4#4112yes—
162C23×C4C2411yescharacteristic
163C22×C8—28no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—48no—
179C22×C8—48no—
180C22×C8—44no—
181C22×C8—44no—
182C22×C8C2×D412yes—
183C22×C8C2×D412yes—
184C22×C8#4114yes—
185C22⋊Q8—28no—
186C22⋊Q8—28no—
187C22⋊Q8—44no—
188C22⋊Q8—44no—
189#117—28no—
190#117—28no—
191#117—28no—
192#117—48no—
193#117—44no—
194#117—44no—
195#117—44no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4#4114yes—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—82no—
243C2×C42—24no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—44no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—42no—
254#107—28no—
255#106—28no—
256C4⋊C8—48no—
257#103—28no—
258#103—28no—
259#103—48no—
260C22⋊C8—28no—
261#95—24no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—42no—
267#95—44no—
268#95—44no—
269#95—82no—
270#95—82no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24#12112yes—
278C24#12112yes—
279C24#12112yes—
280C24C22×D411yescharacteristic
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—42no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
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#12014yes—
363C22×C4#12112yes—
364C22×C4#12112yes—
365C22×C4#12112yes—
366C22×C4#12314yes—
367C22×C4#12314yes—
368C22×C4#12414yes—
369C22×C4#13514yes—
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4112yes—
377C22×C4C2×#4112yes—
378C22×C4H(5,𝔽2)12yes—
379C22×C42−1+412yes—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—416no—
388C2×C8—42no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—42no—
393C2×C8—41no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—48no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C22⋊C4—28no—
420C22⋊C4—44no—
421C22⋊C4—48no—
422C22⋊C4—48no—
423C22⋊C4—48no—
424C22⋊C4—84no—
425C42—48no—
426C42—42no—
427C42—42no—
428C42—42no—
429C42—42no—
430C42—48no—
431C23—22no—
432C23—22no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—22no—
438C23—22no—
439C23—24no—
440C23—22no—
441C23—22no—
442C23—24no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23#39414yes—
452C23C2×#12111yescharacteristic
453C23C2×#12111yescharacteristic
454C23C2×#12111yescharacteristic
455C23#54612yes—
456C23#54712yes—
457C23C2×#13512yes—
458C23#57512yes—
459C23#57712yes—
460C23#57812yes—
461Q8—42no—
462Q8—42no—
463Q8—44no—
464Q8—81no—
465Q8—81no—
466Q8—82no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—28no—
476C2×C4—28no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4#39414yes—
526C2×C4C2×#12011yescharacteristic
527C2×C4C2×#12311yescharacteristic
528C2×C4C2×#12311yescharacteristic
529C2×C4#52511yescharacteristic
530C2×C4#52511yescharacteristic
531C2×C4C2×#12411yescharacteristic
532C2×C4#52912yes—
533C2×C4#52912yes—
534C2×C4#53511yescharacteristic
535C2×C4#53611yescharacteristic
536C2×C4#53912yes—
537C2×C4#53912yes—
538C2×C4#54612yes—
539C2×C4#54612yes—
540C2×C4#54712yes—
541C2×C4#54712yes—
542C2×C4#55112yes—
543C2×C4#55212yes—
544C8—42no—
545C8—44no—
546C8—42no—
547C22—24no—
548C22—24no—
549C22—24no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—44no—
556C22C2×#39411yescharacteristic
557C22#241111yescharacteristic
558C22#241311yescharacteristic
559C22#242512yes—
560C22#242512yes—
561C22#303512yes—
562C22#303512yes—
563C22#304311yescharacteristic
564C22#304411yescharacteristic
565C22#304612yes—
566C22#305011yescharacteristic
567C22#305411yescharacteristic
568C22#306312yes—
569C22#330012yes—
570C22#330312yes—
571C22#331512yes—
572C22#332112yes—
573C22#333712yes—
574C22#334212yes—
575C22#334912yes—
576C22#335012yes—
577C4—24no—
578C4—44no—
579C4—82no—
580C4—82no—
581C4—84no—
582C4—82no—
583C4—82no—
584C4—82no—
585C4—162no—
586C4#240411yescharacteristic
587C4#240611yescharacteristic
588C4#241912yes—
589C4#248812yes—
590C4#262311yescharacteristic
591C4#285311yescharacteristic
592C2—44no—
593C2#1491211yescharacteristic
594C2#2353911yescharacteristic
595C2#2354511yescharacteristic
596C2#2448712yes—
597C2#2458412yes—
598C2#2520112yes—
599C2#2521312yes—
600C2#2523912yes—
601C2#2525312yes—
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
7. #11256C2semidirect
8. #11231C2semidirect
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×#4852
C2×#18922
C2×#18502
C2×#17352
#141382
#139392
#112562
#112312
#111902
#110772
#107312