Finite Group Databasebeta

← #416514

Subgroups — #416514

Order 512.

1414 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2861C211yesmaximal, characteristic
2C2×#1991C211yesmaximal, characteristic, complemented
3C2×#1991C211yesmaximal, characteristic, complemented
4C2×#1958C211yesmaximal, characteristic
5C2×#1958C211yesmaximal, characteristic
6#14707C211yesmaximal, characteristic, complemented
7#11190C212yesmaximal, complemented
8#11189C212yesmaximal
9#11067C212yesmaximal, complemented
10#11066C212yesmaximal
11C4×#336C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×#124C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×C4×C8C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20#2861—24no—
21#2861—24no—
22C2×#391C2211yescharacteristic
23C2×#391C2211yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#390C2211yescharacteristic
26C2×#389C2212yes—
27C2×#388—22no—
28C2×#388—22no—
29C2×#382C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×C43C2211yescharacteristic
32#1991—24no—
33#1991—24no—
34#1958—24no—
35#1958—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1436—22no—
43#1436—22no—
44C4⋊#95C2212yes—
45C23×Q8C2311yescharacteristic
46C23×D4C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×#125—22no—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125—22no—
53C2×#125—22no—
54C2×#125D414yes—
55C2×#124—22no—
56C2×#124—22no—
57C2×#124—22no—
58C2×#124—22no—
59C2×#124—22no—
60C2×#124D414yes—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C42D412yes—
73C22×C42D412yes—
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—24no—
87C2×#102—42no—
88C2×#102—42no—
89C2×C4×C8—22no—
90C2×C4×C8—22no—
91C2×C4×C8—22no—
92C2×C4×C8—22no—
93C2×C4×C8—22no—
94C2×C4×C8—22no—
95#391—22no—
96#391—22no—
97#391—22no—
98#391—22no—
99#390—22no—
100#390—22no—
101#390—22no—
102#390—22no—
103#389—42no—
104#389—42no—
105#388—216no—
106#388—216no—
107#382—22no—
108#382—22no—
109#382—22no—
110#382D414yes—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—22no—
120C43—22no—
121C43—22no—
122#340—24no—
123#340—24no—
124#340—24no—
125#340—24no—
126#336—24no—
127#336—24no—
128C25—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8C2×D412yes—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4C2×D412yes—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2411yescharacteristic
159C22×C8—22no—
160C22×C8—22no—
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#4114yes—
172C22×C8#4114yes—
173#125—28no—
174#125—28no—
175#125—44no—
176#125—44no—
177#124—28no—
178#124—28no—
179#124—44no—
180#124—44no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D414yes—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42#4112yes—
244C2×C42#4112yes—
245C2×C42#4112yes—
246#103—48no—
247#103—48no—
248#102—48no—
249#102—48no—
250C4×C8—44no—
251C4×C8—44no—
252#95—28no—
253#95—28no—
254#95—24no—
255#95—24no—
256#95—44no—
257#95—44no—
258#95—42no—
259#95—42no—
260#95—44no—
261C24—22no—
262C24—41no—
263C24—41no—
264C24—41no—
265C24—41no—
266C24—41no—
267C24—41no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24—42no—
272C24C22×D411yescharacteristic
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
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—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4#12012yes—
353C22×C4#12012yes—
354C22×C4#12112yes—
355C22×C4#12112yes—
356C22×C4#12412yes—
357C22×C4#12412yes—
358C22×C4#13514yes—
359C22×C4#13514yes—
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4112yes—
367C22×C4C2×#4112yes—
368C22×C4H(5,𝔽2)12yes—
369C22×C42−1+412yes—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—48no—
397C4⋊C4—48no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—88no—
403C22⋊C4—416no—
404C22⋊C4—416no—
405C22⋊C4—48no—
406C22⋊C4—48no—
407C42—48no—
408C42—48no—
409C42—42no—
410C42—42no—
411C42—42no—
412C42—42no—
413C42#12114yes—
414C42#12114yes—
415C42#12314yes—
416C42#12314yes—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—44no—
421C23—44no—
422C23—81no—
423C23—81no—
424C23—81no—
425C23—81no—
426C23—81no—
427C23—81no—
428C23—82no—
429C23—82no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23—82no—
434C23—82no—
435C23—82no—
436C23—82no—
437C23—82no—
438C23—81no—
439C23—81no—
440C23C2×#12011yescharacteristic
441C23C2×#12111yescharacteristic
442C23C2×#12411yescharacteristic
443C23#53211yescharacteristic
444C23#53211yescharacteristic
445C23#53511yescharacteristic
446C23#53611yescharacteristic
447C23C2×#13512yes—
448C23C2×#13512yes—
449C23#57512yes—
450C23#57512yes—
451Q8—44no—
452Q8—44no—
453Q8—82no—
454Q8—82no—
455D4—44no—
456D4—44no—
457D4—82no—
458D4—82no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
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—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4#39414yes—
513C2×C4#39414yes—
514C2×C4C2×#12111yescharacteristic
515C2×C4C2×#12111yescharacteristic
516C2×C4C2×#12311yescharacteristic
517C2×C4C2×#12311yescharacteristic
518C2×C4#52912yes—
519C2×C4#52912yes—
520C2×C4#54212yes—
521C2×C4#54212yes—
522C2×C4#54612yes—
523C2×C4#54612yes—
524C2×C4#54712yes—
525C2×C4#54812yes—
526C2×C4#54812yes—
527C2×C4#55012yes—
528C2×C4#55612yes—
529C2×C4#55912yes—
530C8—44no—
531C8—44no—
532C22—84no—
533C22—84no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22—161no—
538C22—161no—
539C22—162no—
540C22—162no—
541C22—161no—
542C22—161no—
543C22C2×#39411yescharacteristic
544C22#243111yescharacteristic
545C22#243211yescharacteristic
546C22#254411yescharacteristic
547C22#254511yescharacteristic
548C22#264011yescharacteristic
549C22#286111yescharacteristic
550C22#300212yes—
551C22#300212yes—
552C22#303512yes—
553C22#303512yes—
554C22#313111yescharacteristic
555C22#313111yescharacteristic
556C22#313511yescharacteristic
557C22#313511yescharacteristic
558C22#316812yes—
559C22#316812yes—
560C22#318312yes—
561C22#318312yes—
562C22#321112yes—
563C22#321212yes—
564C22#323212yes—
565C22#323312yes—
566C4—24no—
567C4—24no—
568C4—44no—
569C4—44no—
570C4—82no—
571C4—84no—
572C4—82no—
573C4—82no—
574C4—162no—
575C4#242612yes—
576C4#242712yes—
577C4#244712yes—
578C4#244812yes—
579C2—162no—
580C2#1522611yescharacteristic
581C2#2350411yescharacteristic
582C2#2350411yescharacteristic
583C2#2560112yes—
584C2#2567512yes—
585C2#2861712yes—
586C2#2865712yes—
587C2#3020711yescharacteristic
588C2#3020811yescharacteristic
589C2#3327911yescharacteristic
590C2#3328811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×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
2. C2×#1991C2semidirect
3. C2×#1991C2semidirect
6. #14707C2semidirect
7. #11190C2semidirect
9. #11067C2semidirect
11. C4×#336C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28612
C2×#19912
C2×#19582
#147072
#111902
#111892
#110672
#110662
C4×#3362