Finite Group Databasebeta

← #308049

Subgroups — #308049

Order 512.

709 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26467C211yesmaximal, characteristic, complemented
2#26106C211yesmaximal, characteristic, complemented
3#22995C211yesmaximal, characteristic
4D4×#107C211yesmaximal, characteristic, complemented
5#10671C211yesmaximal, characteristic, complemented
6#10650C211yesmaximal, characteristic, complemented
7#10591C211yesmaximal, characteristic, complemented
8#10581C211yesmaximal, characteristic, complemented
9#9616C211yesmaximal, characteristic, complemented
10#9536C211yesmaximal, characteristic, complemented
11#9221C211yesmaximal, characteristic
12#8977C211yesmaximal, characteristic
13#8920C211yesmaximal, characteristic, complemented
14#8894C211yesmaximal, characteristic, complemented
15#8843C211yesmaximal, characteristic, complemented
16C2×#478—21no—
17C2×#459C2211yescharacteristic
18C2×C4×D8—21no—
19C2×C8×D4C2211yescharacteristic
20C22×#107—21no—
21#2552C2211yescharacteristic
22#2366C2211yescharacteristic
23D4×C4⋊C4C2211yescharacteristic
24D4×C4⋊C4C2211yescharacteristic
25#2099—21no—
26#2086—21no—
27#2067—21no—
28#2025—21no—
29#1982C2211yescharacteristic
30#1957C2211yescharacteristic
31#1957C2211yescharacteristic
32#1957C2211yescharacteristic
33#1948C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1947C2211yescharacteristic
38#1939C2211yescharacteristic
39#1939C2211yescharacteristic
40#1934—21no—
41#1879—21no—
42#1864—21no—
43#1862—21no—
44#1862—21no—
45#1852—21no—
46#1852—21no—
47#1845C2211yescharacteristic
48#1834C2211yescharacteristic
49#1834C2211yescharacteristic
50#1808—21no—
51#1802—21no—
52#1802C2211yescharacteristic
53C4×#107C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55C4×#102C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#331C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×D8—21no—
68C23×C8—21no—
69C2×#127C2311yescharacteristic
70C2×#125—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4C2311yescharacteristic
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78#478—44no—
79#459—22no—
80#459Q814yes—
81C4×D8—44no—
82C8×D4—22no—
83C8×D4—22no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#107—44no—
91C2×#107D411yescharacteristic
92C2×#107D411yescharacteristic
93C2×#106—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#102—41no—
109C2×#102D411yescharacteristic
110C2×#102D411yescharacteristic
111C2×C22⋊C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116#398—41no—
117#390—21no—
118#390—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#386—41no—
124#385—41no—
125#385—41no—
126#384—21no—
127#384C2311yescharacteristic
128#384C2311yescharacteristic
129#384C2311yescharacteristic
130#380—21no—
131#380—21no—
132C4×C4⋊C4C2311yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—41no—
144#340D411yescharacteristic
145#340D411yescharacteristic
146#340D411yescharacteristic
147#340D411yescharacteristic
148#336—21no—
149#336—21no—
150#336—21no—
151#335—22no—
152#331—22no—
153D4⋊C8—22no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4C2×D411yescharacteristic
158C23×C4—21no—
159C23×C4—41no—
160C23×C4—41no—
161C23×C4—41no—
162C2×D8—24no—
163C2×D8—42no—
164C2×D8—42no—
165C22×C8—24no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—42no—
172C22×C8—42no—
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177#127—21no—
178#127—21no—
179#127—21no—
180#127C2×Q812yes—
181#125—42no—
182#125—81no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—42no—
186C4×D4—42no—
187C4×D4—42no—
188C4×D4—42no—
189C4×D4—81no—
190C4×D4—81no—
191C4×D4—81no—
192C4×D4—81no—
193C4×D4C2×Q812yes—
194C4×D4#4112yes—
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—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—81no—
212C2×C4⋊C4—81no—
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C22⋊C4—21no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42C2411yescharacteristic
230#107—41no—
231#107—42no—
232#107—42no—
233#107—42no—
234#107—42no—
235#107—81no—
236#107—81no—
237#107D814yescomplemented
238#106—42no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—42no—
246C4⋊C8—81no—
247C4⋊C8C2×Q812yes—
248C4⋊C8#4112yes—
249#102—22no—
250#102—22no—
251#102—24no—
252#102—42no—
253#102—42no—
254#102—42no—
255#102—42no—
256#102—42no—
257#102—81no—
258#102—81no—
259#102—82no—
260C22⋊C8—24no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—42no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—41no—
272#95—41no—
273#95—41no—
274C24—21no—
275C24—41no—
276C2×D4—22no—
277C2×D4—21no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—81no—
294C2×D4—81no—
295C2×D4C22⋊Q812yes—
296C2×D4#13512yes—
297C2×D4AGL(1,ℤ/8)12yes—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
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—82no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—161no—
321C22×C4—161no—
322C22×C4—161no—
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4H(5,𝔽2)11yescharacteristic
326D8—48no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—22no—
330C2×C8—21no—
331C2×C8—22no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—48no—
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—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8C2×D812yes—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
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—82no—
379C4⋊C4—82no—
380C4⋊C4—82no—
381C4⋊C4—82no—
382C4⋊C4—81no—
383C4⋊C4—81no—
384C4⋊C4—81no—
385C4⋊C4—161no—
386C4⋊C4C22⋊Q812yes—
387C4⋊C4C2×D812yes—
388C4⋊C4C2×D812yes—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—82no—
392C22⋊C4—82no—
393C22⋊C4—82no—
394C22⋊C4—82no—
395C22⋊C4—161no—
396C42—41no—
397C42—41no—
398C42—41no—
399C42—41no—
400C42—41no—
401C42—41no—
402C42C22×Q811yescharacteristic
403C42C2×#4111yescharacteristic
404C42C2×#4111yescharacteristic
405C422−1+411yescharacteristic
406C23—22no—
407C23—22no—
408C23—22no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—42no—
414C23—81no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23—81no—
420C23D4211yescharacteristic
421D4—22no—
422D4—22no—
423D4—22no—
424D4—42no—
425D4—42no—
426D4—42no—
427D4—42no—
428D4—82no—
429D4—81no—
430D4—81no—
431D4—81no—
432D4—81no—
433D4—161no—
434D4#47814yescomplemented
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
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—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—164no—
472C2×C4—164no—
473C2×C4—164no—
474C2×C4C2×C22⋊Q811yescharacteristic
475C2×C4#52511yescharacteristic
476C2×C4#53911yescharacteristic
477C2×C4#54711yescharacteristic
478C2×C4D4×Q811yescharacteristic
479C2×C4#55411yescharacteristic
480C2×C4C22×D811yescharacteristic
481C2×C4C2×#13511yescharacteristic
482C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
483C2×C4#57511yescharacteristic
484C2×C4#57611yescharacteristic
485C2×C4#57611yescharacteristic
486C2×C4#57811yescharacteristic
487C2×C4#57811yescharacteristic
488C8—22no—
489C8—42no—
490C8—41no—
491C8—41no—
492C8—42no—
493C8—41no—
494C8—81no—
495C8—81no—
496C8—81no—
497C22—24no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—82no—
505C22—82no—
506C22—82no—
507C22—161no—
508C22—161no—
509C22—161no—
510C22—161no—
511C22#244911yescharacteristic
512C22#326411yescharacteristic
513C22#326911yescharacteristic
514C22D4×D811yescharacteristic
515C22#328511yescharacteristic
516C22#328811yescharacteristic
517C22#329011yescharacteristic
518C4—21no—
519C4—21no—
520C4—42no—
521C4—42no—
522C4—42no—
523C4—81no—
524C4—81no—
525C4—81no—
526C4—81no—
527C4—161no—
528C4—161no—
529C4—161no—
530C4C2×#47811yescharacteristic
531C4#308311yescharacteristic
532C4#310011yescharacteristic
533C4#312411yescharacteristic
534C4#333911yescharacteristic
535C4#334311yescharacteristic
536C4Q8×D811yescharacteristic
537C4#338911yescharacteristic
538C2—44no—
539C2—162no—
540C2#1975511yescharacteristic
541C2#1984211yescharacteristic
542C2#1997311yescharacteristic
543C2#2005811yescharacteristic
544C2#2635011yescharacteristic
545C2#2636811yescharacteristic
546C2#2640511yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#459C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #26467C2semidirect
2. #26106C2semidirect
4. D4×#107C2semidirect
5. #10671C2semidirect
6. #10650C2semidirect
7. #10591C2semidirect
8. #10581C2semidirect
9. #9616C2semidirect
10. #9536C2semidirect
13. #8920C2semidirect
14. #8894C2semidirect
15. #8843C2semidirect
237. #107D8semidirect
434. D4#478semidirect

Maximal subgroups

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

subgroupindex
#264672
#261062
#229952
D4×#1072
#106712
#106502
#105912
#105812
#96162
#95362
#92212
#89772
#89202
#88942
#88432