Finite Group Databasebeta

← #406750

Subgroups — #406750

Order 512.

1474 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1853C211yesmaximal, characteristic
2C2×#1850C211yesmaximal, characteristic
3C2×#1824C211yesmaximal, characteristic
4C22×#340C211yesmaximal, characteristic
5#13924C211yesmaximal, characteristic
6#13907C211yesmaximal, characteristic
7#11315C212yesmaximal, complemented
8#11231C212yesmaximal, complemented
9#11227C212yesmaximal, complemented
10#11220C212yesmaximal, complemented
11#10721C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C22×C4⋊C8C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C2×#389C2212yes—
20C2×#380—21no—
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24#1853—24no—
25#1850—24no—
26#1824—24no—
27C2×#340—24no—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37#1445—22no—
38#1441—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C22⋊#95—21no—
43C22⋊#95—21no—
44C22⋊#95—21no—
45C24×C4—21no—
46C24×C4—21no—
47C24×C4C2311yescharacteristic
48C23×C8—22no—
49C23×C8—24no—
50C23×C8—22no—
51C23×C8—22no—
52C23×C8—42no—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2312yes—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—24no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—42no—
67C22×C4⋊C4D414yes—
68C22×C4⋊C4Q814yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4D411yescharacteristic
78C22×C22⋊C4D411yescharacteristic
79C22×C42C2312yes—
80C2×#106—22no—
81C2×#106—22no—
82C2×#106—24no—
83C2×#106—42no—
84C2×C4⋊C8—24no—
85C2×C4⋊C8—24no—
86C2×C4⋊C8—42no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8D414yes—
90#389—24no—
91#389—42no—
92#380—22no—
93#380—22no—
94#380—22no—
95#380—22no—
96#380—44no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95D412yes—
109#340—24no—
110#340—24no—
111#340—24no—
112#340—22no—
113#340—22no—
114#340—24no—
115#340—24no—
116#340—42no—
117#336—24no—
118C25C2×D411yescharacteristic
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—44no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—82no—
137C23×C4—82no—
138C23×C4—82no—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×Q812yes—
145C23×C4#4112yes—
146C23×C4#4112yes—
147C23×C4#4112yes—
148C23×C4C2411yescharacteristic
149C22×C8—28no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—48no—
165C22×C8—48no—
166C22×C8—44no—
167C22×C8—44no—
168C22×C8C2×D412yes—
169C22×C8C2×D412yes—
170C22×C8#4114yes—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
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—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—48no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×Q814yes—
213C2×C4⋊C4#4114yes—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4SD1614yes—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232#106—44no—
233#106—44no—
234C4⋊C8—48no—
235C22⋊C8—28no—
236#95—24no—
237#95—44no—
238#95—42no—
239#95—42no—
240#95—44no—
241#95—82no—
242#95—82no—
243#95—82no—
244#95—82no—
245#95—82no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24—22no—
250C24#12112yes—
251C24#12112yes—
252C24C22⋊Q812yes—
253C24C2×SD1612yes—
254C24#13512yes—
255C24#13712yes—
256C24#13712yes—
257C24C22×D411yescharacteristic
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—48no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—48no—
306C22×C4—48no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—84no—
325C22×C4—88no—
326C22×C4—88no—
327C22×C4—88no—
328C22×C4—84no—
329C22×C4—84no—
330C22×C4—88no—
331C22×C4#12114yes—
332C22×C4#12112yes—
333C22×C4#12112yes—
334C22×C4C22⋊Q814yes—
335C22×C4C22⋊Q814yes—
336C22×C4C22⋊Q812yes—
337C22×C4#12614yes—
338C22×C4C2×SD1614yes—
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×Q811yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4111yescharacteristic
345C22×C4C2×#4112yes—
346C22×C4C2×#4112yes—
347C22×C4H(5,𝔽2)12yes—
348C22×C42−1+412yes—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—416no—
357C2×C8—42no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—42no—
362C2×C8—41no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C4⋊C4—22no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—48no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—48no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—88no—
389C4⋊C4#12314yes—
390C22⋊C4—28no—
391C22⋊C4—44no—
392C22⋊C4—84no—
393C22⋊C4—84no—
394C42—42no—
395C42—42no—
396C42—42no—
397C42—42no—
398C23—24no—
399C23—24no—
400C23—24no—
401C23—24no—
402C23—24no—
403C23—22no—
404C23—22no—
405C23—24no—
406C23—24no—
407C23—24no—
408C23—22no—
409C23—22no—
410C23—24no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23#39614yes—
415C23#46114yes—
416C23#46414yes—
417C23#46514yes—
418C23#47014yes—
419C23#47514yes—
420C23#47914yes—
421C23C2×#12111yescharacteristic
422C23C2×#12111yescharacteristic
423C23C2×C22⋊Q811yescharacteristic
424C23#54612yes—
425C23#54712yes—
426C23C22×SD1611yescharacteristic
427C23C2×#13511yescharacteristic
428C23C2×#13711yescharacteristic
429C23C2×#13711yescharacteristic
430C23#57612yes—
431C23#57712yes—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—28no—
439C2×C4—24no—
440C2×C4—28no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—816no—
468C2×C4—816no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—82no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—82no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4#39614yes—
493C2×C4C2×#12111yescharacteristic
494C2×C4C2×C22⋊Q811yescharacteristic
495C2×C4C2×C22⋊Q811yescharacteristic
496C2×C4C2×#12311yescharacteristic
497C2×C4#52511yescharacteristic
498C2×C4C2×#12611yescharacteristic
499C2×C4#52912yes—
500C2×C4#53312yes—
501C2×C4#53411yescharacteristic
502C2×C4#53611yescharacteristic
503C2×C4#53912yes—
504C2×C4#54412yes—
505C2×C4#54612yes—
506C2×C4#54712yes—
507C2×C4#55112yes—
508C2×C4#55112yes—
509C2×C4#55212yes—
510C2×C4#55212yes—
511C8—42no—
512C8—42no—
513C8—44no—
514C22—24no—
515C22—24no—
516C22—24no—
517C22—24no—
518C22—28no—
519C22—28no—
520C22—28no—
521C22—28no—
522C22—24no—
523C22—24no—
524C22#206618yes—
525C22C2×#39611yescharacteristic
526C22#241211yescharacteristic
527C22#241311yescharacteristic
528C22#242512yes—
529C22#243812yes—
530C22C2×#46111yescharacteristic
531C22C2×#46411yescharacteristic
532C22#304111yescharacteristic
533C22#304311yescharacteristic
534C22C2×#46511yescharacteristic
535C22C2×#47011yescharacteristic
536C22#306212yes—
537C22C2×#47511yescharacteristic
538C22C2×#47911yescharacteristic
539C22#308112yes—
540C22#329512yes—
541C22#330112yes—
542C22#331412yes—
543C22#332012yes—
544C22#333512yes—
545C22#333912yes—
546C22#334412yes—
547C22#334612yes—
548C4—24no—
549C4—44no—
550C4—44no—
551C4—84no—
552C4—82no—
553C4—82no—
554C4—161no—
555C4—162no—
556C4—161no—
557C4#243812yes—
558C4#248611yescharacteristic
559C4#249312yes—
560C4#253411yescharacteristic
561C4#262711yescharacteristic
562C4#285511yescharacteristic
563C2—28no—
564C2#1493511yescharacteristic
565C2C2×#206611yescharacteristic
566C2#2361411yescharacteristic
567C2#2453011yescharacteristic
568C2#2453311yescharacteristic
569C2#2458011yescharacteristic
570C2#2458311yescharacteristic
571C2#2523812yes—
572C2#2524812yes—
573C2#2599212yes—
574C2#2600212yes—
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
7. #11315C2semidirect
8. #11231C2semidirect
9. #11227C2semidirect
10. #11220C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18532
C2×#18502
C2×#18242
C22×#3402
#139242
#139072
#113152
#112312
#112272
#112202
#107212