Finite Group Databasebeta

← #393494

Subgroups — #393494

Order 512.

1820 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2625C211yesmaximal, characteristic
2C2×#2274C211yesmaximal, characteristic
3C2×#1873C211yesmaximal, characteristic
4C2×#1872C211yesmaximal, characteristic
5C2×#1818C211yesmaximal, characteristic
6#10932C212yesmaximal, complemented
7#10932C212yesmaximal, complemented
8#10885C212yesmaximal, complemented
9#10884C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic
11#10730C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2211yescharacteristic
18C22×#103C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21#2625—24no—
22#2625C418yescomplemented
23C2×#400—21no—
24C2×#398—21no—
25C2×#397—22no—
26C2×#391C2212yes—
27C2×#389C2212yes—
28C2×#380C2211yescharacteristic
29#2274—24no—
30#2274—24no—
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4—21no—
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C22⋊C4—22no—
36#1873—24no—
37#1872—24no—
38#1818—24no—
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2211yescharacteristic
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#336C2212yes—
47C2×#336C2212yes—
48C23×Q8C2311yescharacteristic
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8C2×C414yes—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—22no—
65C2×#117—24no—
66C2×#117—24no—
67C2×#117C2×C414yes—
68C2×#117D414yes—
69C2×#117D414yes—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4—24no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4D414yes—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4D412yes—
88C22×C42—21no—
89C22×C42—22no—
90C22×C42—22no—
91C22×C42C2312yes—
92C2×#103—24no—
93C2×#103—22no—
94C2×#103—22no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105#400—28no—
106#398—28no—
107#397—216no—
108#391—24no—
109#391C2×C418yescomplemented
110#389—24no—
111#389C2×C418yescomplemented
112#380—22no—
113#380C2×C414yes—
114C4×C4⋊C4—28no—
115C4×C4⋊C4—28no—
116C4×C4⋊C4—24no—
117C4×C4⋊C4—24no—
118C4×C4⋊C4—24no—
119C4×C4⋊C4—24no—
120C4×C22⋊C4—216no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95—22no—
127#340—24no—
128#340—22no—
129#340—22no—
130#340—24no—
131#340—24no—
132#340—24no—
133#340—24no—
134#336—24no—
135#336—24no—
136C25C2×D411yescharacteristic
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8C22×C412yes—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—28no—
155C23×C4—24no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—81no—
160C23×C4—82no—
161C23×C4—81no—
162C23×C4C22×C412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4#4112yes—
170C23×C4#4112yes—
171C23×C4C2411yescharacteristic
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22⋊Q8—28no—
190C22⋊Q8—28no—
191C22⋊Q8—44no—
192C22⋊Q8—44no—
193#117—216no—
194#117—28no—
195#117—28no—
196#117—28no—
197#117—28no—
198#117—28no—
199#117—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—216no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—82no—
231C2×C4⋊C4—82no—
232C2×C4⋊C4C22⋊C414yes—
233C2×C4⋊C4C22⋊C414yes—
234C2×C4⋊C4C22×C412yes—
235C2×C4⋊C4C22×C414yes—
236C2×C4⋊C4C2×D414yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4#4114yes—
240C2×C4⋊C4#4114yes—
241C2×C4⋊C4#4114yes—
242C2×C4⋊C4#4114yes—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—81no—
252C2×C22⋊C4—81no—
253C2×C22⋊C4C22⋊C418yes—
254C2×C42—28no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—28no—
261C2×C42—28no—
262C2×C42—82no—
263C2×C42—82no—
264C2×C42—82no—
265C2×C42—82no—
266C2×C42C22×C414yes—
267C2×C42C2×D414yes—
268#103—48no—
269#103—44no—
270#103—44no—
271C22⋊C8—44no—
272C22⋊C8—44no—
273#95—28no—
274#95—28no—
275#95—24no—
276#95—28no—
277#95—28no—
278#95—28no—
279C24—22no—
280C24—22no—
281C24—21no—
282C24—21no—
283C24—22no—
284C24—21no—
285C24—22no—
286C24—21no—
287C24C2×C22⋊C412yes—
288C24#12012yes—
289C24#12312yes—
290C24C22×D411yescharacteristic
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—216no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—84no—
375C22×C4—84no—
376C22×C4—88no—
377C22×C4—88no—
378C22×C4C2×C22⋊C414yes—
379C22×C4C2×C22⋊C414yes—
380C22×C4C2×C22⋊C412yes—
381C22×C4C4×D414yes—
382C22×C4C4×D414yes—
383C22×C4#12012yes—
384C22×C4#12114yes—
385C22×C4#12114yes—
386C22×C4#12312yes—
387C22×C4C23×C411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4112yes—
394C22×C4C2×#4112yes—
395C22×C4C2×#4112yes—
396C22×C4C2×#4112yes—
397C2×C8—42no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C4⋊C4—216no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—28no—
423C4⋊C4—28no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—48no—
433C4⋊C4—88no—
434C4⋊C4#12318yes—
435C4⋊C4#12314yes—
436C4⋊C4#12314yes—
437C22⋊C4—216no—
438C22⋊C4—216no—
439C22⋊C4—48no—
440C22⋊C4—84no—
441C42—216no—
442C42—24no—
443C42—24no—
444C42—24no—
445C42—24no—
446C42—88no—
447C42—88no—
448C23—24no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—24no—
454C23—22no—
455C23—24no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—24no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23#38614yes—
471C23C22×C22⋊C411yescharacteristic
472C23C2×#12011yescharacteristic
473C23C2×#12311yescharacteristic
474C23#52512yes—
475C23#52512yes—
476C23#57611yescharacteristic
477C23#57712yes—
478C23#57711yescharacteristic
479C23#57711yescharacteristic
480C23#57812yes—
481C23#57811yescharacteristic
482Q8—44no—
483Q8—42no—
484Q8—42no—
485Q8—82no—
486Q8—81no—
487Q8—81no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—28no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—28no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—816no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4#38614yes—
550C2×C4C2×C4×D411yescharacteristic
551C2×C4C2×C4×D411yescharacteristic
552C2×C4C4×#4112yes—
553C2×C4C4×#4112yes—
554C2×C4C2×#12111yescharacteristic
555C2×C4C2×#12111yescharacteristic
556C2×C4C2×#12311yescharacteristic
557C2×C4C2×#12311yescharacteristic
558C2×C4C2×#12312yes—
559C2×C4#52512yes—
560C2×C4#52511yescharacteristic
561C2×C4#52511yescharacteristic
562C2×C4#52912yes—
563C2×C4#52912yes—
564C2×C4#52912yes—
565C2×C4#52912yes—
566C2×C4#53212yes—
567C2×C4#53212yes—
568C8—82no—
569C8—81no—
570C8—81no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22C2×#38611yescharacteristic
583C22#229712yes—
584C22#284911yescharacteristic
585C22#284911yescharacteristic
586C22#285312yes—
587C22#289711yescharacteristic
588C22#289812yes—
589C22#289811yescharacteristic
590C22#300911yescharacteristic
591C22#301111yescharacteristic
592C22#301211yescharacteristic
593C22#301311yescharacteristic
594C22#309712yes—
595C22#309811yescharacteristic
596C22#309911yescharacteristic
597C22#311812yes—
598C22#312012yes—
599C22#312012yes—
600C22#312112yes—
601C22#312312yes—
602C22#312412yes—
603C22#312512yes—
604C22#312712yes—
605C4—24no—
606C4—44no—
607C4—44no—
608C4—44no—
609C4—44no—
610C4—48no—
611C4—82no—
612C4—84no—
613C4—82no—
614C4C4×#12312yes—
615C4C4×#12311yescharacteristic
616C4C4×#12311yescharacteristic
617C4#285412yes—
618C4#285411yescharacteristic
619C4#285411yescharacteristic
620C2—44no—
621C2#1470611yescharacteristic
622C2#2091311yescharacteristic
623C2#2091511yescharacteristic
624C2#2139712yes—
625C2#2139812yes—
626C2#3301611yescharacteristic
627C2#3301711yescharacteristic
628C2#3301811yescharacteristic
629C2#3301911yescharacteristic
630C2#3310412yes—
631C2#3310512yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×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
6. #10932C2semidirect
7. #10932C2semidirect
8. #10885C2semidirect
9. #10884C2semidirect
22. #2625C4semidirect
109. #391C2×C4semidirect
111. #389C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26252
C2×#22742
C2×#18732
C2×#18722
C2×#18182
#109322
#108852
#108842
#107312
#107302