Finite Group Databasebeta

← #398598

Subgroups — #398598

Order 512.

1556 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic
2C2×#2026C211yesmaximal, characteristic, complemented
3C2×#1933C212yesmaximal, complemented
4C2×C4×C22⋊C8C211yesmaximal, characteristic
5#14702C211yesmaximal, characteristic
6#11385C211yesmaximal, characteristic, complemented
7#11268C211yesmaximal, characteristic, complemented
8#11199C212yesmaximal
9#11052C211yesmaximal, characteristic, complemented
10#10970C212yesmaximal
11#10880C212yesmaximal, complemented
12C22×#125C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C4×C8C2212yes—
19#2851—28no—
20C2×#391C2211yescharacteristic
21C2×#391C2211yescharacteristic
22C2×#389C2212yes—
23C2×#387—22no—
24C2×#385—22no—
25C2×#382C2211yescharacteristic
26C2×#380C2212yes—
27C2×#380C2211yescharacteristic
28#2026—24no—
29#1933—28no—
30C4×C22⋊C8—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39#1436—22no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C22⋊#95—22no—
43C23×Q8C2311yescharacteristic
44C24×C4D412yes—
45C24×C4C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×#125—24no—
49C2×#125—22no—
50C2×#125—22no—
51C2×#125—42no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—24no—
55C2×C22⋊Q8—42no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C22⋊C4—21no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—21no—
67C22×C42—24no—
68C22×C42—22no—
69C22×C42—22no—
70C22×C42—42no—
71C22×C42D412yes—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—42no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—24no—
83C2×C4×C8—24no—
84C2×C4×C8—24no—
85C2×C4×C8—42no—
86#391—24no—
87#391—22no—
88#391—22no—
89#389—24no—
90#389D418yes—
91#387—216no—
92#385—216no—
93#382—24no—
94#380—24no—
95#380—24no—
96#380—24no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—21no—
105#340—24no—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—24no—
110#340—42no—
111#336—22no—
112#336D414yes—
113C25C2×D411yescharacteristic
114C22×Q8—21no—
115C22×Q8—22no—
116C22×Q8—21no—
117C22×Q8—22no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8#4112yes—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4#4114yes—
150C23×C4C2411yescharacteristic
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8C2×D414yes—
164C22×C8#4114yes—
165#125—48no—
166#125—48no—
167C22⋊Q8—48no—
168C22⋊Q8—48no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4C2×D414yes—
200C2×C4⋊C4C2×D414yes—
201C2×C4⋊C4#4112yes—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—44no—
210C2×C42—22no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—48no—
218C2×C42—48no—
219C2×C42—42no—
220C2×C42C2×D414yes—
221C2×C42C2×D412yes—
222C2×C42#4112yes—
223#103—216no—
224#103—48no—
225C22⋊C8—216no—
226C4×C8—48no—
227#95—28no—
228#95—28no—
229#95—28no—
230#95—44no—
231#95—44no—
232#95—44no—
233#95—42no—
234#95—44no—
235C24—21no—
236C24—21no—
237C24—21no—
238C24—21no—
239C24—21no—
240C24—21no—
241C24—22no—
242C24—22no—
243C24—22no—
244C24#12412yes—
245C24#13514yes—
246C24C22×D411yescharacteristic
247C2×Q8—24no—
248C2×Q8—22no—
249C2×Q8—24no—
250C2×Q8—22no—
251C2×Q8—42no—
252C2×Q8—41no—
253C2×Q8—42no—
254C2×Q8—41no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—216no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—48no—
291C22×C4—48no—
292C22×C4—48no—
293C22×C4—48no—
294C22×C4—48no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—48no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4#12014yes—
339C22×C4#12114yes—
340C22×C4#12412yes—
341C22×C4#13514yes—
342C22×C4#13714yes—
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D412yes—
345C22×C4C2×#4112yes—
346C22×C4C2×#4111yescharacteristic
347C22×C4C2×#4112yes—
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4H(5,𝔽2)12yes—
351C22×C42−1+411yescharacteristic
352C22×C42−1+411yescharacteristic
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—48no—
373C4⋊C4—416no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—416no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C22⋊C4—416no—
385C22⋊C4—48no—
386C22⋊C4—84no—
387C42—24no—
388C42—24no—
389C42—24no—
390C42—24no—
391C42—48no—
392C23—22no—
393C23—22no—
394C23—24no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—24no—
399C23—24no—
400C23—22no—
401C23—24no—
402C23—24no—
403C23—24no—
404C23—42no—
405C23—41no—
406C23—42no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23#48814yes—
415C23C2×#12011yescharacteristic
416C23C2×#12111yescharacteristic
417C23#52511yescharacteristic
418C23C2×#12411yescharacteristic
419C23#53211yescharacteristic
420C23#53411yescharacteristic
421C23#53611yescharacteristic
422C23C2×#13512yes—
423C23C2×#13512yes—
424C23C2×#13712yes—
425C23#57512yes—
426Q8—44no—
427Q8—44no—
428Q8—82no—
429Q8—82no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—432no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—82no—
467C2×C4—81no—
468C2×C4—82no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—82no—
487C2×C4—81no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4#45218yes—
492C2×C4#52512yes—
493C2×C4#52512yes—
494C2×C4#52912yes—
495C2×C4#53212yes—
496C2×C4#53912yes—
497C2×C4#54212yes—
498C2×C4D4212yes—
499C2×C4#54612yes—
500C2×C4#54711yescharacteristic
501C2×C4#54812yes—
502C2×C4#54812yes—
503C2×C4#54811yescharacteristic
504C2×C4#55112yes—
505C2×C4#55112yes—
506C2×C4#55211yescharacteristic
507C2×C4#55611yescharacteristic
508C8—44no—
509C8—44no—
510C22—24no—
511C22—24no—
512C22—24no—
513C22—44no—
514C22—42no—
515C22—42no—
516C22—44no—
517C22—44no—
518C22—44no—
519C22—42no—
520C22—42no—
521C22#238811yescharacteristic
522C22#241011yescharacteristic
523C22#241511yescharacteristic
524C22#243611yescharacteristic
525C22#256111yescharacteristic
526C22#266311yescharacteristic
527C22#285011yescharacteristic
528C22C2×#45212yes—
529C22#300212yes—
530C22#305012yes—
531C22#305612yes—
532C22#310212yes—
533C22#310812yes—
534C22C2×#48811yescharacteristic
535C22#313111yescharacteristic
536C22#313911yescharacteristic
537C22#314211yescharacteristic
538C22#316812yes—
539C22#317212yes—
540C22#319012yes—
541C22#319112yes—
542C22#323112yes—
543C22#323212yes—
544C4—28no—
545C4—44no—
546C4—42no—
547C4—84no—
548C4—84no—
549C4—82no—
550C4—82no—
551C4—161no—
552C4—161no—
553C4#246112yes—
554C4#246312yes—
555C4#247712yes—
556C4#256012yes—
557C2—44no—
558C2#1476511yescharacteristic
559C2#2357212yes—
560C2#2438312yes—
561C2#2470512yes—
562C2#2590411yescharacteristic
563C2#2590711yescharacteristic
564C2#2910611yescharacteristic
565C2#2910811yescharacteristic
566C2#3036112yes—
567C2#3304411yescharacteristic
568C2#3305911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2026C2semidirect
3. C2×#1933C2semidirect
6. #11385C2semidirect
7. #11268C2semidirect
9. #11052C2semidirect
11. #10880C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C2×#20262
C2×#19332
C2×C4×C22⋊C82
#147022
#113852
#112682
#111992
#110522
#109702
#108802