Finite Group Databasebeta

← #286175

Subgroups — #286175

Order 512.

704 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27905C211yesmaximal, characteristic, complemented
2#27644C211yesmaximal, characteristic
3#24835C211yesmaximal, characteristic
4#22989C211yesmaximal, characteristic
5#10653C211yesmaximal, characteristic, complemented
6#10619C211yesmaximal, characteristic, complemented
7#10592C211yesmaximal, characteristic, complemented
8#10485C211yesmaximal, characteristic, complemented
9#9617C211yesmaximal, characteristic, complemented
10#9569C211yesmaximal, characteristic, complemented
11#9227C211yesmaximal, characteristic, complemented
12#9131C211yesmaximal, characteristic, complemented
13#8829C211yesmaximal, characteristic, complemented
14#8650C211yesmaximal, characteristic, complemented
15C2×#1296C211yesmaximal, characteristic, complemented
16C2×#483—21no—
17C2×#483—21no—
18C2×#480—21no—
19C2×#474C2211yescharacteristic
20C2×#436C2211yescharacteristic
21C2×#435—21no—
22C22×C4⋊C8C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24#2857C2211yescharacteristic
25C4×#121C2211yescharacteristic
26C2×#384C2211yescharacteristic
27#2096—21no—
28#2096—21no—
29#2086—21no—
30#2083—21no—
31#2071—21no—
32#2067—21no—
33#1988C2211yescharacteristic
34#1956C2211yescharacteristic
35#1956C2211yescharacteristic
36#1954C2211yescharacteristic
37#1947C2211yescharacteristic
38#1947C2211yescharacteristic
39#1947C2211yescharacteristic
40#1939C2211yescharacteristic
41#1924—21no—
42#1893—21no—
43#1869—21no—
44#1851—21no—
45#1844C2211yescharacteristic
46#1844C2211yescharacteristic
47#1823—21no—
48#1808—21no—
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52#1797C2211yescharacteristic
53#1770C2211yescharacteristic
54C4×C22⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68C23×C8—21no—
69C23×C8—21no—
70C2×C4⋊Q8C2311yescharacteristic
71C2×C22⋊Q8—21no—
72C2×#121D411yescharacteristic
73C2×C4×D4—21no—
74C2×C4×D4C2311yescharacteristic
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C42C2311yescharacteristic
79#483—44no—
80#483—44no—
81#480—44no—
82#474—22no—
83#474—41no—
84#436—22no—
85#436—41no—
86#435—44no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#107—21no—
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—21no—
94C2×#106—21no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—41no—
101C2×C4⋊C8—41no—
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×#103—21no—
109C2×#103—21no—
110C2×#103—21no—
111C2×#103—21no—
112C2×#102—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×#102—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8D411yescharacteristic
121C2×C22⋊C8D411yescharacteristic
122C2×#97—21no—
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125#391—21no—
126#390D411yescharacteristic
127#389—21no—
128#389—21no—
129#389D411yescharacteristic
130#388—21no—
131#386—21no—
132#384—21no—
133#384—21no—
134#384—21no—
135#384C2311yescharacteristic
136#384C2311yescharacteristic
137#384C2311yescharacteristic
138#380—21no—
139#380—21no—
140#380D411yescharacteristic
141C4×C4⋊C4C2311yescharacteristic
142C4×C22⋊C4—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336D411yescharacteristic
155#336D411yescharacteristic
156#335—22no—
157#335—41no—
158#332—22no—
159#332—41no—
160C22×Q8—21no—
161C22×D4—21no—
162C22×D4C2×D411yescharacteristic
163C23×C4—21no—
164C23×C4—41no—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4#4111yescharacteristic
168C2×M16—42no—
169C2×M16—42no—
170C2×M16—81no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—42no—
178C22×C8—42no—
179C22×C8—42no—
180C22×C8—42no—
181C22×C8—81no—
182C22×C8—81no—
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C22⋊Q8—42no—
192C22⋊Q8—42no—
193#121—22no—
194#121—22no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—42no—
200C4×D4—42no—
201C2×C4⋊C4—21no—
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⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—41no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42#4111yescharacteristic
239C2×C42C2411yescharacteristic
240#107—42no—
241#107—42no—
242#107—42no—
243#107—81no—
244#106—42no—
245#106—42no—
246#106—42no—
247#106—42no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—42no—
259C4⋊C8—42no—
260C4⋊C8—42no—
261C4⋊C8#4112yes—
262C4⋊C8#4112yes—
263#103—42no—
264#103—42no—
265#103—42no—
266#103—42no—
267#102—42no—
268#102—42no—
269#102—42no—
270#102—81no—
271C22⋊C8—22no—
272C22⋊C8—42no—
273C22⋊C8—42no—
274C22⋊C8—42no—
275C22⋊C8—81no—
276C22⋊C8D814yes—
277#97—42no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—41no—
286#95—41no—
287C24—21no—
288C24#12111yescharacteristic
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
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×C4C2×D812yes—
336C22×C4AGL(1,ℤ/8)12yes—
337C22×C4AGL(1,ℤ/8)12yes—
338C22×C4#13712yes—
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C2×#4111yescharacteristic
342M16—84no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
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×D812yes—
374C2×C8C2×D812yes—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—42no—
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—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—81no—
399C4⋊C4—81no—
400C4⋊C4—81no—
401C4⋊C4—82no—
402C4⋊C4—82no—
403C4⋊C4—82no—
404C4⋊C4—82no—
405C22⋊C4—42no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—81no—
411C22⋊C4—82no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—41no—
419C42—41no—
420C42C2×#4111yescharacteristic
421C42C2×#4111yescharacteristic
422C42H(5,𝔽2)11yescharacteristic
423C422−1+411yescharacteristic
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23#45912yes—
436C23#46812yes—
437C23#47112yes—
438C23C2×#12111yescharacteristic
439Q8—81no—
440Q8—81no—
441Q8—81no—
442Q8—81no—
443D4—41no—
444D4—41no—
445D4—41no—
446D4—41no—
447D4—84no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—164no—
493C2×C4#45912yes—
494C2×C4#46812yes—
495C2×C4#52911yescharacteristic
496C2×C4#53911yescharacteristic
497C2×C4#54611yescharacteristic
498C2×C4#54611yescharacteristic
499C2×C4#54711yescharacteristic
500C2×C4#54711yescharacteristic
501C2×C4C22×D811yescharacteristic
502C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
503C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
504C2×C4C2×#13711yescharacteristic
505C2×C4#57711yescharacteristic
506C2×C4#57711yescharacteristic
507C2×C4#57811yescharacteristic
508C2×C4#57811yescharacteristic
509C8—22no—
510C8—22no—
511C8—41no—
512C8—42no—
513C8—42no—
514C8—41no—
515C8—81no—
516C8—81no—
517C8—81no—
518C8—81no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22#168514yes—
533C22#242511yescharacteristic
534C22C2×#45911yescharacteristic
535C22#304611yescharacteristic
536C22C2×#46811yescharacteristic
537C22#306311yescharacteristic
538C22C2×#47111yescharacteristic
539C22#306911yescharacteristic
540C4—22no—
541C4—22no—
542C4—21no—
543C4—21no—
544C4—41no—
545C4—81no—
546C4—81no—
547C4—81no—
548C4—161no—
549C4—161no—
550C4—161no—
551C4—161no—
552C4#329411yescharacteristic
553C4#330611yescharacteristic
554C4#330811yescharacteristic
555C4#331211yescharacteristic
556C4#333411yescharacteristic
557C4#334311yescharacteristic
558C4#334411yescharacteristic
559C4#334511yescharacteristic
560C2—24no—
561C2—161no—
562C2C2×#168511yescharacteristic
563C2#1923111yescharacteristic
564C2#1945111yescharacteristic
565C2#1958811yescharacteristic
566C2#2520011yescharacteristic
567C2#2525411yescharacteristic
568C2#2526811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #27905C2semidirect
5. #10653C2semidirect
6. #10619C2semidirect
7. #10592C2semidirect
8. #10485C2semidirect
9. #9617C2semidirect
10. #9569C2semidirect
11. #9227C2semidirect
12. #9131C2semidirect
13. #8829C2semidirect
14. #8650C2semidirect
15. C2×#1296C2semidirect

Maximal subgroups

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

subgroupindex
#279052
#276442
#248352
#229892
#106532
#106192
#105922
#104852
#96172
#95692
#92272
#91312
#88292
#86502
C2×#12962