Finite Group Databasebeta

← #209221

Subgroups — #209221

Order 512.

968 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×M16C211yesmaximal, characteristic
2#21596C211yesmaximal, characteristic
3C4×#451C211yesmaximal, characteristic
4C2×#1587C211yesmaximal, characteristic, complemented
5C2×#1579C211yesmaximal, characteristic, complemented
6#8950C211yesmaximal, characteristic, complemented
7#8946C211yesmaximal, characteristic, complemented
8#8942C211yesmaximal, characteristic
9#8850C211yesmaximal, characteristic, complemented
10#8846C211yesmaximal, characteristic, complemented
11#8833C211yesmaximal, characteristic
12#8311C211yesmaximal, characteristic
13#8209C211yesmaximal, characteristic, complemented
14#8205C211yesmaximal, characteristic, complemented
15#8147C211yesmaximal, characteristic
16C22×C4×Q8C2211yescharacteristic
17C2×#451C411yescharacteristic
18Q8×M16—24no—
19Q8×M16—24no—
20C2×#446—21no—
21C2×#446—21no—
22C2×#446C2211yescharacteristic
23C2×#446C2211yescharacteristic
24C2×C8×Q8C2211yescharacteristic
25C2×C8×Q8C2211yescharacteristic
26C2×C8×Q8C2211yescharacteristic
27C2×C4×Q16C2211yescharacteristic
28C2×C4×Q16C2211yescharacteristic
29C2×#423—21no—
30C2×#423C2211yescharacteristic
31C2×C4×M16—21no—
32C2×C4×M16C2211yescharacteristic
33C4×C22⋊Q8C2211yescharacteristic
34#1984C2211yescharacteristic
35#1939C2211yescharacteristic
36#1917—21no—
37#1878C411yescharacteristic
38#1878C411yescharacteristic
39#1872C411yescharacteristic
40#1863C411yescharacteristic
41#1863C411yescharacteristic
42#1852C411yescharacteristic
43#1851—21no—
44#1797C2211yescharacteristic
45#1787C411yescharacteristic
46#1771—21no—
47#1770C2211yescharacteristic
48C4×#103C2211yescharacteristic
49C4×#103C2211yescharacteristic
50#1753—21no—
51C4×C22⋊C8C2211yescharacteristic
52C4×C22⋊C8C2211yescharacteristic
53#1587—24no—
54#1579—24no—
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×Q8⋊C8C2211yescharacteristic
62C2×Q8⋊C8C2211yescharacteristic
63C2×#322C2211yescharacteristic
64C2×#322C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69#1273C2211yescharacteristic
70#1273C2211yescharacteristic
71C23×Q8C2×C411yescharacteristic
72C22×Q16C2×C411yescharacteristic
73C22×Q16C2×C411yescharacteristic
74C22×M16—21no—
75C22×M16D411yescharacteristic
76C22×M16D411yescharacteristic
77C2×C22⋊Q8C2×C411yescharacteristic
78C2×C4×Q8—22no—
79C2×C4×Q8—22no—
80C2×C4×Q8—22no—
81C2×C4×Q8—22no—
82C2×C4×Q8—21no—
83C2×C4×Q8—21no—
84C2×C4×Q8C2×C411yescharacteristic
85C2×C4×Q8C2×C411yescharacteristic
86C2×C4×Q8C2311yescharacteristic
87C2×C4×Q8C2311yescharacteristic
88C2×C4×Q8C2311yescharacteristic
89C22×C4⋊C4—21no—
90C22×C4⋊C4C2×C411yescharacteristic
91C22×C42—21no—
92C22×C42C2311yescharacteristic
93#451—24no—
94#446—28no—
95#446—28no—
96#446—22no—
97#446—22no—
98#446—22no—
99#446D414yes—
100C8×Q8—22no—
101C8×Q8—22no—
102C8×Q8—22no—
103C8×Q8—22no—
104C8×Q8—22no—
105C8×Q8D414yes—
106C4×Q16—22no—
107C4×Q16—22no—
108C4×Q16—22no—
109C4×Q16—22no—
110C2×#107C2×C411yescharacteristic
111C2×#107C2×C411yescharacteristic
112#423—28no—
113#423—22no—
114#423—22no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×#103C2×C411yescharacteristic
127C2×#103C2×C411yescharacteristic
128C2×#103C2×C411yescharacteristic
129C2×#103C2×C411yescharacteristic
130C2×#103C2×C411yescharacteristic
131C2×#103C2×C411yescharacteristic
132C2×#103C2×C411yescharacteristic
133C2×#103C2×C411yescharacteristic
134C2×C22⋊C8—21no—
135C2×C22⋊C8—21no—
136C2×C22⋊C8C2×C411yescharacteristic
137C2×C22⋊C8C2×C411yescharacteristic
138C4×M16—28no—
139C4×M16—22no—
140C4×M16—22no—
141C2×#97—21no—
142C2×#97—21no—
143C2×#97—21no—
144C2×#97D411yescharacteristic
145C2×#97D411yescharacteristic
146C2×C4×C8—21no—
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155#391C2×C411yescharacteristic
156#389C2×C411yescharacteristic
157#387—21no—
158#385—21no—
159#380C2×C411yescharacteristic
160C4×C4⋊C4—21no—
161C4×C4⋊C4C2311yescharacteristic
162C4×C22⋊C4—21no—
163#340C2×C411yescharacteristic
164#340C2×C411yescharacteristic
165#340C2×C411yescharacteristic
166#340C2×C411yescharacteristic
167#336—21no—
168#336—21no—
169#336—21no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336C2×C411yescharacteristic
174#336C2×C411yescharacteristic
175#334—22no—
176#334—22no—
177Q8⋊C8—22no—
178Q8⋊C8—22no—
179Q8⋊C8—22no—
180Q8⋊C8—22no—
181Q8⋊C8—22no—
182Q8⋊C8—22no—
183#322—22no—
184#322—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—21no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—41no—
191C22×Q8—41no—
192C22×Q8C22×C411yescharacteristic
193C22×Q8C22×C411yescharacteristic
194C22×Q8C22×C411yescharacteristic
195C23×C4—21no—
196C23×C4—21no—
197C23×C4C22×C411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4#4111yescharacteristic
200C2×Q16—22no—
201C2×Q16—22no—
202C2×Q16—22no—
203C2×Q16—22no—
204C2×Q16—41no—
205C2×Q16M1614yescomplemented
206C2×M16—22no—
207C2×M16—24no—
208C2×M16—22no—
209C2×M16—22no—
210C2×M16—22no—
211C2×M16—22no—
212C2×M16—41no—
213C2×M16—42no—
214C2×M16—42no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8C22×C411yescharacteristic
222C22×C8C22×C411yescharacteristic
223C22×C8C22×C411yescharacteristic
224C22×C8C22×C411yescharacteristic
225C22×C8C2×D411yescharacteristic
226C22×C8C2×D411yescharacteristic
227C22×C8C2×D411yescharacteristic
228C22×C8C2×D411yescharacteristic
229C22⋊Q8—22no—
230C22⋊Q8—41no—
231C4×Q8—24no—
232C4×Q8—24no—
233C4×Q8—24no—
234C4×Q8—24no—
235C4×Q8—22no—
236C4×Q8—22no—
237C4×Q8—22no—
238C4×Q8—22no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×Q8—21no—
242C4×Q8—21no—
243C4×Q8—21no—
244C4×Q8—21no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×Q8C2×D412yes—
248C4×Q8C2×D412yes—
249C4×Q8#4112yes—
250C4×Q8#4112yes—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C4⋊C4C22×C411yescharacteristic
266C2×C4⋊C4C22×C411yescharacteristic
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42C22×C411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42#4111yescharacteristic
285C2×C42C2411yescharacteristic
286#107—22no—
287#107—22no—
288C4⋊C8—24no—
289C4⋊C8—24no—
290C4⋊C8—24no—
291C4⋊C8—24no—
292C4⋊C8—24no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—42no—
300C4⋊C8—42no—
301C4⋊C8C2×D412yes—
302C4⋊C8C2×D412yes—
303#103—22no—
304#103—22no—
305#103—22no—
306#103—22no—
307#103—22no—
308#103—22no—
309#103—22no—
310#103—41no—
311C22⋊C8—22no—
312C22⋊C8—41no—
313C22⋊C8—42no—
314C22⋊C8—42no—
315#97—24no—
316#97—24no—
317#97—22no—
318#97—22no—
319#97—42no—
320C4×C8—24no—
321C4×C8—24no—
322C4×C8—21no—
323C4×C8—21no—
324C4×C8—21no—
325C4×C8—21no—
326C4×C8—21no—
327C4×C8—21no—
328C4×C8—21no—
329C4×C8—21no—
330C4×C8—21no—
331C4×C8—21no—
332C4×C8—42no—
333C4×C8C2×D412yes—
334C4×C8C2×D412yes—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340C24C4×D411yescharacteristic
341C2×Q8—24no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—44no—
358C2×Q8—44no—
359C2×Q8—44no—
360C2×Q8—44no—
361C2×Q8C4×D412yes—
362C2×Q8C4×D412yes—
363C2×Q8C2×M1612yes—
364C2×Q8C2×M1612yes—
365C2×Q8#13112yes—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—24no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—24no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—81no—
393C22×C4—81no—
394C22×C4C4×D411yescharacteristic
395C22×C4C4×D411yescharacteristic
396C22×C4C4×D411yescharacteristic
397C22×C4#13712yes—
398C22×C4#13712yes—
399C22×C4C23×C411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C2×#4111yescharacteristic
402Q16—28no—
403Q16—44no—
404M16—28no—
405M16—44no—
406M16—48no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—22no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8#12012yes—
446C2×C8#12012yes—
447C2×C8C2×M1612yes—
448C2×C8#13712yes—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—42no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—81no—
468C4⋊C4C4×D412yes—
469C4⋊C4C4×D412yes—
470C22⋊C4—42no—
471C22⋊C4—81no—
472C42—24no—
473C42—24no—
474C42—21no—
475C42—21no—
476C42—22no—
477C42—22no—
478C42—21no—
479C42—22no—
480C42—22no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—41no—
485C42—41no—
486C42—41no—
487C42—41no—
488C42C22×D411yescharacteristic
489C42C22×D411yescharacteristic
490C42C2×#4111yescharacteristic
491C42C2×#4111yescharacteristic
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23#44112yes—
499C23C2×C4×D411yescharacteristic
500Q8—22no—
501Q8—22no—
502Q8—22no—
503Q8—22no—
504Q8—22no—
505Q8—22no—
506Q8—22no—
507Q8—48no—
508Q8—41no—
509Q8—41no—
510Q8—41no—
511Q8—41no—
512Q8#43514yes—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—48no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
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—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—84no—
559C2×C4#44112yes—
560C2×C4C2×C4×D411yescharacteristic
561C2×C4C2×C4×D411yescharacteristic
562C2×C4C2×#12011yescharacteristic
563C2×C4#52511yescharacteristic
564C2×C4#52511yescharacteristic
565C2×C4#52511yescharacteristic
566C2×C4C22×M1611yescharacteristic
567C2×C4C2×#13111yescharacteristic
568C2×C4#56811yescharacteristic
569C2×C4#56811yescharacteristic
570C2×C4C2×#13711yescharacteristic
571C2×C4C2×#13711yescharacteristic
572C2×C4C2×#13711yescharacteristic
573C2×C4#57511yescharacteristic
574C8—22no—
575C8—22no—
576C8—22no—
577C8—22no—
578C8—42no—
579C8—41no—
580C8—41no—
581C8—41no—
582C8—41no—
583C8—42no—
584C8—42no—
585C8—81no—
586C8—81no—
587C8—81no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22C4×#12011yescharacteristic
596C22#293311yescharacteristic
597C22D4×M1611yescharacteristic
598C22C2×#44111yescharacteristic
599C22C4×#13711yescharacteristic
600C22#295611yescharacteristic
601C22#295711yescharacteristic
602C4—24no—
603C4—22no—
604C4—21no—
605C4—21no—
606C4—41no—
607C4—42no—
608C4—42no—
609C4—41no—
610C4—42no—
611C4—42no—
612C4—81no—
613C4—81no—
614C4—81no—
615C4C2×#43511yescharacteristic
616C4#293011yescharacteristic
617C4#293311yescharacteristic
618C4D4×M1611yescharacteristic
619C4#300411yescharacteristic
620C4#300711yescharacteristic
621C4#311311yescharacteristic
622C4#311711yescharacteristic
623C2—42no—
624C2#1800311yescharacteristic
625C2#1801011yescharacteristic
626C2#1804211yescharacteristic
627C2#1804911yescharacteristic
628C2#2143911yescharacteristic
629C2#2145511yescharacteristic
630C2#2146811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1587C2semidirect
5. C2×#1579C2semidirect
6. #8950C2semidirect
7. #8946C2semidirect
9. #8850C2semidirect
10. #8846C2semidirect
13. #8209C2semidirect
14. #8205C2semidirect
205. C2×Q16M16semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×M162
#215962
C4×#4512
C2×#15872
C2×#15792
#89502
#89462
#89422
#88502
#88462
#88332
#83112
#82092
#82052
#81472