Finite Group Databasebeta

← #406784

Subgroups — #406784

Order 512.

1502 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1850C211yesmaximal, characteristic
2C2×#1823C211yesmaximal, characteristic
3C2×#1808C211yesmaximal, characteristic
4C22×#340C211yesmaximal, characteristic
5#14130C211yesmaximal, characteristic
6#13904C211yesmaximal, characteristic
7#11315C212yesmaximal, complemented
8#11232C212yesmaximal, complemented
9#11227C212yesmaximal, complemented
10#11219C212yesmaximal, complemented
11#10730C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C2×#389C2212yes—
20C2×#385—21no—
21C2×#381—21no—
22C2×#381—21no—
23C2×#380C2211yescharacteristic
24C2×#380C2211yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26#1850—24no—
27#1823—24no—
28#1808—24no—
29C2×#340—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39#1445—21no—
40#1445—21no—
41#1444—21no—
42#1442—21no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C22⋊#95—21no—
47C24×C4—21no—
48C24×C4—21no—
49C24×C4C2311yescharacteristic
50C23×C8—22no—
51C23×C8—22no—
52C23×C8—24no—
53C23×C8—42no—
54C23×C8D414yes—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—42no—
71C22×C4⋊C4Q814yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4D411yescharacteristic
79C22×C22⋊C4D411yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×#107—24no—
83C2×#107—24no—
84C2×#107—42no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—24no—
88C2×C4⋊C8—42no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92#389—24no—
93#389—42no—
94#385—44no—
95#381—44no—
96#381—44no—
97#380—22no—
98#380—22no—
99#380—22no—
100#380D414yes—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95D411yescharacteristic
113C2×#95D411yescharacteristic
114#340—24no—
115#340—24no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—42no—
121#336—22no—
122#336—22no—
123C25C2×D411yescharacteristic
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—24no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—44no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4—42no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—82no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2×Q812yes—
152C23×C4#4112yes—
153C23×C4#4112yes—
154C23×C4#4112yes—
155C23×C4C2411yescharacteristic
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—28no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—28no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—44no—
176C22×C8—48no—
177C22×C8—48no—
178C22×C8C2×D412yes—
179C22×C8C2×D412yes—
180C22×C8#4114yes—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—28no—
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—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
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—48no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4C2×D414yes—
218C2×C4⋊C4C2×Q814yes—
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—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4D814yes—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240#107—48no—
241C4⋊C8—44no—
242C4⋊C8—44no—
243C22⋊C8—44no—
244#95—24no—
245#95—22no—
246#95—22no—
247#95—44no—
248#95—44no—
249#95—44no—
250#95—42no—
251#95—44no—
252#95—44no—
253#95—44no—
254#95—82no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24#12112yes—
260C24#12112yes—
261C24C22⋊Q812yes—
262C24C2×D812yes—
263C24#13512yes—
264C24AGL(1,ℤ/8)12yes—
265C24#13712yes—
266C24C22×D411yescharacteristic
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—48no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—48no—
324C22×C4—48no—
325C22×C4—48no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—88no—
334C22×C4—88no—
335C22×C4—84no—
336C22×C4—84no—
337C22×C4—84no—
338C22×C4—84no—
339C22×C4—84no—
340C22×C4—84no—
341C22×C4#12114yes—
342C22×C4#12112yes—
343C22×C4#12112yes—
344C22×C4C22⋊Q814yes—
345C22×C4C22⋊Q814yes—
346C22×C4C22⋊Q812yes—
347C22×C4#12614yes—
348C22×C4C2×D812yes—
349C22×C4C2×D812yes—
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×Q811yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4112yes—
357C22×C4C2×#4112yes—
358C22×C4H(5,𝔽2)12yes—
359C22×C42−1+412yes—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—416no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8#12314yes—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—48no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—48no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—88no—
400C22⋊C4—28no—
401C22⋊C4—48no—
402C22⋊C4—44no—
403C22⋊C4—84no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42—42no—
409C42—42no—
410C23—24no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—24no—
415C23—24no—
416C23—24no—
417C23—22no—
418C23—24no—
419C23—24no—
420C23—22no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—22no—
425C23—22no—
426C23#39614yes—
427C23#45914yes—
428C23#46414yes—
429C23#46614yes—
430C23#46814yes—
431C23#47414yes—
432C23#47814yes—
433C23C2×#12111yescharacteristic
434C23C2×#12111yescharacteristic
435C23C2×C22⋊Q811yescharacteristic
436C23#54612yes—
437C23#54712yes—
438C23C22×D811yescharacteristic
439C23C2×#13511yescharacteristic
440C23C2×AGL(1,ℤ/8)11yescharacteristic
441C23C2×#13711yescharacteristic
442C23#57712yes—
443C23#57812yes—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—28no—
453C2×C4—28no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—41no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—41no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—44no—
483C2×C4—816no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—816no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4#39614yes—
505C2×C4C2×#12111yescharacteristic
506C2×C4C2×C22⋊Q811yescharacteristic
507C2×C4C2×C22⋊Q811yescharacteristic
508C2×C4C2×#12311yescharacteristic
509C2×C4#52511yescharacteristic
510C2×C4C2×#12611yescharacteristic
511C2×C4#52912yes—
512C2×C4#53312yes—
513C2×C4#53411yescharacteristic
514C2×C4#53611yescharacteristic
515C2×C4#53912yes—
516C2×C4#54412yes—
517C2×C4#54612yes—
518C2×C4#54712yes—
519C2×C4#55112yes—
520C2×C4#55112yes—
521C2×C4#55212yes—
522C2×C4#55212yes—
523C8—24no—
524C8—42no—
525C8—81no—
526C8—81no—
527C22—24no—
528C22—24no—
529C22—24no—
530C22—24no—
531C22—28no—
532C22—28no—
533C22—28no—
534C22—28no—
535C22—24no—
536C22—24no—
537C22#206718yes—
538C22C2×#39611yescharacteristic
539C22#242512yes—
540C22#243812yes—
541C22#248611yescharacteristic
542C22#253411yescharacteristic
543C22C2×#45911yescharacteristic
544C22C2×#46411yescharacteristic
545C22#304612yes—
546C22C2×#46611yescharacteristic
547C22C2×#46811yescharacteristic
548C22#305711yescharacteristic
549C22#305911yescharacteristic
550C22C2×#47411yescharacteristic
551C22C2×#47811yescharacteristic
552C22#308212yes—
553C22#329412yes—
554C22#329612yes—
555C22#330812yes—
556C22#331212yes—
557C22#333412yes—
558C22#333612yes—
559C22#334412yes—
560C22#334512yes—
561C4—24no—
562C4—42no—
563C4—42no—
564C4—81no—
565C4—84no—
566C4—84no—
567C4—84no—
568C4—81no—
569C4—162no—
570C4#241211yescharacteristic
571C4#241311yescharacteristic
572C4#243812yes—
573C4#249312yes—
574C4#262711yescharacteristic
575C4#285511yescharacteristic
576C2—28no—
577C2#1493511yescharacteristic
578C2C2×#206711yescharacteristic
579C2#2361311yescharacteristic
580C2#2520012yes—
581C2#2520812yes—
582C2#2598812yes—
583C2#2599612yes—
584C2#2766111yescharacteristic
585C2#2766311yescharacteristic
586C2#2809411yescharacteristic
587C2#2810511yescharacteristic
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. #11232C2semidirect
9. #11227C2semidirect
10. #11219C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18502
C2×#18232
C2×#18082
C22×#3402
#141302
#139042
#113152
#112322
#112272
#112192
#107302