Finite Group Databasebeta

← #368041

Subgroups — #368041

Order 512.

1862 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2323C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic
3C2×#1873C211yesmaximal, characteristic
4C2×#1872C211yesmaximal, characteristic
5C2×#1865C212yesmaximal, complemented
6C2×#1853C211yesmaximal, characteristic
7#11327C212yesmaximal, complemented
8#11067C212yesmaximal, complemented
9#10885C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic
11#10713C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C22×#106C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22#2323—24no—
23#2323—24no—
24#2317—28no—
25C2×#391C2211yescharacteristic
26C2×#389—21no—
27C2×#389C2212yes—
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#388—21no—
31C2×#388—21no—
32C2×#387—21no—
33C2×#385—21no—
34C2×#385—21no—
35C2×#384—21no—
36C2×#382—21no—
37C2×#380C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39#1873C418yescomplemented
40#1872—24no—
41#1865—28no—
42#1853—24no—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47C2×#340C2212yes—
48C2×#336C2212yes—
49C2×#336C2212yes—
50C23×Q8C2311yescharacteristic
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C23×C8C2312yes—
54C23×C8C2312yes—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8C2×C414yes—
59C2×C22⋊Q8D414yes—
60C2×C22⋊Q8Q814yes—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—24no—
64C2×C4×Q8—42no—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—24no—
69C2×#117—24no—
70C2×#117—24no—
71C2×#117—42no—
72C2×#117C2×C414yes—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—42no—
83C22×C4⋊C4D411yescharacteristic
84C22×C4⋊C4D411yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4D411yescharacteristic
94C22×C22⋊C4D411yescharacteristic
95C22×C42—21no—
96C22×C42—21no—
97C22×C42—21no—
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100C2×#106—24no—
101C2×#106—24no—
102C2×#106—42no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—24no—
106C2×#103—24no—
107C2×#103—42no—
108C2×#103C2×C418yescomplemented
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—22no—
114C2×C22⋊C8C2×C414yes—
115#391—22no—
116#391—22no—
117#389—28no—
118#389—24no—
119#389—24no—
120#389—22no—
121#389—22no—
122#389—24no—
123#389D418yes—
124#388—28no—
125#388—28no—
126#387—28no—
127#385—28no—
128#385—28no—
129#384—28no—
130#382—28no—
131#380—22no—
132#380D414yes—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—21no—
142#340—24no—
143#340—24no—
144#340—24no—
145#340—24no—
146#340C2×C418yescomplemented
147#336—24no—
148#336—42no—
149C25C2×D411yescharacteristic
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8C22×C412yes—
158C22×Q8C2×D412yes—
159C22×Q8C2×Q812yes—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—82no—
174C23×C4—82no—
175C23×C4C22×C412yes—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D412yes—
183C23×C4C2×Q812yes—
184C23×C4#4112yes—
185C23×C4#4112yes—
186C23×C4C2411yescharacteristic
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8C22×C414yes—
201C22⋊Q8—28no—
202C22⋊Q8—28no—
203C22⋊Q8—28no—
204C22⋊Q8C4⋊C4116yescomplemented
205C4×Q8—48no—
206C4×Q8—48no—
207#117—216no—
208#117—216no—
209#117—48no—
210#117—48no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
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—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—48no—
241C2×C4⋊C4—48no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—82no—
249C2×C4⋊C4—82no—
250C2×C4⋊C4—82no—
251C2×C4⋊C4—82no—
252C2×C4⋊C4—82no—
253C2×C4⋊C4C22×C412yes—
254C2×C4⋊C4C22×C414yes—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×Q812yes—
261C2×C4⋊C4#4114yes—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—82no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—42no—
289C2×C42—42no—
290C2×C42—82no—
291C2×C42C2×D414yes—
292#106—48no—
293#103—216no—
294#103—48no—
295C22⋊C8—28no—
296C22⋊C8—44no—
297#95—24no—
298#95—28no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—28no—
304C24—21no—
305C24—21no—
306C24—21no—
307C24—21no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24C4×D412yes—
313C24#12012yes—
314C24C22⋊Q812yes—
315C24C22×D411yescharacteristic
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—24no—
322C2×Q8—24no—
323C2×Q8—24no—
324C2×Q8—24no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8C2×C4⋊C414yes—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—48no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—88no—
404C22×C4—84no—
405C22×C4—84no—
406C22×C4—88no—
407C22×C4C2×C4⋊C414yes—
408C22×C4C4×D414yes—
409C22×C4C4×D412yes—
410C22×C4#12012yes—
411C22×C4#12012yes—
412C22×C4#12012yes—
413C22×C4C22⋊Q814yes—
414C22×C4C22⋊Q812yes—
415C22×C4#12314yes—
416C22×C4C23×C411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D412yes—
420C22×C4C22×Q811yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4112yes—
423C22×C4C2×#4111yescharacteristic
424C22×C4H(5,𝔽2)12yes—
425C22×C42−1+412yes—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—216no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—88no—
460C4⋊C4—84no—
461C4⋊C4—88no—
462C4⋊C4C2×C4⋊C414yes—
463C4⋊C4C4×D418yes—
464C22⋊C4—216no—
465C22⋊C4—216no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—84no—
469C42—216no—
470C42—24no—
471C42—24no—
472C42—48no—
473C42—42no—
474C42—42no—
475C42—84no—
476C23—24no—
477C23—24no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—24no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23#38514yes—
499C23C2×C4×D411yescharacteristic
500C23C2×#12011yescharacteristic
501C23C2×#12011yescharacteristic
502C23C2×C22⋊Q811yescharacteristic
503C23#52511yescharacteristic
504C23#53411yescharacteristic
505C23#53611yescharacteristic
506C23#57712yes—
507C23#57712yes—
508C23#57712yes—
509C23#57812yes—
510Q8—28no—
511Q8—44no—
512Q8—44no—
513Q8—44no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—816no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4#38514yes—
582C2×C4C22×C4⋊C411yescharacteristic
583C2×C4C2×C4×D412yes—
584C2×C4C2×C4×D411yescharacteristic
585C2×C4#52012yes—
586C2×C4C2×C22⋊Q811yescharacteristic
587C2×C4C2×#12311yescharacteristic
588C2×C4#52512yes—
589C2×C4#53912yes—
590C2×C4D4212yes—
591C2×C4D4212yes—
592C2×C4#54712yes—
593C2×C4#54712yes—
594C2×C4#54812yes—
595C2×C4D4×Q812yes—
596C2×C4#55312yes—
597C2×C4#55412yes—
598C8—44no—
599C8—82no—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—42no—
604C22—42no—
605C22—44no—
606C22—44no—
607C22—42no—
608C22—42no—
609C22—44no—
610C22—44no—
611C22C2×#38511yescharacteristic
612C22C4×#12011yescharacteristic
613C22#232211yescharacteristic
614C22#240311yescharacteristic
615C22#240611yescharacteristic
616C22#240911yescharacteristic
617C22#257911yescharacteristic
618C22#294712yes—
619C22#295212yes—
620C22#300911yescharacteristic
621C22#300912yes—
622C22#301011yescharacteristic
623C22#301011yescharacteristic
624C22#301111yescharacteristic
625C22#301312yes—
626C22#308212yes—
627C22#308412yes—
628C22#312012yes—
629C22#312512yes—
630C22#320012yes—
631C22#320112yes—
632C22#324212yes—
633C22#324412yes—
634C4—24no—
635C4—42no—
636C4—42no—
637C4—44no—
638C4—44no—
639C4—48no—
640C4—82no—
641C4—81no—
642C4—84no—
643C4—81no—
644C4—82no—
645C4D4×C4⋊C412yes—
646C4#236112yes—
647C4#244312yes—
648C4#244912yes—
649C2—44no—
650C2#1401511yescharacteristic
651C2#2087711yescharacteristic
652C2#2088211yescharacteristic
653C2#2146112yes—
654C2#2209212yes—
655C2#2392611yescharacteristic
656C2#2392711yescharacteristic
657C2#2407111yescharacteristic
658C2#2407411yescharacteristic
659C2#2431012yes—
660C2#2925112yes—
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
5. C2×#1865C2semidirect
7. #11327C2semidirect
8. #11067C2semidirect
9. #10885C2semidirect
39. #1873C4semidirect
108. C2×#103C2×C4semidirect
146. #340C2×C4semidirect
204. C22⋊Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23232
C2×#23172
C2×#18732
C2×#18722
C2×#18652
C2×#18532
#113272
#110672
#108852
#107312
#107132