Finite Group Databasebeta

← #285654

Subgroups — #285654

Order 512.

616 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27892C211yesmaximal, characteristic, complemented
2#27627C211yesmaximal, characteristic
3#24700C211yesmaximal, characteristic
4#23083C211yesmaximal, characteristic
5#10649C211yesmaximal, characteristic, complemented
6#10625C211yesmaximal, characteristic, complemented
7#10600C211yesmaximal, characteristic, complemented
8#10484C211yesmaximal, characteristic, complemented
9#9613C211yesmaximal, characteristic, complemented
10#9561C211yesmaximal, characteristic, complemented
11#9219C211yesmaximal, characteristic, complemented
12#9132C211yesmaximal, characteristic, complemented
13#8831C211yesmaximal, characteristic, complemented
14#8644C211yesmaximal, characteristic, complemented
15#8038C211yesmaximal, characteristic, complemented
16C2×#474C2211yescharacteristic
17C2×C8×D4C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2857C2211yescharacteristic
21C4×#121C2211yescharacteristic
22#2285C2211yescharacteristic
23#2077—21no—
24#2075—21no—
25#2036—21no—
26#2036—21no—
27#2033—21no—
28#2033—21no—
29#1988C2211yescharacteristic
30#1956C2211yescharacteristic
31#1956C2211yescharacteristic
32#1954—21no—
33#1954C2211yescharacteristic
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1942—21no—
38#1939C2211yescharacteristic
39#1934—21no—
40#1928—21no—
41#1928—21no—
42#1926—21no—
43#1923—21no—
44#1921—21no—
45#1869—21no—
46#1845C2211yescharacteristic
47#1834C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51#1797C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53C8×C22⋊C4—21no—
54C4×C22⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×C8—21no—
68C2×C4⋊Q8C2311yescharacteristic
69C2×C22⋊Q8—21no—
70C2×#121D411yescharacteristic
71C2×C4×D4—21no—
72C2×C4×D4C2311yescharacteristic
73C2×#117—21no—
74C22×C42C2311yescharacteristic
75#474—22no—
76#474—41no—
77C8×D4—22no—
78C8×D4—41no—
79#431—22no—
80#431—22no—
81#431—41no—
82#431—41no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#106—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×#103—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—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⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8D411yescharacteristic
111C2×C22⋊C8D411yescharacteristic
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117#391—21no—
118#390D411yescharacteristic
119#389—21no—
120#389—21no—
121#389D411yescharacteristic
122#388—21no—
123#386—21no—
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127#382—21no—
128#380—21no—
129#380—21no—
130#380—21no—
131#380—21no—
132#380D411yescharacteristic
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4C2311yescharacteristic
136C4×C22⋊C4—21no—
137#340—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#340—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336D411yescharacteristic
151#336D411yescharacteristic
152#335—41no—
153#334—22no—
154#332—22no—
155#332—41no—
156C22×Q8—21no—
157C22×D4—21no—
158C22×D4C2×D411yescharacteristic
159C23×C4—21no—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4#4111yescharacteristic
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—42no—
170C22×C8—42no—
171C22×C8—81no—
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C22⋊Q8—81no—
181C22⋊Q8—81no—
182#121—22no—
183#121—22no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—81no—
189C4×D4—81no—
190#117—82no—
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—21no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—41no—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42#4111yescharacteristic
223C2×C42C2411yescharacteristic
224#107—42no—
225#107—42no—
226#107—42no—
227#107—81no—
228#107—81no—
229#106—81no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—81no—
241C4⋊C8#4112yes—
242C4⋊C8#4112yes—
243#103—42no—
244#103—42no—
245#103—81no—
246#103—81no—
247#102—42no—
248#102—42no—
249#102—42no—
250#102—81no—
251C22⋊C8—22no—
252C22⋊C8—41no—
253C22⋊C8—42no—
254C22⋊C8—42no—
255C22⋊C8—42no—
256C22⋊C8—42no—
257C22⋊C8—81no—
258C22⋊C8—81no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—42no—
264C4×C8—42no—
265C4×C8—81no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—41no—
270#95—41no—
271#95—41no—
272#95—41no—
273C24—21no—
274C24#12111yescharacteristic
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×D4—24no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—81no—
288C2×D4—81no—
289C2×D4#13512yes—
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—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4#12111yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4#12111yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
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×C8—41no—
353C2×C8—41no—
354C2×C8—84no—
355C2×C8#13512yes—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
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—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—81no—
378C4⋊C4—81no—
379C4⋊C4—81no—
380C4⋊C4—81no—
381C4⋊C4—82no—
382C4⋊C4#13512yes—
383C22⋊C4—24no—
384C22⋊C4—41no—
385C22⋊C4—41no—
386C22⋊C4—42no—
387C22⋊C4—81no—
388C22⋊C4—81no—
389C22⋊C4—82no—
390C22⋊C4—82no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—41no—
398C42—41no—
399C42—82no—
400C42C2×#4111yescharacteristic
401C42C2×#4111yescharacteristic
402C42H(5,𝔽2)11yescharacteristic
403C422−1+411yescharacteristic
404C23—21no—
405C23—21no—
406C23—21no—
407C23—22no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23C2×#12111yescharacteristic
419Q8—81no—
420Q8—81no—
421Q8—81no—
422Q8—81no—
423D4—22no—
424D4—22no—
425D4—41no—
426D4—41no—
427D4—84no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
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—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#47312yes—
474C2×C4#52911yescharacteristic
475C2×C4#53911yescharacteristic
476C2×C4#54611yescharacteristic
477C2×C4#54611yescharacteristic
478C2×C4#54711yescharacteristic
479C2×C4#54711yescharacteristic
480C2×C4C2×#13511yescharacteristic
481C2×C4C2×#13511yescharacteristic
482C2×C4C2×#13511yescharacteristic
483C2×C4#57511yescharacteristic
484C2×C4#57611yescharacteristic
485C2×C4#57611yescharacteristic
486C2×C4#57711yescharacteristic
487C2×C4#57811yescharacteristic
488C8—22no—
489C8—41no—
490C8—41no—
491C8—42no—
492C8—42no—
493C8—41no—
494C8—81no—
495C8—81no—
496C8—81no—
497C8—81no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—42no—
503C22—41no—
504C22—41no—
505C22—42no—
506C22—41no—
507C22—42no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22#242511yescharacteristic
513C22#303511yescharacteristic
514C22#304111yescharacteristic
515C22#305011yescharacteristic
516C22#306311yescharacteristic
517C22C2×#47311yescharacteristic
518C22#306811yescharacteristic
519C4—21no—
520C4—21no—
521C4—22no—
522C4—41no—
523C4—41no—
524C4—42no—
525C4—81no—
526C4—81no—
527C4—161no—
528C4—161no—
529C4—161no—
530C4—161no—
531C4#329711yescharacteristic
532C4#330111yescharacteristic
533C4#330211yescharacteristic
534C4#330911yescharacteristic
535C4#333611yescharacteristic
536C4#334011yescharacteristic
537C4#334111yescharacteristic
538C4#334711yescharacteristic
539C2—81no—
540C2—82no—
541C2#1912911yescharacteristic
542C2#1924711yescharacteristic
543C2#1938911yescharacteristic
544C2#1955511yescharacteristic
545C2#2520711yescharacteristic
546C2#2523311yescharacteristic
547C2#2526611yescharacteristic
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. #27892C2semidirect
5. #10649C2semidirect
6. #10625C2semidirect
7. #10600C2semidirect
8. #10484C2semidirect
9. #9613C2semidirect
10. #9561C2semidirect
11. #9219C2semidirect
12. #9132C2semidirect
13. #8831C2semidirect
14. #8644C2semidirect
15. #8038C2semidirect

Maximal subgroups

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

subgroupindex
#278922
#276272
#247002
#230832
#106492
#106252
#106002
#104842
#96132
#95612
#92192
#91322
#88312
#86442
#80382