Finite Group Databasebeta

← #227635

Subgroups — #227635

Order 512.

864 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22952C211yesmaximal, characteristic
2Q8×C22⋊C8C211yesmaximal, characteristic
3C4×#451C211yesmaximal, characteristic
4C2×#1579C211yesmaximal, characteristic, complemented
5C22×Q8⋊C8C211yesmaximal, characteristic
6#9658C211yesmaximal, characteristic, complemented
7#9555C211yesmaximal, characteristic, complemented
8#9543C211yesmaximal, characteristic, complemented
9#9071C211yesmaximal, characteristic
10#8946C211yesmaximal, characteristic, complemented
11#8946C211yesmaximal, characteristic, complemented
12#8846C211yesmaximal, characteristic, complemented
13#8830C211yesmaximal, characteristic
14#8201C211yesmaximal, characteristic, complemented
15#8145C211yesmaximal, characteristic
16C22×C4×Q8C2211yescharacteristic
17C2×#451C411yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C2×C4×Q16C2211yescharacteristic
22C2×C4×Q16C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25C4×C22⋊Q8C2211yescharacteristic
26#1984—21no—
27#1984—21no—
28#1984C2211yescharacteristic
29#1984C2211yescharacteristic
30#1984C2211yescharacteristic
31#1939—21no—
32#1939C2211yescharacteristic
33#1916—21no—
34#1878C411yescharacteristic
35#1878C411yescharacteristic
36#1872C411yescharacteristic
37#1863C411yescharacteristic
38#1863C411yescharacteristic
39#1852C411yescharacteristic
40#1850—21no—
41#1797—21no—
42#1797C2211yescharacteristic
43#1787C411yescharacteristic
44C8×C4⋊C4C2211yescharacteristic
45C4×#103C2211yescharacteristic
46C4×#103C2211yescharacteristic
47C8×C22⋊C4—21no—
48C4×C22⋊C8—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51#1579—24no—
52C2×#334C2211yescharacteristic
53C2×#334C2211yescharacteristic
54C2×Q8⋊C8—24no—
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C23×Q8C2×C411yescharacteristic
70C22×Q16C2×C411yescharacteristic
71C22×Q16C2×C411yescharacteristic
72C23×C8—21no—
73C23×C8—21no—
74C2×C22⋊Q8C2×C411yescharacteristic
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×Q8—22no—
78C2×C4×Q8—22no—
79C2×C4×Q8—41no—
80C2×C4×Q8—41no—
81C2×C4×Q8C2×C411yescharacteristic
82C2×C4×Q8C2×C411yescharacteristic
83C2×C4×Q8C2311yescharacteristic
84C2×C4×Q8C2311yescharacteristic
85C2×C4×Q8C2311yescharacteristic
86C22×C4⋊C4—21no—
87C22×C4⋊C4C2×C411yescharacteristic
88C22×C42—21no—
89C22×C42C2311yescharacteristic
90#451C818yescomplemented
91C8×Q8—22no—
92C8×Q8—22no—
93C8×Q8—22no—
94C8×Q8—41no—
95C8×Q8—41no—
96C8×Q8—41no—
97C4×Q16—22no—
98C4×Q16—22no—
99C4×Q16—41no—
100C4×Q16—41no—
101C2×#107C2×C411yescharacteristic
102C2×#107C2×C411yescharacteristic
103C2×C4⋊C8—22no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—41no—
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×C4⋊C8C2311yescharacteristic
114C2×#103C2×C411yescharacteristic
115C2×#103C2×C411yescharacteristic
116C2×#103C2×C411yescharacteristic
117C2×#103C2×C411yescharacteristic
118C2×#103C2×C411yescharacteristic
119C2×#103C2×C411yescharacteristic
120C2×#103C2×C411yescharacteristic
121C2×#103C2×C411yescharacteristic
122C2×C22⋊C8—21no—
123C2×C22⋊C8C2×C411yescharacteristic
124C2×C22⋊C8C2×C411yescharacteristic
125C2×C22⋊C8D411yescharacteristic
126C2×C22⋊C8D411yescharacteristic
127C2×C4×C8—22no—
128C2×C4×C8—21no—
129C2×C4×C8—21no—
130C2×C4×C8—21no—
131C2×C4×C8—21no—
132C2×C4×C8—21no—
133C2×C4×C8—41no—
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140#391C2×C411yescharacteristic
141#389C2×C411yescharacteristic
142#387—21no—
143#385—21no—
144#380C2×C411yescharacteristic
145C4×C4⋊C4—21no—
146C4×C4⋊C4C2311yescharacteristic
147C4×C22⋊C4—21no—
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#340C2×C411yescharacteristic
151#340C2×C411yescharacteristic
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#336—21no—
160#336—21no—
161#336C2×C411yescharacteristic
162#336C2×C411yescharacteristic
163#336D411yescharacteristic
164#336D411yescharacteristic
165#334—22no—
166#334—41no—
167Q8⋊C8—22no—
168Q8⋊C8—22no—
169Q8⋊C8—22no—
170Q8⋊C8—22no—
171Q8⋊C8—41no—
172Q8⋊C8D414yes—
173#322—22no—
174#322—41no—
175C22×Q8—21no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8—41no—
180C22×Q8—41no—
181C22×Q8C2×C814yes—
182C22×Q8C22×C411yescharacteristic
183C22×Q8C22×C411yescharacteristic
184C22×Q8C22×C411yescharacteristic
185C23×C4—21no—
186C23×C4—21no—
187C23×C4C22×C411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4#4111yescharacteristic
190C2×Q16—22no—
191C2×Q16—22no—
192C2×Q16—41no—
193C2×Q16—41no—
194C2×Q16C2×C814yescomplemented
195C2×Q16C2×C814yescomplemented
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—42no—
203C22×C8—42no—
204C22×C8—42no—
205C22×C8—42no—
206C22×C8—42no—
207C22×C8—81no—
208C22×C8C22×C411yescharacteristic
209C22×C8C22×C411yescharacteristic
210C22×C8C22×C411yescharacteristic
211C22×C8C22×C411yescharacteristic
212C22×C8C2×D411yescharacteristic
213C22×C8C2×D411yescharacteristic
214C22×C8C2×D411yescharacteristic
215C22×C8C2×D411yescharacteristic
216C22⋊Q8—22no—
217C22⋊Q8C2×C814yes—
218C4×Q8—21no—
219C4×Q8—21no—
220C4×Q8—22no—
221C4×Q8—22no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8—42no—
231C4×Q8—42no—
232C4×Q8—41no—
233C4×Q8—41no—
234C4×Q8—42no—
235C4×Q8—42no—
236C4×Q8C2×D412yes—
237C4×Q8#4112yes—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C4⋊C4C22×C411yescharacteristic
253C2×C4⋊C4C22×C411yescharacteristic
254C2×C4⋊C4C22×C411yescharacteristic
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42C22×C411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42#4111yescharacteristic
272C2×C42C2411yescharacteristic
273#107—22no—
274#107—41no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—42no—
283C4⋊C8—42no—
284C4⋊C8—42no—
285C4⋊C8—42no—
286C4⋊C8—81no—
287C4⋊C8C2×D412yes—
288#103—22no—
289#103—22no—
290#103—22no—
291#103—41no—
292#103—41no—
293#103—41no—
294#103C2×C814yescomplemented
295#103C2×C814yescomplemented
296C22⋊C8—22no—
297C22⋊C8—22no—
298C22⋊C8—22no—
299C22⋊C8—42no—
300C22⋊C8C2×C814yes—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—42no—
313C4×C8—42no—
314C4×C8—42no—
315C4×C8—42no—
316C4×C8—81no—
317C4×C8C2×D412yes—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323C24C4×D411yescharacteristic
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—24no—
330C2×Q8—28no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—44no—
342C2×Q8—44no—
343C2×Q8—44no—
344C2×Q8C4×D412yes—
345C2×Q8C22×C812yes—
346C2×Q8C22×C812yes—
347C2×Q8C22×C812yes—
348C2×Q8#13112yes—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—81no—
375C22×C4C4×D411yescharacteristic
376C22×C4C4×D411yescharacteristic
377C22×C4C4×D411yescharacteristic
378C22×C4C22×C812yes—
379C22×C4#13112yes—
380C22×C4#13712yes—
381C22×C4#13712yes—
382C22×C4C23×C411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C2×#4111yescharacteristic
385Q16—28no—
386Q16—44no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—22no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—22no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—22no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—48no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—84no—
429C2×C8C22×C812yes—
430C2×C8C22×C812yes—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—24no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—81no—
450C4⋊C4C4×D412yes—
451C4⋊C4C22×C812yes—
452C22⋊C4—24no—
453C22⋊C4—42no—
454C42—21no—
455C42—21no—
456C42—22no—
457C42—21no—
458C42—21no—
459C42—22no—
460C42—21no—
461C42—21no—
462C42—42no—
463C42—42no—
464C42—41no—
465C42—41no—
466C42—41no—
467C42—41no—
468C42—41no—
469C42—41no—
470C42C22×D411yescharacteristic
471C42C2×#4111yescharacteristic
472C42H(5,𝔽2)11yescharacteristic
473C42H(5,𝔽2)11yescharacteristic
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23C8×D412yes—
479C23#44112yes—
480C23#44412yes—
481C23C2×C4×D411yescharacteristic
482Q8—22no—
483Q8—22no—
484Q8—22no—
485Q8—22no—
486Q8—22no—
487Q8—22no—
488Q8—22no—
489Q8—48no—
490Q8—41no—
491Q8—41no—
492Q8—41no—
493Q8—41no—
494Q8C8×D414yes—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—48no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—84no—
540C2×C4C8×D412yes—
541C2×C4#44112yes—
542C2×C4C2×C4×D411yescharacteristic
543C2×C4#51811yescharacteristic
544C2×C4D4211yescharacteristic
545C2×C4#54611yescharacteristic
546C2×C4#54611yescharacteristic
547C2×C4#55111yescharacteristic
548C2×C4C23×C811yescharacteristic
549C2×C4C2×#13111yescharacteristic
550C2×C4C2×#13111yescharacteristic
551C2×C4#56811yescharacteristic
552C2×C4C2×#13711yescharacteristic
553C2×C4C2×#13711yescharacteristic
554C2×C4#57711yescharacteristic
555C2×C4#57811yescharacteristic
556C8—22no—
557C8—22no—
558C8—42no—
559C8—42no—
560C8—41no—
561C8—41no—
562C8—41no—
563C8—42no—
564C8—41no—
565C8—41no—
566C8—41no—
567C8—81no—
568C8—81no—
569C8—81no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22#157914yescomplemented
577C22#235811yescharacteristic
578C22C2×C8×D411yescharacteristic
579C22#293311yescharacteristic
580C22C2×#44111yescharacteristic
581C22#294711yescharacteristic
582C22C2×#44411yescharacteristic
583C22#295611yescharacteristic
584C4—22no—
585C4—24no—
586C4—22no—
587C4—21no—
588C4—21no—
589C4—42no—
590C4—42no—
591C4—41no—
592C4—42no—
593C4—42no—
594C4—81no—
595C4—81no—
596C4—81no—
597C4C2×C8×D411yescharacteristic
598C4#293311yescharacteristic
599C4#297211yescharacteristic
600C4#297711yescharacteristic
601C4#326611yescharacteristic
602C4#327011yescharacteristic
603C4#331211yescharacteristic
604C4#331611yescharacteristic
605C2—24no—
606C2C2×#157911yescharacteristic
607C2#1785311yescharacteristic
608C2#1800311yescharacteristic
609C2#1802311yescharacteristic
610C2#2284211yescharacteristic
611C2#2285911yescharacteristic
612C2#2288611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1579C2semidirect
6. #9658C2semidirect
7. #9555C2semidirect
8. #9543C2semidirect
10. #8946C2semidirect
11. #8946C2semidirect
12. #8846C2semidirect
14. #8201C2semidirect
90. #451C8semidirect
194. C2×Q16C2×C8semidirect
195. C2×Q16C2×C8semidirect
294. #103C2×C8semidirect
295. #103C2×C8semidirect
576. C22#1579semidirect

Maximal subgroups

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

subgroupindex
#229522
Q8×C22⋊C82
C4×#4512
C2×#15792
C22×Q8⋊C82
#96582
#95552
#95432
#90712
#89462
#88462
#88302
#82012
#81452