Finite Group Databasebeta

← #414789

Subgroups — #414789

Order 512.

1744 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic, complemented
2C22×#476C211yesmaximal, characteristic
3C2×#2566C211yesmaximal, characteristic
4C2×#1989C211yesmaximal, characteristic, complemented
5C2×#1762C211yesmaximal, characteristic
6#14155C211yesmaximal, characteristic, complemented
7#11325C212yesmaximal, complemented
8#11323C212yesmaximal, complemented
9#11123C212yesmaximal
10#11121C212yesmaximal
11#10835C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×#501—24no—
16C2×#501—24no—
17C2×#501—24no—
18C2×#476—24no—
19C2×#476—24no—
20C2×#476—24no—
21C22×#107C2212yes—
22C22×#106C2212yes—
23C22×#106C2212yes—
24C22×C4⋊C8C2211yescharacteristic
25C22×#102C2212yes—
26C22×#102C2212yes—
27C22×#97C2211yescharacteristic
28#2566—24no—
29#2566—24no—
30C2×#396—22no—
31C2×#391—22no—
32C2×#390C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#385—22no—
35C2×#384C2211yescharacteristic
36C2×#382C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39#1989—24no—
40#1762—24no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46#1445—22no—
47C4⋊#95C2212yes—
48C4⋊#95C2212yes—
49C23×Q8—21no—
50C23×D4C2311yescharacteristic
51C24×C4—21no—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8D414yes—
60C2×#125—22no—
61C2×#125—22no—
62C2×#125—22no—
63C2×#125—22no—
64C2×#125—22no—
65C2×#125D414yes—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4D414yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4D412yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4—22no—
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#501—232no—
90#476—232no—
91C2×#107—24no—
92C2×#107—24no—
93C2×#107—24no—
94C2×#106—24no—
95C2×#106—24no—
96C2×#106—24no—
97C2×#106—24no—
98C2×#106—24no—
99C2×#106—24no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8D414yes—
103C2×#102—24no—
104C2×#102—24no—
105C2×#102—24no—
106C2×#102—24no—
107C2×#102—24no—
108C2×#102Q818yescomplemented
109C2×#97—22no—
110C2×#97—22no—
111C2×#97D414yes—
112#396—216no—
113#391—48no—
114#390—22no—
115#390—22no—
116#390—22no—
117#390—22no—
118#385—216no—
119#384—22no—
120#384—22no—
121#382—22no—
122#382—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C2×#95—22no—
128C2×#95—24no—
129C2×#95—24no—
130C2×#95—21no—
131C2×#95D412yes—
132#340—24no—
133#340—24no—
134#340—24no—
135#336—24no—
136#336—24no—
137C25—21no—
138C22×Q8—24no—
139C22×Q8—44no—
140C22×Q8—44no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4C2×D412yes—
152C22×D4C2×Q814yes—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—84no—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
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×C8C2×D414yes—
178C22×C8C2×D414yes—
179C22×C8C2×Q814yes—
180C22×C8#4114yes—
181C4⋊Q8—28no—
182C4⋊Q8—28no—
183C4⋊Q8—44no—
184C4⋊Q8—44no—
185#125—28no—
186#125—28no—
187#125—44no—
188#125—44no—
189C4×D4—28no—
190C4×D4—28no—
191C4×D4—44no—
192C4×D4—44no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
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—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—84no—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2×D414yes—
236C2×C4⋊C4C2×Q814yes—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—84no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—44no—
254C2×C42—42no—
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42#4112yes—
261#107—216no—
262#106—216no—
263#106—216no—
264C4⋊C8—28no—
265#102—216no—
266#102—216no—
267#97—28no—
268#95—28no—
269#95—24no—
270#95—48no—
271#95—42no—
272#95—48no—
273C24—22no—
274C24—24no—
275C24—41no—
276C24—41no—
277C24—42no—
278C24—42no—
279C24—42no—
280C24—44no—
281C24C22×D411yescharacteristic
282C2×Q8—416no—
283C2×Q8—48no—
284C2×Q8—84no—
285C2×Q8—84no—
286C2×Q8—88no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—24no—
290C2×D4—22no—
291C2×D4—24no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4C4⋊Q814yes—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—48no—
341C22×C4—88no—
342C22×C4—88no—
343C22×C4—88no—
344C22×C4#12012yes—
345C22×C4#12012yes—
346C22×C4#12112yes—
347C22×C4#12112yes—
348C22×C4#12712yes—
349C22×C4#12712yes—
350C22×C4AGL(1,ℤ/8)14yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×Q812yes—
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4112yes—
359C22×C42−1+412yes—
360C22×C42−1+412yes—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8C22⋊Q818yes—
375C4⋊C4—24no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—48no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—48no—
397C4⋊C4—48no—
398C4⋊C4—48no—
399C4⋊C4—88no—
400C22⋊C4—416no—
401C22⋊C4—48no—
402C22⋊C4—88no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—42no—
408C42—42no—
409C42—84no—
410C42#12014yes—
411C42#12014yes—
412C42#12114yes—
413C42#12114yes—
414C23—24no—
415C23—48no—
416C23—48no—
417C23—42no—
418C23—44no—
419C23—42no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—84no—
424C23—84no—
425C23—81no—
426C23—81no—
427C23—82no—
428C23—82no—
429C23—82no—
430C23C2×#12011yescharacteristic
431C23C2×#12111yescharacteristic
432C23C2×#12711yescharacteristic
433C23#54712yes—
434C23D4×Q812yes—
435C23C2×AGL(1,ℤ/8)12yes—
436C23#57512yes—
437C23#57714yes—
438Q8—816no—
439D4—24no—
440D4—24no—
441D4—42no—
442D4—44no—
443D4—44no—
444D4—42no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—28no—
453C2×C4—24no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—48no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—48no—
471C2×C4—44no—
472C2×C4—816no—
473C2×C4—84no—
474C2×C4—82no—
475C2×C4—81no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—84no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4#39214yes—
487C2×C4#39214yes—
488C2×C4C2×#12011yescharacteristic
489C2×C4C2×#12011yescharacteristic
490C2×C4C2×#12111yescharacteristic
491C2×C4C2×#12111yescharacteristic
492C2×C4C2×C22⋊Q812yes—
493C2×C4C2×C4⋊Q811yescharacteristic
494C2×C4#53211yescharacteristic
495C2×C4#53312yes—
496C2×C4#53612yes—
497C2×C4#53611yescharacteristic
498C2×C4#53611yescharacteristic
499C2×C4#54112yes—
500C2×C4#54712yes—
501C2×C4#54812yes—
502C2×C4#54812yes—
503C2×C4D4×Q812yes—
504C2×C4D4×Q812yes—
505C2×C4D4×Q812yes—
506C8—28no—
507C8—44no—
508C22—44no—
509C22—48no—
510C22—84no—
511C22—88no—
512C22—82no—
513C22—84no—
514C22—82no—
515C22—84no—
516C22C2×#39211yescharacteristic
517C22#242912yes—
518C22#243211yescharacteristic
519C22#243411yescharacteristic
520C22#254212yes—
521C22#300612yes—
522C22#300912yes—
523C22#303612yes—
524C22#304312yes—
525C22#315111yescharacteristic
526C22#315311yescharacteristic
527C22#315412yes—
528C22#334514yes—
529C22#335014yes—
530C22#338414yes—
531C22#338814yes—
532C4—24no—
533C4—24no—
534C4—44no—
535C4—82no—
536C4—82no—
537C4—82no—
538C4—82no—
539C4—84no—
540C4—84no—
541C4#240912yes—
542C4#242912yes—
543C4#243211yescharacteristic
544C4#243411yescharacteristic
545C4#262111yescharacteristic
546C4#285711yescharacteristic
547C2—84no—
548C2#1517811yescharacteristic
549C2#2342311yescharacteristic
550C2#2342811yescharacteristic
551C2#2546114yes—
552C2#2571412yes—
553C2#2585112yes—
554C2#2854712yes—
555C2#2854812yes—
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. C22×#501C2semidirect
4. C2×#1989C2semidirect
6. #14155C2semidirect
7. #11325C2semidirect
8. #11323C2semidirect
11. #10835C2semidirect
108. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5012
C22×#4762
C2×#25662
C2×#19892
C2×#17622
#141552
#113252
#113232
#111232
#111212
#108352