Finite Group Databasebeta

← #365537

Subgroups — #365537

Order 512.

2060 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2366C211yesmaximal, characteristic
2C2×#1982C211yesmaximal, characteristic
3C2×#1961C211yesmaximal, characteristic
4C2×#1949C211yesmaximal, characteristic, complemented
5C2×#1864C212yesmaximal, complemented
6C2×#1862C212yesmaximal, complemented
7#13954C211yesmaximal, characteristic, complemented
8#11066C212yesmaximal, complemented
9#11066C212yesmaximal, complemented
10#10835C211yesmaximal, characteristic, complemented
11#10703C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C22×C4×D4—21no—
14C22×C4×D4C2212yes—
15C22×#107C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×#102C2212yes—
21C22×#102C2212yes—
22C22×#97C2211yescharacteristic
23#2366—24no—
24#2366—24no—
25C2×#390—21no—
26C2×#390C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#386—22no—
30C2×#382C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1982C418yescomplemented
35#1961—24no—
36#1949—24no—
37#1864—28no—
38#1862—28no—
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1444—21no—
46#1444—21no—
47C4⋊#95—22no—
48C23×D4—21no—
49C23×D4—21no—
50C23×D4C2312yes—
51C24×C4—22no—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2312yes—
55C2×#127—22no—
56C2×#127—22no—
57C2×#127—22no—
58C2×#127C2×C414yes—
59C2×#127D414yes—
60C2×#127Q814yes—
61C2×C4×D4—28no—
62C2×C4×D4—24no—
63C2×C4×D4—24no—
64C2×C4×D4—24no—
65C2×C4×D4—24no—
66C2×C4×D4—42no—
67C2×C4×D4—42no—
68C2×C4×D4—48no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C42—21no—
82C22×C42—21no—
83C22×C42C2312yes—
84C22×C42C2311yescharacteristic
85C2×#107—24no—
86C2×#107—22no—
87C2×#106—24no—
88C2×#106—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8C2×C414yescomplemented
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—42no—
101C2×#102—42no—
102C2×#102C2×C418yescomplemented
103C2×#102C2×C418yescomplemented
104C2×#97—22no—
105C2×#97—22no—
106C2×#97—22no—
107#390—28no—
108#390—24no—
109#390—24no—
110#389—24no—
111#389—24no—
112#389—42no—
113#389—42no—
114#386—216no—
115#382—22no—
116#382—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4C2×C414yescomplemented
122C4×C4⋊C4D414yes—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—21no—
129C2×#95D412yes—
130#340—22no—
131#340—22no—
132#340—24no—
133#340D418yescomplemented
134#336—24no—
135#336—24no—
136C25—22no—
137C22×D4—24no—
138C22×D4—24no—
139C22×D4—24no—
140C22×D4—24no—
141C22×D4—24no—
142C22×D4—24no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—44no—
155C22×D4—44no—
156C22×D4—81no—
157C22×D4—81no—
158C22×D4—81no—
159C22×D4—81no—
160C22×D4C22×C414yes—
161C22×D4C2×D414yes—
162C22×D4C2×Q814yes—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4—21no—
166C23×C4—28no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—44no—
170C23×C4—44no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4—82no—
174C23×C4—82no—
175C23×C4C2×D412yes—
176C23×C4C2×D412yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2411yescharacteristic
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8C22×C412yes—
195C22×C8C22×C412yes—
196C22×C8C2×D414yes—
197#127—28no—
198#127—28no—
199#127—28no—
200#127C4⋊C4116yescomplemented
201C4×D4—48no—
202C4×D4—48no—
203C4×D4—432no—
204C4×D4—48no—
205C4×D4—48no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
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⋊C4—82no—
231C2×C4⋊C4—82no—
232C2×C4⋊C4C22×C414yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D414yes—
235C2×C4⋊C4C2×D414yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4#4114yes—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—82no—
246C2×C22⋊C4—82no—
247C2×C42—24no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—24no—
255C2×C42—44no—
256C2×C42—81no—
257C2×C42—81no—
258C2×C42C22×C412yes—
259C2×C42C2×D412yes—
260C2×C42C2×D412yes—
261C2×C42C2×Q812yes—
262C2×C42#4112yes—
263C2×C42#4112yes—
264#107—44no—
265#106—44no—
266C4⋊C8—28no—
267#102—216no—
268#102—216no—
269#102—48no—
270#102—48no—
271#97—44no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—44no—
276#95—44no—
277#95—42no—
278#95—42no—
279C24—24no—
280C24—24no—
281C24—24no—
282C24—42no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24—42no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24C22×D411yescharacteristic
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—24no—
309C2×D4—24no—
310C2×D4—216no—
311C2×D4—216no—
312C2×D4—48no—
313C2×D4—416no—
314C2×D4—416no—
315C2×D4—48no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—84no—
329C2×D4—84no—
330C2×D4—84no—
331C2×D4—84no—
332C2×D4—84no—
333C2×D4—84no—
334C2×D4C2×C4⋊C418yes—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—416no—
361C22×C4—416no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—88no—
384C22×C4—88no—
385C22×C4—88no—
386C22×C4—88no—
387C22×C4—88no—
388C22×C4C4×D412yes—
389C22×C4C4×D412yes—
390C22×C4#12012yes—
391C22×C4#12012yes—
392C22×C4C22⋊Q812yes—
393C22×C4C22⋊Q812yes—
394C22×C4AGL(1,ℤ/8)14yes—
395C22×C4C23×C411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D412yes—
399C22×C4C22×Q811yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4112yes—
402C22×C4C2×#4111yescharacteristic
403C22×C4H(5,𝔽2)12yes—
404C22×C42−1+412yes—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—42no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C4⋊C4—28no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—44no—
431C4⋊C4—42no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—88no—
439C4⋊C4—88no—
440C4⋊C4C4×D418yescomplemented
441C4⋊C4#12714yes—
442C22⋊C4—416no—
443C22⋊C4—416no—
444C22⋊C4—88no—
445C42—48no—
446C42—42no—
447C42—42no—
448C42—42no—
449C42—42no—
450C42—84no—
451C42C2×C4⋊C414yes—
452C42C4×D414yes—
453C42C22⋊Q814yes—
454C42#12314yes—
455C23—28no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23—82no—
490C23C2×C4×D411yescharacteristic
491C23C2×#12011yescharacteristic
492C23C2×C22⋊Q811yescharacteristic
493C23D4212yes—
494C23#54812yes—
495C23C2×AGL(1,ℤ/8)12yes—
496C23#57512yes—
497C23#57611yescharacteristic
498C23#57611yescharacteristic
499C23#57711yescharacteristic
500C23#57711yescharacteristic
501D4—28no—
502D4—28no—
503D4—432no—
504D4—44no—
505D4—44no—
506D4—44no—
507D4—44no—
508D4—44no—
509D4—44no—
510D4—816no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—832no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4#38514yes—
564C2×C4#38514yes—
565C2×C4C22×C4⋊C411yescharacteristic
566C2×C4C2×C4×D411yescharacteristic
567C2×C4C2×C4×D412yes—
568C2×C4#52012yes—
569C2×C4C2×C22⋊Q811yescharacteristic
570C2×C4C2×#12311yescharacteristic
571C2×C4C2×#12711yescharacteristic
572C2×C4#53211yescharacteristic
573C2×C4#53212yes—
574C2×C4#53511yescharacteristic
575C2×C4#53611yescharacteristic
576C2×C4#54212yes—
577C2×C4D4212yes—
578C2×C4#54712yes—
579C2×C4#54712yes—
580C2×C4D4×Q812yes—
581C2×C4#55312yes—
582C2×C4#55412yes—
583C8—42no—
584C8—42no—
585C8—44no—
586C22—48no—
587C22—48no—
588C22—48no—
589C22—84no—
590C22—84no—
591C22—84no—
592C22—84no—
593C22—84no—
594C22—84no—
595C22—84no—
596C22—84no—
597C22—84no—
598C22—84no—
599C22—84no—
600C22—84no—
601C22C2×#38511yescharacteristic
602C22#236112yes—
603C22#243211yescharacteristic
604C22#243311yescharacteristic
605C22#245212yes—
606C22C4×AGL(1,ℤ/8)12yes—
607C22#294811yescharacteristic
608C22#294911yescharacteristic
609C22#300311yescharacteristic
610C22#300511yescharacteristic
611C22#301211yescharacteristic
612C22#301211yescharacteristic
613C22#307712yes—
614C22#308111yescharacteristic
615C22#308111yescharacteristic
616C22#326412yes—
617C22#326512yes—
618C22#326812yes—
619C22#327112yes—
620C22#336612yes—
621C22#336812yes—
622C22#337012yes—
623C22#337112yes—
624C4—24no—
625C4—24no—
626C4—44no—
627C4—44no—
628C4—48no—
629C4—82no—
630C4—82no—
631C4—161no—
632C4—161no—
633C4C4×#12711yescharacteristic
634C4#232611yescharacteristic
635C4D4×C4⋊C412yes—
636C4#242212yes—
637C4#254211yescharacteristic
638C4#256611yescharacteristic
639C2—88no—
640C2#1397711yescharacteristic
641C2#2087311yescharacteristic
642C2#2087911yescharacteristic
643C2#2300812yes—
644C2#2301012yes—
645C2#2565511yescharacteristic
646C2#2565811yescharacteristic
647C2#2576311yescharacteristic
648C2#2576511yescharacteristic
649C2#2647212yes—
650C2#2647512yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
LCSG ▷ C22×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
4. C2×#1949C2semidirect
5. C2×#1864C2semidirect
6. C2×#1862C2semidirect
7. #13954C2semidirect
8. #11066C2semidirect
9. #11066C2semidirect
10. #10835C2semidirect
11. #10703C2semidirect
34. #1982C4semidirect
91. C2×C4⋊C8C2×C4semidirect
102. C2×#102C2×C4semidirect
103. C2×#102C2×C4semidirect
121. C4×C4⋊C4C2×C4semidirect
133. #340D4semidirect
200. #127C4⋊C4semidirect
440. C4⋊C4C4×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#23662
C2×#19822
C2×#19612
C2×#19492
C2×#18642
C2×#18622
#139542
#110662
#108352
#107032