Finite Group Databasebeta

← #224880

Subgroups — #224880

Order 512.

1286 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36576C211yesmaximal, characteristic
2Q8×C22⋊C8C211yesmaximal, characteristic, complemented
3#13935C211yesmaximal, characteristic, complemented
4#9715C212yesmaximal, complemented
5#9712C211yesmaximal, characteristic, complemented
6#9680C211yesmaximal, characteristic, complemented
7#9680C212yesmaximal, complemented
8#8577C212yesmaximal, complemented
9#8571C211yesmaximal, characteristic, complemented
10#8220C212yesmaximal, complemented
11C4×#327C211yesmaximal, characteristic, complemented
12C2×#550C411yescharacteristic, complemented
13C2×#548—21no—
14C22×C4×Q8C2211yescharacteristic
15C2×C8×Q8—21no—
16#2856C2211yescharacteristic
17#2856C2212yes—
18#2548C411yescharacteristic, complemented
19#2548C412yescomplemented
20#2538C411yescharacteristic, complemented
21#2538C412yescomplemented
22#2426—21no—
23#2426—22no—
24#2416—21no—
25#2416—22no—
26Q8×C4⋊C4C411yescharacteristic, complemented
27#2341—21no—
28C4×#121C2211yescharacteristic
29C4×#121C2212yes—
30C2×#391—21no—
31C2×#389—21no—
32#1984—21no—
33#1984—22no—
34#1797C2211yescharacteristic
35#1797C2212yes—
36C4×C22⋊C8C2211yescharacteristic
37C4×C22⋊C8C2212yes—
38C2×#327C2211yescharacteristic
39C2×#327C2211yescharacteristic
40C2×#327C2212yes—
41C2×#327C2212yes—
42C2×#327C2212yes—
43C2×#327C2212yes—
44C2×#327C2212yes—
45#1441—22no—
46C4⋊#95C2211yescharacteristic
47C4⋊#95C2212yes—
48#1434C2212yes—
49C4×#95C2211yescharacteristic
50#1294C2211yescharacteristic
51#1294C2211yescharacteristic
52#1294C2212yes—
53C22×#41—21no—
54C23×Q8D411yescharacteristic
55#550—44no—
56#548—44no—
57#548—44no—
58C2×#127C2×C411yescharacteristic
59C2×#127C2×C412yes—
60C2×#124—21no—
61C2×#124—22no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—22no—
64C2×#121—21no—
65C2×#121—22no—
66C2×#121—22no—
67C2×#121—21no—
68C2×#121—22no—
69C2×#121—21no—
70C2×#121C2312yes—
71C2×#121C2311yescharacteristic
72C2×C4×Q8—21no—
73C2×C4×Q8—44no—
74C2×C4×Q8C2×C411yescharacteristic
75C2×C4×Q8C2×C411yescharacteristic
76C2×C4×Q8C2×C412yes—
77C2×C4×Q8D411yescharacteristic
78C2×C4×Q8D411yescharacteristic
79C2×C4×Q8D412yes—
80C2×C4×D4—21no—
81C2×C4×D4—22no—
82C2×C4×D4C2×C411yescharacteristic
83C2×C4×D4C2×C412yes—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4D411yescharacteristic
87C22×C4⋊C4D412yes—
88C22×C42—21no—
89C22×C42C2311yescharacteristic
90C22×C42C2312yes—
91C8×Q8—44no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—22no—
94C2×C22⋊C8C2312yes—
95C2×C22⋊C8C2311yescharacteristic
96C2×C22⋊C8C2311yescharacteristic
97C2×C4×C8—21no—
98C2×C4×C8—22no—
99#396—22no—
100#396—22no—
101#396—22no—
102#394—22no—
103#394—22no—
104#394—22no—
105#391—44no—
106#390—21no—
107#390—22no—
108#390—21no—
109#390—22no—
110#390—22no—
111#390—21no—
112#390C2×C412yes—
113#390C2×C412yes—
114#390C2×C412yes—
115#390C2×C411yescharacteristic
116#390C2×C412yes—
117#389—22no—
118#389—21no—
119#389—44no—
120#389C2×C411yescharacteristic
121#389C2×C412yes—
122#388—22no—
123#388—21no—
124#386—22no—
125#386—21no—
126#384C2×C411yescharacteristic
127#384C2×C412yes—
128#380—21no—
129#380—22no—
130#380C2311yescharacteristic
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2×C412yes—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—22no—
135C2×#95—21no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95—21no—
140C2×#95C2311yescharacteristic
141C2×#95C2312yes—
142C2×#95C2311yescharacteristic
143#336—22no—
144#336—21no—
145#336—22no—
146#336—21no—
147#327—22no—
148#327—24no—
149#327—22no—
150#327—24no—
151#327—42no—
152#327—42no—
153#327—42no—
154C2×#41—44no—
155C2×#41—44no—
156C2×#41—44no—
157C22×Q8—42no—
158C22×Q8—42no—
159C22×Q8C22⋊C411yescharacteristic
160C22×Q8C22⋊C411yescharacteristic
161C22×Q8C22⋊C412yes—
162C22×Q8D814yes—
163C22×Q8SD1614yes—
164C22×D4—21no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—22no—
170C22×D4C22×C411yescharacteristic
171C22×D4C22×C412yes—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—22no—
175C23×C4—21no—
176C23×C4—22no—
177C23×C4C2×D412yes—
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D412yes—
181C23×C4C2411yescharacteristic
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—22no—
185#127—44no—
186#127—42no—
187#124—44no—
188#124—48no—
189C22⋊Q8—44no—
190C22⋊Q8—48no—
191#121—22no—
192#121—21no—
193#121—22no—
194#121—21no—
195#121—22no—
196#121—21no—
197#121—22no—
198#121—21no—
199#121—48no—
200#121—42no—
201#121—44no—
202#121—42no—
203#121—44no—
204#121—44no—
205#121—82no—
206#121—82no—
207#121—81no—
208#121—81no—
209C4×Q8—24no—
210C4×Q8—24no—
211C4×Q8—44no—
212C4×Q8—44no—
213C4×Q8—42no—
214C4×Q8—42no—
215C4×Q8—44no—
216C4×D4—42no—
217C4×D4—44no—
218C4×D4—42no—
219C4×D4—44no—
220C4×D4—42no—
221C4×D4—44no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—82no—
236C2×C4⋊C4C22⋊C412yes—
237C2×C4⋊C4C22⋊C411yescharacteristic
238C2×C4⋊C4C22⋊C412yes—
239C2×C4⋊C4C22⋊C412yes—
240C2×C4⋊C4C22⋊C412yes—
241C2×C4⋊C4C22⋊C412yes—
242C2×C4⋊C4C22⋊C411yescharacteristic
243C2×C4⋊C4C22×C412yes—
244C2×C4⋊C4C22×C411yescharacteristic
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C4⋊C4C22×C411yescharacteristic
247C2×C22⋊C4—21no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—22no—
262C2×C42—22no—
263C2×C42—21no—
264C2×C42—42no—
265C2×C42—44no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42C22×C411yescharacteristic
270C2×C42C22×C412yes—
271C2×C42C22×C411yescharacteristic
272C2×C42C22×C412yes—
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2×D411yescharacteristic
277C4⋊C8—44no—
278C4⋊C8—42no—
279C22⋊C8—21no—
280C22⋊C8—21no—
281C22⋊C8—22no—
282C22⋊C8—22no—
283C22⋊C8—21no—
284C22⋊C8#4112yescomplemented
285C4×C8—44no—
286C4×C8—42no—
287#95—21no—
288#95—22no—
289#95—21no—
290#95—22no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—22no—
295#95—21no—
296#95—22no—
297#95—42no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—81no—
302#95#4112yescomplemented
303C24—21no—
304C24—22no—
305C24C22×D411yescharacteristic
306#41—816no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—48no—
314C2×Q8—48no—
315C2×Q8#10214yes—
316C2×Q8#10214yes—
317C2×Q8#10218yes—
318C2×D4—21no—
319C2×D4—22no—
320C2×D4—21no—
321C2×D4—22no—
322C2×D4—41no—
323C2×D4—42no—
324C2×D4—41no—
325C2×D4—42no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—41no—
331C2×D4—42no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—42no—
335C2×D4—41no—
336C2×D4—42no—
337C2×D4—41no—
338C2×D4—42no—
339C2×D4—41no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—82no—
347C2×D4—81no—
348C2×D4—82no—
349C2×D4—81no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—44no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C412yes—
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C412yes—
399C22×C4C2×C22⋊C412yes—
400C22×C4C2×C22⋊C412yes—
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×D812yes—
404C22×C4C2×D814yes—
405C22×C4C2×SD1612yes—
406C22×C4C2×SD1614yes—
407C22×C4C23×C411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4H(5,𝔽2)11yescharacteristic
412C22×C42−1+411yescharacteristic
413C2×C8—22no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—42no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—42no—
420C2×C8—41no—
421C2×C8—42no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—42no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—42no—
438C4⋊C4—82no—
439C4⋊C4—81no—
440C4⋊C4—84no—
441C4⋊C4—88no—
442C4⋊C4#9914yes—
443C22⋊C4—42no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C22⋊C4—42no—
447C22⋊C4—41no—
448C22⋊C4—42no—
449C22⋊C4—41no—
450C22⋊C4—44no—
451C22⋊C4—42no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—82no—
457C22⋊C4—81no—
458C22⋊C4—82no—
459C22⋊C4—82no—
460C22⋊C4—82no—
461C42—24no—
462C42—22no—
463C42—24no—
464C42—22no—
465C42—42no—
466C42—41no—
467C42—42no—
468C42—41no—
469C42—44no—
470C42—42no—
471C42—84no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—41no—
477C23—42no—
478C23—41no—
479C23—42no—
480C23—41no—
481C23—42no—
482C23—42no—
483C23—41no—
484C23—41no—
485C23—42no—
486C23—42no—
487C23—41no—
488C23—41no—
489C23—42no—
490C23C22×C22⋊C411yescharacteristic
491C23#54811yescharacteristic
492C23#55011yescharacteristic
493C23C22×D811yescharacteristic
494C23C22×SD1611yescharacteristic
495C23#57611yescharacteristic
496C23#57711yescharacteristic
497Q8—28no—
498Q8—28no—
499Q8—48no—
500D4—81no—
501D4—82no—
502D4—81no—
503D4—82no—
504D4—88no—
505D4—81no—
506D4—82no—
507D4—81no—
508D4—82no—
509D4—81no—
510D4—82no—
511D4—81no—
512D4—82no—
513D4—84no—
514D4—164no—
515D4—162no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—88no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—84no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4C2×#9914yes—
585C2×C4C2×#9912yes—
586C2×C4C2×#10214yes—
587C2×C4C2×#10212yes—
588C2×C4C2×#10214yes—
589C2×C4C2×#10212yes—
590C2×C4#42112yes—
591C2×C4C4×#4111yescharacteristic
592C2×C4#52011yescharacteristic
593C2×C4#54811yescharacteristic
594C2×C4#55011yescharacteristic
595C8—42no—
596C8—82no—
597C8—81no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—81no—
611C22—82no—
612C22—82no—
613C22—81no—
614C22—82no—
615C22—81no—
616C22—82no—
617C22—81no—
618C22—82no—
619C22—81no—
620C22—81no—
621C22—82no—
622C22—82no—
623C22—81no—
624C22#234111yescharacteristic
625C22C22×#9911yescharacteristic
626C22#288411yescharacteristic
627C22C22×#10211yescharacteristic
628C22#289711yescharacteristic
629C22C2×#42111yescharacteristic
630C22#290111yescharacteristic
631C22#335811yescharacteristic
632C22#336211yescharacteristic
633C22#339111yescharacteristic
634C22#339211yescharacteristic
635C4—21no—
636C4—21no—
637C4—21no—
638C4—21no—
639C4—22no—
640C4—24no—
641C4—24no—
642C4—24no—
643C4—42no—
644C4—41no—
645C4—41no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4—81no—
650C4—81no—
651C4—81no—
652C4—162no—
653C4#151914yes—
654C2—21no—
655C2—21no—
656C2—21no—
657C2—21no—
658C2—162no—
659C2—161no—
660C2C2×#151911yescharacteristic
661C2#1760211yescharacteristic
662C2#1760311yescharacteristic
663C2#1761811yescharacteristic
664C2#2253411yescharacteristic
665C2#2254011yescharacteristic
666C2#2255111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#36576C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. Q8×C22⋊C8C2semidirect
3. #13935C2semidirect
4. #9715C2semidirect
5. #9712C2semidirect
6. #9680C2semidirect
7. #9680C2semidirect
8. #8577C2semidirect
9. #8571C2semidirect
10. #8220C2semidirect
11. C4×#327C2semidirect
12. C2×#550C4semidirect
18. #2548C4semidirect
19. #2548C4semidirect
20. #2538C4semidirect
21. #2538C4semidirect
26. Q8×C4⋊C4C4semidirect
284. C22⋊C8#41semidirect
302. #95#41semidirect

Maximal subgroups

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

subgroupindex
#365762
Q8×C22⋊C82
#139352
#97152
#97122
#96802
#85772
#85712
#82202
C4×#3272