Finite Group Databasebeta

← #440810

Subgroups — #440810

Order 512.

1484 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2504C211yesmaximal, characteristic
2C2×#2073C211yesmaximal, characteristic, complemented
3C2×#1835C211yesmaximal, characteristic, complemented
4#15223C211yesmaximal, characteristic, complemented
5#11372C211yesmaximal, characteristic, complemented
6#11208C212yesmaximal, complemented
7#11066C212yesmaximal, complemented
8#10969C212yesmaximal
9#10925C211yesmaximal, characteristic, complemented
10#10921C211yesmaximal, characteristic, complemented
11#10730C212yesmaximal, complemented
12C22×#121C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C22×#102C2212yes—
15C22×#102C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C4×C8C2211yescharacteristic
18#2504—24no—
19#2504—24no—
20C2×#397—22no—
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#389C2212yes—
24C2×#386—22no—
25C2×#382C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×#380C2212yes—
29C2×C4×C22⋊C4C2211yescharacteristic
30#2073—24no—
31#1835—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40#1445—22no—
41#1439—22no—
42C4⋊#95C2211yescharacteristic
43C23×D4—21no—
44C23×D4C2311yescharacteristic
45C24×C4—21no—
46C24×C4C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×#121—22no—
50C2×#121—22no—
51C2×#121—24no—
52C2×#121—42no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—42no—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4D412yes—
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C2×#102—24no—
72C2×#102—24no—
73C2×#102—24no—
74C2×#102—42no—
75C2×#102—42no—
76C2×#102—42no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—24no—
79C2×C22⋊C8—42no—
80C2×C4×C8—24no—
81C2×C4×C8—22no—
82#397—216no—
83#390—22no—
84#390—22no—
85#390—22no—
86#390D414yes—
87#389—24no—
88#389—42no—
89#386—216no—
90#382—22no—
91#382—22no—
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—22no—
114#336—42no—
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#121—48no—
167#121—48no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
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—24no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—48no—
190C2×C4⋊C4—48no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4#4114yes—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—82no—
211C2×C22⋊C4—82no—
212C2×C22⋊C4—82no—
213C2×C22⋊C4—82no—
214C2×C22⋊C4—82no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223#102—48no—
224#102—48no—
225C22⋊C8—48no—
226C4×C8—28no—
227#95—28no—
228#95—28no—
229#95—24no—
230#95—44no—
231#95—44no—
232#95—44no—
233#95—82no—
234C24—21no—
235C24—21no—
236C24—22no—
237C24—22no—
238C24—22no—
239C24—22no—
240C24—22no—
241C24—22no—
242C24—41no—
243C24—41no—
244C24—42no—
245C24—42no—
246C24—42no—
247C24—42no—
248C24—42no—
249C24—42no—
250C24#12412yes—
251C24#13514yes—
252C24C22×D411yescharacteristic
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—48no—
258C2×D4—416no—
259C2×D4—41no—
260C2×D4—42no—
261C2×D4—41no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—84no—
272C2×D4—88no—
273C2×D4—84no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—48no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
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—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—88no—
338C22×C4—88no—
339C22×C4—84no—
340C22×C4—84no—
341C22×C4#12014yes—
342C22×C4#12412yes—
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C2×#4112yes—
346C22×C4C2×#4112yes—
347C22×C4C2×#4112yes—
348C22×C4H(5,𝔽2)12yes—
349C22×C4H(5,𝔽2)12yes—
350C22×C42−1+412yes—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—24no—
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—
363C2×C8—42no—
364C2×C8—42no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—48no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—88no—
381C22⋊C4—48no—
382C22⋊C4—48no—
383C22⋊C4—88no—
384C22⋊C4—88no—
385C22⋊C4—88no—
386C42—22no—
387C42—22no—
388C42—24no—
389C42—41no—
390C42—41no—
391C42—42no—
392C23—24no—
393C23—24no—
394C23—24no—
395C23—22no—
396C23—22no—
397C23—24no—
398C23—24no—
399C23—24no—
400C23—24no—
401C23—24no—
402C23—42no—
403C23—42no—
404C23—41no—
405C23—41no—
406C23—42no—
407C23—41no—
408C23—41no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—44no—
416C23—44no—
417C23—82no—
418C23—82no—
419C23—82no—
420C23—82no—
421C23—81no—
422C23—82no—
423C23—81no—
424C23—82no—
425C23—82no—
426C23—81no—
427C23—81no—
428C23—82no—
429C23—82no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23#48814yes—
434C23C2×#12011yescharacteristic
435C23C2×#12411yescharacteristic
436C23#53511yescharacteristic
437C23#54612yes—
438C23#54712yes—
439C23C2×#13512yes—
440C23#57512yes—
441C23#57612yes—
442C23#57712yes—
443D4—44no—
444D4—44no—
445D4—816no—
446D4—82no—
447D4—82no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—816no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—81no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—81no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4#52511yescharacteristic
502C2×C4#52912yes—
503C2×C4#53211yescharacteristic
504C2×C4#53511yescharacteristic
505C2×C4#53611yescharacteristic
506C2×C4#53912yes—
507C2×C4#53912yes—
508C2×C4#54012yes—
509C2×C4#54612yes—
510C2×C4#54612yes—
511C2×C4#54712yes—
512C2×C4#55012yes—
513C2×C4#55112yes—
514C2×C4#55112yes—
515C2×C4#55212yes—
516C2×C4#56212yes—
517C8—44no—
518C8—82no—
519C22—24no—
520C22—28no—
521C22—42no—
522C22—44no—
523C22—42no—
524C22—44no—
525C22—44no—
526C22—44no—
527C22—44no—
528C22—82no—
529C22—82no—
530C22—84no—
531C22—82no—
532C22—84no—
533C22—162no—
534C22—162no—
535C22—162no—
536C22—162no—
537C22#240611yescharacteristic
538C22#243111yescharacteristic
539C22#243511yescharacteristic
540C22#248812yes—
541C22#270612yes—
542C22#300212yes—
543C22#301212yes—
544C22C2×#48811yescharacteristic
545C22#313612yes—
546C22#314011yescharacteristic
547C22#321112yes—
548C22#322011yescharacteristic
549C22#322211yescharacteristic
550C22#329712yes—
551C22#330112yes—
552C22#332212yes—
553C22#332612yes—
554C22#333712yes—
555C22#333912yes—
556C22#334712yes—
557C22#335012yes—
558C4—44no—
559C4—42no—
560C4—82no—
561C4—82no—
562C4—82no—
563C4—84no—
564C4—82no—
565C4—162no—
566C4#247112yes—
567C4#255312yes—
568C4#269412yes—
569C4#270612yes—
570C2—44no—
571C2—162no—
572C2#1566911yescharacteristic
573C2#2409012yes—
574C2#2557512yes—
575C2#2585911yescharacteristic
576C2#2586711yescharacteristic
577C2#2770112yes—
578C2#2770312yes—
579C2#3194312yes—
580C2#3194512yes—
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×#2073C2semidirect
3. C2×#1835C2semidirect
4. #15223C2semidirect
5. #11372C2semidirect
6. #11208C2semidirect
7. #11066C2semidirect
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×#20732
C2×#18352
#152232
#113722
#112082
#110662
#109692
#109252
#109212
#107302