Finite Group Databasebeta

← #464087

Subgroups — #464087

Order 512.

1594 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×#2538C211yesmaximal, characteristic
2C2×#2083C211yesmaximal, characteristic, complemented
3C2×#2036C212yesmaximal, complemented
4C2×#1865C211yesmaximal, characteristic, complemented
5#15129C211yesmaximal, characteristic
6#11373C211yesmaximal, characteristic, complemented
7#11096C212yesmaximal, complemented
8#11095C212yesmaximal, complemented
9#10970C212yesmaximal, complemented
10#10964C211yesmaximal, characteristic
11#10789C211yesmaximal, characteristic, complemented
12C22×C22⋊Q8C2212yes—
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2211yescharacteristic
19C22×#103C2211yescharacteristic
20C22×C22⋊C8C2212yes—
21C22×C22⋊C8C2212yes—
22#2538—24no—
23#2538—24no—
24C2×#396—22no—
25C2×#394—21no—
26C2×#390—21no—
27C2×#389—21no—
28C2×#389C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33#2083—24no—
34#2036—28no—
35#1865—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1441—21no—
44#1440—22no—
45C4⋊#95C2211yescharacteristic
46C23×Q8C2311yescharacteristic
47C23×D4D411yescharacteristic
48C24×C4C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2312yes—
52C2×C22⋊Q8—24no—
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—24no—
55C2×C22⋊Q8—24no—
56C2×C22⋊Q8—42no—
57C2×C22⋊Q8—42no—
58C2×C4×Q8—22no—
59C2×C4×Q8—42no—
60C2×C4×Q8—42no—
61C2×C4×Q8D414yes—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—42no—
65C2×C4×D4—42no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
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×#107—24no—
85C2×#107—24no—
86C2×#107—42no—
87C2×#106—22no—
88C2×#106—42no—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—22no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—42no—
95C2×#103—42no—
96C2×#103D414yescomplemented
97C2×C22⋊C8—24no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—42no—
101C2×C22⋊C8—42no—
102C2×C22⋊C8—42no—
103#396—216no—
104#394—44no—
105#390—28no—
106#389—28no—
107#389—22no—
108#389—22no—
109#384—22no—
110#384D414yes—
111#380—24no—
112#380—24no—
113#380—24no—
114#380—42no—
115#380—42no—
116#380—42no—
117C2×#95—22no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—21no—
123#340—24no—
124#340—24no—
125#340—22no—
126#340—42no—
127#340—42no—
128#336—22no—
129#336—22no—
130C25C2×D412yes—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8C2×D412yes—
139C22×Q8C2×D412yes—
140C22×Q8#4112yes—
141C22×D4—24no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—42no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
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—21no—
165C23×C4—22no—
166C23×C4C2×D412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2411yescharacteristic
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—21no—
181C22×C8—22no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8C2×D412yes—
186C22×C8#4114yes—
187C22⋊Q8—48no—
188C22⋊Q8—48no—
189C22⋊Q8—48no—
190C22⋊Q8—48no—
191C4×Q8—44no—
192C4×Q8—44no—
193C4×Q8—48no—
194C4×D4—44no—
195C4×D4—44no—
196C4×D4—48no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
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—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—81no—
235C2×C4⋊C4—81no—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D414yes—
238C2×C4⋊C4C2×D412yes—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—82no—
252C2×C22⋊C4—82no—
253C2×C22⋊C4—82no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—24no—
257C2×C42—44no—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260#107—48no—
261#106—44no—
262#103—28no—
263#103—48no—
264#103—44no—
265C22⋊C8—48no—
266C22⋊C8—48no—
267#95—28no—
268#95—44no—
269#95—44no—
270#95—42no—
271#95—42no—
272#95—81no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24#13714yes—
288C24C22×D411yescharacteristic
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8#12114yes—
307C2×D4—28no—
308C2×D4—48no—
309C2×D4—48no—
310C2×D4—44no—
311C2×D4—44no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
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—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—41no—
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—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—41no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4#12012yes—
390C22×C4#12012yes—
391C22×C4#13512yes—
392C22×C4AGL(1,ℤ/8)12yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4112yes—
398C22×C4H(5,𝔽2)12yes—
399C22×C4H(5,𝔽2)12yes—
400C22×C4H(5,𝔽2)11yescharacteristic
401C22×C4H(5,𝔽2)11yescharacteristic
402C22×C42−1+412yes—
403C2×C8—22no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—42no—
419C2×C8—42no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—44no—
427C4⋊C4—48no—
428C4⋊C4—416no—
429C4⋊C4—42no—
430C4⋊C4—41no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—41no—
434C4⋊C4—48no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—84no—
443C4⋊C4—84no—
444C22⋊C4—48no—
445C22⋊C4—416no—
446C22⋊C4—88no—
447C22⋊C4—88no—
448C22⋊C4—88no—
449C42—22no—
450C42—22no—
451C42—24no—
452C42—48no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
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—
487C23—42no—
488C23C2×#12011yescharacteristic
489C23#53412yes—
490C23D4211yescharacteristic
491C23#54611yescharacteristic
492C23#54712yes—
493C23C2×#13511yescharacteristic
494C23C2×AGL(1,ℤ/8)11yescharacteristic
495C23C2×#13712yes—
496C23#57611yescharacteristic
497C23#57711yescharacteristic
498C23#57812yes—
499Q8—24no—
500Q8—24no—
501Q8—42no—
502Q8—42no—
503Q8—44no—
504Q8—44no—
505D4—416no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
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—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—82no—
537C2×C4—81no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—81no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—81no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4#44914yes—
567C2×C4C2×#12111yescharacteristic
568C2×C4#53211yescharacteristic
569C2×C4#53411yescharacteristic
570C2×C4#53411yescharacteristic
571C2×C4#53511yescharacteristic
572C2×C4#53512yes—
573C2×C4#53611yescharacteristic
574C2×C4#53712yes—
575C2×C4#53812yes—
576C2×C4#53912yes—
577C2×C4#54012yes—
578C2×C4D4211yescharacteristic
579C2×C4#54612yes—
580C2×C4#54611yescharacteristic
581C2×C4#54812yes—
582C2×C4#55312yes—
583C2×C4#56212yes—
584C2×C4#56212yes—
585C8—44no—
586C8—42no—
587C8—81no—
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#240811yescharacteristic
604C22#243211yescharacteristic
605C22#261311yescharacteristic
606C22#267112yes—
607C22#270812yes—
608C22C2×#44911yescharacteristic
609C22#300411yescharacteristic
610C22#301111yescharacteristic
611C22#301211yescharacteristic
612C22#319012yes—
613C22#319312yes—
614C22#320112yes—
615C22#320312yes—
616C22#327012yes—
617C22#327411yescharacteristic
618C22#328411yescharacteristic
619C22#330511yescharacteristic
620C22#331212yes—
621C22#333211yescharacteristic
622C22#334212yes—
623C22#334412yes—
624C22#334612yes—
625C22#335312yes—
626C4—24no—
627C4—44no—
628C4—42no—
629C4—82no—
630C4—82no—
631C4—84no—
632C4—82no—
633C4—162no—
634C4—161no—
635C4#254311yescharacteristic
636C4#261811yescharacteristic
637C4#266111yescharacteristic
638C4#267612yes—
639C4#271112yes—
640C4#281611yescharacteristic
641C2—84no—
642C2#1650111yescharacteristic
643C2#2423811yescharacteristic
644C2#2425511yescharacteristic
645C2#2570511yescharacteristic
646C2#2571211yescharacteristic
647C2#2959511yescharacteristic
648C2#2960811yescharacteristic
649C2#3057212yes—
650C2#3057912yes—
651C2#3200912yes—
652C2#3201612yes—
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×#2083C2semidirect
3. C2×#2036C2semidirect
4. C2×#1865C2semidirect
6. #11373C2semidirect
7. #11096C2semidirect
8. #11095C2semidirect
9. #10970C2semidirect
11. #10789C2semidirect
96. C2×#103D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#20832
C2×#20362
C2×#18652
#151292
#113732
#110962
#110952
#109702
#109642
#107892