Finite Group Databasebeta

← #444694

Subgroups — #444694

Order 512.

1370 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2492C211yesmaximal, characteristic
2C2×#2072C211yesmaximal, characteristic, complemented
3C2×#1990C211yesmaximal, characteristic, complemented
4#15135C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic, complemented
6#11199C212yesmaximal
7#11067C212yesmaximal, complemented
8#10970C212yesmaximal
9#10926C211yesmaximal, characteristic, complemented
10#10921C211yesmaximal, characteristic
11#10884C212yesmaximal, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C4×C8C2211yescharacteristic
18#2492—24no—
19#2492—24no—
20C2×#400—21no—
21C2×#399—21no—
22C2×#393—21no—
23C2×#391—21no—
24C2×#391C2211yescharacteristic
25C2×#391C2211yescharacteristic
26C2×#389C2212yes—
27C2×#388—21no—
28C2×#384C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×#380C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33#2072—24no—
34#1990—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43#1445—21no—
44#1444—21no—
45#1439—21no—
46C4⋊#95C2211yescharacteristic
47C23×Q8—21no—
48C23×Q8C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—42no—
55C2×C4⋊Q8D414yes—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—24no—
59C2×C22⋊Q8—42no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4D411yescharacteristic
74C22×C22⋊C4D411yescharacteristic
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103—42no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—42no—
86C2×C22⋊C8—42no—
87C2×C4×C8—22no—
88C2×C4×C8—42no—
89#400—28no—
90#399—44no—
91#393—28no—
92#391—28no—
93#391—22no—
94#391—22no—
95#391—22no—
96#391D414yes—
97#389—24no—
98#389—42no—
99#388—28no—
100#384—22no—
101#384—22no—
102#380—22no—
103#380—22no—
104#380—24no—
105#380—42no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95D411yescharacteristic
121C2×#95D411yescharacteristic
122#340—24no—
123#340—24no—
124#340—24no—
125#340—24no—
126#340—24no—
127#340—42no—
128#336—22no—
129#336—42no—
130C25C2×D411yescharacteristic
131C22×Q8—24no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—44no—
140C22×Q8—42no—
141C22×Q8—42no—
142C22×Q8—82no—
143C22×Q8C2×D412yes—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2411yescharacteristic
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×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8#4114yes—
178C4⋊Q8—28no—
179C4⋊Q8—44no—
180C4⋊Q8—48no—
181C22⋊Q8—48no—
182C22⋊Q8—44no—
183C22⋊Q8—44no—
184C2×C4⋊C4—21no—
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—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
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—22no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4#4114yes—
219C2×C4⋊C4#4114yes—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—82no—
231C2×C22⋊C4—82no—
232C2×C22⋊C4SD1614yes—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—24no—
240C2×C42—82no—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243#103—48no—
244#103—48no—
245C22⋊C8—48no—
246C4×C8—44no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—22no—
254#95—22no—
255#95—44no—
256#95—44no—
257#95—42no—
258#95—42no—
259C24—21no—
260C24—21no—
261C24—21no—
262C24—22no—
263C24—22no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—22no—
268C24—22no—
269C24C2×SD1612yes—
270C24C22×D411yescharacteristic
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—48no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—48no—
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×Q8—84no—
290C2×Q8—88no—
291C2×Q8—88no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
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—41no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
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—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4#12012yes—
361C22×C4#12012yes—
362C22×C4C2×SD1612yes—
363C22×C4C2×SD1612yes—
364C22×C4#13512yes—
365C22×C4#13712yes—
366C22×C4#13712yes—
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C2×#4112yes—
370C22×C4C2×#4112yes—
371C22×C4C2×#4112yes—
372C22×C4H(5,𝔽2)12yes—
373C22×C4H(5,𝔽2)12yes—
374C22×C42−1+412yes—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—22no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—48no—
397C4⋊C4—48no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—88no—
407C4⋊C4—84no—
408C4⋊C4—84no—
409C4⋊C4#12314yes—
410C22⋊C4—28no—
411C22⋊C4—48no—
412C22⋊C4—44no—
413C22⋊C4—84no—
414C22⋊C4—88no—
415C42—22no—
416C42—22no—
417C42—24no—
418C42—41no—
419C42—41no—
420C42—42no—
421C42—84no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—24no—
426C23—24no—
427C23—42no—
428C23—42no—
429C23—41no—
430C23—42no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—42no—
435C23—41no—
436C23—42no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23C2×#12011yescharacteristic
446C23#53211yescharacteristic
447C23#53611yescharacteristic
448C23#54612yes—
449C23#54612yes—
450C23C22×SD1611yescharacteristic
451C23C2×#13511yescharacteristic
452C23C2×#13711yescharacteristic
453C23C2×#13711yescharacteristic
454C23#57612yes—
455C23#57812yes—
456Q8—44no—
457Q8—44no—
458Q8—816no—
459Q8—82no—
460Q8—82no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—82no—
502C2×C4—82no—
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—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4#44814yes—
523C2×C4#45214yes—
524C2×C4C2×#12311yescharacteristic
525C2×C4#52912yes—
526C2×C4#53211yescharacteristic
527C2×C4#53411yescharacteristic
528C2×C4#53511yescharacteristic
529C2×C4#53812yes—
530C2×C4#54212yes—
531C2×C4#54212yes—
532C2×C4#54612yes—
533C2×C4#54712yes—
534C2×C4#54812yes—
535C2×C4#55012yes—
536C2×C4#55112yes—
537C2×C4#55612yes—
538C2×C4#55912yes—
539C2×C4#56212yes—
540C8—44no—
541C8—82no—
542C22—44no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—44no—
547C22—82no—
548C22—82no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22#240511yescharacteristic
554C22#243111yescharacteristic
555C22#243211yescharacteristic
556C22#255712yes—
557C22#270012yes—
558C22C2×#44811yescharacteristic
559C22C2×#45211yescharacteristic
560C22#300811yescharacteristic
561C22#301111yescharacteristic
562C22#316711yescharacteristic
563C22#317512yes—
564C22#318511yescharacteristic
565C22#322911yescharacteristic
566C22#323111yescharacteristic
567C22#324312yes—
568C22#329512yes—
569C22#329712yes—
570C22#329812yes—
571C22#330212yes—
572C22#331212yes—
573C22#331412yes—
574C22#332812yes—
575C22#333312yes—
576C4—24no—
577C4—24no—
578C4—42no—
579C4—44no—
580C4—84no—
581C4—82no—
582C4—81no—
583C4—82no—
584C4—81no—
585C4—82no—
586C4—162no—
587C4#244812yes—
588C4#252112yes—
589C4#271012yes—
590C4#273012yes—
591C2—82no—
592C2#1578511yescharacteristic
593C2#2403911yescharacteristic
594C2#2404111yescharacteristic
595C2#2556911yescharacteristic
596C2#2557011yescharacteristic
597C2#2567211yescharacteristic
598C2#2568111yescharacteristic
599C2#2897912yes—
600C2#2900112yes—
601C2#3174212yes—
602C2#3174512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2072C2semidirect
3. C2×#1990C2semidirect
5. #11384C2semidirect
7. #11067C2semidirect
9. #10926C2semidirect
11. #10884C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24922
C2×#20722
C2×#19902
#151352
#113842
#111992
#110672
#109702
#109262
#109212
#108842