Finite Group Databasebeta

← #197218

Subgroups — #197218

Order 512.

1367 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60484C211yesmaximal, characteristic
2C4⋊C4×#41C211yesmaximal, characteristic
3#17575C211yesmaximal, characteristic, complemented
4C2×#1503C211yesmaximal, characteristic, complemented
5#17437C211yesmaximal, characteristic
6#9176C214yesmaximal, complemented
7#9175C214yesmaximal, complemented
8C2×#1303C211yesmaximal, characteristic, complemented
9#7957C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C2×C4×#41—21no—
12C2×C4×#41C2211yescharacteristic
13#3080—22no—
14#3079C412yescomplemented
15#3079C412yescomplemented
16#3078—21no—
17#3078C2212yes—
18#3077—21no—
19#3077C2212yes—
20#3076—22no—
21#3076C414yescomplemented
22C2×#431C2211yescharacteristic
23C2×#431C2211yescharacteristic
24#2911—21no—
25C2×#423C2211yescharacteristic
26#2896—21no—
27#2351C2212yes—
28Q8×C4⋊C4—21no—
29#2349—22no—
30#2349C2212yes—
31D4×C4⋊C4—22no—
32D4×C4⋊C4—21no—
33#2296—22no—
34#2296—21no—
35#2285—21no—
36#2285C2211yescharacteristic
37C2×C4×C4⋊C4—21no—
38C2×C4×C4⋊C4C2211yescharacteristic
39#1520—22no—
40#1518C2214yes—
41#1503—21no—
42#1503—21no—
43#1503—22no—
44#1503C2214yes—
45C2×#328—22no—
46#1479C2212yes—
47#1478C2212yes—
48#1477—22no—
49#1476—22no—
50#1303C2214yes—
51#1286C2214yes—
52C22×#41D411yescharacteristic
53C22×M16—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533—21no—
59#533—21no—
60#533C2×C412yescomplemented
61#533C2312yes—
62C2×C4⋊Q8C2×C411yescharacteristic, complemented
63C2×C4⋊Q8C2×C411yescharacteristic, complemented
64C2×#125—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C4×#41—21no—
68C4×#41—21no—
69C4×#41—21no—
70C4×#41—21no—
71C4×#41—44no—
72C4×#41—44no—
73C4×#41C2×C412yes—
74C4×#41C2×C412yes—
75C4×#41D412yes—
76C4×#41D412yes—
77C2×C4×Q8—22no—
78C2×C4×Q8—21no—
79C2×C4×Q8—21no—
80C2×C4×Q8D411yescharacteristic
81C2×C4×Q8C2311yescharacteristic
82C2×C4×D4—21no—
83C2×C4×D4—21no—
84C2×C4×D4—24no—
85C2×C4×D4—22no—
86C2×C4×D4D411yescharacteristic
87C2×C4×D4C2311yescharacteristic
88C2×#117—22no—
89C2×#117—21no—
90C2×#117—21no—
91C2×#117—21no—
92C2×#117D411yescharacteristic
93C22×C4⋊C4—22no—
94C22×C4⋊C4—21no—
95C22×C42—21no—
96C22×C42—22no—
97C22×C42—21no—
98C22×C42C2311yescharacteristic
99#479—24no—
100#479—24no—
101#478—24no—
102#478—24no—
103#477—22no—
104#477C2×C414yescomplemented
105#476—22no—
106#476C2×C414yescomplemented
107#475—22no—
108#475C2×C414yescomplemented
109#474—22no—
110#474C2×C414yescomplemented
111#431—21no—
112#431—21no—
113#431—21no—
114#431—21no—
115#431—21no—
116#431—21no—
117#431C2312yes—
118#431C2312yes—
119#428—22no—
120#428—22no—
121#428—22no—
122#428—22no—
123#427—24no—
124#423—21no—
125#423—21no—
126#423C2×C412yescomplemented
127#423C2312yes—
128C2×C4⋊C8—21no—
129C2×C4⋊C8—21no—
130C2×C4⋊C8—21no—
131C2×C4⋊C8C2×C412yescomplemented
132#419—22no—
133#418—22no—
134#417—22no—
135#417—22no—
136#416—24no—
137C2×C22⋊C8—21no—
138C2×C22⋊C8—21no—
139C2×C4×C8—21no—
140C2×C4×C8—21no—
141#389—24no—
142#389—22no—
143#389—22no—
144#389—24no—
145#385—24no—
146#385—24no—
147#385—24no—
148#384—22no—
149#384—21no—
150#382—22no—
151#382C2312yes—
152#380—22no—
153#380—21no—
154#380—22no—
155#380D412yes—
156C4×C4⋊C4—22no—
157C4×C4⋊C4—22no—
158C4×C4⋊C4—22no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4—22no—
163C4×C4⋊C4D411yescharacteristic
164C4×C4⋊C4D411yescharacteristic
165C4×C4⋊C4D412yes—
166C4×C4⋊C4C2312yes—
167C2×#95—22no—
168C2×#95D412yes—
169#328—24no—
170#328—24no—
171#328—28no—
172Q8⋊C8—24no—
173D4⋊C8—24no—
174C2×#41—21no—
175C2×#41—21no—
176C2×#41—22no—
177C2×#41—22no—
178C2×#41C22⋊C412yes—
179C2×#41C22⋊C412yes—
180C2×#41SD1614yes—
181C2×#41Q1614yes—
182C22×Q8—21no—
183C22×Q8—21no—
184C22×Q8C2×D411yescharacteristic
185C22×D4—21no—
186C22×D4C2×D411yescharacteristic
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C2×M16—22no—
196C2×M16—22no—
197C2×M16—42no—
198C2×M16—42no—
199C22×C8—22no—
200C22×C8—41no—
201C22×C8—41no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8C22×C412yes—
209C4⋊Q8C22×C412yes—
210#125—22no—
211#125—22no—
212#125—41no—
213#125—41no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216C22⋊Q8—22no—
217C22⋊Q8—22no—
218C22⋊Q8—41no—
219C22⋊Q8—41no—
220C22⋊Q8—41no—
221C22⋊Q8—41no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—22no—
230C4×Q8—44no—
231C4×Q8—44no—
232C4×Q8—41no—
233C4×Q8—41no—
234C4×Q8—41no—
235C4×Q8—41no—
236C4×Q8C22⋊C412yes—
237C4×Q8C22⋊C412yes—
238C4×Q8C22×C412yes—
239C4×Q8C2×D412yes—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4—22no—
248C4×D4—21no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—416no—
253C4×D4—44no—
254C4×D4—44no—
255C4×D4C22⋊C412yes—
256C4×D4C22⋊C412yes—
257C4×D4C22×C412yes—
258C4×D4C2×D412yes—
259#117—21no—
260#117—21no—
261#117—22no—
262#117—22no—
263#117—22no—
264#117—22no—
265#117—22no—
266#117—22no—
267#117—48no—
268#117—41no—
269#117—41no—
270#117—42no—
271#117—42no—
272#117C22⋊C412yes—
273#117C22⋊C412yes—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—48no—
291C2×C4⋊C4—41no—
292C2×C4⋊C4—41no—
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2×D411yescharacteristic
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—41no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—21no—
310C2×C42—22no—
311C2×C42—21no—
312C2×C42—22no—
313C2×C42—48no—
314C2×C42—41no—
315C2×C42—42no—
316C2×C42—42no—
317C2×C42C22×C411yescharacteristic
318C2×C42C22×C411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42#4111yescharacteristic
325C2×C42#4111yescharacteristic
326C2×C42#4111yescharacteristic
327C2×C42#4111yescharacteristic
328C2×C42C2411yescharacteristic
329#107—24no—
330#107—24no—
331#106—24no—
332#106—24no—
333C4⋊C8—22no—
334C4⋊C8—22no—
335C4⋊C8—22no—
336C4⋊C8—22no—
337C4⋊C8—22no—
338C4⋊C8—41no—
339C4⋊C8—41no—
340C4⋊C8C22×C414yes—
341#103—24no—
342#103—24no—
343#102—24no—
344#102—24no—
345C22⋊C8—22no—
346C22⋊C8—22no—
347C22⋊C8—41no—
348C22⋊C8—41no—
349C4×C8—22no—
350C4×C8—22no—
351C4×C8—41no—
352C4×C8—41no—
353#95—24no—
354#95—24no—
355#95—22no—
356#95—22no—
357#95C2×D412yes—
358#95C2×D412yes—
359#95C2×D414yes—
360C24—21no—
361C24#12011yescharacteristic
362#41—24no—
363#41—24no—
364#41—24no—
365#41—24no—
366#41—24no—
367#41—24no—
368#41#10318yes—
369#41#10318yes—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—41no—
376C2×Q8—41no—
377C2×Q8—82no—
378C2×Q8—82no—
379C2×Q8C2×C22⋊C412yes—
380C2×Q8C2×SD1612yes—
381C2×Q8C2×Q1612yes—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4C2×C22⋊C412yes—
390C2×D4C2×SD1612yes—
391C2×D4C2×Q1612yes—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—42no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—44no—
414C22×C4—81no—
415C22×C4—82no—
416C22×C4—82no—
417C22×C4—84no—
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C4×D411yescharacteristic
421C22×C4C4×D411yescharacteristic
422C22×C4C4×D411yescharacteristic
423C22×C4C4×D411yescharacteristic
424C22×C4#12011yescharacteristic
425C22×C4#12011yescharacteristic
426C22×C4#12011yescharacteristic
427C22×C4#12111yescharacteristic
428C22×C4#12111yescharacteristic
429C22×C4#12111yescharacteristic
430C22×C4#12111yescharacteristic
431C22×C4#12311yescharacteristic
432C22×C4#12311yescharacteristic
433C22×C4C2×SD1612yes—
434C22×C4C2×Q1612yes—
435C22×C4AGL(1,ℤ/8)12yes—
436C22×C4#13712yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440M16—42no—
441M16—42no—
442M16—44no—
443C2×C8—24no—
444C2×C8—44no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—82no—
448C2×C8—82no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—82no—
476C4⋊C4—84no—
477C4⋊C4C2×C22⋊C412yes—
478C4⋊C4C2×C22⋊C412yes—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—48no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—44no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—48no—
496C42—42no—
497C42—42no—
498C42—42no—
499C42—42no—
500C42—42no—
501C42—44no—
502C42—42no—
503C42C2×C22⋊C411yescharacteristic
504C42C2×C22⋊C411yescharacteristic
505C42C4×D411yescharacteristic
506C42C4×D411yescharacteristic
507C42C4×D411yescharacteristic
508C42C4×D411yescharacteristic
509C42#12111yescharacteristic
510C42#12111yescharacteristic
511C42#12111yescharacteristic
512C42#12111yescharacteristic
513C42#12311yescharacteristic
514C42#12311yescharacteristic
515C42C23×C411yescharacteristic
516C42C22×D411yescharacteristic
517C42C2×#4111yescharacteristic
518C42C2×#4111yescharacteristic
519C23—22no—
520C23—22no—
521C23—22no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23#38611yescharacteristic
527C23#38611yescharacteristic
528C23#45012yes—
529C23#45112yes—
530C23C2×#12011yescharacteristic
531Q8—22no—
532Q8—22no—
533Q8—22no—
534Q8—82no—
535Q8—82no—
536Q8—82no—
537Q8—82no—
538Q8C2×#10314yes—
539D4—28no—
540D4—28no—
541D4—22no—
542D4—22no—
543D4—22no—
544D4C2×#10314yes—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—88no—
582C2×C4—816no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4#38611yescharacteristic
586C2×C4#38611yescharacteristic
587C2×C4#38611yescharacteristic
588C2×C4#38611yescharacteristic
589C2×C4#38611yescharacteristic
590C2×C4#38611yescharacteristic
591C2×C4C2×#10312yes—
592C2×C4C2×#10312yes—
593C2×C4#41712yes—
594C2×C4#41712yes—
595C2×C4#45012yes—
596C2×C4#45112yes—
597C2×C4#48312yes—
598C2×C4#48312yes—
599C2×C4#48412yes—
600C2×C4#48412yes—
601C2×C4C22×C22⋊C411yescharacteristic
602C2×C4C2×C4×D411yescharacteristic
603C2×C4C2×C4×D411yescharacteristic
604C2×C4C2×#12111yescharacteristic
605C2×C4C2×#12111yescharacteristic
606C2×C4C2×#12311yescharacteristic
607C2×C4C22×SD1611yescharacteristic
608C2×C4C22×Q1611yescharacteristic
609C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
610C2×C4C2×#13711yescharacteristic
611C8—44no—
612C8—82no—
613C8—82no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—24no—
621C22—42no—
622C22—42no—
623C22—44no—
624C22—44no—
625C22—42no—
626C22—42no—
627C22#187212yes—
628C22#187212yes—
629C22C2×#38611yescharacteristic
630C22C2×#45011yescharacteristic
631C22C2×#45111yescharacteristic
632C4—21no—
633C4—21no—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—21no—
638C4—21no—
639C4—24no—
640C4—44no—
641C4—42no—
642C4—42no—
643C4—84no—
644C4—82no—
645C4—82no—
646C4—82no—
647C4#187212yes—
648C4#187212yes—
649C4#190612yes—
650C4C22×#10311yescharacteristic
651C4C2×#41711yescharacteristic
652C4C2×#48311yescharacteristic
653C4C2×#48411yescharacteristic
654C2—21no—
655C2—21no—
656C2—22no—
657C2—22no—
658C2—44no—
659C2C2×#187211yescharacteristic
660C2#2099811yescharacteristic
661C2C2×#190611yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC22×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17575C2semidirect
4. C2×#1503C2semidirect
6. #9176C2semidirect
7. #9175C2semidirect
8. C2×#1303C2semidirect
9. #7957C2semidirect
14. #3079C4semidirect
15. #3079C4semidirect
21. #3076C4semidirect
60. #533C2×C4semidirect
62. C2×C4⋊Q8C2×C4semidirect
63. C2×C4⋊Q8C2×C4semidirect
104. #477C2×C4semidirect
106. #476C2×C4semidirect
108. #475C2×C4semidirect
110. #474C2×C4semidirect
126. #423C2×C4semidirect
131. C2×C4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#604842
C4⋊C4×#412
#175752
C2×#15032
#174372
#91762
#91752
C2×#13032
#79572