Finite Group Databasebeta

← #411956

Subgroups — #411956

Order 512.

1504 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic
2C2×#2024C212yesmaximal
3C2×#1803C211yesmaximal, characteristic
4C2×#1763C211yesmaximal, characteristic
5#13942C211yesmaximal, characteristic
6#11384C212yesmaximal
7#11292C212yesmaximal
8#11067C212yesmaximal
9#10850C211yesmaximal, characteristic
10#10843C211yesmaximal, characteristic
11#10835C211yesmaximal, characteristic
12C22×C4⋊Q8—21no—
13C22×C4⋊Q8C2212yes—
14C22×C4×Q8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2211yescharacteristic
18C22×#103C2211yescharacteristic
19C22×#97C2211yescharacteristic
20#2551—24no—
21#2551—24no—
22C2×#395—22no—
23C2×#391C2212yes—
24C2×#389—21no—
25C2×#389C2212yes—
26C2×#384C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#2024—28no—
31#1803—24no—
32#1763—24no—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1441—21no—
43#1441—21no—
44C4⋊#95C2212yes—
45#1434—22no—
46C23×Q8—22no—
47C23×Q8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×C4⋊Q8—28no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—42no—
57C2×C4⋊Q8—42no—
58C2×C4⋊Q8—48no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8Q814yescomplemented
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C42—21no—
78C22×C42—21no—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×#103—22no—
85C2×#103—22no—
86C2×#103—22no—
87C2×#103—22no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—42no—
91C2×#103D414yes—
92C2×#103Q814yescomplemented
93C2×#97—22no—
94C2×#97—22no—
95C2×#97—22no—
96#395—216no—
97#391—24no—
98#391—42no—
99#389—28no—
100#389—24no—
101#389—24no—
102#384—22no—
103#384D414yes—
104#382—22no—
105#382—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4D414yes—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95D412yes—
119#340—24no—
120#340—24no—
121#340—22no—
122#340—22no—
123#340—24no—
124#340—24no—
125#340—42no—
126#336—24no—
127#336—42no—
128C22×Q8—28no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—44no—
141C22×Q8—44no—
142C22×Q8—44no—
143C22×Q8—44no—
144C22×Q8—82no—
145C22×Q8—82no—
146C22×Q8C2×D412yes—
147C22×Q8C2×Q812yes—
148C22×Q8C2×Q812yes—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2411yescharacteristic
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8C2×D412yes—
180C22×C8C2×Q812yes—
181C4⋊Q8—48no—
182C4⋊Q8—48no—
183C4⋊Q8—432no—
184C4⋊Q8—48no—
185C4⋊Q8—48no—
186C4×Q8—28no—
187C4×Q8—28no—
188C4×Q8—44no—
189C4×Q8—44no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—82no—
229C2×C4⋊C4C2×D414yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×Q812yes—
234C2×C4⋊C4C2×Q812yes—
235C2×C4⋊C4#4112yes—
236C2×C4⋊C4#4112yes—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—44no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42C2×Q812yes—
251C2×C42#4112yes—
252C2×C42#4112yes—
253C2×C42#4112yes—
254C4⋊C8—44no—
255#103—28no—
256#103—28no—
257#103—48no—
258#97—44no—
259#95—28no—
260#95—24no—
261#95—24no—
262#95—44no—
263#95—44no—
264#95—42no—
265#95—42no—
266#95—44no—
267#95—44no—
268C24C22×D411yescharacteristic
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—416no—
285C2×Q8—416no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—88no—
299C2×Q8—88no—
300C2×Q8—88no—
301C2×Q8—88no—
302C2×Q8—88no—
303C2×Q8C4⋊Q814yescomplemented
304C22×C4—22no—
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—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
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—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4#12012yes—
357C22×C4#12012yes—
358C22×C4C22⋊Q812yes—
359C22×C4C22⋊Q812yes—
360C22×C4#12412yes—
361C22×C4#12412yes—
362C22×C4AGL(1,ℤ/8)14yes—
363C22×C4#13714yes—
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×Q811yescharacteristic
368C22×C4C22×Q811yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4H(5,𝔽2)11yescharacteristic
375C22×C4H(5,𝔽2)11yescharacteristic
376C22×C42−1+411yescharacteristic
377C22×C42−1+411yescharacteristic
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C4⋊C4—28no—
400C4⋊C4—28no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—48no—
412C4⋊C4—416no—
413C4⋊C4—416no—
414C4⋊C4—42no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—42no—
418C4⋊C4—48no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—84no—
424C4⋊C4#12114yes—
425C4⋊C4C22⋊Q814yes—
426C42—48no—
427C42—42no—
428C42—42no—
429C42—42no—
430C42—42no—
431C42—48no—
432C42C22⋊Q814yes—
433C42C22⋊Q814yes—
434C42#12414yes—
435C42#12414yes—
436C23C2×#12011yescharacteristic
437C23C2×C22⋊Q811yescharacteristic
438C23C2×#12411yescharacteristic
439C23D4211yescharacteristic
440C23#54611yescharacteristic
441C23#54811yescharacteristic
442C23D4×Q811yescharacteristic
443C23C2×AGL(1,ℤ/8)12yes—
444C23C2×#13712yes—
445C23#57712yes—
446C23#57812yes—
447Q8—24no—
448Q8—24no—
449Q8—42no—
450Q8—42no—
451Q8—42no—
452Q8—42no—
453Q8—42no—
454Q8—42no—
455Q8—832no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
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—44no—
474C2×C4—44no—
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—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—82no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4#39314yes—
515C2×C4#39314yes—
516C2×C4C2×#12111yescharacteristic
517C2×C4C2×C22⋊Q811yescharacteristic
518C2×C4C2×C22⋊Q811yescharacteristic
519C2×C4C2×C22⋊Q811yescharacteristic
520C2×C4C2×#12411yescharacteristic
521C2×C4C2×#12411yescharacteristic
522C2×C4C2×C4⋊Q811yescharacteristic
523C2×C4#53211yescharacteristic
524C2×C4#53311yescharacteristic
525C2×C4#53311yescharacteristic
526C2×C4#53511yescharacteristic
527C2×C4#53611yescharacteristic
528C2×C4#54211yescharacteristic
529C2×C4#54211yescharacteristic
530C2×C4#54411yescharacteristic
531C2×C4#54411yescharacteristic
532C2×C4D4211yescharacteristic
533C2×C4#54611yescharacteristic
534C2×C4#54711yescharacteristic
535C2×C4#54711yescharacteristic
536C2×C4#54811yescharacteristic
537C2×C4D4×Q811yescharacteristic
538C2×C4#55411yescharacteristic
539C2×C4#55411yescharacteristic
540C2×C4#55511yescharacteristic
541C2×C4#55511yescharacteristic
542C2×C4#55611yescharacteristic
543C2×C4#55611yescharacteristic
544C8—42no—
545C8—42no—
546C8—82no—
547C22C2×#39311yescharacteristic
548C22#240811yescharacteristic
549C22#240911yescharacteristic
550C22#243911yescharacteristic
551C22#244011yescharacteristic
552C22#249111yescharacteristic
553C22#249211yescharacteristic
554C22#300311yescharacteristic
555C22#300411yescharacteristic
556C22#300911yescharacteristic
557C22#301111yescharacteristic
558C22#307912yes—
559C22#308212yes—
560C22#313712yes—
561C22#314112yes—
562C22#326512yes—
563C22#327012yes—
564C22#331012yes—
565C22#331212yes—
566C22#336712yes—
567C22#336912yes—
568C22#338612yes—
569C22#338712yes—
570C4—24no—
571C4—24no—
572C4—44no—
573C4—44no—
574C4—44no—
575C4—44no—
576C4—44no—
577C4—82no—
578C4—84no—
579C4—162no—
580C4#243911yescharacteristic
581C4#244011yescharacteristic
582C4#249111yescharacteristic
583C4#249211yescharacteristic
584C4#254711yescharacteristic
585C4#256311yescharacteristic
586C4#262811yescharacteristic
587C4#285711yescharacteristic
588C2#1512511yescharacteristic
589C2#2345211yescharacteristic
590C2#2345711yescharacteristic
591C2#2425211yescharacteristic
592C2#2425511yescharacteristic
593C2#2434211yescharacteristic
594C2#2434511yescharacteristic
595C2#2603812yes—
596C2#2605612yes—
597C2#2778512yes—
598C2#2782712yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
64. C2×C4×Q8Q8semidirect
92. C2×#103Q8semidirect
303. C2×Q8C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
C2×#20242
C2×#18032
C2×#17632
#139422
#113842
#112922
#110672
#108502
#108432
#108352