Finite Group Databasebeta

← #6798410

Subgroups — #6798410

Order 512.

1704 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2#36127C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic, complemented
4#31357C218yesmaximal, complemented
5#29878C212yesmaximal, complemented
6#29139C211yesmaximal, characteristic, complemented
7#28689C211yesmaximal, characteristic, complemented
8#27962C212yesmaximal, complemented
9#26111C214yesmaximal, complemented
10#25673C212yesmaximal, complemented
11#24968C214yesmaximal, complemented
12#21863C212yesmaximal, complemented
13C4×#486C211yesmaximal, characteristic, complemented
14#21264C211yesmaximal, characteristic, complemented
15C2×#547—21no—
16C2×#532C2212yes—
17C2×#500C2212yes—
18C2×#486C2211yescharacteristic
19C2×#431C2212yes—
20#2868C2211yescharacteristic
21#2868C2211yescharacteristic
22#2857C2212yes—
23#2857C2212yes—
24#2571C2212yes—
25#2563—21no—
26#2547—21no—
27#2538—21no—
28#2537—21no—
29#2537C2212yes—
30#2497—22no—
31#2492—22no—
32#2439C2214yes—
33#2432—22no—
34#2422—22no—
35#2422C2214yes—
36#2369—21no—
37#2355—21no—
38#2354C2212yes—
39D4×C4⋊C4—21no—
40D4×C4⋊C4C2212yes—
41C4×C4⋊Q8C2211yescharacteristic
42C4×#127C2211yescharacteristic
43#2285C2212yes—
44C42×D4C2211yescharacteristic
45#2101C2218yes—
46#2087C2218yes—
47#2083C2214yes—
48#2076C2218yes—
49#2033C2218yes—
50#2023C2214yes—
51#1990C2212yes—
52#1990C2212yes—
53#1990C2214yes—
54#1990C2212yes—
55#1990C2211yescharacteristic
56#1957C2212yes—
57#1957C2211yescharacteristic
58#1954C2212yes—
59#1947C2214yes—
60#1939C2212yes—
61#1934C2214yes—
62#1923C2214yes—
63#1875C2218yes—
64#1862C2218yes—
65#1834C2214yes—
66#1834C2211yescharacteristic
67#1834C2212yes—
68#1825C2218yes—
69#1804C2218yes—
70#1802C2214yes—
71C4×C4⋊C8C2211yescharacteristic
72C4×#102C2214yes—
73C4×C22⋊C8C2212yes—
74C42×C8C2211yescharacteristic
75C22×#41—22no—
76#547—48no—
77#532—28no—
78#532—44no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2312yes—
86C2×#127C2311yescharacteristic
87C2×#124—22no—
88C2×#123—24no—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8—22no—
91C2×#121—22no—
92C2×#121—22no—
93C2×#121C2314yes—
94C2×C4×Q8—21no—
95C2×C4×D4—21no—
96C2×C4×D4—21no—
97C2×C4×D4—22no—
98C2×C4×D4D411yescharacteristic
99C2×C4×D4D411yescharacteristic
100C2×C4×D4C2312yes—
101C2×C4×D4C2314yes—
102C2×C4×D4C2311yescharacteristic
103C2×#117C2314yes—
104C22×C4⋊C4—22no—
105C22×C42C2312yes—
106#500—44no—
107#486—24no—
108#431—44no—
109C2×#107C2318yes—
110C2×C4⋊C8C2312yes—
111C2×C4⋊C8C2314yes—
112C2×#102C2314yes—
113C2×#102C2318yes—
114C2×#102C2314yes—
115C2×C22⋊C8C2314yes—
116C2×C22⋊C8C2314yes—
117C2×C4×C8C2312yes—
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2312yes—
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2312yes—
122#400—24no—
123#397—24no—
124#396—24no—
125#394—24no—
126#394—24no—
127#393—24no—
128#391—21no—
129#391—22no—
130#391—22no—
131#391—24no—
132#390—22no—
133#390—21no—
134#390—21no—
135#390C2312yes—
136#389—22no—
137#389—21no—
138#389—22no—
139#389—22no—
140#389C2312yes—
141#389C2314yes—
142#389C2314yes—
143#388—24no—
144#387—24no—
145#385—24no—
146#385—24no—
147#384—21no—
148#384—21no—
149#384—21no—
150#384—21no—
151#384C2311yescharacteristic
152#384C2311yescharacteristic
153#384C2312yes—
154#384C2312yes—
155#384C2312yes—
156#384C2312yes—
157#382—22no—
158#382C2314yes—
159#380—22no—
160#380—22no—
161#380—22no—
162#380—22no—
163#380—22no—
164#380—22no—
165#380C2314yes—
166#380C2314yes—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4D411yescharacteristic
172C4×C4⋊C4D411yescharacteristic
173C4×C4⋊C4C2312yes—
174C4×C4⋊C4C2311yescharacteristic
175C4×C22⋊C4D412yes—
176C4×C22⋊C4D412yes—
177C4×C22⋊C4C2312yes—
178C43C2311yescharacteristic
179#340C2318yes—
180#340C2318yes—
181#340C2318yes—
182#340C2314yes—
183#340C2314yes—
184#340C2318yes—
185#336C2312yes—
186#336C2314yes—
187C2×#41—48no—
188C2×#41—48no—
189C2×#41—84no—
190C22×Q8—21no—
191C22×Q8—22no—
192C22×Q8—22no—
193C22×D4—24no—
194C22×D4—21no—
195C22×D4—21no—
196C22×D4C2×D411yescharacteristic
197C22×D4C2412yes—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4C2×D412yes—
201C23×C4C2×D412yes—
202C23×C4C2412yes—
203C22×C8C2412yes—
204C22×C8C2414yes—
205C22×C8C2412yes—
206C4⋊Q8—24no—
207C4⋊Q8—22no—
208C4⋊Q8—24no—
209C4⋊Q8—22no—
210C4⋊Q8—44no—
211C4⋊Q8—44no—
212C4⋊Q8—44no—
213C4⋊Q8—82no—
214C4⋊Q8—82no—
215#127—22no—
216#127#4114yes—
217#124—48no—
218#123—416no—
219C22⋊Q8—48no—
220C22⋊Q8—84no—
221#121—28no—
222#121—44no—
223#121—48no—
224#121—48no—
225C4×Q8—44no—
226C4×D4—28no—
227C4×D4—24no—
228C4×D4—24no—
229C4×D4—24no—
230C4×D4—28no—
231C4×D4—44no—
232C4×D4—48no—
233C4×D4—44no—
234C4×D4—44no—
235#117—48no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—48no—
255C2×C4⋊C4C2×D411yescharacteristic
256C2×C4⋊C4C2×D411yescharacteristic
257C2×C4⋊C4C2×D411yescharacteristic
258C2×C4⋊C4C2412yes—
259C2×C4⋊C4C2414yes—
260C2×C4⋊C4C2414yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2412yes—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4C2×D412yes—
272C2×C22⋊C4C2×D412yes—
273C2×C22⋊C4C2×D412yes—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2412yes—
285C2×C42C2411yescharacteristic
286C2×C42C2411yescharacteristic
287C2×C42C2411yescharacteristic
288#107—48no—
289C4⋊C8—24no—
290C4⋊C8—44no—
291#102—28no—
292#102—44no—
293#102—48no—
294C22⋊C8—44no—
295C22⋊C8—44no—
296C4×C8—22no—
297C4×C8—22no—
298C4×C8—24no—
299C4×C8—24no—
300C4×C8—42no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—22no—
305#95—22no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95C2×D412yes—
310#95C2×D412yes—
311#95C2×D412yes—
312C24—22no—
313C24C22×D412yes—
314#41—832no—
315C2×Q8—44no—
316C2×Q8—42no—
317C2×Q8—44no—
318C2×Q8—42no—
319C2×Q8—44no—
320C2×Q8—42no—
321C2×Q8—44no—
322C2×Q8—82no—
323C2×Q8—82no—
324C2×D4—22no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—48no—
333C2×D4—48no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—84no—
339C2×D4C2×#4114yes—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—84no—
361C22×C4C22×D412yes—
362C22×C4C22×D412yes—
363C22×C4C22×D412yes—
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C2511yescharacteristic
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—24no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—24no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—24no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—28no—
390C4⋊C4—48no—
391C4⋊C4—44no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—42no—
403C4⋊C4—84no—
404C4⋊C4—82no—
405C4⋊C4—82no—
406C4⋊C4—82no—
407C4⋊C4—82no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—48no—
412C22⋊C4—48no—
413C22⋊C4—48no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—84no—
418C22⋊C4—84no—
419C42—24no—
420C42—24no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—44no—
427C42—84no—
428C42#13514yes—
429C42AGL(1,ℤ/8)14yes—
430C42C2×#4112yes—
431C42H(5,𝔽2)12yes—
432C422−1+414yes—
433C23—22no—
434C23—24no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23C23×D411yescharacteristic
448Q8—84no—
449Q8—88no—
450Q8—164no—
451D4—28no—
452D4—44no—
453D4—44no—
454D4—48no—
455D4—48no—
456D4—816no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—48no—
475C2×C4—48no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—816no—
492C2×C4—88no—
493C2×C4#53612yes—
494C2×C4#53612yes—
495C2×C4#54612yes—
496C2×C4#54612yes—
497C2×C4C2×#13512yes—
498C2×C4C2×#13512yes—
499C2×C4C2×#13512yes—
500C2×C4C2×AGL(1,ℤ/8)12yes—
501C2×C4C2×AGL(1,ℤ/8)12yes—
502C2×C4C2×AGL(1,ℤ/8)12yes—
503C2×C4#57612yes—
504C2×C4#57812yes—
505C2×C4C22×#4111yescharacteristic
506C2×C4C2×H(5,𝔽2)11yescharacteristic
507C2×C4C2×2−1+412yes—
508C2×C4#58512yes—
509C8—42no—
510C8—42no—
511C8—44no—
512C22—42no—
513C22—44no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—44no—
519C22—42no—
520C22—42no—
521C22—44no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22C2×#53611yescharacteristic
527C22C2×#54611yescharacteristic
528C22#350011yescharacteristic
529C22C22×#13511yescharacteristic
530C22C22×AGL(1,ℤ/8)11yescharacteristic
531C22C2×#57611yescharacteristic
532C22C2×#57811yescharacteristic
533C4—22no—
534C4—22no—
535C4—22no—
536C4—42no—
537C4—42no—
538C4—42no—
539C4—84no—
540C4—82no—
541C4—82no—
542C4—84no—
543C4—82no—
544C4#322814yes—
545C4#330214yes—
546C4#332514yes—
547C4#351512yes—
548C4#354712yes—
549C2—82no—
550C2—84no—
551C2#3895511yescharacteristic
552C2C2×#322811yescharacteristic
553C2#6088011yescharacteristic
554C2C2×#330211yescharacteristic
555C2C2×#332511yescharacteristic
556C2#6252711yescharacteristic
557C2#6253811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36127C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36194C2semidirect
3. #33289C2semidirect
4. #31357C2semidirect
5. #29878C2semidirect
6. #29139C2semidirect
7. #28689C2semidirect
8. #27962C2semidirect
9. #26111C2semidirect
10. #25673C2semidirect
11. #24968C2semidirect
12. #21863C2semidirect
13. C4×#486C2semidirect
14. #21264C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#361272
#332892
#313572
#298782
#291392
#286892
#279622
#261112
#256732
#249682
#218632
C4×#4862
#212642