Finite Group Databasebeta

← #275321

Subgroups — #275321

Order 512.

649 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33107C211yesmaximal, characteristic
2#27890C211yesmaximal, characteristic
3#22599C211yesmaximal, characteristic, complemented
4#21365C211yesmaximal, characteristic, complemented
5#10656C211yesmaximal, characteristic, complemented
6#10654C211yesmaximal, characteristic, complemented
7#10599C211yesmaximal, characteristic, complemented
8#10587C211yesmaximal, characteristic, complemented
9#9616C211yesmaximal, characteristic, complemented
10#9612C211yesmaximal, characteristic, complemented
11#9564C211yesmaximal, characteristic, complemented
12#9563C211yesmaximal, characteristic, complemented
13#9142C211yesmaximal, characteristic
14#8834C211yesmaximal, characteristic, complemented
15#8728C211yesmaximal, characteristic, complemented
16C2×#481—21no—
17C2×#432C2211yescharacteristic
18C2×#431—21no—
19#2856C2211yescharacteristic
20#2285C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22#2034—21no—
23#2034—21no—
24#1982C2211yescharacteristic
25#1982C2211yescharacteristic
26#1957C2211yescharacteristic
27#1957C2211yescharacteristic
28#1948C2211yescharacteristic
29#1948C2211yescharacteristic
30#1948C2211yescharacteristic
31#1948C2211yescharacteristic
32#1947C2211yescharacteristic
33#1947C2211yescharacteristic
34#1939C2211yescharacteristic
35#1939C2211yescharacteristic
36#1926—21no—
37#1926—21no—
38#1926—21no—
39#1926—21no—
40#1916—21no—
41#1916—21no—
42#1874—21no—
43#1853—21no—
44#1852—21no—
45#1844C2211yescharacteristic
46#1834C2211yescharacteristic
47#1817—21no—
48#1797C2211yescharacteristic
49#1797C2211yescharacteristic
50#1776—21no—
51#1776—21no—
52C8×C4⋊C4C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C4×C22⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#331C2211yescharacteristic
58C2×#331C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×#127C2311yescharacteristic
68C2×#121—21no—
69C2×#121—21no—
70C2×#121—21no—
71C2×#117—21no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76#481—44no—
77#432—22no—
78#432—22no—
79#431—44no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#106—21no—
83C2×#106—21no—
84C2×#106—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8D411yescharacteristic
108C2×C22⋊C8D411yescharacteristic
109C2×#97—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#390—21no—
115#390—21no—
116#390—21no—
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#382—21no—
121#380—21no—
122#380—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4D411yescharacteristic
128C4×C4⋊C4D411yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130C4×C4⋊C4C2311yescharacteristic
131C2×#95—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336D411yescharacteristic
147#336D411yescharacteristic
148#335—22no—
149#334—22no—
150#331—22no—
151#331—22no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—41no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4#4111yescharacteristic
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171#127—21no—
172#127—21no—
173#127—21no—
174#127—21no—
175#121—42no—
176#121—42no—
177#121—81no—
178#121—81no—
179#121—82no—
180#117—82no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
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—41no—
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—42no—
213C2×C42—42no—
214C2×C42—41no—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42#4111yescharacteristic
218C2×C42C2411yescharacteristic
219#107—42no—
220#107—42no—
221#106—42no—
222#106—42no—
223#106—42no—
224#106—42no—
225#106—81no—
226#106—81no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—42no—
238C4⋊C8—42no—
239C4⋊C8#4112yes—
240C4⋊C8#4112yes—
241#102—42no—
242#102—42no—
243#102—42no—
244#102—42no—
245#102—42no—
246#102—42no—
247#102—81no—
248#102—81no—
249C22⋊C8—24no—
250C22⋊C8—22no—
251C22⋊C8—41no—
252C22⋊C8—42no—
253C22⋊C8—42no—
254#97—24no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—42no—
260C4×C8—42no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—41no—
266#95—41no—
267C24—41no—
268C24#12111yescharacteristic
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—81no—
284C2×D4—81no—
285C2×D4—82no—
286C2×D4—82no—
287C2×D4—82no—
288C22×C4—24no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
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—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—81no—
317C22×C4#12111yescharacteristic
318C22×C4#12111yescharacteristic
319C22×C4#12111yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C2×#4111yescharacteristic
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
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—41no—
352C2×C8AGL(1,ℤ/8)12yes—
353C4⋊C4—24no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—42no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—81no—
373C4⋊C4—81no—
374C4⋊C4—81no—
375C4⋊C4#13512yes—
376C4⋊C4#13512yes—
377C4⋊C4AGL(1,ℤ/8)12yes—
378C22⋊C4—82no—
379C22⋊C4—82no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—42no—
389C42—42no—
390C42—41no—
391C42—41no—
392C42—41no—
393C42—41no—
394C42—82no—
395C42C2×#4111yescharacteristic
396C42C2×#4111yescharacteristic
397C422−1+411yescharacteristic
398C422−1+411yescharacteristic
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23#47112yes—
413C23C2×#12111yescharacteristic
414D4—81no—
415D4—81no—
416D4—81no—
417D4—81no—
418D4—81no—
419D4—81no—
420D4—81no—
421D4—81no—
422D4—161no—
423D4—161no—
424D4—164no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—48no—
442C2×C4—41no—
443C2×C4—41no—
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—41no—
458C2×C4—41no—
459C2×C4—84no—
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#52511yescharacteristic
470C2×C4#53611yescharacteristic
471C2×C4#54711yescharacteristic
472C2×C4#54711yescharacteristic
473C2×C4#55211yescharacteristic
474C2×C4#55211yescharacteristic
475C2×C4C2×#13511yescharacteristic
476C2×C4C2×#13511yescharacteristic
477C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
478C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
479C2×C4#57511yescharacteristic
480C2×C4#57511yescharacteristic
481C2×C4#57711yescharacteristic
482C2×C4#57711yescharacteristic
483C8—22no—
484C8—41no—
485C8—41no—
486C8—41no—
487C8—42no—
488C8—42no—
489C8—81no—
490C8—81no—
491C8—81no—
492C8—81no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—161no—
501C22—161no—
502C22—161no—
503C22—161no—
504C22—161no—
505C22—161no—
506C22—161no—
507C22#241311yescharacteristic
508C22#303611yescharacteristic
509C22#304311yescharacteristic
510C22#305011yescharacteristic
511C22#305511yescharacteristic
512C22C2×#47111yescharacteristic
513C22#306811yescharacteristic
514C4—21no—
515C4—21no—
516C4—41no—
517C4—41no—
518C4—41no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—81no—
523C4—161no—
524C4—161no—
525C4—161no—
526C4#310211yescharacteristic
527C4#310611yescharacteristic
528C4#322811yescharacteristic
529C4#323311yescharacteristic
530C4#334511yescharacteristic
531C4#334811yescharacteristic
532C4#335111yescharacteristic
533C4#335411yescharacteristic
534C2—42no—
535C2—321no—
536C2#1925711yescharacteristic
537C2#1930611yescharacteristic
538C2#1940111yescharacteristic
539C2#1946811yescharacteristic
540C2#2458911yescharacteristic
541C2#2460511yescharacteristic
542C2#2462311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
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
3. #22599C2semidirect
4. #21365C2semidirect
5. #10656C2semidirect
6. #10654C2semidirect
7. #10599C2semidirect
8. #10587C2semidirect
9. #9616C2semidirect
10. #9612C2semidirect
11. #9564C2semidirect
12. #9563C2semidirect
14. #8834C2semidirect
15. #8728C2semidirect

Maximal subgroups

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

subgroupindex
#331072
#278902
#225992
#213652
#106562
#106542
#105992
#105872
#96162
#96122
#95642
#95632
#91422
#88342
#87282