Finite Group Databasebeta

← #163959

Subgroups — #163959

Order 512.

1630 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34259C211yesmaximal, characteristic
2#22471C212yesmaximal, complemented
3#7645C211yesmaximal, characteristic, complemented
4#7643C211yesmaximal, characteristic, complemented
5#7631C211yesmaximal, characteristic, complemented
6#7629C211yesmaximal, characteristic, complemented
7#7612C212yesmaximal, complemented
8#7609C212yesmaximal, complemented
9#7595C211yesmaximal, characteristic, complemented
10#7578C212yesmaximal, complemented
11C2×#1272C211yesmaximal, characteristic, complemented
12C22×2−1+4C411yescharacteristic
13C2×#520C411yescharacteristic
14C2×C4×#41C2211yescharacteristic
15C2×#435—22no—
16Q8×C22⋊C4—22no—
17#2339—22no—
18#2339—22no—
19#2339—22no—
20#2311C411yescharacteristic
21#2311C411yescharacteristic
22#2311C411yescharacteristic
23#2310C411yescharacteristic
24#2310C411yescharacteristic
25#2310C411yescharacteristic
26#2297C2211yescharacteristic
27#2297C2211yescharacteristic
28#2297C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#391C2211yescharacteristic
32#1985—22no—
33#1870—22no—
34#1869—22no—
35#1817C2212yes—
36#1817C2212yes—
37#1797C2212yes—
38#1749C2212yes—
39#1728C2212yes—
40C2×#336C2212yes—
41#1272C2212yes—
42#1272C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1272C2211yescharacteristic
48#1272C2211yescharacteristic
49#1270C2212yes—
50#1270C2212yes—
51C2×2−1+4—24no—
52C2×2−1+4—24no—
53C2×2−1+4—44no—
54C2×2−1+4—44no—
55C22×#41—21no—
56C22×#41—21no—
57C22×#41—21no—
58C22×#41C2×C411yescharacteristic
59C22×#41C2×C411yescharacteristic
60C22×#41C2×C411yescharacteristic
61C22×#41C2×C411yescharacteristic
62C23×Q8—21no—
63C23×Q8C2×C411yescharacteristic
64C23×Q8C2×C411yescharacteristic
65C23×Q8C2×C411yescharacteristic
66C22×M16—22no—
67C23×C8—22no—
68#520—44no—
69C4×#41—44no—
70C2×C4×Q8—21no—
71C2×C4×Q8D412yes—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4D412yes—
76C2×C4×D4D412yes—
77C2×C4×D4D412yes—
78C2×#117C2×C411yescharacteristic
79C2×#117C2×C411yescharacteristic
80C2×#117C2×C411yescharacteristic
81C2×#117C2×C411yescharacteristic
82C2×#117C2×C411yescharacteristic
83C2×#117C2×C411yescharacteristic
84C22×C4⋊C4C2×C411yescharacteristic
85C22×C4⋊C4C2×C411yescharacteristic
86C22×C4⋊C4C2×C411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#435—48no—
91C2×C4⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×#97—22no—
97#391—22no—
98#391—22no—
99#391—22no—
100#391—22no—
101#391—22no—
102#391—22no—
103#391C2×C411yescharacteristic
104#391C2×C411yescharacteristic
105#391C2×C411yescharacteristic
106#391C2×C411yescharacteristic
107#391C2×C411yescharacteristic
108#391C2×C411yescharacteristic
109#391C2×C411yescharacteristic
110#391C2×C411yescharacteristic
111#391C2×C411yescharacteristic
112#391C2311yescharacteristic
113#391C2311yescharacteristic
114#391C2311yescharacteristic
115#390C2×C411yescharacteristic
116#390C2×C411yescharacteristic
117#390C2×C411yescharacteristic
118#390C2311yescharacteristic
119#386—22no—
120#386—22no—
121#386—22no—
122#386—22no—
123#386—22no—
124#386—22no—
125#381—22no—
126#381—22no—
127#381—22no—
128#380—22no—
129#380—22no—
130#380—22no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137#336—22no—
138#336—22no—
139#336—22no—
140#336—22no—
141#336C2312yes—
142#336C2312yes—
143#336C2312yes—
144#336C2312yes—
1452−1+4—432no—
1462−1+4—432no—
147C2×#41—28no—
148C2×#41—28no—
149C2×#41—28no—
150C2×#41—22no—
151C2×#41—22no—
152C2×#41—24no—
153C2×#41—24no—
154C2×#41—22no—
155C2×#41—22no—
156C2×#41—22no—
157C2×#41—22no—
158C2×#41—22no—
159C2×#41—42no—
160C2×#41—42no—
161C2×#41—42no—
162C2×#41—42no—
163C2×#41—42no—
164C2×#41—42no—
165C2×#41—44no—
166C2×#41—44no—
167C2×#41—44no—
168C2×#41—44no—
169C2×#41—44no—
170C2×#41—44no—
171C2×#41M1614yescomplemented
172C22×Q8—28no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—21no—
178C22×Q8—21no—
179C22×Q8—21no—
180C22×Q8—22no—
181C22×Q8—42no—
182C22×Q8—42no—
183C22×Q8—42no—
184C22×Q8—42no—
185C22×Q8—42no—
186C22×Q8—42no—
187C22×Q8—44no—
188C22×Q8—44no—
189C22×Q8C22⋊C411yescharacteristic
190C22×Q8M1614yescomplemented
191C22×Q8C22×C411yescharacteristic
192C22×Q8C22×C411yescharacteristic
193C22×Q8C22×C411yescharacteristic
194C22×D4—21no—
195C22×D4—21no—
196C22×D4—21no—
197C22×D4C22⋊C411yescharacteristic
198C22×D4C22⋊C411yescharacteristic
199C22×D4C22⋊C411yescharacteristic
200C22×D4C22×C411yescharacteristic
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4C22×C411yescharacteristic
211C23×C4C22×C411yescharacteristic
212C23×C4C22×C411yescharacteristic
213C23×C4C2×D412yes—
214C23×C4C2×D412yes—
215C23×C4C2×D412yes—
216C2×M16—44no—
217C2×M16—44no—
218C2×M16—82no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—44no—
224C22×C8—44no—
225C22×C8—82no—
226C4×Q8—44no—
227C4×Q8—44no—
228C4×D4—28no—
229C4×D4—44no—
230C4×D4—44no—
231C4×D4—44no—
232C4×D4—44no—
233C4×D4—44no—
234#117—42no—
235#117—42no—
236#117—42no—
237#117—42no—
238#117—42no—
239#117—42no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4C22⋊C412yes—
247C2×C4⋊C4C22⋊C412yes—
248C2×C4⋊C4C22⋊C412yes—
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C4⋊C4C22×C411yescharacteristic
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4C22⋊C412yes—
261C2×C22⋊C4C22⋊C412yes—
262C2×C22⋊C4C22⋊C412yes—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42C22×C411yescharacteristic
270C2×C42C22×C411yescharacteristic
271C2×C42C22×C411yescharacteristic
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42C2411yescharacteristic
276C4⋊C8—44no—
277C22⋊C8—44no—
278C22⋊C8—44no—
279C22⋊C8—44no—
280C22⋊C8—82no—
281#97—44no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294C24—21no—
295C24C2×C22⋊C411yescharacteristic
296C24C2×C22⋊C411yescharacteristic
297C24C2×C22⋊C411yescharacteristic
298#41—216no—
299#41—216no—
300#41—48no—
301#41—48no—
302#41—48no—
303#41—48no—
304#41—48no—
305#41—48no—
306#41—432no—
307#41—432no—
308#41—432no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—22no—
315C2×Q8—24no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—48no—
323C2×Q8—48no—
324C2×Q8—48no—
325C2×Q8—48no—
326C2×Q8—48no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—48no—
334C2×Q8—432no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8C2×M1612yes—
338C2×Q8C2×M1612yes—
339C2×Q8C2×M1612yes—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—24no—
349C2×D4—21no—
350C2×D4—22no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4#9914yes—
360C2×D4#10014yes—
361C2×D4C2×M1612yes—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—42no—
394C22×C4—42no—
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—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—82no—
413C22×C4—82no—
414C22×C4—82no—
415C22×C4C2×C22⋊C412yes—
416C22×C4C2×C22⋊C412yes—
417C22×C4C2×C22⋊C412yes—
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×M1612yes—
422C22×C4C2×M1612yes—
423C22×C4C23×C411yescharacteristic
424C22×C4C22×D412yes—
425M16—88no—
426C2×C8—24no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—88no—
439C4⋊C4—24no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—82no—
458C22⋊C4—82no—
459C22⋊C4—82no—
460C42—24no—
461C42—41no—
462C42—41no—
463C42—41no—
464C42—41no—
465C42—42no—
466C42—42no—
467C42—42no—
468C42—42no—
469C42—42no—
470C23—22no—
471C23—22no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—42no—
485C23#40712yes—
486C23C2×#9912yes—
487C23C2×#9912yes—
488C23C2×#10012yes—
489C23C2×#10012yes—
490C23C22×C22⋊C411yescharacteristic
491Q8—28no—
492Q8—42no—
493Q8—42no—
494Q8—42no—
495Q8—42no—
496Q8—42no—
497Q8—42no—
498Q8—48no—
499Q8—48no—
500Q8—48no—
501D4—28no—
502D4—28no—
503D4—28no—
504D4—42no—
505D4—42no—
506D4—48no—
507D4—48no—
508D4—48no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—48no—
542C2×C4—48no—
543C2×C4—48no—
544C2×C4—48no—
545C2×C4—48no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—48no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—162no—
564C2×C4—162no—
565C2×C4—162no—
566C2×C4#40712yes—
567C2×C4C2×#9912yes—
568C2×C4C2×#10012yes—
569C2×C4C22×M1611yescharacteristic
570C2×C4C22×M1611yescharacteristic
571C2×C4#57712yes—
572C2×C4#57812yes—
573C8—44no—
574C8—82no—
575C8—82no—
576C8—82no—
577C22—22no—
578C22—24no—
579C22—22no—
580C22—24no—
581C22—22no—
582C22—22no—
583C22—24no—
584C22—22no—
585C22—22no—
586C22—41no—
587C22—42no—
588C22—42no—
589C22—41no—
590C22—42no—
591C22—44no—
592C22—42no—
593C22—81no—
594C22—81no—
595C22—82no—
596C22#146214yes—
597C22C2×#40711yescharacteristic
598C22C22×#9911yescharacteristic
599C22C22×#10011yescharacteristic
600C22#289812yes—
601C22#290112yes—
602C4—22no—
603C4—22no—
604C4—21no—
605C4—21no—
606C4—22no—
607C4—22no—
608C4—22no—
609C4—42no—
610C4—42no—
611C4—42no—
612C4—42no—
613C4—42no—
614C4—42no—
615C4—81no—
616C4—81no—
617C4—81no—
618C4—81no—
619C2—24no—
620C2—42no—
621C2—42no—
622C2—82no—
623C2C2×#146211yescharacteristic
624C2#1745412yes—
625C2#1760812yes—
626C2#1768511yescharacteristic
627C2#1777911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×2−1+4C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22471C2semidirect
3. #7645C2semidirect
4. #7643C2semidirect
5. #7631C2semidirect
6. #7629C2semidirect
7. #7612C2semidirect
8. #7609C2semidirect
9. #7595C2semidirect
10. #7578C2semidirect
11. C2×#1272C2semidirect
171. C2×#41M16semidirect
190. C22×Q8M16semidirect

Maximal subgroups

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

subgroupindex
#342592
#224712
#76452
#76432
#76312
#76292
#76122
#76092
#75952
#75782
C2×#12722