Finite Group Databasebeta

← #4453181

Subgroups — #4453181

Order 512.

4534 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#538C211yesmaximal, characteristic
2C2×#2487C212yesmaximal, complemented
3C22×#399C211yesmaximal, characteristic, complemented
4#16674C214yesmaximal, complemented
5#16510C214yesmaximal, complemented
6#16363C214yesmaximal, complemented
7#16360C214yesmaximal, complemented
8#14720C212yesmaximal, complemented
9#14719C212yesmaximal, complemented
10#14718C214yesmaximal, complemented
11#14338C211yesmaximal, characteristic, complemented
12#13960C212yesmaximal, complemented
13C2×#538—216no—
14C2×#538—28no—
15C22×#126C2212yes—
16C22×#124C2212yes—
17C22×#123C2212yes—
18C22×C22⋊Q8C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×#121C2212yes—
21C22×#120C2212yes—
22C22×C4×D4C2212yes—
23C23×C4⋊C4C2211yescharacteristic
24C23×C4⋊C4C2212yes—
25C23×C22⋊C4C2211yescharacteristic
26C23×C22⋊C4C2212yes—
27#2487—216no—
28C2×#400C2214yescomplemented
29C2×#399—28no—
30C2×#399C2214yes—
31C2×#399C2214yes—
32C2×#399C2214yescomplemented
33C2×#395C2214yescomplemented
34C2×#394C2218yes—
35C2×#394C2214yescomplemented
36C2×#389C2214yescomplemented
37C2×#386C2214yes—
38C2×#386C2214yes—
39C2×#385C2214yes—
40C2×#380C2212yes—
41C2×#380C2214yescomplemented
42C2×#379C2212yes—
43C22×#95C2211yescharacteristic
44#1444C2218yescomplemented
45#1441C2214yescomplemented
46#1441C2214yescomplemented
47#1441C2214yescomplemented
48#1440C2218yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2218yes—
52#1439C2218yescomplemented
53#1439C2214yescomplemented
54#1438C2218yes—
55#1434C2214yescomplemented
56C22⋊#95C2212yes—
57C22⋊#95C2214yes—
58C22⋊#95C2214yescomplemented
59C26C2311yescharacteristic
60C23×Q8C2311yescharacteristic
61C23×D4C2312yes—
62C23×D4C2312yes—
63C23×D4C2311yescharacteristic
64C23×D4C2312yes—
65C24×C4C2312yes—
66C24×C4C2312yes—
67C24×C4C2311yescharacteristic
68C24×C4C2312yes—
69C24×C4C2311yescharacteristic
70#538—2128no—
71C2×#126—28no—
72C2×#126—28no—
73C2×#126—28no—
74C2×#124—28no—
75C2×#124—28no—
76C2×#124—28no—
77C2×#123—28no—
78C2×#123—28no—
79C2×#123—28no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—28no—
82C2×#121—28no—
83C2×#121—28no—
84C2×#121—24no—
85C2×#121—28no—
86C2×#121—28no—
87C2×#120—28no—
88C2×#120—28no—
89C2×#120—28no—
90C2×C4×D4—28no—
91C2×C4×D4—28no—
92C2×C4×D4D4116yes—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—28no—
96C22×C4⋊C4—48no—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2318yes—
105C22×C4⋊C4C2314yes—
106C22×C22⋊C4—28no—
107C22×C22⋊C4—28no—
108C22×C22⋊C4—48no—
109C22×C22⋊C4D418yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2318yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2318yes—
120C22×C22⋊C4C2318yes—
121C22×C22⋊C4C2318yes—
122C22×C42C2312yes—
123#400—216no—
124#399—216no—
125#399—216no—
126#399—216no—
127#395—48no—
128#394—416no—
129#394—48no—
130#389—216no—
131#386—216no—
132#386—48no—
133#385—216no—
134#380—28no—
135#380—216no—
136#379—44no—
137C2×#95—24no—
138C2×#95C2318yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2318yes—
142C2×#95C2318yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2318yes—
146C2×#95C2318yes—
147C25—24no—
148C25—24no—
149C25—22no—
150C25—22no—
151C25—44no—
152C25—44no—
153C25C2×D414yes—
154C25C2412yes—
155C25C2412yes—
156C25C2411yescharacteristic
157C22×Q8—22no—
158C22×Q8—24no—
159C22×Q8—44no—
160C22×D4—24no—
161C22×D4—24no—
162C22×D4—24no—
163C22×D4—24no—
164C22×D4—24no—
165C22×D4—24no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—22no—
174C22×D4—48no—
175C22×D4—48no—
176C22×D4—44no—
177C22×D4C2×D418yes—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—28no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—28no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—48no—
193C23×C4C2×D414yes—
194C23×C4C2×D418yes—
195C23×C4C2×D418yes—
196C23×C4C2×D414yes—
197C23×C4C2418yes—
198C23×C4C2414yes—
199C23×C4C2411yescharacteristic
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206#126—264no—
207#124—264no—
208#123—264no—
209C22⋊Q8—232no—
210#121—232no—
211#121—264no—
212#120—264no—
213C4×D4—264no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—216no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—48no—
235C2×C4⋊C4—432no—
236C2×C4⋊C4—416no—
237C2×C4⋊C4—48no—
238C2×C4⋊C4—48no—
239C2×C4⋊C4C2×D418yes—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—232no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—216no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—216no—
263C2×C22⋊C4—216no—
264C2×C22⋊C4—216no—
265C2×C22⋊C4—48no—
266C2×C22⋊C4—432no—
267C2×C22⋊C4—48no—
268C2×C22⋊C4—48no—
269C2×C22⋊C4—48no—
270C2×C22⋊C4—48no—
271C2×C22⋊C4—48no—
272C2×C22⋊C4—48no—
273C2×C22⋊C4—48no—
274C2×C22⋊C4C2×D418yes—
275C2×C22⋊C4C2×D418yes—
276C2×C22⋊C4C2×D418yes—
277C2×C22⋊C4C2×D418yes—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42C2×D418yes—
281#95—28no—
282#95—216no—
283#95—28no—
284#95—216no—
285#95—216no—
286#95—48no—
287#95—48no—
288#95—44no—
289#95—44no—
290C24—24no—
291C24—24no—
292C24—28no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—28no—
298C24—28no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—28no—
308C24—28no—
309C24—28no—
310C24—28no—
311C24—48no—
312C24—48no—
313C24—432no—
314C24—48no—
315C24—48no—
316C24—88no—
317C24—88no—
318C24—88no—
319C24—88no—
320C24C22×D414yes—
321C24C22×D414yes—
322C24C22×D414yes—
323C24C22×D412yes—
324C24H(5,𝔽2)12yes—
325C242−1+414yes—
326C242−1+412yes—
327C24C2511yescharacteristic
328C2×Q8—28no—
329C2×Q8—84no—
330C2×Q8—88no—
331C2×D4—216no—
332C2×D4—216no—
333C2×D4—216no—
334C2×D4—216no—
335C2×D4—216no—
336C2×D4—28no—
337C2×D4—48no—
338C2×D4—48no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—84no—
342C2×D4—88no—
343C2×D4—88no—
344C2×D4—88no—
345C2×D4—88no—
346C2×D4—88no—
347C2×D4—88no—
348C2×D4—88no—
349C22×C4—28no—
350C22×C4—216no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—28no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—28no—
372C22×C4—24no—
373C22×C4—28no—
374C22×C4—216no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—24no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—416no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—416no—
391C22×C4—416no—
392C22×C4—416no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—88no—
399C22×C4—88no—
400C22×C4—88no—
401C22×C4—816no—
402C22×C4#120116yes—
403C22×C4C22×D414yes—
404C22×C4C22×D414yes—
405C22×C4C22×D414yes—
406C22×C4C22×D412yes—
407C22×C4C22×D412yes—
408C22×C4C22×D414yes—
409C22×C4C22×D414yes—
410C22×C4C22×D414yes—
411C22×C4H(5,𝔽2)18yes—
412C4⋊C4—216no—
413C4⋊C4—28no—
414C4⋊C4—28no—
415C4⋊C4—216no—
416C4⋊C4—416no—
417C4⋊C4—416no—
418C4⋊C4—416no—
419C4⋊C4—416no—
420C4⋊C4—816no—
421C22⋊C4—216no—
422C22⋊C4—216no—
423C22⋊C4—216no—
424C22⋊C4—216no—
425C22⋊C4—216no—
426C22⋊C4—216no—
427C22⋊C4—216no—
428C22⋊C4—432no—
429C22⋊C4—416no—
430C22⋊C4—816no—
431C22⋊C4—816no—
432C22⋊C4—816no—
433C42—216no—
434C23—24no—
435C23—24no—
436C23—28no—
437C23—28no—
438C23—28no—
439C23—28no—
440C23—216no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—28no—
448C23—28no—
449C23—28no—
450C23—28no—
451C23—432no—
452C23—416no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—48no—
470C23—48no—
471C23—48no—
472C23—48no—
473C23—48no—
474C23—48no—
475C23—48no—
476C23—48no—
477C23—816no—
478C23—88no—
479C23—88no—
480C23—832no—
481C23—832no—
482C23C2×#12014yes—
483C23#53618yes—
484C23#53614yes—
485C23#53718yes—
486C23#53714yes—
487C23C23×D411yescharacteristic
488C23C23×D412yes—
489C23C2×H(5,𝔽2)11yescharacteristic
490C23C2×H(5,𝔽2)14yes—
491C23C2×2−1+411yescharacteristic
492C23C2×2−1+412yes—
493C23#58514yes—
494Q8—88no—
495D4—432no—
496D4—816no—
497D4—88no—
498D4—816no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—28no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—28no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—28no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—416no—
517C2×C4—432no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—48no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—48no—
537C2×C4—48no—
538C2×C4—48no—
539C2×C4—48no—
540C2×C4—48no—
541C2×C4—48no—
542C2×C4—48no—
543C2×C4—48no—
544C2×C4—816no—
545C2×C4—88no—
546C2×C4—816no—
547C2×C4C2×#12018yes—
548C22—28no—
549C22—28no—
550C22—24no—
551C22—216no—
552C22—216no—
553C22—24no—
554C22—44no—
555C22—44no—
556C22—48no—
557C22—48no—
558C22—44no—
559C22—44no—
560C22—48no—
561C22—48no—
562C22—44no—
563C22—44no—
564C22—832no—
565C22—84no—
566C22—84no—
567C22—84no—
568C22—84no—
569C22—84no—
570C22—84no—
571C22—84no—
572C22—84no—
573C22#261618yes—
574C22C22×#12011yescharacteristic
575C22C2×#53611yescharacteristic
576C22C2×#53612yes—
577C22C2×#53711yescharacteristic
578C22C2×#53712yes—
579C22#348412yes—
580C22#348512yes—
581C22#348514yes—
582C22#349814yes—
583C22#353414yes—
584C22#353514yes—
585C22#354118yes—
586C4—28no—
587C4—44no—
588C4—48no—
589C4—44no—
590C4—42no—
591C4—42no—
592C4—44no—
593C4—88no—
594C4—84no—
595C2—28no—
596C2—84no—
597C2—84no—
598C2C2×#261611yescharacteristic
599C2#3544611yescharacteristic
600C2#3552711yescharacteristic
601C2#3670512yes—
602C2#3670912yes—
603C2#4136814yes—
604C2#4137014yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#538C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2487C2semidirect
3. C22×#399C2semidirect
4. #16674C2semidirect
5. #16510C2semidirect
6. #16363C2semidirect
7. #16360C2semidirect
8. #14720C2semidirect
9. #14719C2semidirect
10. #14718C2semidirect
11. #14338C2semidirect
12. #13960C2semidirect
28. C2×#400C22semidirect
32. C2×#399C22semidirect
33. C2×#395C22semidirect
35. C2×#394C22semidirect
36. C2×#389C22semidirect
41. C2×#380C22semidirect
44. #1444C22semidirect
45. #1441C22semidirect
46. #1441C22semidirect
47. #1441C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
55. #1434C22semidirect
58. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5382
C2×#24872
C22×#3992
#166742
#165102
#163632
#163602
#147202
#147192
#147182
#143382
#139602