Finite Group Databasebeta

← #211158

Subgroups — #211158

Order 512.

800 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#492C211yesmaximal, characteristic
2#21615C211yesmaximal, characteristic, complemented
3#21505C211yesmaximal, characteristic
4C4×C8×Q8C211yesmaximal, characteristic, complemented
5C2×#1666C211yesmaximal, characteristic
6#9732C211yesmaximal, characteristic
7#9611C211yesmaximal, characteristic, complemented
8#9563C211yesmaximal, characteristic
9#9561C211yesmaximal, characteristic, complemented
10#9222C211yesmaximal, characteristic
11#9220C211yesmaximal, characteristic, complemented
12#8643C211yesmaximal, characteristic, complemented
13#8273C211yesmaximal, characteristic, complemented
14#8237C211yesmaximal, characteristic, complemented
15#8234C211yesmaximal, characteristic
16C2×#492C2211yescharacteristic
17C2×#475C2211yescharacteristic
18C2×#460C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C2×#440—21no—
22C2×C4×SD16—21no—
23C2×C4×SD16—21no—
24C4×C4⋊Q8C2211yescharacteristic
25C4×#127C2211yescharacteristic
26C42×Q8C2211yescharacteristic
27#1988C2211yescharacteristic
28#1982C2211yescharacteristic
29#1968—21no—
30#1968—21no—
31#1957C2211yescharacteristic
32#1956C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1925—21no—
36#1920—21no—
37#1880—21no—
38#1863—21no—
39#1844C2211yescharacteristic
40#1803C2211yescharacteristic
41#1803C2211yescharacteristic
42#1776—21no—
43C8×C4⋊C4—21no—
44C8×C4⋊C4—21no—
45C8×C4⋊C4C2211yescharacteristic
46C8×C4⋊C4C2211yescharacteristic
47#1767C2211yescharacteristic
48#1767C2211yescharacteristic
49C4×C4⋊C8—21no—
50C4×C4⋊C8C2211yescharacteristic
51#1763—21no—
52#1762—21no—
53C42×C8—21no—
54C42×C8C2211yescharacteristic
55#1666C418yescomplemented
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#331C2211yescharacteristic
61C2×#331C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×SD16—21no—
69C22×SD16—21no—
70C22×SD16—21no—
71C2×C4⋊Q8C2311yescharacteristic
72C2×#127C2311yescharacteristic
73C2×C4×Q8—21no—
74C2×C4×Q8D411yescharacteristic
75C2×C4×Q8D411yescharacteristic
76C2×C4×Q8C2311yescharacteristic
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×C4×D4—21no—
80#492—22no—
81#492C2×C414yes—
82#475—22no—
83#475C2×C414yescomplemented
84#460—22no—
85#460C2×C414yes—
86C8×Q8—22no—
87C8×Q8—22no—
88C8×Q8—22no—
89C8×Q8C2×C414yescomplemented
90#440—44no—
91C4×SD16—44no—
92C4×SD16—44no—
93C2×#107—21no—
94C2×#107—21no—
95C2×#106—21no—
96C2×#106—21no—
97C2×#106—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×#103—21no—
107C2×#103—21no—
108C2×#103—21no—
109C2×#103—21no—
110C2×#103—21no—
111C2×#102—21no—
112C2×#102—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×#102—21no—
116C2×#97—21no—
117C2×#97—21no—
118C2×C4×C8—21no—
119C2×C4×C8—21no—
120C2×C4×C8—21no—
121C2×C4×C8—21no—
122C2×C4×C8—21no—
123C2×C4×C8D411yescharacteristic
124C2×C4×C8D411yescharacteristic
125C2×C4×C8D411yescharacteristic
126C2×C4×C8D411yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129#391—21no—
130#390—21no—
131#390—21no—
132#390—21no—
133#389—21no—
134#389—21no—
135#384C2311yescharacteristic
136#384C2311yescharacteristic
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4D411yescharacteristic
143C4×C4⋊C4D411yescharacteristic
144C4×C4⋊C4C2311yescharacteristic
145C43—21no—
146C43C2311yescharacteristic
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#334—22no—
159#334C2×C414yescomplemented
160#332—22no—
161#332C2×C414yescomplemented
162#331—22no—
163#331C2×C414yes—
164C22×Q8—21no—
165C22×Q8C2×D411yescharacteristic
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C23×C4—41no—
170C2×SD16—24no—
171C2×SD16—24no—
172C2×SD16—24no—
173C2×SD16—42no—
174C2×SD16—42no—
175C2×SD16—42no—
176C2×SD16—42no—
177C2×SD16—42no—
178C2×SD16—42no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C22×C8C2×D411yescharacteristic
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8C22×C412yes—
193#127—21no—
194#127—21no—
195#127—21no—
196#127C22×C412yes—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—42no—
203C4×Q8—42no—
204C4×Q8SD1614yescomplemented
205C4×Q8C22×C412yes—
206C4×Q8#4112yes—
207C4×D4—42no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—82no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—41no—
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42#4111yescharacteristic
243C2×C42C2411yescharacteristic
244#107—42no—
245#107—42no—
246#106—24no—
247#106—42no—
248#106—42no—
249C4⋊C8—24no—
250C4⋊C8—24no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8C22×C412yes—
260C4⋊C8C22×C412yes—
261C4⋊C8C22×C412yes—
262C4⋊C8#4112yes—
263#103—24no—
264#103—42no—
265#103—42no—
266#103—42no—
267#103—42no—
268#102—24no—
269#102—42no—
270#102—42no—
271#102—42no—
272#102—42no—
273#97—42no—
274#97—42no—
275C4×C8—24no—
276C4×C8—24no—
277C4×C8—22no—
278C4×C8—22no—
279C4×C8—22no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—41no—
283C4×C8—42no—
284C4×C8—42no—
285C4×C8—42no—
286C4×C8C22×C412yes—
287C4×C8#4112yes—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—41no—
292#95—41no—
293C24—41no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—22no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8C4×D412yes—
304C2×Q8C2×SD1612yes—
305C2×Q8#13512yes—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—81no—
321C2×D4—81no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4#12111yescharacteristic
333C22×C4#12111yescharacteristic
334C22×C4#12111yescharacteristic
335C22×C4#12111yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C2×#4111yescharacteristic
339SD16—48no—
340SD16—48no—
341SD16—48no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—21no—
350C2×C8—22no—
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—41no—
366C2×C8—41no—
367C2×C8C4×D412yes—
368C2×C8C4×D412yes—
369C2×C8AGL(1,ℤ/8)12yes—
370C2×C8AGL(1,ℤ/8)12yes—
371C2×C8#13712yes—
372C4⋊C4—24no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—81no—
392C4⋊C4—81no—
393C4⋊C4—82no—
394C4⋊C4C4×D412yes—
395C4⋊C4C2×SD1612yes—
396C4⋊C4#13512yes—
397C22⋊C4—82no—
398C22⋊C4—82no—
399C22⋊C4—82no—
400C42—24no—
401C42—22no—
402C42—22no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—24no—
410C42—22no—
411C42—41no—
412C42—42no—
413C42—41no—
414C42—41no—
415C42—42no—
416C42—82no—
417C42C2×SD1612yes—
418C42#13512yes—
419C42C23×C411yescharacteristic
420C42C2×#4111yescharacteristic
421C42C2×#4111yescharacteristic
422C42C2×#4111yescharacteristic
423C23—42no—
424C23—81no—
425C23—81no—
426C23—81no—
427C23—81no—
428C23—81no—
429C23—81no—
430C23C2×#12111yescharacteristic
431Q8—22no—
432Q8—22no—
433Q8—22no—
434Q8—42no—
435Q8—42no—
436Q8—81no—
437Q8—81no—
438Q8C4×SD1614yescomplemented
439D4—42no—
440D4—42no—
441D4—42no—
442D4—42no—
443D4—82no—
444D4—82no—
445D4—81no—
446D4—81no—
447D4—81no—
448D4—81no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—164no—
482C2×C4#46512yes—
483C2×C4#46512yes—
484C2×C4#47112yes—
485C2×C4C2×C4×D411yescharacteristic
486C2×C4C2×C4×D411yescharacteristic
487C2×C4C4×#4111yescharacteristic
488C2×C4#52511yescharacteristic
489C2×C4#52911yescharacteristic
490C2×C4#53211yescharacteristic
491C2×C4C22×SD1611yescharacteristic
492C2×C4C2×#13511yescharacteristic
493C2×C4C2×#13511yescharacteristic
494C2×C4C2×#13511yescharacteristic
495C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
496C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
497C2×C4C2×#13711yescharacteristic
498C2×C4#57511yescharacteristic
499C8—22no—
500C8—22no—
501C8—41no—
502C8—42no—
503C8—42no—
504C8—42no—
505C8—42no—
506C8—81no—
507C8—81no—
508C8#44014yescomplemented
509C22—82no—
510C22—82no—
511C22—82no—
512C22—161no—
513C22—161no—
514C22—161no—
515C22—161no—
516C22C4×#12111yescharacteristic
517C22#303711yescharacteristic
518C22#303811yescharacteristic
519C22C2×#46511yescharacteristic
520C22#305011yescharacteristic
521C22C2×#47111yescharacteristic
522C22#306711yescharacteristic
523C4—24no—
524C4—21no—
525C4—21no—
526C4—22no—
527C4—22no—
528C4—22no—
529C4—42no—
530C4—42no—
531C4—41no—
532C4—42no—
533C4—82no—
534C4—161no—
535C4—161no—
536C4C2×C4×SD1611yescharacteristic
537C4C4×#13511yescharacteristic
538C4C2×#44011yescharacteristic
539C4C4×AGL(1,ℤ/8)11yescharacteristic
540C4#311211yescharacteristic
541C4#311511yescharacteristic
542C4#316711yescharacteristic
543C4#316811yescharacteristic
544C2—162no—
545C2#1926011yescharacteristic
546C2#1926311yescharacteristic
547C2#1926611yescharacteristic
548C2#1926911yescharacteristic
549C2#2150511yescharacteristic
550C2C4×#46511yescharacteristic
551C2C4×#47111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21615C2semidirect
4. C4×C8×Q8C2semidirect
7. #9611C2semidirect
9. #9561C2semidirect
11. #9220C2semidirect
12. #8643C2semidirect
13. #8273C2semidirect
14. #8237C2semidirect
55. #1666C4semidirect
83. #475C2×C4semidirect
89. C8×Q8C2×C4semidirect
159. #334C2×C4semidirect
161. #332C2×C4semidirect
204. C4×Q8SD16semidirect
438. Q8C4×SD16semidirect
508. C8#440semidirect

Maximal subgroups

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

subgroupindex
C4×#4922
#216152
#215052
C4×C8×Q82
C2×#16662
#97322
#96112
#95632
#95612
#92222
#92202
#86432
#82732
#82372
#82342