Finite Group Databasebeta

← #6386120

Subgroups — #6386120

Order 512.

3112 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#552C211yesmaximal, characteristic
2#60230C212yesmaximal
3#22480C2116yesmaximal, complemented
4#21830C212yesmaximal, complemented
5#21753C212yesmaximal, complemented
6#21753C212yesmaximal, complemented
7#20727C212yesmaximal, complemented
8#20694C212yesmaximal, complemented
9C2×#1821C212yesmaximal, complemented
10C2×#552—22no—
11C2×#552—22no—
12C2×#552—22no—
13C2×#552—28no—
14C2×#552—22no—
15C2×#552C418yes—
16C2×#552C2218yes—
17C22×#126C412yes—
18C22×#125C412yes—
19C22×#125C2211yescharacteristic
20C22×#123C412yes—
21C22×C22⋊Q8C2211yescharacteristic
22C22×C22⋊Q8C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24C22×C4×D4C2211yescharacteristic
25C22×#117C412yes—
26C22×#117C2211yescharacteristic
27C23×C4⋊C4C2211yescharacteristic
28#2975—22no—
29#2974—28no—
30#2973—22no—
31#2973—22no—
32#2972—22no—
33D4×M16C22116yes—
34#2920C2214yes—
35C2×#407C2214yes—
36C2×#407C2214yes—
37#1998C22116yescomplemented
38#1901C22116yescomplemented
39#1901C22116yescomplemented
40#1822C2218yescomplemented
41#1821C2214yes—
42#1821C2214yes—
43#1821C2214yes—
44#1821C2218yescomplemented
45#1801C2218yescomplemented
46#1784C2214yes—
47C4×#101C2218yescomplemented
48#1742C2218yescomplemented
49C2×#343C2218yescomplemented
50C23×Q8C2311yescharacteristic
51C23×D4C2×C411yescharacteristic
52C24×C4C2×C412yes—
53C24×C4C2311yescharacteristic
54C24×C4C2311yescharacteristic
55C22×M16C2318yes—
56C22×M16C2318yes—
57#552—216no—
58#552—216no—
59#552—216no—
60#552—216no—
61#552—216no—
62#552—216no—
63#552—216no—
64#552—216no—
65C2×#126—28no—
66C2×#126—24no—
67C2×#126—24no—
68C2×#126C2×C4116yes—
69C2×#125—24no—
70C2×#125—24no—
71C2×#125—28no—
72C2×#125—22no—
73C2×#125—24no—
74C2×#125—22no—
75C2×#125C2×C4116yes—
76C2×#125C2×C414yes—
77C2×#125C2314yes—
78C2×#123—24no—
79C2×#123—28no—
80C2×#123—24no—
81C2×#123C2×C4116yes—
82C2×C22⋊Q8—22no—
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8—22no—
85C2×C22⋊Q8—22no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—22no—
88C2×C22⋊Q8C2×C414yes—
89C2×C22⋊Q8C2×C414yes—
90C2×C22⋊Q8C2314yes—
91C2×C22⋊Q8C2314yes—
92C2×C4×Q8—22no—
93C2×C4×Q8—24no—
94C2×C4×Q8—22no—
95C2×C4×Q8C2×C414yes—
96C2×C4×Q8C2314yes—
97C2×C4×D4C2×C414yes—
98C2×C4×D4D412yes—
99C2×C4×D4D412yes—
100C2×C4×D4D412yes—
101C2×C4×D4D418yes—
102C2×C4×D4D412yes—
103C2×C4×D4C2314yes—
104C2×#117—22no—
105C2×#117—24no—
106C2×#117—22no—
107C2×#117—28no—
108C2×#117—22no—
109C2×#117—22no—
110C2×#117—22no—
111C2×#117—22no—
112C2×#117C2×C4116yes—
113C2×#117C2×C414yes—
114C2×#117C2314yes—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—24no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4C2×C412yes—
119C22×C4⋊C4C2×C414yes—
120C22×C4⋊C4C2×C412yes—
121C22×C4⋊C4C2×C412yes—
122C22×C4⋊C4C2×C412yes—
123C22×C4⋊C4C2×C412yes—
124C22×C4⋊C4C2×C412yes—
125C22×C4⋊C4C2×C411yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2314yes—
130C22×C22⋊C4C2×C412yes—
131C22×C22⋊C4C2×C412yes—
132C22×C22⋊C4C2×C412yes—
133C22×C22⋊C4C2×C411yescharacteristic
134C22×C22⋊C4C2×C411yescharacteristic
135C22×C42C2×C412yes—
136C22×C42C2×C412yes—
137C22×C42C2311yescharacteristic
138#436—216no—
139#435—216no—
140#435—216no—
141C8×D4—216no—
142#432—24no—
143#432—24no—
144#431—24no—
145#431—24no—
146#423C2318yes—
147C2×#101C23116yes—
148#407—24no—
149#407—24no—
150#407—24no—
151#407—24no—
152#407C2318yes—
153#407C2318yes—
154#407C2318yes—
155C2×C22⋊C8—24no—
156C2×C22⋊C8—24no—
157C2×C22⋊C8—24no—
158C2×C22⋊C8—24no—
159C4×M16C2318yes—
160#343C23116yes—
161#343C23116yes—
162#343C23116yes—
163C25C22×C411yescharacteristic
164C25C22×C411yescharacteristic
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—44no—
168C22×Q8C22×C412yes—
169C22×Q8C2412yes—
170C22×D4—44no—
171C22×D4C22⋊C412yes—
172C22×D4C22⋊C412yes—
173C22×D4C22⋊C414yes—
174C22×D4C22×C414yes—
175C23×C4—24no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4—48no—
181C23×C4C22×C412yes—
182C23×C4C22×C412yes—
183C23×C4C22×C412yes—
184C23×C4C22×C412yes—
185C23×C4C22×C418yes—
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C412yes—
191C23×C4C22×C412yes—
192C23×C4C2×D412yes—
193C23×C4C2×D414yes—
194C23×C4C2×D412yes—
195C23×C4C2×D412yes—
196C23×C4C2×D414yes—
197C23×C4C2×D412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C23×C4C2412yes—
201C2×M16—28no—
202C2×M16—28no—
203C2×M16—416no—
204C2×M16—416no—
205C2×M16C24116yes—
206C22×C8—28no—
207C22×C8—28no—
208#126—232no—
209#126—232no—
210#125—232no—
211#125—232no—
212#125—28no—
213#125—28no—
214#125—28no—
215#125—28no—
216#123—232no—
217#123—232no—
218C22⋊Q8—28no—
219C22⋊Q8—28no—
220C22⋊Q8—28no—
221C22⋊Q8—28no—
222C22⋊Q8—28no—
223C22⋊Q8—28no—
224C22⋊Q8—28no—
225C22⋊Q8—28no—
226C4×Q8—28no—
227C4×Q8—28no—
228C4×Q8—28no—
229C4×Q8—28no—
230C4×D4C22⋊C418yes—
231C4×D4C22⋊C418yes—
232C4×D4C22⋊C418yes—
233C4×D4C22⋊C418yes—
234C4×D4C2×D418yes—
235C4×D4C2×D418yes—
236C4×D4C2×D418yes—
237C4×D4C2×D418yes—
238#117—28no—
239#117—28no—
240#117—28no—
241#117—28no—
242#117—216no—
243#117—216no—
244#117—216no—
245#117—216no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4C22⋊C412yes—
271C2×C4⋊C4C22⋊C412yes—
272C2×C4⋊C4C22⋊C414yes—
273C2×C4⋊C4C22×C412yes—
274C2×C4⋊C4C22×C412yes—
275C2×C4⋊C4C22×C418yes—
276C2×C4⋊C4C22×C418yes—
277C2×C4⋊C4C22×C418yes—
278C2×C4⋊C4C22×C418yes—
279C2×C4⋊C4C22×C418yes—
280C2×C4⋊C4C22×C418yes—
281C2×C4⋊C4C22×C412yes—
282C2×C4⋊C4C22×C414yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4C22⋊C414yes—
294C2×C22⋊C4C22⋊C411yescharacteristic
295C2×C22⋊C4C22⋊C411yescharacteristic
296C2×C22⋊C4C22⋊C411yescharacteristic
297C2×C22⋊C4C22⋊C411yescharacteristic
298C2×C22⋊C4C22⋊C414yes—
299C2×C22⋊C4C22⋊C412yes—
300C2×C22⋊C4C22⋊C412yes—
301C2×C22⋊C4C22×C418yes—
302C2×C22⋊C4C22×C418yes—
303C2×C22⋊C4C22×C418yes—
304C2×C22⋊C4C22×C414yes—
305C2×C22⋊C4C22×C414yes—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42C22×C418yes—
313C2×C42C22×C418yes—
314C2×C42C22×C412yes—
315C2×C42C2×D412yes—
316C2×C42C2×D414yes—
317C2×C42C2×D412yes—
318C2×C42C2412yes—
319C4⋊C8—28no—
320C4⋊C8—28no—
321#101—416no—
322C22⋊C8—28no—
323C22⋊C8—28no—
324C22⋊C8—28no—
325C22⋊C8—28no—
326C22⋊C8—28no—
327C22⋊C8—28no—
328#97—28no—
329C4×C8—28no—
330C24—24no—
331C24—42no—
332C24—42no—
333C24—42no—
334C24—42no—
335C24—42no—
336C24C2×C22⋊C412yes—
337C24C2×C22⋊C412yes—
338C24C2×C22⋊C412yes—
339C24C2×C22⋊C411yescharacteristic
340C24C2×C22⋊C411yescharacteristic
341C24C23×C411yescharacteristic
342C24C23×C412yes—
343C24C23×C412yes—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—28no—
347C2×Q8—44no—
348C2×Q8—48no—
349C2×Q8—48no—
350C2×D4—28no—
351C2×D4—44no—
352C2×D4—48no—
353C2×D4—48no—
354C2×D4C2×C22⋊C418yes—
355C2×D4C2×C22⋊C418yes—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—44no—
382C22×C4—416no—
383C22×C4—48no—
384C22×C4—48no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—416no—
388C22×C4—44no—
389C22×C4C2×C22⋊C412yes—
390C22×C4C2×C22⋊C412yes—
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C412yes—
396C22×C4C2×C22⋊C412yes—
397C22×C4C2×C22⋊C412yes—
398C22×C4C2×C22⋊C412yes—
399C22×C4C2×C22⋊C414yes—
400C22×C4C2×C22⋊C414yes—
401C22×C4C2×C22⋊C414yes—
402C22×C4C2×C22⋊C414yes—
403C22×C4C23×C414yes—
404C22×C4C23×C412yes—
405C22×C4C23×C411yescharacteristic
406C22×C4C23×C412yes—
407C22×C4C23×C414yes—
408C22×C4C23×C414yes—
409C22×C4C23×C414yes—
410C22×C4C23×C412yes—
411C22×C4C23×C412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D414yes—
415C22×C4C22×D414yes—
416C22×C4C22×D414yes—
417C22×C4C22×D414yes—
418C22×C4C2511yescharacteristic
419M16—232no—
420M16—88no—
421M16—88no—
422C2×C8—216no—
423C2×C8—88no—
424C2×C8—88no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—216no—
428C4⋊C4—24no—
429C4⋊C4—28no—
430C4⋊C4—28no—
431C4⋊C4—28no—
432C4⋊C4—28no—
433C4⋊C4—216no—
434C4⋊C4—216no—
435C4⋊C4—216no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4C2×C22⋊C418yes—
440C4⋊C4C2×C22⋊C418yes—
441C22⋊C4—28no—
442C22⋊C4—28no—
443C22⋊C4—216no—
444C22⋊C4—216no—
445C22⋊C4C2×C22⋊C414yes—
446C22⋊C4C2×C22⋊C414yes—
447C22⋊C4C2×C22⋊C414yes—
448C22⋊C4C2×C22⋊C414yes—
449C22⋊C4C2×C22⋊C418yes—
450C22⋊C4C2×C22⋊C418yes—
451C42—28no—
452C42—28no—
453C42—28no—
454C42—28no—
455C42C2×C22⋊C414yes—
456C42C2×C22⋊C414yes—
457C42C22×D414yes—
458C42C22×D414yes—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—24no—
463C23—22no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—41no—
471C23—42no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—416no—
483C23—88no—
484C23#41314yes—
485C23C22×C22⋊C414yes—
486C23C22×C22⋊C412yes—
487C23C22×C22⋊C412yes—
488C23C22×C22⋊C411yescharacteristic
489C23C22×C22⋊C411yescharacteristic
490C23C24×C411yescharacteristic
491Q8—48no—
492Q8—48no—
493D4—216no—
494D4—48no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—28no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—416no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—416no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4C22×C22⋊C412yes—
539C2×C4C22×C22⋊C412yes—
540C2×C4C22×C22⋊C412yes—
541C2×C4C22×C22⋊C412yes—
542C2×C4C22×C22⋊C412yes—
543C2×C4C22×C22⋊C414yes—
544C2×C4C22×C22⋊C414yes—
545C2×C4C23×D412yes—
546C2×C4#58514yes—
547C8—416no—
548C22—28no—
549C22—21no—
550C22—21no—
551C22—22no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—22no—
557C22—22no—
558C22—24no—
559C22—22no—
560C22—24no—
561C22—22no—
562C22—24no—
563C22—44no—
564C22—48no—
565C22—48no—
566C22—44no—
567C22—84no—
568C22—84no—
569C22—84no—
570C22C2×#41312yes—
571C22C2×#41314yes—
572C22#288912yes—
573C22C23×C22⋊C411yescharacteristic
574C22#342812yes—
575C4—22no—
576C4—22no—
577C4—24no—
578C4—22no—
579C4—24no—
580C4—22no—
581C4—22no—
582C4—44no—
583C4—44no—
584C4—44no—
585C4—44no—
586C4#348514yes—
587C2—22no—
588C2—22no—
589C2—22no—
590C2—24no—
591C2—42no—
592C2—44no—
593C2#3417211yescharacteristic
594C2C22×#41311yescharacteristic
595C2C2×#288911yescharacteristic
subgroupG/H
center, socleC22C23×C22⋊C4
commutatorC23C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #22480C2semidirect
4. #21830C2semidirect
5. #21753C2semidirect
6. #21753C2semidirect
7. #20727C2semidirect
8. #20694C2semidirect
9. C2×#1821C2semidirect
37. #1998C22semidirect
38. #1901C22semidirect
39. #1901C22semidirect
40. #1822C22semidirect
44. #1821C22semidirect
45. #1801C22semidirect
47. C4×#101C22semidirect
48. #1742C22semidirect
49. C2×#343C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5522
#602302
#224802
#218302
#217532
#207272
#206942
C2×#18212