Finite Group Databasebeta

← #460468

Subgroups — #460468

Order 512.

1262 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2C2×#2084C211yesmaximal, characteristic, complemented
3C2×#2083C211yesmaximal, characteristic, complemented
4#14340C211yesmaximal, characteristic
5#11384C212yesmaximal, complemented
6#11325C212yesmaximal, complemented
7#11208C211yesmaximal, characteristic, complemented
8#11208C211yesmaximal, characteristic, complemented
9#11096C212yesmaximal
10#11095C212yesmaximal
11#10713C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16C22×C22⋊C8C2212yes—
17#2628—24no—
18#2628—24no—
19C2×#400—21no—
20C2×#399—21no—
21C2×#395—21no—
22C2×#393—21no—
23C2×#389C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×#380C2212yes—
31#2084—24no—
32#2083—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2211yescharacteristic
42#1441—21no—
43#1441—21no—
44#1439—21no—
45#1439—21no—
46C4⋊#95C2212yes—
47C23×Q8—21no—
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—42no—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—24no—
58C2×#117—42no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4D411yescharacteristic
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—21no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4D411yescharacteristic
76C22×C22⋊C4D411yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C2×#107—24no—
80C2×#107—42no—
81C2×#107—42no—
82C2×#106—24no—
83C2×#106—42no—
84C2×#106—42no—
85C2×C22⋊C8—42no—
86C2×C22⋊C8—42no—
87C2×C22⋊C8Q818yescomplemented
88#400—28no—
89#399—28no—
90#395—28no—
91#393—28no—
92#389—22no—
93#389—22no—
94#389D414yes—
95#389D414yes—
96#384—22no—
97#384—22no—
98#382—22no—
99#382—22no—
100#382—42no—
101#380—22no—
102#380—22no—
103#380—22no—
104#380—22no—
105#380—42no—
106#380—42no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118#340—24no—
119#340—24no—
120#340—42no—
121#340—42no—
122#340—42no—
123#340—42no—
124#336—22no—
125#336—22no—
126#336—24no—
127C25C2×D411yescharacteristic
128C22×Q8—24no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8—82no—
132C22×Q8—82no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×Q814yes—
154C23×C4C2411yescharacteristic
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—22no—
166C22×C8—22no—
167C22×C8C2×Q814yes—
168C22×C8C2×Q814yes—
169C4⋊Q8—48no—
170C4⋊Q8—48no—
171#117—48no—
172#117—48no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—82no—
232C2×C22⋊C4—82no—
233C2×C22⋊C4—82no—
234C2×C22⋊C4—82no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42#4114yes—
243#107—48no—
244#106—48no—
245C22⋊C8—48no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—44no—
253#95—42no—
254#95—42no—
255#95—82no—
256#95—82no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24C4⋊Q812yes—
268C24C22×D411yescharacteristic
269C2×Q8—48no—
270C2×Q8—88no—
271C2×Q8—84no—
272C2×Q8—84no—
273C2×Q8—88no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
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#12012yes—
347C22×C4#12012yes—
348C22×C4#12312yes—
349C22×C4#12312yes—
350C22×C4C4⋊Q812yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×Q812yes—
354C22×C4C2×#4112yes—
355C22×C4H(5,𝔽2)11yescharacteristic
356C22×C4H(5,𝔽2)11yescharacteristic
357C22×C4H(5,𝔽2)11yescharacteristic
358C22×C4H(5,𝔽2)12yes—
359C22×C42−1+411yescharacteristic
360C22×C42−1+412yes—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—48no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—84no—
399C4⋊C4—84no—
400C4⋊C4—84no—
401C4⋊C4—84no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—84no—
405C22⋊C4—84no—
406C22⋊C4—88no—
407C42—24no—
408C42—24no—
409C42—24no—
410C42—24no—
411C23—24no—
412C23—24no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—41no—
417C23—42no—
418C23—42no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23C2×#12011yescharacteristic
435C23C2×#12311yescharacteristic
436C23C2×C4⋊Q811yescharacteristic
437C23#53411yescharacteristic
438C23#53511yescharacteristic
439C23#53611yescharacteristic
440C23#53711yescharacteristic
441C23#57612yes—
442C23#57712yes—
443C23#57712yes—
444C23#57812yes—
445Q8—816no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
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—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4#53511yescharacteristic
512C2×C4#53511yescharacteristic
513C2×C4#53711yescharacteristic
514C2×C4#53711yescharacteristic
515C2×C4#54112yes—
516C2×C4#54212yes—
517C2×C4#54312yes—
518C2×C4#54412yes—
519C2×C4#54612yes—
520C2×C4#54612yes—
521C2×C4D4×Q812yes—
522C2×C4D4×Q812yes—
523C2×C4#55412yes—
524C2×C4#55412yes—
525C2×C4#56212yes—
526C2×C4#56212yes—
527C8—44no—
528C8—44no—
529C22—44no—
530C22—44no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—82no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—82no—
540C22#240511yescharacteristic
541C22#243411yescharacteristic
542C22#250211yescharacteristic
543C22#261511yescharacteristic
544C22#262511yescharacteristic
545C22#265911yescharacteristic
546C22#281611yescharacteristic
547C22#300811yescharacteristic
548C22#300911yescharacteristic
549C22#301011yescharacteristic
550C22#301111yescharacteristic
551C22#309612yes—
552C22#309712yes—
553C22#316312yes—
554C22#316412yes—
555C22#319812yes—
556C22#320112yes—
557C22#322412yes—
558C22#322512yes—
559C22#324312yes—
560C22#324412yes—
561C22#325712yes—
562C22#325812yes—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—82no—
569C4—82no—
570C4—82no—
571C4—82no—
572C4—162no—
573C4—162no—
574C4#269212yes—
575C4#269212yes—
576C4#270412yes—
577C4#270412yes—
578C2—82no—
579C2#1636511yescharacteristic
580C2#2401311yescharacteristic
581C2#2401411yescharacteristic
582C2#2582011yescharacteristic
583C2#2582111yescharacteristic
584C2#2795912yes—
585C2#2966911yescharacteristic
586C2#2967211yescharacteristic
587C2#2994512yes—
588C2#3029912yes—
589C2#3278112yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2084C2semidirect
3. C2×#2083C2semidirect
5. #11384C2semidirect
6. #11325C2semidirect
7. #11208C2semidirect
8. #11208C2semidirect
87. C2×C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#20842
C2×#20832
#143402
#113842
#113252
#112082
#110962
#110952
#107132