Finite Group Databasebeta

← #416715

Subgroups — #416715

Order 512.

1328 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2571C211yesmaximal, characteristic, complemented
2C2×#1983C211yesmaximal, characteristic
3C2×#1959C211yesmaximal, characteristic
4#14153C211yesmaximal, characteristic
5#11067C212yesmaximal, complemented
6#11066C212yesmaximal
7#11052C212yesmaximal, complemented
8#11051C212yesmaximal
9#10849C211yesmaximal, characteristic, complemented
10#10835C211yesmaximal, characteristic, complemented
11#10703C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#102C2212yes—
18C22×#102C2211yescharacteristic
19C22×#102C2211yescharacteristic
20C22×#97C2211yescharacteristic
21#2571—24no—
22#2571—24no—
23C2×#400—22no—
24C2×#391C2212yes—
25C2×#390C2212yes—
26C2×#389—21no—
27C2×#389C2212yes—
28C2×#384C2211yescharacteristic
29C2×#382—21no—
30C2×#382C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1983—24no—
35#1959—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1439—21no—
43#1439—21no—
44#1437—22no—
45C23×Q8C2311yescharacteristic
46C23×D4C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
50C2×#124—22no—
51C2×#124—22no—
52C2×#124—22no—
53C2×#124—22no—
54C2×#124—22no—
55C2×#124Q814yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C42—21no—
70C22×C42—21no—
71C22×C42C2312yes—
72C22×C42C2311yescharacteristic
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×#103—24no—
77C2×#103—24no—
78C2×#103—22no—
79C2×#103—22no—
80C2×#103—42no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—22no—
86C2×#102—22no—
87C2×#102—42no—
88C2×#102—42no—
89C2×#102—42no—
90C2×#97—22no—
91C2×#97—22no—
92C2×#97—22no—
93#400—216no—
94#391—24no—
95#391—42no—
96#390—24no—
97#390—42no—
98#389—28no—
99#389—24no—
100#389—24no—
101#384—22no—
102#384D414yes—
103#382—28no—
104#382—24no—
105#382—24no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4D414yes—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—21no—
119C2×#95D412yes—
120#340—24no—
121#340—24no—
122#340—42no—
123#340—42no—
124#336—42no—
125#336—42no—
126C25—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8C2×Q812yes—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—21no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4C2×Q812yes—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2411yescharacteristic
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8#4114yes—
174#124—28no—
175#124—28no—
176#124—44no—
177#124—44no—
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—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—81no—
203C2×C4⋊C4—81no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D414yes—
209C2×C4⋊C4#4114yes—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—82no—
221C2×C22⋊C4—82no—
222C2×C42—24no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—44no—
230C2×C42—42no—
231C2×C42—81no—
232C2×C42—81no—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2×Q812yes—
236C2×C42#4112yes—
237C2×C42#4112yes—
238C2×C42#4112yes—
239C4⋊C8—44no—
240#103—48no—
241#103—44no—
242#103—44no—
243#102—48no—
244#102—44no—
245#102—44no—
246#97—44no—
247#95—28no—
248#95—28no—
249#95—28no—
250#95—24no—
251#95—24no—
252#95—44no—
253#95—44no—
254#95—42no—
255C24—22no—
256C24—41no—
257C24—41no—
258C24—42no—
259C24—42no—
260C24—42no—
261C24—42no—
262C24—41no—
263C24—41no—
264C24—41no—
265C24—41no—
266C24C22×D411yescharacteristic
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
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—41no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4#12012yes—
353C22×C4#12012yes—
354C22×C4C22⋊Q812yes—
355C22×C4C22⋊Q812yes—
356C22×C4#12412yes—
357C22×C4#12412yes—
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×Q811yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4112yes—
365C22×C4C2×#4112yes—
366C22×C4H(5,𝔽2)12yes—
367C22×C4H(5,𝔽2)12yes—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—48no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—84no—
403C4⋊C4—88no—
404C22⋊C4—48no—
405C22⋊C4—48no—
406C22⋊C4—88no—
407C22⋊C4—88no—
408C42—48no—
409C42—42no—
410C42—42no—
411C42—42no—
412C42—42no—
413C42—84no—
414C42C22⋊Q814yes—
415C42C22⋊Q814yes—
416C42#12414yes—
417C42#12414yes—
418C23—44no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—44no—
423C23—81no—
424C23—81no—
425C23—82no—
426C23—82no—
427C23—82no—
428C23—82no—
429C23—82no—
430C23—81no—
431C23—81no—
432C23—82no—
433C23—81no—
434C23—81no—
435C23—81no—
436C23—81no—
437C23—82no—
438C23—82no—
439C23—82no—
440C23—82no—
441C23C2×#12011yescharacteristic
442C23C2×C22⋊Q811yescharacteristic
443C23C2×#12411yescharacteristic
444C23#54612yes—
445C23#55012yes—
446C23#57512yes—
447C23#57512yes—
448C23#57611yescharacteristic
449C23#57711yescharacteristic
450C23#57711yescharacteristic
451C23#57811yescharacteristic
452Q8—42no—
453Q8—42no—
454Q8—44no—
455Q8—81no—
456Q8—81no—
457Q8—82no—
458D4—42no—
459D4—42no—
460D4—44no—
461D4—81no—
462D4—81no—
463D4—82no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
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—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
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—81no—
513C2×C4—81no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4#39314yes—
518C2×C4#39314yes—
519C2×C4C2×C22⋊Q811yescharacteristic
520C2×C4C2×C22⋊Q811yescharacteristic
521C2×C4C2×#12411yescharacteristic
522C2×C4C2×#12411yescharacteristic
523C2×C4#52912yes—
524C2×C4#52912yes—
525C2×C4#53211yescharacteristic
526C2×C4#53211yescharacteristic
527C2×C4#53511yescharacteristic
528C2×C4#53511yescharacteristic
529C2×C4#54012yes—
530C2×C4#54012yes—
531C2×C4#54612yes—
532C2×C4#55012yes—
533C2×C4#55312yes—
534C2×C4#55312yes—
535C2×C4#55412yes—
536C2×C4#55412yes—
537C8—44no—
538C8—81no—
539C8—81no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22—84no—
544C22—84no—
545C22—161no—
546C22—161no—
547C22—162no—
548C22—162no—
549C22—161no—
550C22—161no—
551C22C2×#39311yescharacteristic
552C22#243111yescharacteristic
553C22#243111yescharacteristic
554C22#245412yes—
555C22#249012yes—
556C22#300511yescharacteristic
557C22#300511yescharacteristic
558C22#301211yescharacteristic
559C22#301311yescharacteristic
560C22#308012yes—
561C22#308311yescharacteristic
562C22#308411yescharacteristic
563C22#313611yescharacteristic
564C22#313711yescharacteristic
565C22#314212yes—
566C22#331112yes—
567C22#331312yes—
568C22#331912yes—
569C22#332112yes—
570C22#340112yes—
571C22#340112yes—
572C22#340212yes—
573C22#340312yes—
574C4—24no—
575C4—24no—
576C4—82no—
577C4—84no—
578C4—84no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4—82no—
583C4#245412yes—
584C4#249012yes—
585C4#256611yescharacteristic
586C4#256611yescharacteristic
587C4#264111yescharacteristic
588C4#286111yescharacteristic
589C2—162no—
590C2#1522911yescharacteristic
591C2#2345311yescharacteristic
592C2#2345811yescharacteristic
593C2#2557811yescharacteristic
594C2#2558011yescharacteristic
595C2#2563811yescharacteristic
596C2#2563911yescharacteristic
597C2#2654312yes—
598C2#2654512yes—
599C2#2775912yes—
600C2#2776112yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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
1. C2×#2571C2semidirect
5. #11067C2semidirect
7. #11052C2semidirect
9. #10849C2semidirect
10. #10835C2semidirect
11. #10703C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25712
C2×#19832
C2×#19592
#141532
#110672
#110662
#110522
#110512
#108492
#108352
#107032