Finite Group Databasebeta

← #415016

Subgroups — #415016

Order 512.

1318 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#479C211yesmaximal, characteristic
2C2×#1989C211yesmaximal, characteristic
3C2×#1842C211yesmaximal, characteristic
4C2×#1805C211yesmaximal, characteristic
5#14709C211yesmaximal, characteristic
6C4×#391C211yesmaximal, characteristic
7#11232C212yesmaximal
8#11231C212yesmaximal
9#11077C212yesmaximal
10#11077C212yesmaximal
11#10850C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#479—24no—
15C2×#479—24no—
16C2×#479—24no—
17C22×#107C2211yescharacteristic
18C22×#106C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2212yes—
21C22×C4⋊C8C2212yes—
22C22×C4⋊C8C2211yescharacteristic
23C22×#103C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2212yes—
26C2×#391C2212yes—
27C2×#382C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C43C2211yescharacteristic
30#1989—24no—
31#1842—24no—
32#1805—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38#1436—21no—
39#1436—22no—
40#1436—21no—
41#1436—21no—
42#1436—21no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C4×#95—22no—
47C23×Q8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×#125—22no—
52C2×#125—22no—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125D414yes—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8D414yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C42—21no—
72C22×C42—21no—
73C22×C42D411yescharacteristic
74C22×C42D412yes—
75C22×C42D411yescharacteristic
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78#479—232no—
79C2×#107—22no—
80C2×#107—22no—
81C2×#107—22no—
82C2×#106—22no—
83C2×#106—22no—
84C2×#106—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—24no—
88C2×C4⋊C8—24no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—24no—
93C2×C4⋊C8—24no—
94C2×C4⋊C8—42no—
95C2×C4⋊C8—42no—
96C2×C4⋊C8D414yes—
97C2×#103—22no—
98C2×#103—22no—
99C2×#103—22no—
100C2×#103—22no—
101C2×#103—22no—
102C2×#103—22no—
103C2×#103—24no—
104C2×#103—42no—
105C2×#103—42no—
106#391—24no—
107#391Q818yes—
108#382—22no—
109#382—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C43—22no—
124C43—22no—
125#340—24no—
126#340—24no—
127#340—24no—
128#340—42no—
129#340—42no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8C2×D412yes—
139C22×Q8C2×Q814yes—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2411yescharacteristic
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8C2×D412yes—
167C22×C8C2×D412yes—
168#125—28no—
169#125—28no—
170#125—44no—
171#125—44no—
172C4×Q8—28no—
173C4×Q8—28no—
174C4×Q8—44no—
175C4×Q8—44no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
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—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×Q814yes—
215C2×C42—22no—
216C2×C42—24no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—22no—
224C2×C42—24no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42—44no—
230C2×C42—42no—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×Q814yes—
235C2×C42#4114yes—
236C2×C42#4112yes—
237C2×C42#4112yes—
238C2×C42#4112yes—
239#107—28no—
240#106—28no—
241C4⋊C8—28no—
242C4⋊C8—48no—
243C4⋊C8—48no—
244C4⋊C8—44no—
245#103—28no—
246#103—28no—
247#103—48no—
248#95—28no—
249#95—28no—
250#95—44no—
251#95—44no—
252#95—44no—
253#95—42no—
254#95—42no—
255#95—42no—
256#95—44no—
257#95—44no—
258#95—44no—
259C24C22×D411yescharacteristic
260C2×Q8—22no—
261C2×Q8—24no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8C4⋊Q814yes—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
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—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4#12112yes—
327C22×C4#12112yes—
328C22×C4#12112yes—
329C22×C4#12112yes—
330C22×C4#12112yes—
331C22×C4#12112yes—
332C22×C4C22⋊Q814yes—
333C22×C4C22⋊Q814yes—
334C22×C4#13514yes—
335C22×C4#13714yes—
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×Q812yes—
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4112yes—
344C22×C4H(5,𝔽2)12yes—
345C22×C42−1+412yes—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—48no—
380C4⋊C4—48no—
381C4⋊C4—48no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C42—28no—
387C42—24no—
388C42—24no—
389C42—24no—
390C42—24no—
391C42—48no—
392C42—44no—
393C42—48no—
394C42—84no—
395C42#12014yes—
396C42#12314yes—
397C42#12314yes—
398C42#12414yes—
399C23C2×#12111yescharacteristic
400C23C2×#12111yescharacteristic
401C23C2×#12111yescharacteristic
402C23C2×C22⋊Q812yes—
403C23#53712yes—
404C23C2×#13512yes—
405C23C2×#13712yes—
406C23#57512yes—
407C23#57512yes—
408Q8—24no—
409Q8—24no—
410Q8—44no—
411Q8—42no—
412Q8—44no—
413Q8—42no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4#39414yes—
470C2×C4#39414yes—
471C2×C4#47918yescomplemented
472C2×C4C2×#12011yescharacteristic
473C2×C4C2×#12311yescharacteristic
474C2×C4C2×#12311yescharacteristic
475C2×C4C2×#12411yescharacteristic
476C2×C4C2×C4⋊Q811yescharacteristic
477C2×C4#53211yescharacteristic
478C2×C4#53312yes—
479C2×C4#53511yescharacteristic
480C2×C4#53611yescharacteristic
481C2×C4#54212yes—
482C2×C4#54612yes—
483C2×C4D4×Q812yes—
484C2×C4D4×Q812yes—
485C2×C4#55012yes—
486C2×C4#55212yes—
487C2×C4#55412yes—
488C2×C4#55412yes—
489C2×C4#55612yes—
490C8—42no—
491C8—42no—
492C8—82no—
493C22C2×#39411yescharacteristic
494C22#241712yes—
495C22#253812yes—
496C22#262011yescharacteristic
497C22#285711yescharacteristic
498C22#303512yes—
499C22#303512yes—
500C22#303712yes—
501C22#303911yescharacteristic
502C22#304011yescharacteristic
503C22#305011yescharacteristic
504C22#305411yescharacteristic
505C22#305612yes—
506C22C2×#47912yes—
507C22#307612yes—
508C22#307812yes—
509C22#308012yes—
510C22#324812yes—
511C22#325012yes—
512C22#325212yes—
513C22#325612yes—
514C4—28no—
515C4—24no—
516C4—24no—
517C4—44no—
518C4—44no—
519C4—84no—
520C4—82no—
521C4—82no—
522C4—82no—
523C4—162no—
524C4#242312yes—
525C4#243111yescharacteristic
526C4#243411yescharacteristic
527C4#249212yes—
528C4#249911yescharacteristic
529C4#250211yescharacteristic
530C2#1518211yescharacteristic
531C2#2353911yescharacteristic
532C2#2354711yescharacteristic
533C2#2484012yes—
534C2#2484212yes—
535C2#2828512yes—
536C2#2828712yes—
537C2#2983312yes—
538C2#3320012yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
471. C2×C4#479semidirect

Maximal subgroups

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

subgroupindex
C22×#4792
C2×#19892
C2×#18422
C2×#18052
#147092
C4×#3912
#112322
#112312
#110772
#108502