Finite Group Databasebeta

← #219922

Subgroups — #219922

Order 512.

1323 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36126C211yesmaximal, characteristic
2#22599C211yesmaximal, characteristic, complemented
3C2×#1517C212yesmaximal, complemented
4#13918C211yesmaximal, characteristic, complemented
5#9712C211yesmaximal, characteristic, complemented
6#9680C211yesmaximal, characteristic, complemented
7#9156C212yesmaximal, complemented
8#9114C212yesmaximal, complemented
9#8571C211yesmaximal, characteristic, complemented
10#8332C212yesmaximal, complemented
11C4×#327C211yesmaximal, characteristic, complemented
12C2×#547—21no—
13C2×#532C2212yes—
14C2×#431C2212yes—
15#2856C2211yescharacteristic
16#2854C2212yes—
17#2551C411yescharacteristic, complemented
18#2541C411yescharacteristic, complemented
19#2491—21no—
20#2461—22no—
21#2452C412yescomplemented
22#2445C412yescomplemented
23#2430—22no—
24#2416—21no—
25#2366C411yescharacteristic, complemented
26#2361—21no—
27D4×C4⋊C4C411yescharacteristic, complemented
28C4×#121C2211yescharacteristic
29C2×#389—21no—
30C2×#382C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32#1939—21no—
33#1916—22no—
34#1817C2212yes—
35#1797C2211yescharacteristic
36C8×C4⋊C4—21no—
37C4×C22⋊C8C2211yescharacteristic
38#1517—28no—
39C2×#327C2211yescharacteristic
40C2×#327C2212yes—
41C2×#327C2212yes—
42C2×#327C2212yes—
43C2×#327C2212yes—
44C2×#327C2212yes—
45C2×#327C2211yescharacteristic
46#1442—22no—
47C4⋊#95—21no—
48C4⋊#95C2211yescharacteristic
49#1434C2212yes—
50C4×#95C2211yescharacteristic
51#1294C2211yescharacteristic
52#1294C2211yescharacteristic
53#1294C2212yes—
54C22×#41—22no—
55#547—44no—
56#547—44no—
57#532—24no—
58#532—24no—
59#532—42no—
60#532—42no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8C2×C412yes—
63C2×#127C2×C412yes—
64C2×#127C2×C411yescharacteristic
65C2×#124—22no—
66C2×#123—22no—
67C2×#123—22no—
68C2×C22⋊Q8—22no—
69C2×#121—21no—
70C2×#121—22no—
71C2×#121—21no—
72C2×#121—21no—
73C2×#121—21no—
74C2×#121C2312yes—
75C2×#121C2311yescharacteristic
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4—22no—
79C2×C4×D4C2×C412yes—
80C2×C4×D4C2×C412yes—
81C2×C4×D4C2×C411yescharacteristic
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4D411yescharacteristic
86C22×C4⋊C4D411yescharacteristic
87C22×C42—21no—
88C22×C42C2312yes—
89C22×C42C2311yescharacteristic
90#431—24no—
91#431—42no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—22no—
94C2×C22⋊C8C2312yes—
95C2×C22⋊C8C2311yescharacteristic
96C2×C22⋊C8C2311yescharacteristic
97C2×C4×C8—22no—
98C2×C4×C8—21no—
99#396—22no—
100#395—22no—
101#394—22no—
102#394—22no—
103#393—22no—
104#392—22no—
105#390—21no—
106#390—21no—
107#390—21no—
108#390C2×C412yes—
109#390C2×C412yes—
110#390C2×C412yes—
111#390C2×C411yescharacteristic
112#389—21no—
113#389—44no—
114#389C2×C412yes—
115#389C2×C412yes—
116#389C2×C411yescharacteristic
117#388—22no—
118#388—22no—
119#388—21no—
120#386—22no—
121#386—21no—
122#386—22no—
123#386—22no—
124#385—22no—
125#385—22no—
126#384C2×C411yescharacteristic
127#382—24no—
128#382—42no—
129#380—21no—
130#380C2311yescharacteristic
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4C2×C412yes—
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C4⋊C4D411yescharacteristic
136C4×C4⋊C4D412yes—
137C4×C4⋊C4D411yescharacteristic
138C4×C22⋊C4—21no—
139C2×#95—21no—
140C2×#95—22no—
141C2×#95—22no—
142C2×#95—22no—
143C2×#95—21no—
144C2×#95D412yes—
145C2×#95C2311yescharacteristic
146C2×#95C2312yes—
147C2×#95C2311yescharacteristic
148#336—21no—
149#336—22no—
150#336—22no—
151#336—21no—
152#327—24no—
153#327—22no—
154#327—24no—
155#327—24no—
156#327—42no—
157#327—42no—
158#327D414yescomplemented
159C2×#41—44no—
160C2×#41—44no—
161C2×#41—44no—
162C2×#41—44no—
163C2×#41—82no—
164C2×#41—82no—
165C22×Q8—22no—
166C22×D4—21no—
167C22×D4—22no—
168C22×D4—21no—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4—21no—
172C22×D4C22×C412yes—
173C22×D4C22×C411yescharacteristic
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D412yes—
182C23×C4C2×D412yes—
183C23×C4C2×D411yescharacteristic
184C23×C4C2411yescharacteristic
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—22no—
188C4⋊Q8—24no—
189C4⋊Q8—42no—
190C4⋊Q8—44no—
191#127—24no—
192#127—42no—
193#127—42no—
194#124—44no—
195#124—44no—
196#123—44no—
197#123—44no—
198#123—48no—
199C22⋊Q8—48no—
200#121—21no—
201#121—21no—
202#121—22no—
203#121—22no—
204#121—22no—
205#121—22no—
206#121—44no—
207#121—44no—
208#121—44no—
209#121—42no—
210#121—42no—
211#121—44no—
212#121—81no—
213#121—81no—
214#121C2×D412yes—
215#121#4112yes—
216C4×D4—24no—
217C4×D4—24no—
218C4×D4—44no—
219C4×D4—44no—
220C4×D4—42no—
221C4×D4—42no—
222C4×D4—42no—
223C4×D4—42no—
224C4×D4—42no—
225C4×D4—44no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—82no—
240C2×C4⋊C4C22⋊C411yescharacteristic
241C2×C4⋊C4C22⋊C411yescharacteristic
242C2×C4⋊C4C22⋊C412yes—
243C2×C4⋊C4C22⋊C411yescharacteristic
244C2×C4⋊C4C22⋊C411yescharacteristic
245C2×C4⋊C4C22⋊C412yes—
246C2×C4⋊C4D814yes—
247C2×C4⋊C4SD1614yes—
248C2×C4⋊C4C22×C412yes—
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—21no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42—42no—
275C2×C42C22×C412yes—
276C2×C42C22×C411yescharacteristic
277C2×C42C22×C412yes—
278C2×C42C22×C411yescharacteristic
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C4⋊C8—44no—
284C4⋊C8—42no—
285C22⋊C8—21no—
286C22⋊C8—22no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—22no—
290C22⋊C8C2×D412yes—
291C4×C8—44no—
292C4×C8—42no—
293#95—21no—
294#95—21no—
295#95—22no—
296#95—22no—
297#95—21no—
298#95—21no—
299#95—22no—
300#95—21no—
301#95—21no—
302#95—22no—
303#95—44no—
304#95—42no—
305#95—44no—
306#95—44no—
307#95—81no—
308#95C22⋊C412yes—
309#95C22⋊C412yes—
310#95C22⋊C412yes—
311#95C22⋊C412yes—
312#95C2×D412yes—
313C24—21no—
314C24—22no—
315C24C22×D411yescharacteristic
316#41—88no—
317#41—88no—
318#41—88no—
319#41—88no—
320C2×Q8—44no—
321C2×Q8—44no—
322C2×Q8—82no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—21no—
327C2×D4—22no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—44no—
335C2×D4—42no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—44no—
349C2×D4—42no—
350C2×D4—82no—
351C2×D4—82no—
352C2×D4—81no—
353C2×D4—81no—
354C2×D4C4×D412yes—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—82no—
404C22×C4—82no—
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C412yes—
408C22×C4C2×C22⋊C412yes—
409C22×C4C2×C22⋊C412yes—
410C22×C4C2×C22⋊C412yes—
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×D814yes—
414C22×C4C2×D812yes—
415C22×C4C2×SD1612yes—
416C22×C4C2×SD1614yes—
417C22×C4C23×C411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4H(5,𝔽2)11yescharacteristic
422C22×C4H(5,𝔽2)11yescharacteristic
423C2×C8—22no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—42no—
428C2×C8—41no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—41no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—42no—
438C4⋊C4—44no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—42no—
442C4⋊C4—44no—
443C4⋊C4—42no—
444C4⋊C4—44no—
445C4⋊C4—81no—
446C4⋊C4—88no—
447C4⋊C4—84no—
448C4⋊C4#10214yes—
449C4⋊C4#10214yes—
450C4⋊C4#10218yes—
451C4⋊C4C4×D412yes—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—42no—
457C22⋊C4—44no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—81no—
467C22⋊C4—82no—
468C22⋊C4—82no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—41no—
480C42—41no—
481C42—42no—
482C42—84no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—22no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23C22×C22⋊C411yescharacteristic
502C23D4211yescharacteristic
503C23#54611yescharacteristic
504C23C22×D811yescharacteristic
505C23C22×SD1611yescharacteristic
506C23#57611yescharacteristic
507C23#57711yescharacteristic
508Q8—88no—
509D4—42no—
510D4—42no—
511D4—42no—
512D4—42no—
513D4—42no—
514D4—42no—
515D4—88no—
516D4—82no—
517D4—82no—
518D4—88no—
519D4—82no—
520D4—82no—
521D4—82no—
522D4—82no—
523D4—84no—
524D4—162no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—88no—
576C2×C4—88no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—84no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4#41012yes—
598C2×C4C2×#10212yes—
599C2×C4C2×#10214yes—
600C2×C4C2×#10214yes—
601C2×C4C2×#10212yes—
602C2×C4#42112yes—
603C2×C4C2×C4×D411yescharacteristic
604C2×C4#51811yescharacteristic
605C2×C4D4211yescharacteristic
606C2×C4#54611yescharacteristic
607C8—42no—
608C8—82no—
609C8—81no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22D4×C22⋊C411yescharacteristic
637C22C2×#41011yescharacteristic
638C22#288411yescharacteristic
639C22C22×#10211yescharacteristic
640C22#289711yescharacteristic
641C22C2×#42111yescharacteristic
642C22#290111yescharacteristic
643C22D4×D811yescharacteristic
644C22D4×SD1611yescharacteristic
645C22#329411yescharacteristic
646C22#329511yescharacteristic
647C4—21no—
648C4—21no—
649C4—21no—
650C4—21no—
651C4—24no—
652C4—24no—
653C4—22no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—41no—
658C4—41no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—81no—
663C4—81no—
664C4—162no—
665C4#151714yescomplemented
666C2—21no—
667C2—21no—
668C2—21no—
669C2—21no—
670C2—82no—
671C2—162no—
672C2C2×#151711yescharacteristic
673C2#1760211yescharacteristic
674C2#1761011yescharacteristic
675C2#1761811yescharacteristic
676C2#2240611yescharacteristic
677C2D4×#10211yescharacteristic
678C2#2243511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#36126C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22599C2semidirect
3. C2×#1517C2semidirect
4. #13918C2semidirect
5. #9712C2semidirect
6. #9680C2semidirect
7. #9156C2semidirect
8. #9114C2semidirect
9. #8571C2semidirect
10. #8332C2semidirect
11. C4×#327C2semidirect
17. #2551C4semidirect
18. #2541C4semidirect
21. #2452C4semidirect
22. #2445C4semidirect
25. #2366C4semidirect
27. D4×C4⋊C4C4semidirect
158. #327D4semidirect
665. C4#1517semidirect

Maximal subgroups

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

subgroupindex
#361262
#225992
C2×#15172
#139182
#97122
#96802
#91562
#91142
#85712
#83322
C4×#3272