Finite Group Databasebeta

← #365728

Subgroups — #365728

Order 512.

1634 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#124C211yesmaximal, characteristic
2C2×#1958C211yesmaximal, characteristic
3C2×#1958C211yesmaximal, characteristic
4C2×#1835C211yesmaximal, characteristic, complemented
5#14707C211yesmaximal, characteristic, complemented
6#11067C212yesmaximal, complemented
7#11066C212yesmaximal
8#10842C211yesmaximal, characteristic, complemented
9#10789C212yesmaximal, complemented
10#10788C212yesmaximal
11C4×#336C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×C4×C8C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21C4×#124—24no—
22C4×#124—24no—
23C2×#391C2211yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#389C2212yes—
26C2×#388—22no—
27C2×#382C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×C4×C22⋊C4—22no—
31C2×C43C2211yescharacteristic
32#1958—24no—
33#1958C418yescomplemented
34#1835—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1436—22no—
42#1436—22no—
43C4⋊#95C2212yes—
44C23×Q8C2311yescharacteristic
45C23×D4C2311yescharacteristic
46C24×C4—21no—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×#124—22no—
50C2×#124—22no—
51C2×#124—22no—
52C2×#124C2×C414yes—
53C2×#124D414yes—
54C2×#124Q814yes—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—24no—
58C2×C4×Q8—42no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—24no—
62C2×C4×D4—42no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C22⋊C4—22no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C42—21no—
73C22×C42D412yes—
74C22×C42D412yes—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—42no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103C2×C418yescomplemented
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—42no—
87C2×#102—42no—
88C2×#102—42no—
89C2×#102C2×C418yes—
90C2×C4×C8—22no—
91C2×C4×C8—22no—
92C2×C4×C8—22no—
93C2×C4×C8—22no—
94C2×C4×C8—22no—
95C2×C4×C8C2×C414yescomplemented
96#391—22no—
97#391—22no—
98#390—22no—
99#390—22no—
100#389—42no—
101#389—42no—
102#388—216no—
103#382—22no—
104#382—22no—
105#382—22no—
106#382—22no—
107#382C2×C414yescomplemented
108#382D414yes—
109C4×C22⋊C4—216no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C43—22no—
118C43—22no—
119#340—24no—
120#340—24no—
121#340—42no—
122#340—42no—
123#336—24no—
124#336—24no—
125C25—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8C22×C412yes—
134C22×Q8C2×D412yes—
135C22×Q8C2×Q812yes—
136C22×D4—21no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4C22×C412yes—
144C22×D4C2×D412yes—
145C22×D4C2×Q812yes—
146C23×C4—21no—
147C23×C4—24no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—42no—
153C23×C4—44no—
154C23×C4—42no—
155C23×C4—44no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2411yescharacteristic
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8C22×C414yes—
174C22×C8#4114yes—
175#124—28no—
176#124—28no—
177#124—28no—
178#124C4⋊C4116yescomplemented
179C4×Q8—48no—
180C4×Q8—48no—
181C4×D4—48no—
182C4×D4—48no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4C22×C414yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D412yes—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—44no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—22no—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42C22×C412yes—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2×Q812yes—
238C2×C42#4114yes—
239C2×C42#4112yes—
240C2×C42#4112yes—
241#103—48no—
242#103—48no—
243#102—48no—
244#102—48no—
245C4×C8—28no—
246C4×C8—44no—
247#95—28no—
248#95—28no—
249#95—28no—
250#95—44no—
251#95—44no—
252#95—44no—
253#95—44no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—41no—
258C24—41no—
259C24—41no—
260C24—41no—
261C24—42no—
262C24—42no—
263C24—42no—
264C24—42no—
265C24C22×D411yescharacteristic
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8C2×C4⋊C414yes—
282C2×D4—22no—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4C2×C4⋊C414yes—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—48no—
322C22×C4—416no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—84no—
349C22×C4—88no—
350C22×C4—84no—
351C22×C4C4×D414yes—
352C22×C4#12014yes—
353C22×C4C22⋊Q814yes—
354C22×C4#12412yes—
355C22×C4#12412yes—
356C22×C4#13514yes—
357C22×C4#13514yes—
358C22×C4C23×C411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×Q811yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4112yes—
364C22×C4C2×#4112yes—
365C22×C4C2×#4111yescharacteristic
366C22×C4H(5,𝔽2)12yes—
367C22×C4H(5,𝔽2)12yes—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—48no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—88no—
396C22⋊C4—232no—
397C22⋊C4—416no—
398C22⋊C4—48no—
399C42—24no—
400C42—24no—
401C42—48no—
402C42—48no—
403C42—48no—
404C42—42no—
405C42—42no—
406C42C2×C4⋊C414yes—
407C42C4×D414yes—
408C42C22⋊Q814yes—
409C42#12314yes—
410C23—24no—
411C23—42no—
412C23—44no—
413C23—44no—
414C23—42no—
415C23—42no—
416C23—44no—
417C23—42no—
418C23—44no—
419C23—42no—
420C23—44no—
421C23—44no—
422C23—42no—
423C23—81no—
424C23—81no—
425C23—81no—
426C23—81no—
427C23—82no—
428C23—82no—
429C23—82no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23C2×C4×D411yescharacteristic
434C23C2×#12011yescharacteristic
435C23C2×C22⋊Q811yescharacteristic
436C23C2×#12411yescharacteristic
437C23#53211yescharacteristic
438C23#53511yescharacteristic
439C23#53511yescharacteristic
440C23C2×#13512yes—
441C23C2×#13512yes—
442C23#57512yes—
443C23#57512yes—
444Q8—28no—
445Q8—44no—
446Q8—44no—
447Q8—44no—
448D4—28no—
449D4—44no—
450D4—44no—
451D4—44no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—816no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4#38518yes—
507C2×C4#48814yes—
508C2×C4C22×C4⋊C411yescharacteristic
509C2×C4C2×C4×D411yescharacteristic
510C2×C4C4×#4112yes—
511C2×C4#51812yes—
512C2×C4C2×C22⋊Q811yescharacteristic
513C2×C4C2×#12311yescharacteristic
514C2×C4#52912yes—
515C2×C4#54012yes—
516C2×C4#54612yes—
517C2×C4#54612yes—
518C2×C4#54612yes—
519C2×C4#55012yes—
520C2×C4#55312yes—
521C2×C4#55412yes—
522C2×C4#55412yes—
523C2×C4#55512yes—
524C8—44no—
525C8—44no—
526C22—48no—
527C22—44no—
528C22—84no—
529C22—82no—
530C22—82no—
531C22—84no—
532C22—82no—
533C22—82no—
534C22—82no—
535C22—82no—
536C22—84no—
537C22C2×#38511yescharacteristic
538C22C4×#12411yescharacteristic
539C22#232811yescharacteristic
540C22#243111yescharacteristic
541C22#243511yescharacteristic
542C22#256311yescharacteristic
543C22#256611yescharacteristic
544C22C4×#13512yes—
545C22#294312yes—
546C22#300212yes—
547C22#300212yes—
548C22#307612yes—
549C22#307612yes—
550C22C2×#48811yescharacteristic
551C22#313111yescharacteristic
552C22#314011yescharacteristic
553C22#314211yescharacteristic
554C22#316812yes—
555C22#318312yes—
556C22#321112yes—
557C22#321112yes—
558C22#321212yes—
559C22#321212yes—
560C4—24no—
561C4—28no—
562C4—44no—
563C4—42no—
564C4—44no—
565C4—82no—
566C4—82no—
567C4—84no—
568C4—162no—
569C4#234912yes—
570C4#236312yes—
571C4#242012yes—
572C4#245412yes—
573C2—84no—
574C2#1397911yescharacteristic
575C2#2087212yes—
576C2C4×#48811yescharacteristic
577C2#2164811yescharacteristic
578C2#2227112yes—
579C2#2560112yes—
580C2#2585811yescharacteristic
581C2#2585911yescharacteristic
582C2#2914011yescharacteristic
583C2#2914211yescharacteristic
584C2#2919412yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1835C2semidirect
5. #14707C2semidirect
6. #11067C2semidirect
8. #10842C2semidirect
9. #10789C2semidirect
11. C4×#336C2semidirect
33. #1958C4semidirect
83. C2×#103C2×C4semidirect
95. C2×C4×C8C2×C4semidirect
107. #382C2×C4semidirect
178. #124C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1242
C2×#19582
C2×#18352
#147072
#110672
#110662
#108422
#107892
#107882
C4×#3362