Finite Group Databasebeta

← #448420

Subgroups — #448420

Order 512.

1258 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2497C211yesmaximal, characteristic
2C2×#2057C211yesmaximal, characteristic, complemented
3C2×#1990C211yesmaximal, characteristic, complemented
4#15125C211yesmaximal, characteristic
5#11371C211yesmaximal, characteristic, complemented
6#11293C212yesmaximal, complemented
7#11200C212yesmaximal
8#11096C212yesmaximal
9#10921C211yesmaximal, characteristic
10#10907C212yesmaximal, complemented
11#10880C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×C22⋊C8C2212yes—
15C22×C4×C8C2211yescharacteristic
16#2497—24no—
17#2497—24no—
18C2×#400—21no—
19C2×#396—21no—
20C2×#393—21no—
21C2×#389C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#388—21no—
24C2×#387—21no—
25C2×#384C2211yescharacteristic
26C2×#380C2211yescharacteristic
27C2×#380C2211yescharacteristic
28C2×#380C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#2057—24no—
32#1990—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42#1446—21no—
43#1442—21no—
44#1439—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C23×Q8—21no—
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—42no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4D411yescharacteristic
70C22×C22⋊C4D411yescharacteristic
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×#107—24no—
75C2×#107—24no—
76C2×#107—42no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—24no—
79C2×C22⋊C8—42no—
80C2×C4×C8—22no—
81C2×C4×C8—42no—
82#400—28no—
83#396—28no—
84#393—44no—
85#389—22no—
86#389—22no—
87#389—22no—
88#389D414yes—
89#388—28no—
90#387—28no—
91#384—22no—
92#384D414yes—
93#380—22no—
94#380—22no—
95#380—22no—
96#380—22no—
97#380—24no—
98#380—42no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—22no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95D411yescharacteristic
114C2×#95D411yescharacteristic
115#340—24no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—42no—
122#336—24no—
123#336—22no—
124C25C2×D411yescharacteristic
125C22×Q8—44no—
126C22×Q8—42no—
127C22×Q8—42no—
128C22×Q8—42no—
129C22×Q8—82no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2411yescharacteristic
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—
163C22×C8#4114yes—
164C4⋊Q8—28no—
165C4⋊Q8—48no—
166C4⋊Q8—44no—
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—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
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—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4#4114yes—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—82no—
216C2×C22⋊C4—82no—
217C2×C22⋊C4—82no—
218C2×C22⋊C4Q1614yes—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—24no—
226C2×C42—82no—
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229#107—48no—
230C22⋊C8—48no—
231C4×C8—44no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—22no—
238#95—22no—
239#95—44no—
240#95—44no—
241#95—42no—
242#95—42no—
243#95—82no—
244C24—21no—
245C24—21no—
246C24—21no—
247C24—22no—
248C24—22no—
249C24—21no—
250C24—21no—
251C24—21no—
252C24—22no—
253C24—22no—
254C24C2×Q1612yes—
255C24C22×D411yescharacteristic
256C2×Q8—48no—
257C2×Q8—88no—
258C2×Q8—88no—
259C2×Q8—84no—
260C2×Q8—84no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
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—42no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
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#12012yes—
330C22×C4#12012yes—
331C22×C4C2×Q1612yes—
332C22×C4C2×Q1612yes—
333C22×C4#13512yes—
334C22×C4AGL(1,ℤ/8)12yes—
335C22×C4#13712yes—
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C2×#4112yes—
339C22×C4C2×#4112yes—
340C22×C4C2×#4112yes—
341C22×C4H(5,𝔽2)12yes—
342C22×C42−1+412yes—
343C22×C42−1+412yes—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—48no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—48no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—84no—
379C4⋊C4—84no—
380C4⋊C4—84no—
381C22⋊C4—28no—
382C22⋊C4—44no—
383C22⋊C4—84no—
384C22⋊C4—84no—
385C22⋊C4—88no—
386C42—22no—
387C42—22no—
388C42—24no—
389C42—41no—
390C42—41no—
391C42—42no—
392C42—84no—
393C23—24no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—24no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—41no—
402C23—42no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—42no—
407C23—41no—
408C23—42no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23C2×#12011yescharacteristic
417C23#53211yescharacteristic
418C23#53611yescharacteristic
419C23#54612yes—
420C23#54712yes—
421C23C22×Q1611yescharacteristic
422C23C2×#13511yescharacteristic
423C23C2×AGL(1,ℤ/8)11yescharacteristic
424C23C2×#13711yescharacteristic
425C23#57612yes—
426C23#57712yes—
427Q8—816no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—41no—
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—42no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
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#44914yes—
490C2×C4#45114yes—
491C2×C4#52511yescharacteristic
492C2×C4#52912yes—
493C2×C4#53211yescharacteristic
494C2×C4#53511yescharacteristic
495C2×C4#53611yescharacteristic
496C2×C4#53912yes—
497C2×C4#54112yes—
498C2×C4#54212yes—
499C2×C4#54612yes—
500C2×C4#54712yes—
501C2×C4#54812yes—
502C2×C4#54812yes—
503C2×C4#55112yes—
504C2×C4#55612yes—
505C2×C4#55612yes—
506C2×C4#56312yes—
507C8—44no—
508C8—44no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—44no—
513C22—44no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22—81no—
518C22—82no—
519C22—82no—
520C22#240411yescharacteristic
521C22#243211yescharacteristic
522C22#243211yescharacteristic
523C22#252112yes—
524C22#268912yes—
525C22C2×#45111yescharacteristic
526C22C2×#44911yescharacteristic
527C22#300811yescharacteristic
528C22#300911yescharacteristic
529C22#317011yescharacteristic
530C22#317912yes—
531C22#318211yescharacteristic
532C22#323011yescharacteristic
533C22#323111yescharacteristic
534C22#324112yes—
535C22#329712yes—
536C22#329912yes—
537C22#331612yes—
538C22#331812yes—
539C22#333812yes—
540C22#334012yes—
541C22#335412yes—
542C22#335712yes—
543C4—24no—
544C4—24no—
545C4—42no—
546C4—81no—
547C4—82no—
548C4—82no—
549C4—81no—
550C4—82no—
551C4—82no—
552C4—84no—
553C4—162no—
554C4#246912yes—
555C4#256012yes—
556C4#271612yes—
557C4#272312yes—
558C2—82no—
559C2#1591011yescharacteristic
560C2#2396311yescharacteristic
561C2#2397011yescharacteristic
562C2#2567411yescharacteristic
563C2#2568511yescharacteristic
564C2#2570011yescharacteristic
565C2#2570611yescharacteristic
566C2#2806912yes—
567C2#2809112yes—
568C2#3142812yes—
569C2#3142912yes—
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×#2057C2semidirect
3. C2×#1990C2semidirect
5. #11371C2semidirect
6. #11293C2semidirect
10. #10907C2semidirect
11. #10880C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24972
C2×#20572
C2×#19902
#151252
#113712
#112932
#112002
#110962
#109212
#109072
#108802