Finite Group Databasebeta

← #225478

Subgroups — #225478

Order 512.

717 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22701C211yesmaximal, characteristic, complemented
2#22651C211yesmaximal, characteristic, complemented
3#21277C211yesmaximal, characteristic
4C2×#1594C211yesmaximal, characteristic
5C2×C8×D8C211yesmaximal, characteristic
6#9596C211yesmaximal, characteristic, complemented
7#9593C211yesmaximal, characteristic, complemented
8#8849C211yesmaximal, characteristic, complemented
9#8847C211yesmaximal, characteristic, complemented
10#8845C211yesmaximal, characteristic, complemented
11#8843C211yesmaximal, characteristic, complemented
12#8808C211yesmaximal, characteristic, complemented
13#8569C211yesmaximal, characteristic, complemented
14#8567C211yesmaximal, characteristic, complemented
15#8262C211yesmaximal, characteristic, complemented
16C2×C4×D8C2211yescharacteristic
17C2×#436C2211yescharacteristic
18C2×#436C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C2×#424—21no—
22#2293C2211yescharacteristic
23#2293C2211yescharacteristic
24#1972C411yescharacteristic
25#1972C411yescharacteristic
26#1943C411yescharacteristic
27#1943C411yescharacteristic
28#1939C2211yescharacteristic
29#1939C2211yescharacteristic
30#1939C2211yescharacteristic
31#1939C2211yescharacteristic
32#1936C411yescharacteristic
33#1936C411yescharacteristic
34#1936C411yescharacteristic
35#1936C411yescharacteristic
36#1851—21no—
37#1851—21no—
38#1850—21no—
39#1850—21no—
40#1842C2211yescharacteristic
41#1831C2211yescharacteristic
42#1798—21no—
43#1798—21no—
44C4×#106C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×#102C2211yescharacteristic
49#1729—21no—
50C4×#97C2211yescharacteristic
51#1594—24no—
52C8×D8—24no—
53C2×#335C2211yescharacteristic
54C2×#335C2211yescharacteristic
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×C82C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×D8C2×C411yescharacteristic
70C22×M16—21no—
71C22×M16—21no—
72C23×C8—21no—
73C23×C8—21no—
74C2×C4×D4C2311yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C2×#117—21no—
77C2×#117—21no—
78C4×D8D414yes—
79C4×D8Q814yes—
80#436—22no—
81#436—22no—
82#436—41no—
83#436—41no—
84C8×D4—22no—
85C8×D4—22no—
86C8×D4—41no—
87C8×D4—41no—
88C2×#107C2×C411yescharacteristic
89C2×#106C2×C411yescharacteristic
90C2×#106C2×C411yescharacteristic
91#424—44no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8D411yescharacteristic
100C2×C4⋊C8D411yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108C2×#102C2×C411yescharacteristic
109C2×#102C2×C411yescharacteristic
110C2×#102C2×C411yescharacteristic
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×#97C2×C411yescharacteristic
116C2×#97C2×C411yescharacteristic
117C2×#97C2×C411yescharacteristic
118C2×#97C2×C411yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125#390C2×C411yescharacteristic
126#390C2×C411yescharacteristic
127#390C2×C411yescharacteristic
128#390C2×C411yescharacteristic
129#389C2×C411yescharacteristic
130#389C2×C411yescharacteristic
131#389C2×C411yescharacteristic
132#389C2×C411yescharacteristic
133#388—21no—
134#388—21no—
135#388—21no—
136#388—21no—
137C4×C4⋊C4C2311yescharacteristic
138C4×C4⋊C4C2311yescharacteristic
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C43C2311yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336D411yescharacteristic
156#336D411yescharacteristic
157#335—22no—
158#335—41no—
159#334—22no—
160#334—41no—
161D4⋊C8—22no—
162D4⋊C8—22no—
163D4⋊C8—22no—
164D4⋊C8—22no—
165#322—22no—
166C82—22no—
167C22×D4C22×C411yescharacteristic
168C22×D4C22×C411yescharacteristic
169C23×C4—21no—
170C23×C4—21no—
171C2×D8—22no—
172C2×D8—22no—
173C2×D8C4⋊C414yes—
174C2×M16—42no—
175C2×M16—42no—
176C2×M16—42no—
177C2×M16—42no—
178C2×M16—81no—
179C2×M16—81no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—42no—
187C22×C8—42no—
188C22×C8—42no—
189C22×C8—42no—
190C22×C8—81no—
191C22×C8—81no—
192C22×C8C22×C411yescharacteristic
193C22×C8C22×C411yescharacteristic
194C22×C8C22×C411yescharacteristic
195C22×C8C22×C411yescharacteristic
196C22×C8C2×D411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C4×D4—21no—
201C4×D4—21no—
202C4×D4—21no—
203C4×D4—21no—
204C4×D4C2×D412yes—
205C4×D4C2×D412yes—
206C4×D4C2×Q812yes—
207C4×D4C2×Q812yes—
208#117—42no—
209#117—42no—
210#117—81no—
211#117—81no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42C22×C411yescharacteristic
235C2×C42C22×C411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42#4111yescharacteristic
239C2×C42#4111yescharacteristic
240C2×C42C2411yescharacteristic
241#107C4⋊C414yes—
242#106—22no—
243#106—22no—
244C4⋊C8—22no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—41no—
254C4⋊C8—42no—
255C4⋊C8—42no—
256C4⋊C8—42no—
257C4⋊C8—42no—
258C4⋊C8—81no—
259C4⋊C8—81no—
260C4⋊C8—81no—
261#102—22no—
262#102—22no—
263#102—41no—
264#102—41no—
265#102C4⋊C414yes—
266#102C4⋊C414yes—
267C22⋊C8—42no—
268C22⋊C8—42no—
269C22⋊C8—42no—
270C22⋊C8—42no—
271#97—22no—
272#97—22no—
273#97—22no—
274#97—41no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8C2×D412yes—
286C4×C8C2×Q812yes—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293C24—21no—
294C24—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4C2×C4⋊C412yes—
306C2×D4C2×C4⋊C412yes—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4C4×D411yescharacteristic
326C22×C4C4×D411yescharacteristic
327C22×C4C4×D411yescharacteristic
328C22×C4C4×D411yescharacteristic
329C22×C4C23×C411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C2×#4111yescharacteristic
332D8—28no—
333M16—84no—
334M16—84no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—22no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—84no—
372C2×C8—84no—
373C2×C8C2×C4⋊C412yes—
374C2×C8C2×C4⋊C412yes—
375C2×C8#13112yes—
376C2×C8#13112yes—
377C2×C8#13112yes—
378C2×C8AGL(1,ℤ/8)12yes—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—81no—
392C4⋊C4—81no—
393C4⋊C4C2×C4⋊C412yes—
394C4⋊C4C2×C4⋊C412yes—
395C22⋊C4—22no—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—81no—
404C22⋊C4—81no—
405C22⋊C4—81no—
406C22⋊C4—81no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42C22×D411yescharacteristic
422C42C22×Q811yescharacteristic
423C42H(5,𝔽2)11yescharacteristic
424C422−1+411yescharacteristic
425C23—22no—
426C23—22no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23C2×C4×D411yescharacteristic
440D4—22no—
441D4—22no—
442D4—22no—
443D4—22no—
444D4—41no—
445D4—41no—
446D4—41no—
447D4—41no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4C22×C4⋊C411yescharacteristic
489C2×C4#52011yescharacteristic
490C2×C4D4211yescharacteristic
491C2×C4#54711yescharacteristic
492C2×C4D4×Q811yescharacteristic
493C2×C4#55411yescharacteristic
494C2×C4C2×#13111yescharacteristic
495C2×C4C2×#13111yescharacteristic
496C2×C4C2×#13111yescharacteristic
497C2×C4#56811yescharacteristic
498C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
499C2×C4#57511yescharacteristic
500C2×C4#57611yescharacteristic
501C2×C4#57711yescharacteristic
502C8—22no—
503C8—22no—
504C8—22no—
505C8—42no—
506C8—41no—
507C8—41no—
508C8—41no—
509C8—41no—
510C8—81no—
511C8—81no—
512C8—81no—
513C8—81no—
514C8—81no—
515C8#42414yescomplemented
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22D4×C4⋊C411yescharacteristic
531C22#293111yescharacteristic
532C22#293511yescharacteristic
533C22C4×AGL(1,ℤ/8)11yescharacteristic
534C22#294611yescharacteristic
535C22#295511yescharacteristic
536C22#295611yescharacteristic
537C4—21no—
538C4—21no—
539C4—41no—
540C4—41no—
541C4—42no—
542C4—42no—
543C4—41no—
544C4—41no—
545C4—81no—
546C4—81no—
547C4—81no—
548C4—81no—
549C4C2×#42411yescharacteristic
550C4#290611yescharacteristic
551C4#299311yescharacteristic
552C4#299411yescharacteristic
553C4#326411yescharacteristic
554C4#326711yescharacteristic
555C4#338411yescharacteristic
556C4#338911yescharacteristic
557C2—82no—
558C2—82no—
559C2#1796411yescharacteristic
560C2#1796811yescharacteristic
561C2#1807611yescharacteristic
562C2#1808011yescharacteristic
563C2#2265111yescharacteristic
564C2#2267611yescharacteristic
565C2#2268711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D8C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22701C2semidirect
2. #22651C2semidirect
6. #9596C2semidirect
7. #9593C2semidirect
8. #8849C2semidirect
9. #8847C2semidirect
10. #8845C2semidirect
11. #8843C2semidirect
12. #8808C2semidirect
13. #8569C2semidirect
14. #8567C2semidirect
15. #8262C2semidirect
515. C8#424semidirect

Maximal subgroups

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

subgroupindex
#227012
#226512
#212772
C2×#15942
C2×C8×D82
#95962
#95932
#88492
#88472
#88452
#88432
#88082
#85692
#85672
#82622