Finite Group Databasebeta

← #314767

Subgroups — #314767

Order 512.

677 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27678C211yesmaximal, characteristic
2#27625C211yesmaximal, characteristic
3#21486C211yesmaximal, characteristic
4#20800C211yesmaximal, characteristic, complemented
5#10654C211yesmaximal, characteristic, complemented
6#10651C211yesmaximal, characteristic, complemented
7#10619C211yesmaximal, characteristic, complemented
8#10616C211yesmaximal, characteristic, complemented
9#9621C211yesmaximal, characteristic, complemented
10#9618C211yesmaximal, characteristic, complemented
11#8727C211yesmaximal, characteristic, complemented
12#8724C211yesmaximal, characteristic, complemented
13#8644C211yesmaximal, characteristic, complemented
14#8642C211yesmaximal, characteristic, complemented
15#8597C211yesmaximal, characteristic, complemented
16C2×#435—21no—
17C2×C8×D4C2211yescharacteristic
18C2×#428—21no—
19C2×#428—21no—
20C2×C4×M16C2211yescharacteristic
21C4×#121C2211yescharacteristic
22#2285C2211yescharacteristic
23C2×#384C2211yescharacteristic
24#2100—21no—
25#2094—21no—
26#2077—21no—
27#2034—21no—
28#1954C2211yescharacteristic
29#1948C2211yescharacteristic
30#1948C2211yescharacteristic
31#1948C2211yescharacteristic
32#1948C2211yescharacteristic
33#1939C2211yescharacteristic
34#1926—21no—
35#1924—21no—
36#1919—21no—
37#1874—21no—
38#1870—21no—
39#1853—21no—
40#1847—21no—
41#1847—21no—
42#1845C2211yescharacteristic
43#1845C2211yescharacteristic
44#1844C2211yescharacteristic
45#1844C2211yescharacteristic
46#1802C2211yescharacteristic
47#1802C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1797C2211yescharacteristic
51#1797C2211yescharacteristic
52#1797C2211yescharacteristic
53#1770C2211yescharacteristic
54#1752—21no—
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57#1296C2211yescharacteristic
58#1296C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16D411yescharacteristic
68C22×M16D411yescharacteristic
69C23×C8—21no—
70C2×#121D411yescharacteristic
71C2×C4×D4—21no—
72C2×C4×D4C2311yescharacteristic
73C2×#117—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C42C2311yescharacteristic
78#435—44no—
79C8×D4—22no—
80C8×D4—22no—
81#428—44no—
82#428—44no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—21no—
94C2×#106—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C4×M16—22no—
113C4×M16—22no—
114C2×#97—21no—
115C2×#97D411yescharacteristic
116C2×#97D411yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119#390D411yescharacteristic
120#389D411yescharacteristic
121#388—21no—
122#386—21no—
123#384—21no—
124#384—21no—
125#384—21no—
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#384C2311yescharacteristic
129#384C2311yescharacteristic
130#382—21no—
131#380—21no—
132#380—21no—
133#380D411yescharacteristic
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4C2311yescharacteristic
137C4×C22⋊C4—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#335—22no—
152#334—22no—
153C22×D4—21no—
154C22×D4C2×D411yescharacteristic
155C23×C4—21no—
156C23×C4—41no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4#4111yescharacteristic
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16SD1614yes—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—42no—
173C22×C8—42no—
174C22×C8—42no—
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179#121—22no—
180#121—41no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—42no—
186C4×D4—42no—
187#117—42no—
188#117—42no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—81no—
208C2×C4⋊C4—81no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—41no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—41no—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42#4111yescharacteristic
228C2×C42C2411yescharacteristic
229#107—42no—
230#107—42no—
231#107—42no—
232#107—42no—
233#106—42no—
234#106—42no—
235#106—42no—
236#106—42no—
237#106—42no—
238#106—81no—
239#106—81no—
240#106—81no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
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—42no—
254#102—42no—
255#102—81no—
256#102—81no—
257#102—81no—
258C22⋊C8—42no—
259C22⋊C8—42no—
260C22⋊C8—42no—
261C22⋊C8—42no—
262C22⋊C8—42no—
263C22⋊C8—81no—
264#97—22no—
265#97—22no—
266#97—42no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8#4112yes—
270C4×C8#4112yes—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—41no—
275#95—41no—
276C24—21no—
277C24#12111yescharacteristic
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—81no—
286C2×D4#13712yes—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4#12111yescharacteristic
318C22×C4#12111yescharacteristic
319C22×C4#12111yescharacteristic
320C22×C4C2×SD1612yes—
321C22×C4#13712yes—
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C2×#4111yescharacteristic
325M16—28no—
326M16—28no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—84no—
352C2×C8#12312yes—
353C2×C8#12312yes—
354C2×C8C2×SD1612yes—
355C2×C8C2×SD1612yes—
356C2×C8AGL(1,ℤ/8)12yes—
357C2×C8AGL(1,ℤ/8)12yes—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—42no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—81no—
382C4⋊C4—81no—
383C4⋊C4—81no—
384C4⋊C4—81no—
385C4⋊C4—82no—
386C4⋊C4—82no—
387C4⋊C4—82no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—42no—
393C22⋊C4—81no—
394C22⋊C4—82no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—41no—
402C42—41no—
403C42—82no—
404C42C2×#4111yescharacteristic
405C42C2×#4111yescharacteristic
406C422−1+411yescharacteristic
407C422−1+411yescharacteristic
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—22no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23#46112yes—
422C23C2×#12111yescharacteristic
423D4—22no—
424D4—22no—
425D4—41no—
426D4—41no—
427D4—84no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—164no—
474C2×C4#46112yes—
475C2×C4C2×#12311yescharacteristic
476C2×C4#53611yescharacteristic
477C2×C4#54711yescharacteristic
478C2×C4#54711yescharacteristic
479C2×C4#55211yescharacteristic
480C2×C4#55211yescharacteristic
481C2×C4C22×SD1611yescharacteristic
482C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
483C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
484C2×C4C2×#13711yescharacteristic
485C2×C4C2×#13711yescharacteristic
486C2×C4#57511yescharacteristic
487C2×C4#57611yescharacteristic
488C2×C4#57711yescharacteristic
489C8—22no—
490C8—22no—
491C8—22no—
492C8—81no—
493C8—81no—
494C8—81no—
495C8—81no—
496C8—81no—
497C8—81no—
498C8#48114yes—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22#248611yescharacteristic
514C22C2×#46111yescharacteristic
515C22#303811yescharacteristic
516C22#305611yescharacteristic
517C22#306211yescharacteristic
518C22#306811yescharacteristic
519C22#306911yescharacteristic
520C4—21no—
521C4—21no—
522C4—22no—
523C4—41no—
524C4—41no—
525C4—81no—
526C4—81no—
527C4—81no—
528C4—161no—
529C4—161no—
530C4—161no—
531C4—161no—
532C4C2×#48111yescharacteristic
533C4#309211yescharacteristic
534C4#322711yescharacteristic
535C4#323911yescharacteristic
536C4#334411yescharacteristic
537C4#334611yescharacteristic
538C4#334711yescharacteristic
539C4#335011yescharacteristic
540C2—42no—
541C2—82no—
542C2#1916511yescharacteristic
543C2#1923211yescharacteristic
544C2#1931311yescharacteristic
545C2#1937611yescharacteristic
546C2#2764511yescharacteristic
547C2#2766811yescharacteristic
548C2#2767611yescharacteristic
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
4. #20800C2semidirect
5. #10654C2semidirect
6. #10651C2semidirect
7. #10619C2semidirect
8. #10616C2semidirect
9. #9621C2semidirect
10. #9618C2semidirect
11. #8727C2semidirect
12. #8724C2semidirect
13. #8644C2semidirect
14. #8642C2semidirect
15. #8597C2semidirect

Maximal subgroups

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

subgroupindex
#276782
#276252
#214862
#208002
#106542
#106512
#106192
#106162
#96212
#96182
#87272
#87242
#86442
#86422
#85972