Finite Group Databasebeta

← #440856

Subgroups — #440856

Order 512.

1290 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2442C211yesmaximal, characteristic
2C2×#2099C211yesmaximal, characteristic, complemented
3C2×#1836C211yesmaximal, characteristic, complemented
4#15195C211yesmaximal, characteristic
5#11385C212yesmaximal, complemented
6#11358C211yesmaximal, characteristic, complemented
7#11292C212yesmaximal, complemented
8#11200C212yesmaximal
9#10922C211yesmaximal, characteristic
10#10881C211yesmaximal, characteristic, complemented
11#10730C212yesmaximal
12C22×#125C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15C22×#97C2211yescharacteristic
16#2442—24no—
17#2442—24no—
18C2×#400—21no—
19C2×#398—21no—
20C2×#389C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#385—21no—
23C2×#384C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#380—21no—
26C2×#380C2212yes—
27C2×#380C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29#2099—24no—
30#1836—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1442—21no—
42#1442—21no—
43#1439—21no—
44#1439—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×#125—22no—
52C2×#125—22no—
53C2×#125—24no—
54C2×#125—42no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—24no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—42no—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4D411yescharacteristic
75C22×C22⋊C4D411yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×C22⋊C8—24no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—42no—
81C2×#97—24no—
82C2×#97—22no—
83#400—28no—
84#398—28no—
85#389—22no—
86#389—22no—
87#389—22no—
88#389D414yes—
89#385—28no—
90#384—22no—
91#384—22no—
92#382—22no—
93#382D414yes—
94#380—28no—
95#380—22no—
96#380—22no—
97#380—24no—
98#380—24no—
99C4×C22⋊C4—22no—
100C4×C22⋊C4—22no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95D411yescharacteristic
112C2×#95D411yescharacteristic
113#340—24no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—42no—
120#340—42no—
121#336—22no—
122#336—42no—
123C25C2×D411yescharacteristic
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—24no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—44no—
139C23×C4—42no—
140C23×C4—42no—
141C23×C4—82no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4#4114yes—
148C23×C4C2411yescharacteristic
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
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#4114yes—
163#125—48no—
164#125—44no—
165#125—44no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
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—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
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—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—48no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4#4114yes—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—82no—
222C2×C22⋊C4—82no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C22⋊C8—48no—
232#97—28no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—22no—
239#95—22no—
240#95—42no—
241#95—42no—
242#95—44no—
243#95—82no—
244#95—82no—
245C24—21no—
246C24—21no—
247C24—22no—
248C24—22no—
249C24—21no—
250C24—21no—
251C24—22no—
252C24—22no—
253C24#12412yes—
254C24#13712yes—
255C24#13712yes—
256C24C22×D411yescharacteristic
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—48no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—48no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
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—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—84no—
332C22×C4—88no—
333C22×C4—88no—
334C22×C4#12012yes—
335C22×C4#12012yes—
336C22×C4#12412yes—
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C2×#4112yes—
340C22×C4C2×#4112yes—
341C22×C4C2×#4112yes—
342C22×C4H(5,𝔽2)12yes—
343C22×C42−1+412yes—
344C22×C42−1+412yes—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—48no—
365C4⋊C4—48no—
366C4⋊C4—48no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—84no—
382C4⋊C4—84no—
383C4⋊C4—84no—
384C4⋊C4—84no—
385C4⋊C4#12414yes—
386C22⋊C4—44no—
387C22⋊C4—44no—
388C22⋊C4—48no—
389C22⋊C4—88no—
390C42—22no—
391C42—22no—
392C42—24no—
393C42—41no—
394C42—41no—
395C42—42no—
396C23—24no—
397C23—24no—
398C23—24no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—24no—
404C23—22no—
405C23—22no—
406C23—24no—
407C23—24no—
408C23—42no—
409C23—42no—
410C23—41no—
411C23—41no—
412C23—42no—
413C23—42no—
414C23—41no—
415C23—41no—
416C23—42no—
417C23—42no—
418C23#49114yes—
419C23C2×#12011yescharacteristic
420C23C2×#12411yescharacteristic
421C23#53611yescharacteristic
422C23#54612yes—
423C23#54712yes—
424C23C2×#13711yescharacteristic
425C23C2×#13711yescharacteristic
426C23#57511yescharacteristic
427C23#57511yescharacteristic
428C23#57612yes—
429C23#57712yes—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—816no—
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#52511yescharacteristic
495C2×C4C2×#12411yescharacteristic
496C2×C4#52912yes—
497C2×C4#53511yescharacteristic
498C2×C4#53611yescharacteristic
499C2×C4#53912yes—
500C2×C4#54112yes—
501C2×C4#54212yes—
502C2×C4#54612yes—
503C2×C4#54712yes—
504C2×C4#54812yes—
505C2×C4#54812yes—
506C2×C4#55112yes—
507C2×C4#55212yes—
508C2×C4#55212yes—
509C2×C4#56312yes—
510C8—44no—
511C8—82no—
512C22—24no—
513C22—28no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—44no—
521C22—44no—
522C22—44no—
523C22#240611yescharacteristic
524C22#243511yescharacteristic
525C22#243611yescharacteristic
526C22#248812yes—
527C22#269812yes—
528C22#300711yescharacteristic
529C22#300711yescharacteristic
530C22#300811yescharacteristic
531C22#301011yescharacteristic
532C22C2×#49111yescharacteristic
533C22#313612yes—
534C22#314211yescharacteristic
535C22#323611yescharacteristic
536C22#323611yescharacteristic
537C22#324112yes—
538C22#331412yes—
539C22#331612yes—
540C22#331912yes—
541C22#332112yes—
542C22#334412yes—
543C22#334612yes—
544C22#335112yes—
545C22#335512yes—
546C4—42no—
547C4—44no—
548C4—42no—
549C4—42no—
550C4—82no—
551C4—81no—
552C4—82no—
553C4—81no—
554C4—82no—
555C4—162no—
556C4—162no—
557C4#247712yes—
558C4#248912yes—
559C4#269012yes—
560C4#272312yes—
561C2—44no—
562C2#1567011yescharacteristic
563C2#2410211yescharacteristic
564C2#2411111yescharacteristic
565C2#2589511yescharacteristic
566C2#2589711yescharacteristic
567C2#2591211yescharacteristic
568C2#2591511yescharacteristic
569C2#2771512yes—
570C2#2772612yes—
571C2#3171012yes—
572C2#3171212yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
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×#2099C2semidirect
3. C2×#1836C2semidirect
5. #11385C2semidirect
6. #11358C2semidirect
7. #11292C2semidirect
10. #10881C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24422
C2×#20992
C2×#18362
#151952
#113852
#113582
#112922
#112002
#109222
#108812
#107302