Finite Group Databasebeta

← #410014

Subgroups — #410014

Order 512.

2138 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#484C211yesmaximal, characteristic
2C2×#2855C211yesmaximal, characteristic
3C22×#389C211yesmaximal, characteristic
4C2×#1872C211yesmaximal, characteristic
5C2×#1824C211yesmaximal, characteristic
6C2×#1817C211yesmaximal, characteristic
7#11328C214yesmaximal, complemented
8#11067C214yesmaximal, complemented
9#10730C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11C23×C4⋊C4—21no—
12C23×C4⋊C4C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C2×#484—24no—
16C2×#484—24no—
17C2×#484—24no—
18C22×#107C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23#2855—24no—
24#2855—24no—
25C2×#391C2212yes—
26C2×#389—22no—
27C2×#389—24no—
28C2×#389—24no—
29C2×#389C2212yes—
30C2×#389C2214yes—
31C2×#387—22no—
32C2×#385—21no—
33C2×#381—21no—
34C2×#380C2211yescharacteristic
35C22×#95—21no—
36#1872—24no—
37#1824—24no—
38#1817—24no—
39C2×#340C2214yes—
40C2×#340C2214yes—
41C2×#340C2212yes—
42C2×#336C2214yes—
43C23×Q8C2311yescharacteristic
44C24×C4—21no—
45C24×C4—21no—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8D414yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—28no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—28no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—48no—
71C22×C4⋊C4D414yes—
72C22×C4⋊C4D414yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C42—22no—
80C22×C42—22no—
81C22×C42—22no—
82C22×C42—22no—
83C22×C42—22no—
84C22×C42—22no—
85C22×C42D412yes—
86C22×C42C2312yes—
87#484—232no—
88C2×#107—24no—
89C2×#107—24no—
90C2×#107—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—28no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8D414yes—
102#391—24no—
103#391—24no—
104#389—216no—
105#389—24no—
106#389—24no—
107#389—28no—
108#389—28no—
109#387—216no—
110#385—28no—
111#381—28no—
112#380—22no—
113#380D414yes—
114C2×#95—28no—
115C2×#95—24no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120#340—28no—
121#340—24no—
122#340—28no—
123#336—28no—
124C25C2×D411yescharacteristic
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—24no—
131C22×Q8—24no—
132C22×Q8C2×D412yes—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—28no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—42no—
154C23×C4—44no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4Q1614yes—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4#4112yes—
171C23×C4C2411yescharacteristic
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—24no—
175C22×C8—24no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—24no—
180C22×C8—24no—
181C22×C8C2×D414yes—
182C22⋊Q8—28no—
183C22⋊Q8—28no—
184C22⋊Q8—44no—
185C22⋊Q8—44no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—216no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—216no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—216no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—432no—
213C2×C4⋊C4—48no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—48no—
222C2×C4⋊C4C2×D414yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D414yes—
225C2×C4⋊C4C2×D414yes—
226C2×C4⋊C4#4118yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—44no—
243C2×C42—44no—
244C2×C42—44no—
245C2×C42—44no—
246#107—216no—
247#103—216no—
248#103—48no—
249C22⋊C8—28no—
250C22⋊C8—44no—
251#95—216no—
252#95—28no—
253#95—28no—
254#95—28no—
255#95—24no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24#12012yes—
261C24#12012yes—
262C24#12112yes—
263C24C2×Q1612yes—
264C24#13512yes—
265C24#13712yes—
266C24#13712yes—
267C24C22×D411yescharacteristic
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—28no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—44no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—44no—
279C2×Q8—44no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—28no—
287C22×C4—28no—
288C22×C4—28no—
289C22×C4—28no—
290C22×C4—216no—
291C22×C4—28no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—28no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—28no—
307C22×C4—28no—
308C22×C4—28no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—48no—
314C22×C4—48no—
315C22×C4—48no—
316C22×C4—416no—
317C22×C4—416no—
318C22×C4—44no—
319C22×C4—42no—
320C22×C4—44no—
321C22×C4—42no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—48no—
327C22×C4—44no—
328C22×C4—48no—
329C22×C4—48no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—44no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—44no—
348C22×C4—48no—
349C22×C4—88no—
350C22×C4—88no—
351C22×C4#12014yes—
352C22×C4#12012yes—
353C22×C4#12012yes—
354C22×C4#12114yes—
355C22×C4#12114yes—
356C22×C4#12112yes—
357C22×C4#12714yes—
358C22×C4C2×Q1614yes—
359C22×C4AGL(1,ℤ/8)14yes—
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4114yes—
367C22×C42−1+414yes—
368C2×C8—24no—
369C2×C8—28no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—44no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—44no—
377C2×C8—44no—
378C4⋊C4—216no—
379C4⋊C4—24no—
380C4⋊C4—28no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4—28no—
386C4⋊C4—416no—
387C4⋊C4—432no—
388C4⋊C4—42no—
389C4⋊C4—41no—
390C4⋊C4—42no—
391C4⋊C4—41no—
392C4⋊C4—44no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4#12318yes—
396C22⋊C4—48no—
397C22⋊C4—48no—
398C42—42no—
399C42—42no—
400C42—42no—
401C42—42no—
402C42—48no—
403C23—24no—
404C23—22no—
405C23—24no—
406C23—24no—
407C23—24no—
408C23—22no—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—24no—
415C23—22no—
416C23—22no—
417C23—24no—
418C23—24no—
419C23#39214yes—
420C23#45114yes—
421C23#45114yes—
422C23#45214yes—
423C23#45214yes—
424C23#46714yes—
425C23#47014yes—
426C23C2×#12011yescharacteristic
427C23C2×#12011yescharacteristic
428C23C2×#12111yescharacteristic
429C23#52512yes—
430C23#53612yes—
431C23C22×Q1611yescharacteristic
432C23C2×#13511yescharacteristic
433C23C2×AGL(1,ℤ/8)12yes—
434C23C2×#13711yescharacteristic
435C23C2×#13711yescharacteristic
436C23#57512yes—
437Q8—44no—
438Q8—44no—
439Q8—82no—
440Q8—82no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—28no—
446C2×C4—28no—
447C2×C4—28no—
448C2×C4—28no—
449C2×C4—28no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—416no—
457C2×C4—432no—
458C2×C4—48no—
459C2×C4—48no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—416no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—816no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—81no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—84no—
492C2×C4—82no—
493C2×C4—84no—
494C2×C4#39214yes—
495C2×C4C2×#12011yescharacteristic
496C2×C4C2×#12111yescharacteristic
497C2×C4C2×#12111yescharacteristic
498C2×C4C2×#12312yes—
499C2×C4#52512yes—
500C2×C4C2×#12711yescharacteristic
501C2×C4#53612yes—
502C2×C4#53612yes—
503C2×C4#54714yes—
504C2×C4#54714yes—
505C2×C4#54714yes—
506C2×C4#55214yes—
507C8—44no—
508C8—82no—
509C22—28no—
510C22—28no—
511C22—28no—
512C22—28no—
513C22—24no—
514C22—24no—
515C22—24no—
516C22—24no—
517C22—24no—
518C22—24no—
519C22#200118yes—
520C22C2×#39211yescharacteristic
521C22#240612yes—
522C22#241312yes—
523C22#261411yescharacteristic
524C22#284911yescharacteristic
525C22C2×#45111yescharacteristic
526C22C2×#45111yescharacteristic
527C22C2×#45211yescharacteristic
528C22C2×#45211yescharacteristic
529C22#300311yescharacteristic
530C22#300511yescharacteristic
531C22#300612yes—
532C22C2×#46711yescharacteristic
533C22C2×#47011yescharacteristic
534C22#305512yes—
535C22#310412yes—
536C22#310512yes—
537C22#311012yes—
538C22#311712yes—
539C22#323012yes—
540C22#323212yes—
541C22#323612yes—
542C22#323912yes—
543C4—24no—
544C4—44no—
545C4—44no—
546C4—48no—
547C4—84no—
548C4—82no—
549C4—84no—
550C4—82no—
551C4#240612yes—
552C4#241312yes—
553C4#248612yes—
554C4#250112yes—
555C2—28no—
556C2#1506911yescharacteristic
557C2C2×#200111yescharacteristic
558C2#2339111yescharacteristic
559C2#2406412yes—
560C2#2406712yes—
561C2#2460212yes—
562C2#2460612yes—
563C2#2964211yescharacteristic
564C2#2964311yescharacteristic
565C2#3300911yescharacteristic
566C2#3301011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #11328C2semidirect
8. #11067C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4842
C2×#28552
C22×#3892
C2×#18722
C2×#18242
C2×#18172
#113282
#110672
#107302