Finite Group Databasebeta

← #212696

Subgroups — #212696

Order 512.

661 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22309C211yesmaximal, characteristic, complemented
2#22262C211yesmaximal, characteristic
3C4×#486C211yesmaximal, characteristic
4#9877C211yesmaximal, characteristic, complemented
5#9648C211yesmaximal, characteristic
6#9560C211yesmaximal, characteristic, complemented
7#9560C211yesmaximal, characteristic, complemented
8#9554C211yesmaximal, characteristic, complemented
9#9554C211yesmaximal, characteristic, complemented
10#8785C211yesmaximal, characteristic, complemented
11#8212C211yesmaximal, characteristic, complemented
12#8206C211yesmaximal, characteristic, complemented
13#8204C211yesmaximal, characteristic, complemented
14C8×#102C211yesmaximal, characteristic, complemented
15C8×#97C211yesmaximal, characteristic, complemented
16C2×#486C411yescharacteristic
17C2×#436C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C4×C4⋊Q8C2211yescharacteristic
20C4×#127C2211yescharacteristic
21#1990C411yescharacteristic
22#1985—21no—
23#1984—21no—
24#1957C411yescharacteristic
25#1954C2211yescharacteristic
26#1954C2211yescharacteristic
27#1939C2211yescharacteristic
28#1939C2211yescharacteristic
29#1934C411yescharacteristic
30#1934C411yescharacteristic
31#1934C411yescharacteristic
32#1934C411yescharacteristic
33#1917—21no—
34#1916—21no—
35#1870—21no—
36#1870—21no—
37#1869—21no—
38#1869—21no—
39#1834C411yescharacteristic
40#1831C2211yescharacteristic
41#1831C2211yescharacteristic
42#1770C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44C4×#102C2211yescharacteristic
45C4×#102C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1744C2211yescharacteristic
49C4×#97C2211yescharacteristic
50C42×C8C2211yescharacteristic
51C2×D4⋊C8C2211yescharacteristic
52C2×D4⋊C8C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×#322C2211yescharacteristic
57C2×C82C2211yescharacteristic
58C2×C82C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68C23×C8—21no—
69C2×C4⋊Q8C2×C411yescharacteristic
70C2×#127C2×C411yescharacteristic
71C2×C4×Q8—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4C2311yescharacteristic
75C2×C4×D4C2311yescharacteristic
76#486—24no—
77#436—22no—
78#436—22no—
79C8×D4—22no—
80C8×D4—22no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×#102C2×C411yescharacteristic
88C2×#102C2×C411yescharacteristic
89C2×#102C2×C411yescharacteristic
90C2×#102C2×C411yescharacteristic
91C2×#102C2×C411yescharacteristic
92C2×#102C2×C411yescharacteristic
93C2×#102C2×C411yescharacteristic
94C2×#102C2×C411yescharacteristic
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×#97—21no—
102C2×#97—21no—
103C2×#97—21no—
104C2×C4×C8—21no—
105C2×C4×C8C2×C411yescharacteristic
106C2×C4×C8C2×C411yescharacteristic
107C2×C4×C8C2×C411yescharacteristic
108C2×C4×C8C2×C411yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#391—21no—
116#390—21no—
117#390—21no—
118#390C2×C411yescharacteristic
119#390C2×C411yescharacteristic
120#389—21no—
121#389C2×C411yescharacteristic
122#389C2×C411yescharacteristic
123#384C2×C411yescharacteristic
124#384C2×C411yescharacteristic
125C4×C4⋊C4—21no—
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C43C2311yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#336—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336D411yescharacteristic
146#336D411yescharacteristic
147#336D411yescharacteristic
148#336D411yescharacteristic
149D4⋊C8—22no—
150D4⋊C8—22no—
151D4⋊C8—22no—
152D4⋊C8—22no—
153#322—22no—
154#322—22no—
155C82—22no—
156C82—22no—
157C22×Q8—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4C22×C411yescharacteristic
161C22×D4C22×C411yescharacteristic
162C23×C4—21no—
163C23×C4—21no—
164C2×M16—42no—
165C2×M16—81no—
166C2×M16—81no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—42no—
174C22×C8—81no—
175C22×C8—81no—
176C22×C8C22×C411yescharacteristic
177C22×C8C22×C411yescharacteristic
178C22×C8C22×C411yescharacteristic
179C22×C8C22×C411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C4⋊Q8—22no—
185C4⋊Q8—41no—
186#127—22no—
187#127—41no—
188C4×Q8—81no—
189C4×Q8—81no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—81no—
199C4×D4—81no—
200C4×D4—81no—
201C4×D4—81no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42C22×C411yescharacteristic
224C2×C42C22×C411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42#4111yescharacteristic
228C2×C42#4111yescharacteristic
229C2×C42C2411yescharacteristic
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—81no—
239C4⋊C8—81no—
240#102—22no—
241#102—22no—
242#102—22no—
243#102—22no—
244#102—22no—
245#102—41no—
246#102—41no—
247#102M1614yescomplemented
248C22⋊C8—42no—
249C22⋊C8—42no—
250C22⋊C8—81no—
251C22⋊C8—81no—
252C22⋊C8—81no—
253C22⋊C8—81no—
254#97—24no—
255#97—24no—
256#97—81no—
257C4×C8—22no—
258C4×C8—22no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—41no—
268C4×C8—41no—
269C4×C8—81no—
270C4×C8#4112yes—
271C4×C8#4112yes—
272C4×C8#4112yes—
273C4×C8#4112yes—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—21no—
278C24—21no—
279C24—21no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—81no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—81no—
299C2×D4—81no—
300C2×D4C2×M1612yes—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4C4×D411yescharacteristic
320C22×C4C4×D411yescharacteristic
321C22×C4C4×D411yescharacteristic
322C22×C4C4×D411yescharacteristic
323C22×C4C23×C411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C2×#4111yescharacteristic
326M16—84no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
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—41no—
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—84no—
366C2×C8#12412yes—
367C2×C8#12412yes—
368C2×C8C2×M1612yes—
369C2×C8#13112yes—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—81no—
384C4⋊C4—81no—
385C4⋊C4—81no—
386C4⋊C4—81no—
387C4⋊C4—81no—
388C4⋊C4C2×M1612yes—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—81no—
394C22⋊C4—81no—
395C22⋊C4—81no—
396C22⋊C4—81no—
397C42—21no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—22no—
406C42—22no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42C2×#4111yescharacteristic
412C42C2×#4111yescharacteristic
413C42C2×#4111yescharacteristic
414C42C2×#4111yescharacteristic
415C23—22no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23C2×C4×D411yescharacteristic
430Q8—84no—
431D4—22no—
432D4—22no—
433D4—41no—
434D4—41no—
435D4—41no—
436D4—41no—
437D4—41no—
438D4—41no—
439D4—84no—
440D4—84no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4#44312yes—
480C2×C4#44312yes—
481C2×C4C4×#4111yescharacteristic
482C2×C4C4×#4111yescharacteristic
483C2×C4C2×#12411yescharacteristic
484C2×C4#52911yescharacteristic
485C2×C4#52911yescharacteristic
486C2×C4#53211yescharacteristic
487C2×C4C22×M1611yescharacteristic
488C2×C4C2×#13111yescharacteristic
489C2×C4#56811yescharacteristic
490C2×C4#56811yescharacteristic
491C2×C4#57611yescharacteristic
492C2×C4#57611yescharacteristic
493C2×C4#57711yescharacteristic
494C2×C4#57711yescharacteristic
495C8—24no—
496C8—24no—
497C8—22no—
498C8—22no—
499C8—22no—
500C8—22no—
501C8—41no—
502C8—41no—
503C8—41no—
504C8—41no—
505C8—81no—
506C8—81no—
507C8—81no—
508C8—81no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22—81no—
523C22C4×#12411yescharacteristic
524C22D4×M1611yescharacteristic
525C22#293511yescharacteristic
526C22#294811yescharacteristic
527C22#294911yescharacteristic
528C22C2×#44311yescharacteristic
529C22C2×#44311yescharacteristic
530C4—22no—
531C4—22no—
532C4—21no—
533C4—21no—
534C4—42no—
535C4—41no—
536C4—41no—
537C4—81no—
538C4—81no—
539C4—81no—
540C4—81no—
541C4—81no—
542C4#296511yescharacteristic
543C4#296711yescharacteristic
544C4#296811yescharacteristic
545C4#297011yescharacteristic
546C4#313611yescharacteristic
547C4#313611yescharacteristic
548C4#317911yescharacteristic
549C4#317911yescharacteristic
550C2—82no—
551C2—161no—
552C2#1805411yescharacteristic
553C2#1805611yescharacteristic
554C2#1808411yescharacteristic
555C2#1808511yescharacteristic
556C2#2163211yescharacteristic
557C2#2164411yescharacteristic
558C2#2165811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#127C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ #1744 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22309C2semidirect
4. #9877C2semidirect
6. #9560C2semidirect
7. #9560C2semidirect
8. #9554C2semidirect
9. #9554C2semidirect
10. #8785C2semidirect
11. #8212C2semidirect
12. #8206C2semidirect
13. #8204C2semidirect
14. C8×#102C2semidirect
15. C8×#97C2semidirect
247. #102M16semidirect

Maximal subgroups

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

subgroupindex
#223092
#222622
C4×#4862
#98772
#96482
#95602
#95542
#87852
#82122
#82062
#82042
C8×#1022
C8×#972