Finite Group Databasebeta

← #636799

Subgroups — #636799

Order 512.

3913 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#125C211yesmaximal, characteristic, complemented
2C2×#2853C211yesmaximal, characteristic
3C2×#2413C212yesmaximal
4#16605C214yesmaximal, complemented
5#16604C214yesmaximal, complemented
6#15092C211yesmaximal, characteristic, complemented
7#15091C211yesmaximal, characteristic
8#15073C211yesmaximal, characteristic, complemented
9#15069C211yesmaximal, characteristic
10#14933C214yesmaximal, complemented
11#14929C214yesmaximal, complemented
12#14263C212yesmaximal, complemented
13#14256C212yesmaximal
14#13870C212yesmaximal, complemented
15#13756C211yesmaximal, characteristic, complemented
16C22×#125—28no—
17C22×#125—216no—
18C22×#125C2214yes—
19C22×#125C2214yes—
20C22×#123C2211yescharacteristic
21C22×#123C2212yes—
22C22×#117C2212yes—
23C23×C4⋊C4C2212yes—
24C23×C4⋊C4C2211yescharacteristic
25C23×C4⋊C4C2211yescharacteristic
26C23×C4⋊C4C2212yes—
27C23×C42C2211yescharacteristic
28#2853—28no—
29#2413—216no—
30C2×#398C2214yes—
31C2×#396C2214yes—
32C2×#389C2214yes—
33C2×#388C2212yes—
34C2×#388C2212yes—
35C2×#388C2214yes—
36C2×#387C2212yes—
37C2×#387C2212yes—
38C2×#386C2211yescharacteristic
39C2×#386C2211yescharacteristic
40C2×#386C2212yes—
41C2×#385C2211yescharacteristic
42C2×#385C2211yescharacteristic
43C2×#385C2212yes—
44C2×#381C2211yescharacteristic
45C2×#381C2212yes—
46C2×#381C2212yes—
47C2×#381C2212yes—
48#1444C2218yes—
49#1444C2214yes—
50#1444C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1442C2218yes—
54#1441C2218yes—
55#1441C2218yes—
56#1441C2214yes—
57#1441C2214yes—
58#1439C2218yes—
59#1439C2214yes—
60#1439C2214yes—
61#1436C2218yes—
62C4⋊#95C2214yes—
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C22⋊#95C2212yes—
66C22⋊#95C2211yescharacteristic
67C22⋊#95C2211yescharacteristic
68C4×#95C2214yes—
69C23×D4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C24×C4C2312yes—
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2312yes—
75C2×#125—2128no—
76C2×#125—232no—
77C2×#125—232no—
78C2×#125—216no—
79C2×#125—216no—
80C2×#123—24no—
81C2×#123—28no—
82C2×#123—28no—
83C2×#123—28no—
84C2×#123—44no—
85C2×#117—28no—
86C2×#117—28no—
87C2×#117—44no—
88C22×C4⋊C4—28no—
89C22×C4⋊C4—28no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—28no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—28no—
95C22×C4⋊C4—28no—
96C22×C4⋊C4—28no—
97C22×C4⋊C4D4116yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2318yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C42—28no—
122C22×C42D418yes—
123C22×C42C2314yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126#398—216no—
127#396—216no—
128#389—216no—
129#388—216no—
130#388—28no—
131#388—28no—
132#387—28no—
133#387—28no—
134#386—24no—
135#386—24no—
136#386—28no—
137#385—24no—
138#385—24no—
139#385—28no—
140#381—24no—
141#381—28no—
142#381—28no—
143#381—44no—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2318yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2318yes—
158C2×#95C2318yes—
159C2×#95C2318yes—
160C2×#95C2318yes—
161C25C2411yescharacteristic
162C25C2411yescharacteristic
163C22×D4—24no—
164C22×D4—22no—
165C22×D4—24no—
166C22×D4—22no—
167C22×D4—22no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—28no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—44no—
183C23×C4—48no—
184C23×C4C2×D418yes—
185C23×C4C2×D414yes—
186C23×C4C2×D414yes—
187C23×C4C2×D418yes—
188C23×C4C2×D418yes—
189C23×C4C2×D414yes—
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2414yes—
195C23×C4C2411yescharacteristic
196C23×C4C2414yes—
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203#125—2128no—
204#125—2128no—
205#123—432no—
206#123—416no—
207#117—432no—
208C2×C4⋊C4—232no—
209C2×C4⋊C4—232no—
210C2×C4⋊C4—264no—
211C2×C4⋊C4—264no—
212C2×C4⋊C4—216no—
213C2×C4⋊C4—216no—
214C2×C4⋊C4—216no—
215C2×C4⋊C4—216no—
216C2×C4⋊C4—216no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—48no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4C2×D4116yes—
240C2×C4⋊C4C2×D4116yes—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—48no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4#4118yes—
267C2×C42—232no—
268C2×C42—28no—
269C2×C42—28no—
270C2×C42—28no—
271C2×C42—28no—
272C2×C42—44no—
273C2×C42C2×D418yes—
274C2×C42C2×D418yes—
275#95—24no—
276#95—24no—
277#95—28no—
278#95—28no—
279#95—24no—
280#95—24no—
281#95—28no—
282#95—24no—
283#95—24no—
284#95—28no—
285#95—216no—
286#95—28no—
287#95—28no—
288#95—48no—
289#95—48no—
290#95—48no—
291#95—48no—
292C24—24no—
293C24—24no—
294C24—21no—
295C24—22no—
296C24—22no—
297C24—21no—
298C24—24no—
299C24—21no—
300C24—24no—
301C24C22×D412yes—
302C24C22×D414yes—
303C24C2×#4114yes—
304C24H(5,𝔽2)12yes—
305C242−1+412yes—
306C24C2511yescharacteristic
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—48no—
311C2×D4—48no—
312C22×C4—232no—
313C22×C4—28no—
314C22×C4—28no—
315C22×C4—28no—
316C22×C4—216no—
317C22×C4—216no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—24no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—216no—
327C22×C4—28no—
328C22×C4—216no—
329C22×C4—216no—
330C22×C4—216no—
331C22×C4—216no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—28no—
343C22×C4—24no—
344C22×C4—28no—
345C22×C4—28no—
346C22×C4—24no—
347C22×C4—28no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—24no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—28no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—21no—
370C22×C4—28no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—416no—
375C22×C4—48no—
376C22×C4—416no—
377C22×C4—416no—
378C22×C4—416no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—48no—
383C22×C4#120116yes—
384C22×C4C22×D414yes—
385C22×C4C22×D418yes—
386C22×C4C22×D418yes—
387C22×C4C22×D412yes—
388C22×C4C22×D412yes—
389C22×C4C22×D414yes—
390C22×C4C22×D414yes—
391C22×C4C22×D414yes—
392C22×C4C2×#4118yes—
393C22×C42−1+418yes—
394C4⋊C4—232no—
395C4⋊C4—232no—
396C4⋊C4—264no—
397C4⋊C4—232no—
398C4⋊C4—232no—
399C4⋊C4—416no—
400C4⋊C4—416no—
401C4⋊C4—416no—
402C4⋊C4—48no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—48no—
406C22⋊C4—216no—
407C22⋊C4—416no—
408C22⋊C4—48no—
409C22⋊C4—416no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—48no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—48no—
417C42—216no—
418C42—216no—
419C42—416no—
420C23—24no—
421C23—24no—
422C23—28no—
423C23—28no—
424C23—216no—
425C23—28no—
426C23—28no—
427C23—41no—
428C23—41no—
429C23—42no—
430C23—41no—
431C23—42no—
432C23—44no—
433C23—42no—
434C23—42no—
435C23—44no—
436C23—48no—
437C23—44no—
438C23—44no—
439C23C2×#12014yes—
440C23#53214yes—
441C23#53714yes—
442C23#54618yes—
443C23#54718yes—
444C23C23×D411yescharacteristic
445C23C23×D412yes—
446C23C22×#4112yes—
447C23C2×H(5,𝔽2)11yescharacteristic
448C23C2×2−1+414yes—
449C23C2×2−1+411yescharacteristic
450C23#58514yes—
451D4—88no—
452C2×C4—232no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—232no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—216no—
465C2×C4—24no—
466C2×C4—28no—
467C2×C4—28no—
468C2×C4—24no—
469C2×C4—28no—
470C2×C4—24no—
471C2×C4—28no—
472C2×C4—28no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—216no—
476C2×C4—416no—
477C2×C4—432no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—48no—
482C2×C4—48no—
483C2×C4—48no—
484C2×C4—432no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—48no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—48no—
510C2×C4—48no—
511C2×C4—48no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4C2×#12014yes—
515C2×C4C2×#12014yes—
516C2×C4#53618yes—
517C22—28no—
518C22—24no—
519C22—216no—
520C22—216no—
521C22—24no—
522C22—28no—
523C22—82no—
524C22—81no—
525C22—84no—
526C22—84no—
527C22—81no—
528C22—81no—
529C22—82no—
530C22#243018yes—
531C22C22×#12011yescharacteristic
532C22C2×#53211yescharacteristic
533C22C2×#53612yes—
534C22C2×#53711yescharacteristic
535C22#344912yes—
536C22#344914yes—
537C22C2×#54612yes—
538C22C2×#54712yes—
539C22#348614yes—
540C22#351518yes—
541C22#355914yes—
542C22#356314yes—
543C4—24no—
544C4—24no—
545C4—28no—
546C4—44no—
547C4—48no—
548C4—48no—
549C4—48no—
550C4—82no—
551C4—81no—
552C4—84no—
553C4—81no—
554C4—81no—
555C4—82no—
556C2—28no—
557C2—161no—
558C2#3463211yescharacteristic
559C2C2×#243011yescharacteristic
560C2#3513912yes—
561C2#3524211yescharacteristic
562C2#3754912yes—
563C2#3802914yes—
564C2#3804514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×#125C2semidirect
4. #16605C2semidirect
5. #16604C2semidirect
6. #15092C2semidirect
8. #15073C2semidirect
10. #14933C2semidirect
11. #14929C2semidirect
12. #14263C2semidirect
14. #13870C2semidirect
15. #13756C2semidirect

Maximal subgroups

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

subgroupindex
C23×#1252
C2×#28532
C2×#24132
#166052
#166042
#150922
#150912
#150732
#150692
#149332
#149292
#142632
#142562
#138702
#137562