Finite Group Databasebeta

← #4420102

Subgroups — #4420102

Order 512.

4262 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#551C211yesmaximal, characteristic
2C2×#2658C211yesmaximal, characteristic, complemented
3C2×#2628C212yesmaximal, complemented
4#16674C214yesmaximal, complemented
5#16360C214yesmaximal, complemented
6#15265C214yesmaximal, complemented
7#15262C212yesmaximal, complemented
8#15259C212yesmaximal
9#14807C214yesmaximal, complemented
10#14734C214yesmaximal, complemented
11#14662C211yesmaximal, characteristic, complemented
12#13957C212yesmaximal, complemented
13C2×#551—28no—
14C2×#551—216no—
15C22×C4⋊Q8C2211yescharacteristic, complemented
16C22×C4⋊Q8C2212yescomplemented
17C22×#126C2212yes—
18C22×#124C2211yescharacteristic
19C22×#124C2212yes—
20C22×#123C2211yescharacteristic
21C22×#123C2212yes—
22C22×C22⋊Q8C2212yes—
23C22×#120C2211yescharacteristic
24C22×C4×D4C2212yes—
25C22×#117C2212yes—
26C23×C22⋊C4C2212yes—
27#2658—28no—
28#2628—216no—
29C2×#400C2214yescomplemented
30C2×#399C2214yes—
31C2×#399C2214yes—
32C2×#399C2214yescomplemented
33C2×#395C2214yescomplemented
34C2×#393C2214yes—
35C2×#389C2214yescomplemented
36C2×#385C2218yes—
37C2×#385C2214yescomplemented
38C2×#382C2212yescomplemented
39C2×#380C2214yes—
40C2×#379C2214yes—
41C2×#376C2211yescharacteristic, complemented
42#1446C2212yescomplemented
43#1444C2218yescomplemented
44#1441C2214yescomplemented
45#1441C2218yescomplemented
46#1440C2214yes—
47#1440C2214yescomplemented
48#1439C2214yes—
49#1439C2218yes—
50#1439C2214yes—
51#1439C2218yes—
52#1439C2214yescomplemented
53#1438C2212yes—
54#1438C2218yes—
55#1437C2214yes—
56#1436C2214yescomplemented
57#1434C2214yescomplemented
58C22⋊#95C2214yes—
59C26C2311yescharacteristic
60C23×Q8C2312yes—
61C23×Q8C2311yescharacteristic
62C23×D4C2312yes—
63C23×D4C2311yescharacteristic
64C24×C4C2312yes—
65C24×C4C2312yes—
66C24×C4C2311yescharacteristic
67#551—2128no—
68C2×C4⋊Q8—28no—
69C2×C4⋊Q8—28no—
70C2×C4⋊Q8—24no—
71C2×C4⋊Q8—48no—
72C2×#126—28no—
73C2×#126—28no—
74C2×#126—28no—
75C2×#124—28no—
76C2×#124—28no—
77C2×#124—48no—
78C2×#124D418yes—
79C2×#123—24no—
80C2×#123—28no—
81C2×#123—28no—
82C2×#123—28no—
83C2×#123—28no—
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8—28no—
86C2×C22⋊Q8—28no—
87C2×#120—24no—
88C2×#120—28no—
89C2×C4×D4—28no—
90C2×C4×D4—28no—
91C2×C4×D4—28no—
92C2×#117—28no—
93C2×#117—28no—
94C2×#117D4116yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2318yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2318yes—
102C22×C4⋊C4C2314yes—
103C22×C22⋊C4—28no—
104C22×C22⋊C4—28no—
105C22×C22⋊C4—28no—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2318yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2318yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123#400—216no—
124#399—216no—
125#399—216no—
126#399—216no—
127#395—216no—
128#393—216no—
129#389—216no—
130#385—48no—
131#385—416no—
132#382—28no—
133#380—216no—
134#379—48no—
135#376—24no—
136C2×#95C2314yes—
137C2×#95C2318yes—
138C2×#95C2318yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2318yes—
142C2×#95C2318yes—
143C2×#95C2318yes—
144C2×#95C2318yes—
145C25—22no—
146C25—24no—
147C25—28no—
148C25—44no—
149C25—44no—
150C25C2411yescharacteristic
151C25C2411yescharacteristic
152C25C2412yes—
153C22×Q8—24no—
154C22×Q8—24no—
155C22×Q8—28no—
156C22×Q8—44no—
157C22×Q8—44no—
158C22×Q8—44no—
159C22×Q8C2×D414yes—
160C22×D4—24no—
161C22×D4—24no—
162C22×D4—28no—
163C22×D4—24no—
164C22×D4—24no—
165C22×D4—44no—
166C22×D4—44no—
167C22×D4C2×D414yes—
168C23×C4—28no—
169C23×C4—24no—
170C23×C4—22no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—48no—
179C23×C4C2×D418yes—
180C23×C4C2414yes—
181C23×C4C2418yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2414yes—
187C23×C4C2412yes—
188C23×C4C2411yescharacteristic
189C4⋊Q8—432no—
190C4⋊Q8—416no—
191#126—264no—
192#124—232no—
193#124—432no—
194#123—264no—
195#123—232no—
196C22⋊Q8—264no—
197#120—232no—
198C4×D4—264no—
199#117—264no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—216no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—216no—
216C2×C4⋊C4—216no—
217C2×C4⋊C4—48no—
218C2×C4⋊C4—48no—
219C2×C4⋊C4—48no—
220C2×C4⋊C4—48no—
221C2×C4⋊C4C2×D418yes—
222C2×C4⋊C4C2×D418yes—
223C2×C22⋊C4—264no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—216no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—216no—
248C2×C22⋊C4—216no—
249C2×C22⋊C4—48no—
250C2×C22⋊C4—48no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—48no—
255C2×C22⋊C4—48no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—48no—
258C2×C22⋊C4C2×D418yes—
259C2×C22⋊C4C2×D418yes—
260C2×C22⋊C4C2×D418yes—
261C2×C22⋊C4C2×D418yes—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—28no—
266C2×C42C2×D418yes—
267C2×C42C2×D414yes—
268C2×C42C2×D418yes—
269#95—28no—
270#95—216no—
271#95—28no—
272#95—28no—
273#95—216no—
274#95—216no—
275#95—48no—
276#95—48no—
277#95—48no—
278C24—22no—
279C24—22no—
280C24—28no—
281C24—24no—
282C24—24no—
283C24—24no—
284C24—28no—
285C24—22no—
286C24—22no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24—24no—
292C24—24no—
293C24—416no—
294C24—416no—
295C24—416no—
296C24—48no—
297C24—48no—
298C24—88no—
299C24—816no—
300C24—84no—
301C24—88no—
302C24C22×D412yes—
303C24C22×D412yes—
304C24C22×D414yes—
305C242−1+418yes—
306C24C2511yescharacteristic
307C2×Q8—28no—
308C2×Q8—48no—
309C2×Q8—48no—
310C2×Q8—48no—
311C2×Q8—48no—
312C2×Q8—88no—
313C2×Q8—816no—
314C2×D4—216no—
315C2×D4—28no—
316C2×D4—416no—
317C2×D4—48no—
318C2×D4—416no—
319C2×D4—48no—
320C2×D4—48no—
321C2×D4—88no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—216no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—24no—
338C22×C4—28no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—28no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—216no—
345C22×C4—28no—
346C22×C4—216no—
347C22×C4—28no—
348C22×C4—28no—
349C22×C4—24no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—28no—
359C22×C4—48no—
360C22×C4—416no—
361C22×C4—416no—
362C22×C4—88no—
363C22×C4—88no—
364C22×C4—88no—
365C22×C4—84no—
366C22×C4—88no—
367C22×C4C22×D414yes—
368C22×C4C22×D414yes—
369C22×C4C22×D414yes—
370C22×C4C22×D414yes—
371C22×C4C22×D414yes—
372C22×C4C22×D414yes—
373C22×C4C22×D414yes—
374C22×C4C22×D412yes—
375C22×C4C22×D414yes—
376C22×C4H(5,𝔽2)18yes—
377C4⋊C4—216no—
378C4⋊C4—216no—
379C4⋊C4—216no—
380C4⋊C4—216no—
381C4⋊C4—432no—
382C4⋊C4—416no—
383C4⋊C4—416no—
384C4⋊C4—816no—
385C22⋊C4—216no—
386C22⋊C4—216no—
387C22⋊C4—28no—
388C22⋊C4—28no—
389C22⋊C4—216no—
390C22⋊C4—216no—
391C22⋊C4—216no—
392C22⋊C4—216no—
393C22⋊C4—432no—
394C22⋊C4—416no—
395C22⋊C4—416no—
396C22⋊C4—416no—
397C22⋊C4—816no—
398C22⋊C4—88no—
399C42—216no—
400C42—216no—
401C42#120116yes—
402C23—24no—
403C23—24no—
404C23—28no—
405C23—28no—
406C23—28no—
407C23—28no—
408C23—216no—
409C23—216no—
410C23—22no—
411C23—22no—
412C23—24no—
413C23—24no—
414C23—28no—
415C23—28no—
416C23—432no—
417C23—42no—
418C23—42no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—48no—
423C23—48no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—48no—
433C23—48no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—832no—
439C23—88no—
440C23—832no—
441C23—832no—
442C23#53618yes—
443C23C23×D412yes—
444C23C23×D411yescharacteristic
445C23C2×H(5,𝔽2)14yes—
446C23C2×2−1+414yes—
447C23#58514yes—
448Q8—88no—
449Q8—816no—
450D4—432no—
451D4—88no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—28no—
456C2×C4—28no—
457C2×C4—28no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—28no—
467C2×C4—28no—
468C2×C4—216no—
469C2×C4—432no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—48no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—48no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—48no—
487C2×C4—48no—
488C2×C4—48no—
489C2×C4—48no—
490C2×C4—48no—
491C2×C4—48no—
492C2×C4—48no—
493C2×C4—48no—
494C2×C4—88no—
495C2×C4—816no—
496C2×C4C2×#12018yes—
497C2×C4C2×#12014yes—
498C22—216no—
499C22—28no—
500C22—416no—
501C22—44no—
502C22—48no—
503C22—48no—
504C22—48no—
505C22—44no—
506C22—48no—
507C22—48no—
508C22—42no—
509C22—42no—
510C22—44no—
511C22—44no—
512C22—832no—
513C22—84no—
514C22—84no—
515C22—84no—
516C22—84no—
517C22C22×#12011yescharacteristic
518C22C2×#53612yes—
519C22#344912yes—
520C22#348412yes—
521C22#348914yes—
522C22#348914yes—
523C22#349814yes—
524C22#353518yes—
525C22#353814yes—
526C22#354014yes—
527C4—24no—
528C4—28no—
529C4—48no—
530C4—44no—
531C4—44no—
532C4—44no—
533C4—44no—
534C4—84no—
535C4—88no—
536C2—48no—
537C2—82no—
538C2—82no—
539C2#3530512yes—
540C2#3552711yescharacteristic
541C2#4195314yes—
542C2#4196014yes—
543C2#5153014yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
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×#2658C2semidirect
3. C2×#2628C2semidirect
4. #16674C2semidirect
5. #16360C2semidirect
6. #15265C2semidirect
7. #15262C2semidirect
9. #14807C2semidirect
10. #14734C2semidirect
11. #14662C2semidirect
12. #13957C2semidirect
15. C22×C4⋊Q8C22semidirect
16. C22×C4⋊Q8C22semidirect
29. C2×#400C22semidirect
32. C2×#399C22semidirect
33. C2×#395C22semidirect
35. C2×#389C22semidirect
37. C2×#385C22semidirect
38. C2×#382C22semidirect
41. C2×#376C22semidirect
42. #1446C22semidirect
43. #1444C22semidirect
44. #1441C22semidirect
45. #1441C22semidirect
47. #1440C22semidirect
52. #1439C22semidirect
56. #1436C22semidirect
57. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5512
C2×#26582
C2×#26282
#166742
#163602
#152652
#152622
#152592
#148072
#147342
#146622
#139572