Finite Group Databasebeta

← #471770

Subgroups — #471770

Order 512.

1542 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2076C212yesmaximal, complemented
2C2×#2034C212yesmaximal
3#14698C211yesmaximal, characteristic
4#14686C211yesmaximal, characteristic, complemented
5#11378C212yesmaximal, complemented
6#11324C211yesmaximal, characteristic, complemented
7#11323C211yesmaximal, characteristic, complemented
8#11123C212yesmaximal, complemented
9#11049C211yesmaximal, characteristic, complemented
10#11049C211yesmaximal, characteristic, complemented
11#10712C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×C22⋊C8C2212yes—
19C2×#390C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#386—22no—
22C2×#384C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#380—22no—
25C2×#380C2211yescharacteristic
26C2×#380C2212yes—
27#2076—28no—
28#2034—28no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36#1437—22no—
37C4⋊#95—22no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C23×D4—21no—
42C23×D4C2311yescharacteristic
43C24×C4D412yes—
44C24×C4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×#121—22no—
48C2×#121—22no—
49C2×#121—24no—
50C2×#121—42no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C22⋊C4—21no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C42—22no—
64C22×C42—22no—
65C22×C42—42no—
66C22×C42—42no—
67C22×C42D412yes—
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×#107—24no—
71C2×#107—42no—
72C2×#107—42no—
73C2×#106—24no—
74C2×#106—24no—
75C2×#106—42no—
76C2×#102—24no—
77C2×#102—24no—
78C2×#102—24no—
79C2×#102—42no—
80C2×#102—42no—
81C2×#102—42no—
82C2×C22⋊C8—42no—
83C2×C22⋊C8—42no—
84C2×C22⋊C8D418yes—
85#390—22no—
86#390—22no—
87#390—42no—
88#386—48no—
89#384—42no—
90#384—42no—
91#380—24no—
92#380—24no—
93#380—24no—
94#380—48no—
95C2×#95—22no—
96C2×#95—24no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—24no—
104#336—22no—
105#336—22no—
106#336—42no—
107C25—21no—
108C25C2×D411yescharacteristic
109C22×D4—24no—
110C22×D4—22no—
111C22×D4—21no—
112C22×D4—21no—
113C22×D4—21no—
114C22×D4—22no—
115C22×D4—22no—
116C22×D4—21no—
117C22×D4—21no—
118C22×D4—22no—
119C22×D4—42no—
120C22×D4—44no—
121C22×D4—42no—
122C22×D4—44no—
123C22×D4#4112yes—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D414yes—
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2411yescharacteristic
150C22×C8—22no—
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×C8C2×D414yes—
163C22×C8C2×D414yes—
164#121—48no—
165#121—48no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
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—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4#4112yes—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—48no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—24no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—48no—
217C2×C42—48no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42#4112yes—
221C2×C42#4112yes—
222#107—48no—
223#106—48no—
224#102—48no—
225#102—48no—
226C22⋊C8—48no—
227#95—44no—
228#95—44no—
229#95—42no—
230#95—48no—
231#95—44no—
232C24—21no—
233C24—21no—
234C24—21no—
235C24—21no—
236C24—21no—
237C24—21no—
238C24—22no—
239C24—22no—
240C24—22no—
241C24—22no—
242C24—41no—
243C24—41no—
244C24—41no—
245C24—41no—
246C24—41no—
247C24—41no—
248C24—42no—
249C24—42no—
250C24—42no—
251C24—42no—
252C24#12712yes—
253C24#13714yes—
254C24C22×D411yescharacteristic
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—48no—
260C2×D4—416no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—84no—
274C2×D4—88no—
275C2×D4—84no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—216no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—48no—
301C22×C4—48no—
302C22×C4—48no—
303C22×C4—48no—
304C22×C4—48no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—48no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4#12314yes—
341C22×C4#12712yes—
342C22×C4#13514yes—
343C22×C4AGL(1,ℤ/8)14yes—
344C22×C4AGL(1,ℤ/8)14yes—
345C22×C4C22×D412yes—
346C22×C4C2×#4111yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4H(5,𝔽2)11yescharacteristic
351C22×C4H(5,𝔽2)12yes—
352C22×C4H(5,𝔽2)12yes—
353C22×C42−1+411yescharacteristic
354C22×C42−1+412yes—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C4⋊C4—24no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—88no—
384C4⋊C4—88no—
385C22⋊C4—48no—
386C22⋊C4—88no—
387C22⋊C4—88no—
388C22⋊C4—84no—
389C22⋊C4—88no—
390C42—24no—
391C42—24no—
392C42—48no—
393C42—42no—
394C42—42no—
395C23—24no—
396C23—24no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—24no—
402C23—24no—
403C23—24no—
404C23—24no—
405C23—22no—
406C23—22no—
407C23—42no—
408C23—42no—
409C23—41no—
410C23—41no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—41no—
415C23—42no—
416C23—41no—
417C23—44no—
418C23—44no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—82no—
423C23—82no—
424C23—81no—
425C23—81no—
426C23—81no—
427C23—81no—
428C23—81no—
429C23—81no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23—82no—
434C23—82no—
435C23—82no—
436C23—82no—
437C23—82no—
438C23—81no—
439C23—81no—
440C23#49714yes—
441C23C2×#12311yescharacteristic
442C23#52511yescharacteristic
443C23C2×#12711yescharacteristic
444C23#53411yescharacteristic
445C23#53411yescharacteristic
446C23#53611yescharacteristic
447C23#53711yescharacteristic
448C23C2×#13512yes—
449C23C2×AGL(1,ℤ/8)12yes—
450C23C2×AGL(1,ℤ/8)12yes—
451C23C2×#13712yes—
452D4—44no—
453D4—44no—
454D4—816no—
455D4—82no—
456D4—82no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—432no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—84no—
489C2×C4—81no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—81no—
497C2×C4—82no—
498C2×C4—81no—
499C2×C4—84no—
500C2×C4—82no—
501C2×C4—84no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—84no—
505C2×C4—81no—
506C2×C4—82no—
507C2×C4#48218yes—
508C2×C4#53412yes—
509C2×C4#53712yes—
510C2×C4#53812yes—
511C2×C4#53812yes—
512C2×C4#53912yes—
513C2×C4#54212yes—
514C2×C4#54612yes—
515C2×C4#54612yes—
516C2×C4#54712yes—
517C2×C4#54712yes—
518C2×C4#55211yescharacteristic
519C2×C4#55311yescharacteristic
520C2×C4#55511yescharacteristic
521C2×C4#55611yescharacteristic
522C2×C4#56212yes—
523C2×C4#56312yes—
524C8—44no—
525C8—44no—
526C22—24no—
527C22—24no—
528C22—24no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—44no—
534C22—44no—
535C22—44no—
536C22—44no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—84no—
541C22—84no—
542C22—161no—
543C22—161no—
544C22—161no—
545C22—161no—
546C22—162no—
547C22—162no—
548C22#248511yescharacteristic
549C22#250111yescharacteristic
550C22#253911yescharacteristic
551C22#262211yescharacteristic
552C22#263811yescharacteristic
553C22#265611yescharacteristic
554C22#281411yescharacteristic
555C22C2×#48212yes—
556C22#309212yes—
557C22#310612yes—
558C22#311012yes—
559C22#314711yescharacteristic
560C22C2×#49711yescharacteristic
561C22#315111yescharacteristic
562C22#315111yescharacteristic
563C22#318812yes—
564C22#319012yes—
565C22#319212yes—
566C22#319312yes—
567C22#322812yes—
568C22#323912yes—
569C22#324812yes—
570C22#325312yes—
571C4—28no—
572C4—42no—
573C4—44no—
574C4—82no—
575C4—161no—
576C4—161no—
577C4—162no—
578C4—162no—
579C4#268612yes—
580C4#269512yes—
581C4#270812yes—
582C4#271512yes—
583C2—44no—
584C2—162no—
585C2#1680711yescharacteristic
586C2#2762612yes—
587C2#2790711yescharacteristic
588C2#2791311yescharacteristic
589C2#2830711yescharacteristic
590C2#2831111yescharacteristic
591C2#2989112yes—
592C2#3017212yes—
593C2#3024411yescharacteristic
594C2#3024711yescharacteristic
595C2#3273912yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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
1. C2×#2076C2semidirect
4. #14686C2semidirect
5. #11378C2semidirect
6. #11324C2semidirect
7. #11323C2semidirect
8. #11123C2semidirect
9. #11049C2semidirect
10. #11049C2semidirect
11. #10712C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20762
C2×#20342
#146982
#146862
#113782
#113242
#113232
#111232
#110492
#107122