Finite Group Databasebeta

← #470229

Subgroups — #470229

Order 512.

1634 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2C2×#2024C211yesmaximal, characteristic, complemented
3C2×#1927C211yesmaximal, characteristic, complemented
4#15177C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic, complemented
6#11200C212yesmaximal, complemented
7#11095C212yesmaximal, complemented
8#10970C212yesmaximal, complemented
9#10970C212yesmaximal, complemented
10#10964C211yesmaximal, characteristic
11#10806C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124—21no—
14C22×C22⋊Q8C2212yes—
15C22×C4×D4C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C22×#103C2211yescharacteristic
18C22×#103C2211yescharacteristic
19C22×#103C2212yes—
20C22×C22⋊C8C2212yes—
21C22×C22⋊C8C2212yes—
22#2547—24no—
23#2547—24no—
24C2×#397—22no—
25C2×#391—21no—
26C2×#391C2211yescharacteristic
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31#2024—24no—
32#1927—24no—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41#1444—21no—
42#1439—22no—
43#1438—21no—
44C4⋊#95C2211yescharacteristic
45C23×Q8—21no—
46C23×Q8C2311yescharacteristic
47C23×D4D411yescharacteristic
48C24×C4C2312yes—
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—42no—
54C2×C4⋊Q8—42no—
55C2×C4⋊Q8D414yes—
56C2×#124—28no—
57C2×#124—44no—
58C2×#124—44no—
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—42no—
62C2×C22⋊Q8—42no—
63C2×C22⋊Q8—42no—
64C2×C22⋊Q8—42no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—42no—
68C2×C4×D4—42no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4D412yes—
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84C2×#106—22no—
85C2×#106—42no—
86C2×#103—22no—
87C2×#103—22no—
88C2×#103—22no—
89C2×#103—22no—
90C2×#103—24no—
91C2×#103—42no—
92C2×#103—42no—
93C2×#103—42no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—42no—
96C2×C22⋊C8—42no—
97C2×C22⋊C8—42no—
98C2×C22⋊C8—42no—
99C2×C22⋊C8—42no—
100#397—216no—
101#391—28no—
102#391—22no—
103#391—22no—
104#380—24no—
105#380—24no—
106#380—24no—
107#380—24no—
108#380—24no—
109#380—42no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4D414yes—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—21no—
120#340—22no—
121#340—24no—
122#340—24no—
123#340—42no—
124#340—42no—
125#340—42no—
126#336—22no—
127#336—22no—
128C25C2×D412yes—
129C22×Q8—24no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—21no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—21no—
139C22×Q8—44no—
140C22×Q8—44no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8#4112yes—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—42no—
147C22×D4—42no—
148C22×D4SD1614yes—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—21no—
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2411yescharacteristic
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—21no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—21no—
189C22×C8#4112yes—
190C4⋊Q8—48no—
191C4⋊Q8—44no—
192C4⋊Q8—44no—
193#124—432no—
194C22⋊Q8—48no—
195C22⋊Q8—48no—
196C22⋊Q8—48no—
197C22⋊Q8—48no—
198C4×D4—48no—
199C4×D4—44no—
200C4×D4—44no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
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—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4C2×D414yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4#4114yes—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—82no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—24no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256#106—44no—
257#103—28no—
258#103—48no—
259#103—44no—
260C22⋊C8—48no—
261C22⋊C8—48no—
262#95—28no—
263#95—28no—
264#95—24no—
265#95—44no—
266#95—42no—
267#95—42no—
268#95—42no—
269#95—81no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24C2×SD1614yes—
285C24C22×D411yescharacteristic
286C2×Q8—22no—
287C2×Q8—24no—
288C2×Q8—22no—
289C2×Q8—24no—
290C2×Q8—48no—
291C2×Q8—48no—
292C2×Q8—41no—
293C2×Q8—42no—
294C2×Q8—41no—
295C2×Q8—42no—
296C2×Q8—48no—
297C2×Q8—41no—
298C2×Q8—42no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—88no—
308C2×Q8—88no—
309C2×D4—28no—
310C2×D4—48no—
311C2×D4—44no—
312C2×D4—48no—
313C2×D4—44no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
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—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
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—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4#12012yes—
392C22×C4#12012yes—
393C22×C4C2×SD1612yes—
394C22×C4AGL(1,ℤ/8)12yes—
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4112yes—
400C22×C4H(5,𝔽2)12yes—
401C22×C4H(5,𝔽2)12yes—
402C22×C4H(5,𝔽2)12yes—
403C22×C4H(5,𝔽2)11yescharacteristic
404C22×C42−1+411yescharacteristic
405C2×C8—22no—
406C2×C8—24no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—44no—
431C4⋊C4—48no—
432C4⋊C4—416no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—42no—
443C22⋊C4—48no—
444C22⋊C4—416no—
445C22⋊C4—416no—
446C22⋊C4—416no—
447C22⋊C4—88no—
448C42—24no—
449C42—22no—
450C42—22no—
451C42—48no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23C2×#12011yescharacteristic
488C23#53412yes—
489C23#54612yes—
490C23#54611yescharacteristic
491C23#54711yescharacteristic
492C23C22×SD1611yescharacteristic
493C23C2×AGL(1,ℤ/8)11yescharacteristic
494C23#57512yes—
495C23#57711yescharacteristic
496C23#57812yes—
497C23#57811yescharacteristic
498Q8—42no—
499Q8—42no—
500Q8—44no—
501Q8—816no—
502Q8—81no—
503Q8—81no—
504Q8—82no—
505D4—416no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—82no—
537C2×C4—82no—
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—82no—
548C2×C4—81no—
549C2×C4—82no—
550C2×C4—81no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4#45014yes—
567C2×C4#52511yescharacteristic
568C2×C4#53211yescharacteristic
569C2×C4#53412yes—
570C2×C4#53511yescharacteristic
571C2×C4#53511yescharacteristic
572C2×C4#53511yescharacteristic
573C2×C4#53711yescharacteristic
574C2×C4#53812yes—
575C2×C4#53812yes—
576C2×C4#53812yes—
577C2×C4#54212yes—
578C2×C4#54811yescharacteristic
579C2×C4#55012yes—
580C2×C4#55011yescharacteristic
581C2×C4#55112yes—
582C2×C4#55512yes—
583C2×C4H(3,𝔽4)12yes—
584C2×C4#56212yes—
585C8—42no—
586C8—81no—
587C8—82no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22#240711yescharacteristic
604C22#243111yescharacteristic
605C22#261311yescharacteristic
606C22#268412yes—
607C22#268612yes—
608C22C2×#45011yescharacteristic
609C22#300511yescharacteristic
610C22#301111yescharacteristic
611C22#301311yescharacteristic
612C22#318712yes—
613C22#319412yes—
614C22#320112yes—
615C22#320212yes—
616C22#329811yescharacteristic
617C22#329812yes—
618C22#331112yes—
619C22#331312yes—
620C22#331312yes—
621C22#333012yes—
622C22#333211yescharacteristic
623C22#333511yescharacteristic
624C22#334912yes—
625C22#335311yescharacteristic
626C4—24no—
627C4—42no—
628C4—82no—
629C4—82no—
630C4—84no—
631C4—82no—
632C4—84no—
633C4—82no—
634C4—161no—
635C4#261011yescharacteristic
636C4#263111yescharacteristic
637C4#266211yescharacteristic
638C4#273012yes—
639C4#273312yes—
640C4#281911yescharacteristic
641C2—84no—
642C2#1668011yescharacteristic
643C2#2412011yescharacteristic
644C2#2413511yescharacteristic
645C2#2557111yescharacteristic
646C2#2558011yescharacteristic
647C2#2959611yescharacteristic
648C2#2960511yescharacteristic
649C2#3126212yes—
650C2#3127812yes—
651C2#3132012yes—
652C2#3134312yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ 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×#1927C2semidirect
5. #11384C2semidirect
6. #11200C2semidirect
7. #11095C2semidirect
8. #10970C2semidirect
9. #10970C2semidirect
11. #10806C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#20242
C2×#19272
#151772
#113842
#112002
#110952
#109702
#109642
#108062