Finite Group Databasebeta

← #360166

Subgroups — #360166

Order 512.

1972 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#525C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic, complemented
3C2×#1897C212yesmaximal
4C2×#1806C212yesmaximal, complemented
5#20626C211yesmaximal, characteristic, complemented
6#11226C212yesmaximal, complemented
7#11075C212yesmaximal, complemented
8#10852C212yesmaximal, complemented
9#10720C212yesmaximal, complemented
10C2×#525C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12C22×#117C2212yes—
13C23×C42C2211yescharacteristic
14C2×#423C2212yes—
15C22×#104C2214yes—
16C2×#407C2212yes—
17C4×C22⋊Q8C412yescomplemented
18C4×C22⋊Q8C412yescomplemented
19C4×#123—21no—
20C4×#123—21no—
21C4×#121C412yescomplemented
22C4×#121C412yescomplemented
23C4×#120—21no—
24C4×#120—21no—
25#2297C2211yescharacteristic
26#2296C2211yescharacteristic
27#2290—22no—
28#2285—28no—
29C2×#384—21no—
30C2×#382—21no—
31#2280C2211yescharacteristic
32C2×#380—21no—
33C2×#380—21no—
34C42×D4—22no—
35C4×#117C2211yescharacteristic
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2212yes—
38#1897—28no—
39#1806—28no—
40#1797—22no—
41#1797—22no—
42C2×#339C2214yes—
43C2×#339C2214yes—
44C2×#339C2214yes—
45#1728C2212yes—
46C22×#41C2311yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4C2311yescharacteristic
50C22×M16C2314yes—
51#525—22no—
52#525—22no—
53#525D414yes—
54#525Q814yes—
55C2×#123—21no—
56C2×C22⋊Q8C2×C412yes—
57C2×#121C2×C412yes—
58C2×#120—21no—
59C4×#41—24no—
60C4×#41—24no—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4C2×C411yescharacteristic
70C2×#117—24no—
71C2×#117—24no—
72C2×#117—28no—
73C2×#117—21no—
74C2×#117—44no—
75C2×#117C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C42—24no—
83C22×C42—24no—
84C22×C42—21no—
85C22×C42C2×C412yes—
86C22×C42C2×C412yes—
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D412yes—
90C22×C42D412yes—
91C22×C42C2314yes—
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94#423—24no—
95#423—24no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×#104—28no—
99C2×#104—28no—
100C2×#104—44no—
101#407—24no—
102#407—24no—
103C2×C22⋊C8—24no—
104#391C2×C412yes—
105#390C2×C412yes—
106#389C2×C412yes—
107#389C2×C412yes—
108#388—22no—
109#388—22no—
110#387—22no—
111#387—22no—
112#386—22no—
113#386—22no—
114#386—21no—
115#386—21no—
116#385—22no—
117#385—22no—
118#385—21no—
119#385—21no—
120#384—28no—
121#382—28no—
122#381—21no—
123#380—28no—
124#380—28no—
125#380C2×C412yes—
126#380C2×C412yes—
127#379—21no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—28no—
131C4×C4⋊C4—24no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4C2×C412yes—
134C4×C4⋊C4C2×C412yes—
135C4×C4⋊C4D418yescomplemented
136C4×C22⋊C4—22no—
137C4×C22⋊C4—22no—
138C4×C22⋊C4—21no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4—22no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4—22no—
143C4×C22⋊C4—22no—
144#376—21no—
145C2×#95—22no—
146C2×#95—22no—
147C43—21no—
148#339—28no—
149#339—28no—
150#339—44no—
151#336—24no—
152C25C2×D411yescharacteristic
153C2×#41—21no—
154C2×#41—21no—
155C2×#41—22no—
156C2×#41—22no—
157C2×#41—22no—
158C2×#41—22no—
159C2×#41C2×D412yes—
160C2×#41C2×Q812yes—
161C22×Q8C22×C411yescharacteristic
162C22×D4—22no—
163C22×D4—21no—
164C22×D4C22×C411yescharacteristic
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—21no—
174C23×C4—24no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4C22⋊C412yes—
180C23×C4C22⋊C412yes—
181C23×C4C22⋊C412yes—
182C23×C4C22⋊C412yes—
183C23×C4C22×C412yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D412yes—
188C23×C4C2×D412yes—
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×Q812yes—
192C23×C4#4112yes—
193C23×C4#4112yes—
194C23×C4C2411yescharacteristic
195C2×M16—24no—
196C2×M16—24no—
197C2×M16—24no—
198C2×M16—24no—
199C2×M16—24no—
200C2×M16—24no—
201C22×C8—24no—
202#123—24no—
203#123—24no—
204C22⋊Q8—24no—
205C22⋊Q8C4⋊C418yescomplemented
206#121—24no—
207#121C4⋊C418yescomplemented
208#120—24no—
209#120—24no—
210C4×Q8—24no—
211C4×Q8—24no—
212C4×D4—28no—
213C4×D4—28no—
214C4×D4—24no—
215C4×D4—24no—
216C4×D4—48no—
217C4×D4—48no—
218C4×D4—42no—
219C4×D4—42no—
220C4×D4—42no—
221C4×D4—42no—
222#117—21no—
223#117—21no—
224#117—22no—
225#117—22no—
226#117—22no—
227#117—22no—
228#117—416no—
229#117—416no—
230#117—44no—
231#117C2×D412yes—
232#117C2×Q812yes—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—48no—
242C2×C4⋊C4—48no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C4⋊C4C22×C411yescharacteristic
247C2×C4⋊C4C22×C412yes—
248C2×C4⋊C4C22×C412yes—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—21no—
269C2×C42—22no—
270C2×C42—28no—
271C2×C42—28no—
272C2×C42—28no—
273C2×C42—28no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—44no—
286C2×C42C22⋊C412yes—
287C2×C42C22⋊C412yes—
288C2×C42C22⋊C412yes—
289C2×C42C22⋊C412yes—
290C2×C42C22×C412yes—
291C2×C42C22×C411yescharacteristic
292C2×C42C22×C411yescharacteristic
293C2×C42C2×D418yes—
294C2×C42C2×D412yes—
295C2×C42C2×D414yes—
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42#4114yes—
299C4⋊C8—44no—
300C4⋊C8—44no—
301#104—416no—
302C22⋊C8—48no—
303#95—28no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—28no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—42no—
319C24—42no—
320C24C2×C22⋊C412yes—
321C24#12112yes—
322C24C22⋊Q812yes—
323C24C22×D411yescharacteristic
324#41—24no—
325#41—24no—
326#41—24no—
327#41—24no—
328#41—42no—
329#41—42no—
330#41—42no—
331#41—42no—
332C2×Q8—21no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8C2×C4⋊C412yes—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—28no—
339C2×D4—21no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4C2×C4⋊C412yes—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—28no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—44no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—44no—
407C22×C4C2×C22⋊C412yes—
408C22×C4C2×C22⋊C412yes—
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C412yes—
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C4⋊C414yes—
415C22×C4#11714yes—
416C22×C4#12012yes—
417C22×C4#12012yes—
418C22×C4#12114yes—
419C22×C4#12112yes—
420C22×C4C22⋊Q814yes—
421C22×C4C22⋊Q812yes—
422C22×C4#12312yes—
423C22×C4#12312yes—
424C22×C4C23×C411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D412yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×Q811yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4112yes—
432C22×C4H(5,𝔽2)12yes—
433C22×C42−1+412yes—
434M16—44no—
435M16—44no—
436M16—44no—
437M16—44no—
438C2×C8—44no—
439C2×C8—44no—
440C2×C8—44no—
441C4⋊C4—22no—
442C4⋊C4—28no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—416no—
447C4⋊C4—416no—
448C4⋊C4—416no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—44no—
453C4⋊C4C2×C4⋊C414yes—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—28no—
457C22⋊C4—28no—
458C22⋊C4—44no—
459C22⋊C4—416no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—44no—
469C42—24no—
470C42—24no—
471C42—22no—
472C42—22no—
473C42—24no—
474C42—24no—
475C42—24no—
476C42—24no—
477C42—28no—
478C42—28no—
479C42—44no—
480C42—42no—
481C42—42no—
482C42—44no—
483C42—44no—
484C42—42no—
485C42—42no—
486C42—44no—
487C42C4×D414yes—
488C42C4×D414yes—
489C42#12114yes—
490C42#12114yes—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—44no—
505C23—44no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—44no—
511C23—44no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23—41no—
516C23#38014yes—
517C23C22×C22⋊C411yescharacteristic
518C23C2×#12011yescharacteristic
519C23C2×#12111yescharacteristic
520C23C2×C22⋊Q811yescharacteristic
521C23C2×#12311yescharacteristic
522C23#53511yescharacteristic
523C23#53611yescharacteristic
524Q8—24no—
525Q8—42no—
526D4—24no—
527D4—416no—
528D4—42no—
529D4—42no—
530D4—42no—
531D4—42no—
532D4—42no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—48no—
560C2×C4—48no—
561C2×C4—48no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—48no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—48no—
572C2×C4—48no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—48no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—84no—
582C2×C4—88no—
583C2×C4—88no—
584C2×C4—84no—
585C2×C4#38014yes—
586C2×C4#38612yes—
587C2×C4#38612yes—
588C2×C4#38612yes—
589C2×C4#38612yes—
590C2×C4#42114yes—
591C2×C4#42114yes—
592C2×C4C22×C4⋊C411yescharacteristic
593C2×C4C2×#11711yescharacteristic
594C2×C4C2×C4×D412yes—
595C2×C4#52012yes—
596C2×C4C2×#12111yescharacteristic
597C2×C4C2×#12112yes—
598C2×C4C2×C22⋊Q811yescharacteristic
599C2×C4#53712yes—
600C2×C4D4212yes—
601C2×C4#54612yes—
602C2×C4#54712yes—
603C2×C4#54812yes—
604C2×C4D4×Q812yes—
605C2×C4#55212yes—
606C2×C4#55312yes—
607C2×C4#55412yes—
608C8—84no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—44no—
630C22—82no—
631C22—81no—
632C22—81no—
633C22C2×#38011yescharacteristic
634C22C2×#38611yescharacteristic
635C22#231111yescharacteristic
636C22#240811yescharacteristic
637C22#240911yescharacteristic
638C22#248611yescharacteristic
639C22#248711yescharacteristic
640C22C2×#42112yes—
641C22C2×#42112yes—
642C4—24no—
643C4—22no—
644C4—24no—
645C4—22no—
646C4—22no—
647C4—42no—
648C4—44no—
649C4—42no—
650C4—88no—
651C4—82no—
652C4—81no—
653C4—81no—
654C4#189714yescomplemented
655C4D4×C4⋊C412yes—
656C4#235212yes—
657C4#241612yes—
658C4#241712yes—
659C2—21no—
660C2—21no—
661C2—21no—
662C2—21no—
663C2—44no—
664C2—82no—
665C2#1387211yescharacteristic
666C2#2065412yes—
667C2C2×#189711yescharacteristic
668C2#2097911yescharacteristic
669C2#2190012yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2285C2semidirect
4. C2×#1806C2semidirect
5. #20626C2semidirect
6. #11226C2semidirect
7. #11075C2semidirect
8. #10852C2semidirect
9. #10720C2semidirect
17. C4×C22⋊Q8C4semidirect
18. C4×C22⋊Q8C4semidirect
21. C4×#121C4semidirect
22. C4×#121C4semidirect
135. C4×C4⋊C4D4semidirect
205. C22⋊Q8C4⋊C4semidirect
207. #121C4⋊C4semidirect
654. C4#1897semidirect

Maximal subgroups

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

subgroupindex
C4×#5252
C2×#22852
C2×#18972
C2×#18062
#206262
#112262
#110752
#108522
#107202