Finite Group Databasebeta

← #401261

Subgroups — #401261

Order 512.

1582 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2066C212yesmaximal, complemented
2C2×#2055C212yesmaximal
3C2×#2013C211yesmaximal, characteristic
4C2×#1968C212yesmaximal
5C2×C4×C4⋊C8C211yesmaximal, characteristic, complemented
6#14714C211yesmaximal, characteristic, complemented
7#14700C211yesmaximal, characteristic
8#11323C211yesmaximal, characteristic, complemented
9#11049C211yesmaximal, characteristic
10#11010C211yesmaximal, characteristic, complemented
11#10926C212yesmaximal, complemented
12C23×SD16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×#106C2212yes—
15C22×C4⋊C8C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#102C2212yes—
20C22×C4×C8C2212yes—
21C2×#391C2212yes—
22C2×#391C2212yes—
23C2×#390C2211yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#390C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#388—22no—
28C2×#384C2211yescharacteristic
29C2×C43C2211yescharacteristic
30#2066—28no—
31#2055—28no—
32#2013—24no—
33#1968—28no—
34C4×C4⋊C8—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39#1437—22no—
40C4⋊#95—22no—
41C4⋊#95—22no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C23×Q8C2312yes—
45C23×D4C2311yescharacteristic
46C22×SD16—24no—
47C22×SD16—24no—
48C22×SD16—24no—
49C22×SD16—24no—
50C22×SD16—24no—
51C22×SD16—42no—
52C22×SD16—42no—
53C23×C8C2312yes—
54C23×C8C2312yes—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—42no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C22⋊C4—22no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—21no—
70C22×C42D412yes—
71C22×C42D412yes—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C2×#106—24no—
76C2×#106—24no—
77C2×#106—42no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8D418yes—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—42no—
89C2×#103—42no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—42no—
93C2×C4×C8—24no—
94C2×C4×C8—24no—
95C2×C4×C8—42no—
96#391—24no—
97#391—24no—
98#391—24no—
99#391—24no—
100#390—24no—
101#390—22no—
102#390—22no—
103#390—22no—
104#390D414yes—
105#389—24no—
106#388—48no—
107#384—42no—
108C2×#95—24no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C43—22no—
115C43—22no—
116#340—24no—
117#340—24no—
118#336—22no—
119#336D414yes—
120C25—21no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8#4114yes—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—22no—
143C22×D4—21no—
144C22×D4C2×D412yes—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2411yescharacteristic
155C2×SD16—216no—
156C2×SD16—48no—
157C2×SD16—48no—
158C2×SD16—48no—
159C2×SD16—48no—
160C2×SD16—48no—
161C2×SD16—48no—
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×C8C2×D414yes—
173C22×C8C2×D414yes—
174C22×C8C2×D414yes—
175C22×C8#4114yes—
176C4⋊Q8—48no—
177C4⋊Q8—48no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4C2×D412yes—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C42—24no—
215C2×C42—24no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—42no—
228C2×C42SD1618yescomplemented
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D414yes—
232C2×C42#4112yes—
233C2×C42#4112yes—
234#106—48no—
235C4⋊C8—216no—
236#103—48no—
237#103—48no—
238#103—48no—
239#102—48no—
240C4×C8—48no—
241#95—48no—
242#95—42no—
243#95—44no—
244#95—44no—
245#95—44no—
246#95—44no—
247C24—22no—
248C24—41no—
249C24—41no—
250C24—41no—
251C24—41no—
252C24—41no—
253C24—41no—
254C24—42no—
255C24—42no—
256C24—42no—
257C24—42no—
258C24C22×D411yescharacteristic
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
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×D4—24no—
288C2×D4—24no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—41no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—41no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—44no—
343C22×C4—42no—
344C22×C4—44no—
345C22×C4#12114yes—
346C22×C4#12114yes—
347C22×C4#12712yes—
348C22×C4#12712yes—
349C22×C4C2×SD1614yes—
350C22×C4C2×SD1614yes—
351C22×C4C2×SD1614yes—
352C22×C4#13514yes—
353C22×C4#13714yes—
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D412yes—
356C22×C4C22×D411yescharacteristic
357C22×C4C2×#4112yes—
358C22×C4C2×#4112yes—
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4H(5,𝔽2)11yescharacteristic
362C22×C4H(5,𝔽2)12yes—
363C22×C42−1+411yescharacteristic
364SD16—432no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
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#12118yes—
379C4⋊C4—24no—
380C4⋊C4—416no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—88no—
391C4⋊C4—84no—
392C22⋊C4—416no—
393C22⋊C4—84no—
394C22⋊C4—88no—
395C22⋊C4—84no—
396C42—24no—
397C42—24no—
398C42—22no—
399C42—24no—
400C42—22no—
401C42—22no—
402C42—24no—
403C42—22no—
404C42—216no—
405C42—48no—
406C23—42no—
407C23—42no—
408C23—44no—
409C23—44no—
410C23—42no—
411C23—82no—
412C23—82no—
413C23—81no—
414C23—81no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23—82no—
419C23—82no—
420C23—81no—
421C23—81no—
422C23—82no—
423C23—82no—
424C23—82no—
425C23—82no—
426C23—82no—
427C23—81no—
428C23—82no—
429C23C2×#12111yescharacteristic
430C23C2×#12111yescharacteristic
431C23#52511yescharacteristic
432C23C2×#12711yescharacteristic
433C23#53211yescharacteristic
434C23#53511yescharacteristic
435C23#53711yescharacteristic
436C23C22×SD1612yes—
437C23C2×#13512yes—
438C23C2×#13712yes—
439C23#57512yes—
440Q8—44no—
441Q8—44no—
442Q8—44no—
443Q8—44no—
444Q8—82no—
445Q8—82no—
446Q8—82no—
447Q8—82no—
448D4—44no—
449D4—44no—
450D4—82no—
451D4—82no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—82no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—82no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—84no—
495C2×C4—82no—
496C2×C4—84no—
497C2×C4—82no—
498C2×C4—81no—
499C2×C4—82no—
500C2×C4#46118yes—
501C2×C4#46518yes—
502C2×C4#49214yes—
503C2×C4C2×#12112yes—
504C2×C4#52512yes—
505C2×C4#52912yes—
506C2×C4#53212yes—
507C2×C4#53812yes—
508C2×C4#54212yes—
509C2×C4D4211yescharacteristic
510C2×C4D4212yes—
511C2×C4#54612yes—
512C2×C4#54612yes—
513C2×C4#54711yescharacteristic
514C2×C4#54811yescharacteristic
515C2×C4#54812yes—
516C2×C4#55012yes—
517C2×C4#55112yes—
518C2×C4#55311yescharacteristic
519C8—28no—
520C8—44no—
521C22—82no—
522C22—82no—
523C22—84no—
524C22—84no—
525C22—82no—
526C22—162no—
527C22—161no—
528C22—161no—
529C22—162no—
530C22—161no—
531C22—161no—
532C22#239211yescharacteristic
533C22#241111yescharacteristic
534C22#241611yescharacteristic
535C22#254011yescharacteristic
536C22#254111yescharacteristic
537C22#266211yescharacteristic
538C22#285611yescharacteristic
539C22C2×#46112yes—
540C22#303512yes—
541C22C2×#46512yes—
542C22#305612yes—
543C22#310312yes—
544C22#310812yes—
545C22C2×#49211yescharacteristic
546C22#314711yescharacteristic
547C22#315211yescharacteristic
548C22#315311yescharacteristic
549C22#316712yes—
550C22#317212yes—
551C22#320612yes—
552C22#321112yes—
553C22#324612yes—
554C22#324812yes—
555C4—24no—
556C4—28no—
557C4—28no—
558C4—42no—
559C4—82no—
560C4—84no—
561C4—162no—
562C4—161no—
563C4—161no—
564C4#242412yes—
565C4#242612yes—
566C4#246112yes—
567C4#255412yes—
568C2—162no—
569C2#1481511yescharacteristic
570C2#2367712yes—
571C2#2450312yes—
572C2#2475112yes—
573C2#2837411yescharacteristic
574C2#2837711yescharacteristic
575C2#2843811yescharacteristic
576C2#2844211yescharacteristic
577C2#3033912yes—
578C2#3317011yescharacteristic
579C2#3318611yescharacteristic
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
1. C2×#2066C2semidirect
5. C2×C4×C4⋊C8C2semidirect
6. #14714C2semidirect
8. #11323C2semidirect
10. #11010C2semidirect
11. #10926C2semidirect
228. C2×C42SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#20662
C2×#20552
C2×#20132
C2×#19682
C2×C4×C4⋊C82
#147142
#147002
#113232
#110492
#110102
#109262