Finite Group Databasebeta

← #224752

Subgroups — #224752

Order 512.

1182 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2362C211yesmaximal, characteristic
2#22510C212yesmaximal, complemented
3#9654C212yesmaximal, complemented
4#9652C212yesmaximal
5#9651C212yesmaximal
6#8560C211yesmaximal, characteristic, complemented
7#8559C211yesmaximal, characteristic
8#8558C211yesmaximal, characteristic, complemented
9#8153C211yesmaximal, characteristic, complemented
10#8152C211yesmaximal, characteristic
11C4×#324C211yesmaximal, characteristic, complemented
12C22×C22⋊Q8C411yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×#117C411yescharacteristic
15C2×#446—22no—
16#2362—22no—
17#2362—22no—
18#2362—22no—
19#2362—22no—
20C2×#391C411yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#388C411yescharacteristic
23C2×#388C411yescharacteristic
24C2×#388C2211yescharacteristic
25C2×#388C2211yescharacteristic
26C2×#387C411yescharacteristic
27C2×#387C411yescharacteristic
28C2×C4×C4⋊C4C411yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2211yescharacteristic
32#1985—22no—
33#1985—22no—
34#1984—22no—
35#1797C2212yes—
36#1797C2212yes—
37#1797C2212yes—
38#1749C2212yes—
39#1749C2212yes—
40C4×C22⋊C8C2212yes—
41C2×#324C2212yes—
42C2×#324C2212yes—
43C2×#324C2212yes—
44C2×#324C2212yes—
45C2×#323C2212yes—
46C2×#323C2212yes—
47#1280C2211yescharacteristic
48#1280C2211yescharacteristic
49#1280C2211yescharacteristic
50#1280C2211yescharacteristic
51C23×Q8C2×C411yescharacteristic
52C24×C4C2×C411yescharacteristic
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—24no—
56C2×C22⋊Q8—24no—
57C2×C4×Q8—44no—
58C2×C4×Q8D412yes—
59C2×C4×Q8D412yes—
60C2×C4×Q8D412yes—
61C2×C4×Q8D412yes—
62C2×#117—21no—
63C2×#117—21no—
64C2×#117—21no—
65C2×#117—21no—
66C2×#117—44no—
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4C2×C411yescharacteristic
77C22×C22⋊C4C2×C411yescharacteristic
78C22×C22⋊C4C2×C411yescharacteristic
79C22×C22⋊C4C2×C411yescharacteristic
80C22×C22⋊C4C2×C411yescharacteristic
81C22×C22⋊C4C2311yescharacteristic
82C22×C42C2×C411yescharacteristic
83C22×C42C2×C411yescharacteristic
84C22×C42C2×C411yescharacteristic
85C22×C42C2×C411yescharacteristic
86C22×C42C2×C411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#446—48no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C22⋊C8C2312yes—
95C2×C22⋊C8C2312yes—
96C2×C22⋊C8C2312yes—
97C2×C22⋊C8C2312yes—
98C2×#97—22no—
99C2×#97—22no—
100C2×C4×C8—22no—
101#391—22no—
102#391—22no—
103#389—22no—
104#389—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#387—22no—
114#387—22no—
115#387—22no—
116#387—22no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C2×#95C2×C411yescharacteristic
128C2×#95C2×C411yescharacteristic
129C2×#95C2×C411yescharacteristic
130C2×#95C2×C411yescharacteristic
131C2×#95C2×C411yescharacteristic
132C2×#95C2×C411yescharacteristic
133C2×#95C2×C411yescharacteristic
134C2×#95C2×C411yescharacteristic
135C2×#95C2×C411yescharacteristic
136C2×#95C2311yescharacteristic
137#336—22no—
138#336—22no—
139#336—22no—
140#336—22no—
141#336—22no—
142#336—22no—
143#324—24no—
144#324—24no—
145#324—42no—
146#324—42no—
147#323—24no—
148#323—42no—
149C25C22×C411yescharacteristic
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—42no—
153C22×Q8—42no—
154C22×Q8C22⋊C412yes—
155C22×Q8C22⋊C412yes—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C411yescharacteristic
175C23×C4C2411yescharacteristic
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22⋊Q8—216no—
181C22⋊Q8—216no—
182C4×Q8—28no—
183C4×Q8—44no—
184C4×Q8—44no—
185C4×Q8—44no—
186#117—44no—
187#117—44no—
188#117—44no—
189#117—44no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4C22⋊C411yescharacteristic
214C2×C4⋊C4C22⋊C411yescharacteristic
215C2×C4⋊C4C22⋊C411yescharacteristic
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22⋊C412yes—
219C2×C4⋊C4C22⋊C412yes—
220C2×C4⋊C4C22⋊C411yescharacteristic
221C2×C4⋊C4M1614yes—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4M1614yes—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—41no—
257C2×C42—41no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C4⋊C8—44no—
268C4⋊C8—44no—
269C4⋊C8—44no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272C22⋊C8—22no—
273C22⋊C8—22no—
274C22⋊C8—22no—
275C22⋊C8—22no—
276C22⋊C8—22no—
277C22⋊C8#4114yes—
278#97—44no—
279#97—44no—
280C4×C8—44no—
281#95—22no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—41no—
300C24—41no—
301C24—41no—
302C24—41no—
303C24C2×M1612yes—
304C24C23×C411yescharacteristic
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—48no—
312C2×Q8—48no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—48no—
366C22×C4—48no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—41no—
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×C4C2×C22⋊C412yes—
381C22×C4C2×C22⋊C412yes—
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C412yes—
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×M1612yes—
390C22×C4C2×M1612yes—
391C22×C4C2×M1612yes—
392C22×C4C2×M1612yes—
393C22×C4C2×M1612yes—
394C22×C4#13112yes—
395C22×C4C22×D412yes—
396C22×C4C2×#4112yes—
397C22×C4H(5,𝔽2)11yescharacteristic
398C22×C42−1+411yescharacteristic
399C2×C8—24no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—28no—
414C4⋊C4—28no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4#9914yes—
429C22⋊C4—28no—
430C22⋊C4—28no—
431C22⋊C4—44no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C42—24no—
439C42—24no—
440C42—24no—
441C42—42no—
442C42—42no—
443C42—42no—
444C42—42no—
445C42—42no—
446C42—42no—
447C42—42no—
448C42—42no—
449C42—42no—
450C42—42no—
451C42—42no—
452C42—42no—
453C42—42no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—81no—
474C23—81no—
475C23—81no—
476C23—81no—
477C23—81no—
478C23—81no—
479C23—81no—
480C23—81no—
481C23C22×C22⋊C411yescharacteristic
482C23C4×#4111yescharacteristic
483C23#52011yescharacteristic
484C23C22×M1611yescharacteristic
485C23C22×M1611yescharacteristic
486C23C2×#13111yescharacteristic
487C23#56811yescharacteristic
488Q8—48no—
489Q8—48no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—48no—
534C2×C4—48no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4#40714yes—
571C2×C4C2×#9912yes—
572C2×C4C2×#9912yes—
573C2×C4C2×#9912yes—
574C2×C4#41312yes—
575C2×C4#54812yes—
576C2×C4#55012yes—
577C8—44no—
578C8—82no—
579C8—82no—
580C8—82no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22#234111yescharacteristic
605C22C2×#40711yescharacteristic
606C22#288011yescharacteristic
607C22C22×#9911yescharacteristic
608C22#288411yescharacteristic
609C22C2×#41311yescharacteristic
610C22#288911yescharacteristic
611C22#296511yescharacteristic
612C22#297011yescharacteristic
613C22#298911yescharacteristic
614C22#299011yescharacteristic
615C4—22no—
616C4—22no—
617C4—22no—
618C4—22no—
619C4—22no—
620C4—22no—
621C4—24no—
622C4—42no—
623C4—42no—
624C4—41no—
625C4—41no—
626C4—42no—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—81no—
631C4—81no—
632C4—81no—
633C2—21no—
634C2—21no—
635C2—21no—
636C2—21no—
637C2—82no—
638C2#1736211yescharacteristic
639C2#1737111yescharacteristic
640C2#1741611yescharacteristic
641C2#1742311yescharacteristic
642C2#2253211yescharacteristic
643C2#2253411yescharacteristic
644C2#2253711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ 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. #22510C2semidirect
3. #9654C2semidirect
6. #8560C2semidirect
8. #8558C2semidirect
9. #8153C2semidirect
11. C4×#324C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23622
#225102
#96542
#96522
#96512
#85602
#85592
#85582
#81532
#81522
C4×#3242