Finite Group Databasebeta

← #460192

Subgroups — #460192

Order 512.

1246 conjugacy classes of subgroups.

The subgroup lattice has 609 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2625C211yesmaximal, characteristic
2C2×#2084C211yesmaximal, characteristic, complemented
3C2×#2072C211yesmaximal, characteristic, complemented
4#14340C211yesmaximal, characteristic
5#11293C212yesmaximal, complemented
6#11208C211yesmaximal, characteristic, complemented
7#11122C212yesmaximal, complemented
8#11095C212yesmaximal
9#11052C211yesmaximal, characteristic, complemented
10#10970C212yesmaximal
11#10713C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#106C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×C22⋊C8C2212yes—
18#2625—24no—
19#2625—24no—
20C2×#400—21no—
21C2×#398—21no—
22C2×#397—21no—
23C2×#397—21no—
24C2×#391C2211yescharacteristic
25C2×#391C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#380C2211yescharacteristic
31C2×#380C2212yes—
32#2084—24no—
33#2072—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42#1441—21no—
43#1441—21no—
44#1439—21no—
45#1439—21no—
46C4⋊#95C2212yes—
47C23×Q8C2311yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—42no—
55C2×#117—24no—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—42no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4D411yescharacteristic
75C22×C22⋊C4D411yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×#106—24no—
79C2×#106—42no—
80C2×#106—42no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#103—42no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—42no—
90#400—28no—
91#398—28no—
92#397—28no—
93#397—28no—
94#391—22no—
95#391—22no—
96#391—22no—
97#391D414yes—
98#389—22no—
99#389—22no—
100#389—22no—
101#389D414yes—
102#384—22no—
103#384—22no—
104#382—42no—
105#380—22no—
106#380—22no—
107#380—42no—
108#380—42no—
109C2×#95—21no—
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×#95—22no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340—42no—
125#340—42no—
126#336—22no—
127#336—22no—
128#336—42no—
129C25C2×D411yescharacteristic
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—22no—
138C22×Q8—21no—
139C22×Q8C2×D412yes—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4#4114yes—
161C23×C4C2411yescharacteristic
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×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8#4114yes—
176C22⋊Q8—48no—
177C22⋊Q8—48no—
178#117—48no—
179#117—48no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
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—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—82no—
235C2×C22⋊C4—82no—
236C2×C22⋊C4—82no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245#106—48no—
246#103—48no—
247#103—48no—
248C22⋊C8—48no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—44no—
257#95—42no—
258#95—42no—
259#95—82no—
260#95—82no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24#12312yes—
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—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
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—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4#12012yes—
366C22×C4#12012yes—
367C22×C4#12312yes—
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C2×#4112yes—
371C22×C4C2×#4112yes—
372C22×C4H(5,𝔽2)11yescharacteristic
373C22×C4H(5,𝔽2)11yescharacteristic
374C22×C4H(5,𝔽2)11yescharacteristic
375C22×C4H(5,𝔽2)12yes—
376C22×C42−1+411yescharacteristic
377C22×C42−1+412yes—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—48no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—84no—
412C4⋊C4—84no—
413C4⋊C4—84no—
414C4⋊C4—84no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—48no—
418C22⋊C4—84no—
419C22⋊C4—88no—
420C42—24no—
421C42—24no—
422C42—42no—
423C42—42no—
424C23—24no—
425C23—24no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—41no—
430C23—41no—
431C23—42no—
432C23—42no—
433C23—41no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23C2×#12011yescharacteristic
448C23C2×#12311yescharacteristic
449C23#53211yescharacteristic
450C23#53411yescharacteristic
451C23#53511yescharacteristic
452C23#53611yescharacteristic
453C23#53711yescharacteristic
454C23#57712yes—
455C23#57712yes—
456C23#57712yes—
457C23#57812yes—
458Q8—44no—
459Q8—44no—
460Q8—82no—
461Q8—82no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—82no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4#53511yescharacteristic
530C2×C4#53511yescharacteristic
531C2×C4#53711yescharacteristic
532C2×C4#53711yescharacteristic
533C2×C4#53812yes—
534C2×C4#53912yes—
535C2×C4#53912yes—
536C2×C4#54012yes—
537C2×C4#54612yes—
538C2×C4#54612yes—
539C2×C4#54712yes—
540C2×C4#54812yes—
541C2×C4#55112yes—
542C2×C4#55212yes—
543C2×C4#56212yes—
544C2×C4#56312yes—
545C8—44no—
546C8—82no—
547C22—42no—
548C22—42no—
549C22—44no—
550C22—44no—
551C22—42no—
552C22—81no—
553C22—81no—
554C22—82no—
555C22—81no—
556C22—81no—
557C22—82no—
558C22#240511yescharacteristic
559C22#243211yescharacteristic
560C22#249811yescharacteristic
561C22#261511yescharacteristic
562C22#262311yescharacteristic
563C22#266111yescharacteristic
564C22#281411yescharacteristic
565C22#300911yescharacteristic
566C22#301011yescharacteristic
567C22#301011yescharacteristic
568C22#301111yescharacteristic
569C22#309712yes—
570C22#309912yes—
571C22#317312yes—
572C22#317812yes—
573C22#320112yes—
574C22#320412yes—
575C22#322212yes—
576C22#322512yes—
577C22#324212yes—
578C22#324412yes—
579C22#325812yes—
580C22#326012yes—
581C4—42no—
582C4—42no—
583C4—42no—
584C4—82no—
585C4—81no—
586C4—82no—
587C4—81no—
588C4—82no—
589C4—82no—
590C4—162no—
591C4—162no—
592C4#269512yes—
593C4#269612yes—
594C4#270612yes—
595C4#270712yes—
596C2—82no—
597C2#1635911yescharacteristic
598C2#2401211yescharacteristic
599C2#2401411yescharacteristic
600C2#2568011yescharacteristic
601C2#2568611yescharacteristic
602C2#2787312yes—
603C2#2967111yescharacteristic
604C2#2967211yescharacteristic
605C2#2991912yes—
606C2#3033412yes—
607C2#3275512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2084C2semidirect
3. C2×#2072C2semidirect
5. #11293C2semidirect
6. #11208C2semidirect
7. #11122C2semidirect
9. #11052C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26252
C2×#20842
C2×#20722
#143402
#112932
#112082
#111222
#110952
#110522
#109702
#107132