Finite Group Databasebeta

← #370650

Subgroups — #370650

Order 512.

1160 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
1#34948C211yesmaximal, characteristic
2C4⋊C4×M16C211yesmaximal, characteristic
3C4×#389C211yesmaximal, characteristic
4#11370C212yesmaximal
5#11226C212yesmaximal
6#10886C212yesmaximal
7#10852C211yesmaximal, characteristic
8#10851C212yesmaximal
9#10851C211yesmaximal, characteristic
10#10839C211yesmaximal, characteristic
11C4×#339C211yesmaximal, characteristic
12C2×#423C2211yescharacteristic
13C2×C4×M16C2212yes—
14C2×C4×M16C2211yescharacteristic
15#2571C411yescharacteristic, complemented
16#2571C411yescharacteristic, complemented
17#2570C412yescomplemented
18#2570C412yescomplemented
19#2569—22no—
20#2552C411yescharacteristic, complemented
21#2552C411yescharacteristic, complemented
22#2442—21no—
23#2442—21no—
24#2402—22no—
25C4×#125—21no—
26C4×#125—21no—
27#2296C2212yes—
28C2×#389—21no—
29#2285C2211yescharacteristic
30#2285C2212yes—
31C4×#117C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C43C2211yescharacteristic
36#1771—22no—
37#1770—21no—
38C8×C4⋊C4—21no—
39C2×#339C2212yes—
40C2×#339C2212yes—
41C2×#339C2211yescharacteristic
42C2×#339C2211yescharacteristic
43C2×#339C2211yescharacteristic
44C2×#339C2211yescharacteristic
45C2×#339C2212yes—
46C2×#339C2212yes—
47C2×#339C2212yes—
48C2×#339C2212yes—
49#1728C2212yes—
50#1436—22no—
51C4⋊#95—21no—
52C4×#95C2212yes—
53C22×M16C2312yes—
54C22×M16C2311yescharacteristic
55C22×M16C2311yescharacteristic
56C2×#125—21no—
57C2×#125—21no—
58C2×#117C2311yescharacteristic
59C2×#117C2311yescharacteristic
60C2×#117C2312yes—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C42—21no—
66C22×C42—21no—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2312yes—
71C22×C42C2311yescharacteristic
72#423—22no—
73#423Q814yes—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C4×M16—22no—
77C4×M16—24no—
78C4×M16—42no—
79C4×M16Q814yes—
80C2×#97—21no—
81C2×#97—22no—
82C2×C4×C8—21no—
83C2×C4×C8—22no—
84#400—22no—
85#400—22no—
86#400—22no—
87#400—22no—
88#398—22no—
89#398—22no—
90#398—22no—
91#398—22no—
92#389—28no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389C2×C412yes—
98#389C2×C412yes—
99#389C2×C412yes—
100#389C2×C412yes—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#385—22no—
106#385—22no—
107#384C2×C412yes—
108#384C2×C412yes—
109#384C2×C411yescharacteristic
110#384C2×C411yescharacteristic
111#382—21no—
112#382—21no—
113#382C2×C412yes—
114#382C2×C412yes—
115#382C2×C412yes—
116#380—21no—
117#380—22no—
118#380—22no—
119#380—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—24no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—42no—
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C4⋊C4C2×C412yes—
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C4⋊C4C2×C411yescharacteristic
137C4×C4⋊C4D411yescharacteristic
138C4×C4⋊C4D411yescharacteristic
139C4×C4⋊C4D412yes—
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142#376—21no—
143C2×#95—22no—
144C2×#95—22no—
145C2×#95—22no—
146C2×#95—21no—
147C2×#95D412yes—
148C43—21no—
149C43—22no—
150C43—22no—
151#339—22no—
152#339—22no—
153#339—24no—
154#339—24no—
155#339—22no—
156#339—22no—
157#339—42no—
158#339—42no—
159#339—42no—
160#339—42no—
161#336—22no—
162#336—22no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4C2×D412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2411yescharacteristic
171C2×M16—21no—
172C2×M16—21no—
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—22no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—21no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—21no—
184C2×M16C2×Q812yes—
185C2×M16C2×Q814yes—
186C2×M16C2×Q812yes—
187C2×M16#4112yes—
188C2×M16#4112yes—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—22no—
192#125—44no—
193#125—44no—
194#117—21no—
195#117—21no—
196#117—21no—
197#117—22no—
198#117—21no—
199#117—22no—
200#117—21no—
201#117—21no—
202#117—22no—
203#117—21no—
204#117—21no—
205#117—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22⋊C411yescharacteristic
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22⋊C412yes—
230C2×C4⋊C4C22⋊C412yes—
231C2×C4⋊C4C22⋊C411yescharacteristic
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C4⋊C4C22×C411yescharacteristic
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C22×C412yes—
237C2×C4⋊C4C22×C412yes—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—24no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—22no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42C22×C412yes—
275C2×C42C22×C412yes—
276C2×C42C22×C411yescharacteristic
277C2×C42C22×C411yescharacteristic
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×Q812yes—
283C2×C42C2×Q812yes—
284C2×C42#4112yes—
285C2×C42#4112yes—
286C2×C42#4112yes—
287C2×C42#4112yes—
288C4⋊C8—24no—
289C4⋊C8—24no—
290#97—24no—
291#97—44no—
292C4×C8—24no—
293C4×C8—44no—
294#95—28no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—44no—
310#95—44no—
311#95—42no—
312#95C22⋊C412yes—
313#95C22⋊C412yes—
314#95C22⋊C412yes—
315#95C22⋊C412yes—
316C24C22×D411yescharacteristic
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4C2×C22⋊C412yes—
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C22⋊C412yes—
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C412yes—
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4#12412yes—
365C22×C4#12412yes—
366C22×C4C4⋊Q812yes—
367C22×C4C4⋊Q812yes—
368C22×C4C23×C411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×Q811yescharacteristic
371C22×C4C22×Q811yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4H(5,𝔽2)11yescharacteristic
379C22×C4H(5,𝔽2)11yescharacteristic
380C22×C42−1+411yescharacteristic
381C22×C42−1+411yescharacteristic
382M16—22no—
383M16—22no—
384M16—22no—
385M16—22no—
386M16—22no—
387M16—24no—
388M16—24no—
389M16—42no—
390M16—42no—
391M16—41no—
392M16—41no—
393M16#12514yescomplemented
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—24no—
399C2×C8—42no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—42no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—42no—
420C4⋊C4—48no—
421C4⋊C4—416no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—82no—
429C4⋊C4—81no—
430C4⋊C4—81no—
431C4⋊C4—82no—
432C4⋊C4—82no—
433C4⋊C4—82no—
434C4⋊C4#10414yes—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—42no—
439C22⋊C4—41no—
440C22⋊C4—42no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—48no—
454C42—48no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42—42no—
461C42—41no—
462C42—42no—
463C42—41no—
464C42—41no—
465C42—42no—
466C42—42no—
467C42—42no—
468C42—42no—
469C42—42no—
470C42—42no—
471C42—81no—
472C42—81no—
473C42C4×Q812yes—
474C42C4×Q812yes—
475C42C4×Q812yes—
476C42C4×Q812yes—
477C42C22⋊Q812yes—
478C42C22⋊Q812yes—
479C42C22⋊Q812yes—
480C42C22⋊Q812yes—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23C22×C22⋊C411yescharacteristic
486C23C2×#12411yescharacteristic
487C23C2×C4⋊Q811yescharacteristic
488C23#54611yescharacteristic
489C23#54711yescharacteristic
490C23#54811yescharacteristic
491C23#55011yescharacteristic
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4#39112yes—
557C2×C4#39112yes—
558C2×C4#39112yes—
559C2×C4#39112yes—
560C2×C4C2×#10414yes—
561C2×C4C2×#10412yes—
562C2×C4#42112yes—
563C2×C4C2×C4×Q811yescharacteristic
564C2×C4C2×C4×Q811yescharacteristic
565C2×C4C4×#4111yescharacteristic
566C2×C4C4×#4111yescharacteristic
567C2×C4#52011yescharacteristic
568C2×C4#52011yescharacteristic
569C2×C4C2×C22⋊Q811yescharacteristic
570C2×C4C2×C22⋊Q811yescharacteristic
571C2×C4C2×#12511yescharacteristic
572C2×C4#52911yescharacteristic
573C2×C4#52911yescharacteristic
574C2×C4#53311yescharacteristic
575C2×C4#54011yescharacteristic
576C2×C4#54111yescharacteristic
577C2×C4#54411yescharacteristic
578C2×C4#54411yescharacteristic
579C2×C4#54611yescharacteristic
580C2×C4#54711yescharacteristic
581C2×C4#54811yescharacteristic
582C2×C4#55011yescharacteristic
583C2×C4#55211yescharacteristic
584C2×C4#55211yescharacteristic
585C2×C4#55311yescharacteristic
586C2×C4#55311yescharacteristic
587C2×C4#55411yescharacteristic
588C2×C4#55411yescharacteristic
589C2×C4#55711yescharacteristic
590C2×C4#55711yescharacteristic
591C8—24no—
592C8—42no—
593C8—44no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22C2×#39111yescharacteristic
607C22#233911yescharacteristic
608C22#234111yescharacteristic
609C22#248911yescharacteristic
610C22#249011yescharacteristic
611C22#255011yescharacteristic
612C22#255211yescharacteristic
613C22C22×#10411yescharacteristic
614C22C2×#42111yescharacteristic
615C22#290111yescharacteristic
616C22#290111yescharacteristic
617C4—22no—
618C4—22no—
619C4—22no—
620C4—24no—
621C4—44no—
622C4—44no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—84no—
627C4—81no—
628C4—82no—
629C4—81no—
630C4—82no—
631C4—81no—
632C4—81no—
633C4#198614yes—
634C4C4×#12511yescharacteristic
635C4#233011yescharacteristic
636C4#237011yescharacteristic
637C4#237011yescharacteristic
638C4#244111yescharacteristic
639C4#244211yescharacteristic
640C4#245311yescharacteristic
641C4#245411yescharacteristic
642C2—21no—
643C2—21no—
644C2—21no—
645C2—21no—
646C2#1405411yescharacteristic
647C2C2×#198611yescharacteristic
648C2#2130711yescharacteristic
649C2#2250511yescharacteristic
650C2#2250711yescharacteristic
651C2#2255011yescharacteristic
652C2#2255111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #2571C4semidirect
16. #2571C4semidirect
17. #2570C4semidirect
18. #2570C4semidirect
20. #2552C4semidirect
21. #2552C4semidirect
393. M16#125semidirect

Maximal subgroups

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

subgroupindex
#349482
C4⋊C4×M162
C4×#3892
#113702
#112262
#108862
#108522
#108512
#108392
C4×#3392