Finite Group Databasebeta

← #440737

Subgroups — #440737

Order 512.

1508 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2504C211yesmaximal, characteristic
2C2×#2025C211yesmaximal, characteristic, complemented
3C2×#1991C211yesmaximal, characteristic, complemented
4#15216C211yesmaximal, characteristic, complemented
5#11372C211yesmaximal, characteristic, complemented
6#11268C212yesmaximal, complemented
7#11189C212yesmaximal, complemented
8#10969C212yesmaximal
9#10925C211yesmaximal, characteristic, complemented
10#10921C211yesmaximal, characteristic, complemented
11#10730C212yesmaximal, complemented
12C22×#125C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C4×C8C2211yescharacteristic
19#2504—24no—
20#2504—24no—
21C2×#397—22no—
22C2×#390C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#386—22no—
25C2×#382C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#380C2212yes—
28C2×C4×C22⋊C4C2211yescharacteristic
29#2025—24no—
30#1991—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39#1442—22no—
40#1439—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2211yescharacteristic
43C23×D4—21no—
44C23×D4C2311yescharacteristic
45C24×C4—21no—
46C24×C4C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×#125—24no—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125—42no—
53C2×#121—22no—
54C2×#121—22no—
55C2×#121—24no—
56C2×#121—42no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—24no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—42no—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4D412yes—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C2×#102—24no—
76C2×#102—24no—
77C2×#102—24no—
78C2×#102—42no—
79C2×#102—42no—
80C2×#102—42no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—42no—
84C2×C4×C8—22no—
85C2×C4×C8—42no—
86#397—216no—
87#390—22no—
88#390—22no—
89#390—22no—
90#390D414yes—
91#386—216no—
92#382—22no—
93#382—22no—
94#382—22no—
95#382D414yes—
96#380—24no—
97#380—42no—
98C4×C22⋊C4—22no—
99C4×C22⋊C4—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95D412yes—
107#340—24no—
108#340—24no—
109#340—24no—
110#340—42no—
111#340—42no—
112#340—42no—
113#336—24no—
114#336—22no—
115C25—21no—
116C25C2×D411yescharacteristic
117C22×D4—24no—
118C22×D4—21no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—42no—
126C22×D4—44no—
127C22×D4—42no—
128C22×D4—44no—
129C22×D4C2×D412yes—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—24no—
142C23×C4—22no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—82no—
146C23×C4—82no—
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4#4114yes—
151C23×C4C2411yescharacteristic
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
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#4114yes—
166#125—28no—
167#125—48no—
168#125—44no—
169#121—48no—
170#121—48no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
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—48no—
193C2×C4⋊C4—48no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4#4114yes—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—82no—
214C2×C22⋊C4—82no—
215C2×C22⋊C4—82no—
216C2×C22⋊C4—82no—
217C2×C22⋊C4—82no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226#102—48no—
227#102—48no—
228C22⋊C8—48no—
229C4×C8—44no—
230#95—28no—
231#95—28no—
232#95—24no—
233#95—44no—
234#95—44no—
235#95—82no—
236#95—82no—
237C24—21no—
238C24—21no—
239C24—22no—
240C24—22no—
241C24—22no—
242C24—22no—
243C24—22no—
244C24—22no—
245C24—41no—
246C24—41no—
247C24—42no—
248C24—42no—
249C24—42no—
250C24—42no—
251C24—42no—
252C24—42no—
253C24#12412yes—
254C24#13514yes—
255C24C22×D411yescharacteristic
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—48no—
261C2×D4—416no—
262C2×D4—41no—
263C2×D4—42no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—84no—
275C2×D4—88no—
276C2×D4—84no—
277C22×C4—21no—
278C22×C4—21no—
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—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
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—48no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
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—88no—
341C22×C4—88no—
342C22×C4—84no—
343C22×C4—84no—
344C22×C4#12014yes—
345C22×C4#12412yes—
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C2×#4112yes—
349C22×C4C2×#4112yes—
350C22×C4C2×#4112yes—
351C22×C4H(5,𝔽2)12yes—
352C22×C4H(5,𝔽2)12yes—
353C22×C42−1+412yes—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—48no—
374C4⋊C4—416no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C22⋊C4—48no—
385C22⋊C4—48no—
386C22⋊C4—88no—
387C22⋊C4—88no—
388C22⋊C4—88no—
389C42—22no—
390C42—22no—
391C42—24no—
392C42—41no—
393C42—41no—
394C42—42no—
395C23—24no—
396C23—24no—
397C23—24no—
398C23—22no—
399C23—22no—
400C23—24no—
401C23—24no—
402C23—24no—
403C23—24no—
404C23—24no—
405C23—42no—
406C23—42no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—42no—
411C23—41no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—44no—
419C23—44no—
420C23—82no—
421C23—82no—
422C23—82no—
423C23—82no—
424C23—81no—
425C23—82no—
426C23—81no—
427C23—82no—
428C23—81no—
429C23—82no—
430C23—81no—
431C23—82no—
432C23—82no—
433C23—82no—
434C23—82no—
435C23—82no—
436C23#48814yes—
437C23C2×#12011yescharacteristic
438C23C2×#12411yescharacteristic
439C23#53511yescharacteristic
440C23#54612yes—
441C23#54712yes—
442C23C2×#13512yes—
443C23#57512yes—
444C23#57612yes—
445C23#57712yes—
446D4—44no—
447D4—44no—
448D4—816no—
449D4—82no—
450D4—82no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—816no—
484C2×C4—81no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—81no—
488C2×C4—82no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
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#52511yescharacteristic
505C2×C4#52912yes—
506C2×C4#53211yescharacteristic
507C2×C4#53511yescharacteristic
508C2×C4#53611yescharacteristic
509C2×C4#53812yes—
510C2×C4#53912yes—
511C2×C4#54212yes—
512C2×C4#54612yes—
513C2×C4#54612yes—
514C2×C4#54612yes—
515C2×C4#54812yes—
516C2×C4#55112yes—
517C2×C4#55212yes—
518C2×C4#55912yes—
519C2×C4#56212yes—
520C8—44no—
521C8—82no—
522C22—24no—
523C22—28no—
524C22—42no—
525C22—42no—
526C22—44no—
527C22—44no—
528C22—44no—
529C22—44no—
530C22—44no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22—84no—
535C22—84no—
536C22—162no—
537C22—162no—
538C22—162no—
539C22—162no—
540C22#240411yescharacteristic
541C22#243211yescharacteristic
542C22#243511yescharacteristic
543C22#248812yes—
544C22#271012yes—
545C22#300212yes—
546C22#301212yes—
547C22C2×#48811yescharacteristic
548C22#313612yes—
549C22#314011yescharacteristic
550C22#321112yes—
551C22#322011yescharacteristic
552C22#322211yescharacteristic
553C22#329712yes—
554C22#330112yes—
555C22#330712yes—
556C22#330912yes—
557C22#333712yes—
558C22#333912yes—
559C22#335112yes—
560C22#335512yes—
561C4—42no—
562C4—44no—
563C4—82no—
564C4—82no—
565C4—82no—
566C4—82no—
567C4—84no—
568C4—162no—
569C4#245812yes—
570C4#255712yes—
571C4#268712yes—
572C4#269612yes—
573C2—44no—
574C2—162no—
575C2#1566811yescharacteristic
576C2#2396212yes—
577C2#2570712yes—
578C2#2585911yescharacteristic
579C2#2586711yescharacteristic
580C2#2770112yes—
581C2#2772512yes—
582C2#3205612yes—
583C2#3206012yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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
2. C2×#2025C2semidirect
3. C2×#1991C2semidirect
4. #15216C2semidirect
5. #11372C2semidirect
6. #11268C2semidirect
7. #11189C2semidirect
9. #10925C2semidirect
10. #10921C2semidirect
11. #10730C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25042
C2×#20252
C2×#19912
#152162
#113722
#112682
#111892
#109692
#109252
#109212
#107302