Finite Group Databasebeta

← #460482

Subgroups — #460482

Order 512.

1506 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2452C211yesmaximal, characteristic
2C2×#2083C211yesmaximal, characteristic, complemented
3C2×#1802C211yesmaximal, characteristic
4#15151C211yesmaximal, characteristic, complemented
5#11384C212yesmaximal, complemented
6#11328C212yesmaximal, complemented
7#11227C211yesmaximal, characteristic, complemented
8#11202C211yesmaximal, characteristic, complemented
9#11123C212yesmaximal
10#11066C212yesmaximal
11#10788C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8—21no—
14C22×C4×D4C2211yescharacteristic
15C22×#107C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×#102C2212yes—
18C22×#102C2212yes—
19#2452—24no—
20#2452—24no—
21C2×#398—21no—
22C2×#398—21no—
23C2×#396—21no—
24C2×#393—21no—
25C2×#390C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#385—21no—
31C2×#384C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#2083—24no—
34#1802—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1445—21no—
44#1445—21no—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2212yes—
47C23×Q8—21no—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—42no—
56C2×C22⋊Q8—28no—
57C2×C22⋊Q8—44no—
58C2×C22⋊Q8—44no—
59C2×C4×D4—24no—
60C2×C4×D4—22no—
61C2×C4×D4—42no—
62C2×C4×D4D414yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C2×#107—24no—
85C2×#107—42no—
86C2×#107—42no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—42no—
89C2×#102—24no—
90C2×#102—42no—
91C2×#102—42no—
92C2×#102—42no—
93C2×#102—42no—
94C2×#102Q818yescomplemented
95#398—44no—
96#398—44no—
97#396—28no—
98#393—28no—
99#390—22no—
100#390—22no—
101#390—22no—
102#390—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106#389—42no—
107#389—42no—
108#389D414yes—
109#385—28no—
110#384—22no—
111#384—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95D411yescharacteristic
122C2×#95D411yescharacteristic
123#340—24no—
124#340—24no—
125#340—24no—
126#340—42no—
127#340—42no—
128#336—22no—
129#336—42no—
130#336—42no—
131C25—21no—
132C22×Q8—24no—
133C22×Q8—44no—
134C22×Q8—42no—
135C22×Q8—42no—
136C22×Q8—82no—
137C22×D4—21no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—22no—
145C22×D4C2×D412yes—
146C22×D4C2×Q814yes—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—44no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—82no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2411yescharacteristic
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×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8C2×Q814yes—
178C4⋊Q8—48no—
179C4⋊Q8—44no—
180C4⋊Q8—44no—
181C22⋊Q8—432no—
182C4×D4—28no—
183C4×D4—44no—
184C4×D4—48no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
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—22no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4D814yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×Q814yes—
233C2×C4⋊C4#4114yes—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—82no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42C2×D412yes—
260C2×C42C2×D412yes—
261#107—48no—
262C4⋊C8—44no—
263#102—48no—
264#102—48no—
265#95—24no—
266#95—24no—
267#95—22no—
268#95—22no—
269#95—44no—
270#95—42no—
271#95—42no—
272#95—44no—
273#95—44no—
274C24—22no—
275C24—24no—
276C24—41no—
277C24—41no—
278C24—42no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—41no—
283C24—42no—
284C24—42no—
285C24C22×D411yescharacteristic
286C2×Q8—48no—
287C2×Q8—48no—
288C2×Q8—88no—
289C2×Q8—84no—
290C2×Q8—88no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4C4⋊Q814yes—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—48no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—48no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—84no—
368C22×C4—88no—
369C22×C4—88no—
370C22×C4#12012yes—
371C22×C4#12012yes—
372C22×C4C2×D812yes—
373C22×C4C2×D812yes—
374C22×C4C2×D812yes—
375C22×C4AGL(1,ℤ/8)12yes—
376C22×C4AGL(1,ℤ/8)12yes—
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×Q812yes—
380C22×C4C2×#4112yes—
381C22×C4H(5,𝔽2)12yes—
382C22×C4H(5,𝔽2)11yescharacteristic
383C22×C4H(5,𝔽2)11yescharacteristic
384C22×C4H(5,𝔽2)11yescharacteristic
385C22×C42−1+412yes—
386C22×C42−1+411yescharacteristic
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—28no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—44no—
410C4⋊C4—48no—
411C4⋊C4—48no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—48no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—88no—
424C4⋊C4—84no—
425C4⋊C4—84no—
426C4⋊C4#12314yes—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—48no—
431C22⋊C4—84no—
432C42—22no—
433C42—22no—
434C42—24no—
435C42—41no—
436C42—41no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—42no—
442C23—24no—
443C23—44no—
444C23—42no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—42no—
449C23—44no—
450C23—44no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—81no—
456C23—81no—
457C23—82no—
458C23—81no—
459C23—81no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—82no—
465C23C2×#12011yescharacteristic
466C23#53511yescharacteristic
467C23#53711yescharacteristic
468C23#54612yes—
469C23D4×Q812yes—
470C23C22×D811yescharacteristic
471C23C2×AGL(1,ℤ/8)11yescharacteristic
472C23C2×AGL(1,ℤ/8)11yescharacteristic
473C23#57511yescharacteristic
474C23#57712yes—
475C23#57812yes—
476Q8—816no—
477D4—24no—
478D4—24no—
479D4—42no—
480D4—42no—
481D4—44no—
482D4—44no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—816no—
529C2×C4—82no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—82no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4#44714yes—
547C2×C4C2×#12311yescharacteristic
548C2×C4C2×C4⋊Q811yescharacteristic
549C2×C4#53411yescharacteristic
550C2×C4#53511yescharacteristic
551C2×C4#53511yescharacteristic
552C2×C4#53611yescharacteristic
553C2×C4#53711yescharacteristic
554C2×C4#53711yescharacteristic
555C2×C4#54112yes—
556C2×C4#54212yes—
557C2×C4#54312yes—
558C2×C4#54412yes—
559C2×C4#54612yes—
560C2×C4D4×Q812yes—
561C2×C4#55412yes—
562C2×C4#55412yes—
563C2×C4#56212yes—
564C2×C4#56212yes—
565C8—44no—
566C8—82no—
567C22—44no—
568C22—44no—
569C22—44no—
570C22—84no—
571C22—84no—
572C22—84no—
573C22—84no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22#240511yescharacteristic
579C22#243411yescharacteristic
580C22#261511yescharacteristic
581C22#269212yes—
582C22#270412yes—
583C22C2×#44711yescharacteristic
584C22#300611yescharacteristic
585C22#300911yescharacteristic
586C22#301111yescharacteristic
587C22#320511yescharacteristic
588C22#321411yescharacteristic
589C22#322512yes—
590C22#324511yescharacteristic
591C22#325311yescharacteristic
592C22#325812yes—
593C22#329412yes—
594C22#330812yes—
595C22#331012yes—
596C22#332712yes—
597C22Q8×D812yes—
598C22#338412yes—
599C22#338612yes—
600C22#338812yes—
601C4—24no—
602C4—42no—
603C4—42no—
604C4—44no—
605C4—42no—
606C4—42no—
607C4—84no—
608C4—82no—
609C4—82no—
610C4—81no—
611C4—162no—
612C4#250211yescharacteristic
613C4#262511yescharacteristic
614C4#265911yescharacteristic
615C4#269212yes—
616C4#270412yes—
617C4#281611yescharacteristic
618C2—84no—
619C2#1636511yescharacteristic
620C2#2403211yescharacteristic
621C2#2405111yescharacteristic
622C2#2583811yescharacteristic
623C2#2585111yescharacteristic
624C2#2966011yescharacteristic
625C2#2967211yescharacteristic
626C2#3151812yes—
627C2#3154612yes—
628C2#3186612yes—
629C2#3188612yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2083C2semidirect
4. #15151C2semidirect
5. #11384C2semidirect
6. #11328C2semidirect
7. #11227C2semidirect
8. #11202C2semidirect
94. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24522
C2×#20832
C2×#18022
#151512
#113842
#113282
#112272
#112022
#111232
#110662
#107882