Finite Group Databasebeta

← #469666

Subgroups — #469666

Order 512.

1430 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2001C211yesmaximal, characteristic
2C2×#1817C211yesmaximal, characteristic
3#14702C212yesmaximal
4#11316C212yesmaximal, complemented
5#11190C212yesmaximal, complemented
6#11124C212yesmaximal, complemented
7#11096C211yesmaximal, characteristic
8#11024C212yesmaximal, complemented
9#10970C212yesmaximal
10C23×Q16C2211yescharacteristic
11C22×C22⋊Q8C2212yes—
12C23×C42C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C2×#391C2212yes—
20C2×#391C2212yes—
21C2×#380C2212yes—
22#2001—24no—
23#1817—24no—
24C2×#340C2211yescharacteristic
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#336C2212yes—
30C2×#336C2212yes—
31#1436—22no—
32#1436—22no—
33C4⋊#95C2212yes—
34C4⋊#95C2212yes—
35C22⋊#95—22no—
36C22⋊#95—22no—
37C23×Q8C2312yes—
38C24×C4D411yescharacteristic
39C24×C4D411yescharacteristic
40C24×C4C2311yescharacteristic
41C22×Q16—24no—
42C22×Q16—22no—
43C22×Q16—42no—
44C22×Q16—41no—
45C22×Q16—41no—
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×C22⋊Q8—24no—
50C2×C22⋊Q8—24no—
51C2×C22⋊Q8—24no—
52C2×C22⋊Q8—24no—
53C2×C22⋊Q8—42no—
54C2×C22⋊Q8—42no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C42—22no—
62C22×C42—22no—
63C22×C42—41no—
64C22×C42—41no—
65C22×C42—41no—
66C22×C42—41no—
67C22×C42D412yes—
68C22×C42C2312yes—
69C2×#107—42no—
70C2×#107—41no—
71C2×#103—24no—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—42no—
75C2×#103—42no—
76C2×#103—42no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#103—42no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—41no—
82C2×C22⋊C8—41no—
83C2×C22⋊C8—41no—
84C2×C22⋊C8—41no—
85C2×C22⋊C8D414yes—
86#391—42no—
87#391—42no—
88#391—42no—
89#391—42no—
90#380—42no—
91#380—42no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101#340—22no—
102#340—24no—
103#340—42no—
104#340—42no—
105#340—41no—
106#336—24no—
107#336—24no—
108C25C2×D411yescharacteristic
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8#4114yes—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—24no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—41no—
140C23×C4—41no—
141C23×C4—42no—
142C23×C4—41no—
143C23×C4—41no—
144C23×C4—41no—
145C23×C4—41no—
146C23×C4Q1614yescomplemented
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4#4112yes—
152C23×C4C2411yescharacteristic
153C2×Q16—48no—
154C2×Q16—44no—
155C2×Q16—44no—
156C2×Q16—48no—
157C2×Q16—44no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8C2×D412yes—
174C22×C8C2×D412yes—
175C22⋊Q8—48no—
176C22⋊Q8—48no—
177C22⋊Q8—48no—
178C22⋊Q8—48no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
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—22no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4#4114yes—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C42—24no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—42no—
214C2×C42—42no—
215C2×C42—44no—
216C2×C42—44no—
217C2×C42—44no—
218C2×C42—44no—
219#107—44no—
220#103—48no—
221#103—48no—
222#103—48no—
223C22⋊C8—44no—
224C22⋊C8—44no—
225#95—44no—
226#95—44no—
227#95—44no—
228#95—44no—
229#95—44no—
230#95—44no—
231#95—44no—
232#95—44no—
233#95—44no—
234C24—22no—
235C24—21no—
236C24—21no—
237C24—22no—
238C24—22no—
239C24—22no—
240C24—21no—
241C24—21no—
242C24#12112yes—
243C24C2×Q1612yes—
244C24#13512yes—
245C24C22×D411yescharacteristic
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—28no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—48no—
305C22×C4—48no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4#12112yes—
349C22×C4C2×Q1614yes—
350C22×C4#13714yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C2×#4112yes—
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4112yes—
355C22×C4H(5,𝔽2)11yescharacteristic
356C22×C4H(5,𝔽2)11yescharacteristic
357C22×C4H(5,𝔽2)11yescharacteristic
358C22×C4H(5,𝔽2)11yescharacteristic
359C22×C4H(5,𝔽2)12yes—
360C22×C4H(5,𝔽2)12yes—
361Q16—416no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—416no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C22⋊C4—416no—
393C22⋊C4—88no—
394C42—48no—
395C42—42no—
396C42—42no—
397C42—42no—
398C42—42no—
399C23—24no—
400C23—24no—
401C23—22no—
402C23—24no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—24no—
408C23—24no—
409C23—22no—
410C23—24no—
411C23—42no—
412C23—41no—
413C23—42no—
414C23—42no—
415C23—41no—
416C23—42no—
417C23—41no—
418C23—42no—
419C23—42no—
420C23—41no—
421C23#46714yes—
422C23C2×#12111yescharacteristic
423C23#52512yes—
424C23#53411yescharacteristic
425C23#53411yescharacteristic
426C23#53512yes—
427C23C22×Q1611yescharacteristic
428C23C2×#13511yescharacteristic
429C23C2×#13712yes—
430C23#57512yes—
431C23#57511yescharacteristic
432C23#57511yescharacteristic
433Q8—44no—
434Q8—44no—
435Q8—44no—
436Q8—44no—
437Q8—82no—
438Q8—82no—
439Q8—82no—
440Q8—82no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—416no—
450C2×C4—416no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
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—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
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#53411yescharacteristic
497C2×C4#53511yescharacteristic
498C2×C4#53812yes—
499C2×C4#53812yes—
500C2×C4#53811yescharacteristic
501C2×C4#53811yescharacteristic
502C2×C4#54012yes—
503C2×C4#54012yes—
504C2×C4#54612yes—
505C2×C4#54612yes—
506C2×C4#55111yescharacteristic
507C2×C4#55112yes—
508C2×C4#55112yes—
509C2×C4#55312yes—
510C2×C4#55911yescharacteristic
511C2×C4#56011yescharacteristic
512C2×C4#56012yes—
513C2×C4#56011yescharacteristic
514C8—42no—
515C8—42no—
516C8—82no—
517C22—24no—
518C22—28no—
519C22—42no—
520C22—42no—
521C22—44no—
522C22—44no—
523C22—44no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22#241112yes—
529C22#260011yescharacteristic
530C22#261711yescharacteristic
531C22#263512yes—
532C22#281511yescharacteristic
533C22C2×#46711yescharacteristic
534C22#305411yescharacteristic
535C22#305612yes—
536C22#310412yes—
537C22#310812yes—
538C22#311112yes—
539C22#311612yes—
540C22#318911yescharacteristic
541C22#318911yescharacteristic
542C22#319111yescharacteristic
543C22#319111yescharacteristic
544C22#319511yescharacteristic
545C22#319612yes—
546C22#319711yescharacteristic
547C22#320912yes—
548C22#321112yes—
549C22#321612yes—
550C22#321912yes—
551C4—42no—
552C4—44no—
553C4—44no—
554C4—84no—
555C4—84no—
556C4—162no—
557C4—162no—
558C4#269711yescharacteristic
559C4#270312yes—
560C4#270911yescharacteristic
561C4#274912yes—
562C4#275211yescharacteristic
563C4#276011yescharacteristic
564C2—44no—
565C2#1666011yescharacteristic
566C2#2450212yes—
567C2#2451112yes—
568C2#2941511yescharacteristic
569C2#2941611yescharacteristic
570C2#2973111yescharacteristic
571C2#2974311yescharacteristic
572C2#3010912yes—
573C2#3011012yes—
574C2#3276211yescharacteristic
575C2#3276311yescharacteristic
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
4. #11316C2semidirect
5. #11190C2semidirect
6. #11124C2semidirect
8. #11024C2semidirect
146. C23×C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#20012
C2×#18172
#147022
#113162
#111902
#111242
#110962
#110242
#109702