Finite Group Databasebeta

← #6452902

Subgroups — #6452902

Order 512.

3440 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#533C211yesmaximal, characteristic
2C2×#3095C212yesmaximal
3C2×#2296C211yesmaximal, characteristic
4#24065C218yesmaximal, complemented
5#24064C214yesmaximal, complemented
6#24063C214yesmaximal, complemented
7#23552C212yesmaximal, complemented
8#23507C212yesmaximal, complemented
9#20911C212yesmaximal
10#20874C214yesmaximal, complemented
11#20682C211yesmaximal, characteristic
12C2×#533—22no—
13C2×#533—22no—
14C2×#533—216no—
15C2×#533C2214yes—
16C2×#533C2214yes—
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125—21no—
20C22×C22⋊Q8—21no—
21C22×C22⋊Q8C2212yes—
22C22×C4×Q8—21no—
23C22×C4×Q8C2212yes—
24C22×#117—21no—
25C22×#117C2211yescharacteristic
26C22×#117C2211yescharacteristic
27C23×C42C2211yescharacteristic
28#3095—216no—
29C2×#484C2214yes—
30C2×#485C2218yes—
31C2×#483C2214yes—
32C2×#428C2214yes—
33C2×#419C2214yes—
34C2×#419C2214yes—
35C2×#407C2212yes—
36#2296—21no—
37#2296—21no—
38#2296—22no—
39#2296C2212yes—
40#2296C2214yes—
41#2296C2212yes—
42C2×#389C2212yes—
43C2×#389C2212yes—
44C2×#385—21no—
45C2×#385—21no—
46C2×C4×C4⋊C4—22no—
47#2058C2218yescomplemented
48#2057C2218yescomplemented
49#2043C2218yescomplemented
50#2043C2218yescomplemented
51#1873C2218yes—
52#1872C2218yes—
53#1865C22116yescomplemented
54#1863C22116yescomplemented
55#1817C2214yes—
56#1817C2214yes—
57#1736C2214yes—
58#1728C2212yes—
59#1728C2212yes—
60C23×Q8—21no—
61C23×Q8C2312yes—
62C24×C4D411yescharacteristic
63C24×C4D411yescharacteristic
64C24×C4C2311yescharacteristic
65C22×M16C2314yes—
66#533—232no—
67#533—232no—
68#533—232no—
69#533—232no—
70C2×C4⋊Q8—28no—
71C2×C4⋊Q8—28no—
72C2×C4⋊Q8—22no—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8C2314yes—
75C2×C4⋊Q8C2314yes—
76C2×#125—216no—
77C2×#125—24no—
78C2×#125—24no—
79C2×C22⋊Q8—216no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—28no—
83C2×C22⋊Q8—22no—
84C2×C22⋊Q8—22no—
85C2×C22⋊Q8D4116yes—
86C2×C22⋊Q8C2314yes—
87C2×C22⋊Q8C2314yes—
88C2×C4×Q8—216no—
89C2×C4×Q8—28no—
90C2×C4×Q8—28no—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8C2318yes—
95C2×#117—216no—
96C2×#117—24no—
97C2×#117—24no—
98C2×#117—21no—
99C2×#117—21no—
100C2×#117—28no—
101C2×#117—21no—
102C2×#117—21no—
103C2×#117—28no—
104C2×#117C2312yes—
105C2×#117C2312yes—
106C2×#117C2312yes—
107C2×#117C2312yes—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4—22no—
113C22×C4⋊C4—22no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4—22no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C42—28no—
123C22×C42—22no—
124C22×C42—21no—
125C22×C42—21no—
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42D412yes—
129C22×C42D412yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#485—232no—
135#484—216no—
136#483—216no—
137#428—216no—
138C2×#107C2318yes—
139C2×#106C2318yes—
140#419—216no—
141#419—216no—
142C2×#103C2318yes—
143C2×#103C2318yes—
144C2×#103C23116yes—
145#407—28no—
146C2×C22⋊C8C2314yes—
147C2×C22⋊C8C2314yes—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389C2314yes—
153#389C2314yes—
154#389C2314yes—
155#389C2314yes—
156#385—22no—
157#385—22no—
158#385—24no—
159#385—22no—
160#385—24no—
161#385—22no—
162C4×C4⋊C4—24no—
163C4×C4⋊C4—24no—
164C4×C4⋊C4—24no—
165C4×C4⋊C4—24no—
166C2×#95—22no—
167C2×#95—22no—
168#340C23116yes—
169#336C2318yes—
170#336C2318yes—
171C25C2×D411yescharacteristic
172C22×Q8—28no—
173C22×Q8—24no—
174C22×Q8—22no—
175C22×Q8—24no—
176C22×Q8—28no—
177C22×Q8—28no—
178C22×Q8—48no—
179C22×Q8C2×D418yes—
180C22×Q8C2414yes—
181C23×C4—24no—
182C23×C4—28no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—44no—
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D412yes—
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D412yes—
197C23×C4C2×D412yes—
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D418yes—
202C23×C4C2×D412yes—
203C23×C4C2×D411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C2×M16—28no—
208C2×M16—28no—
209C2×M16—28no—
210C22×C8C2418yes—
211C4⋊Q8—216no—
212C4⋊Q8—216no—
213C4⋊Q8—216no—
214C4⋊Q8—216no—
215#125—232no—
216#125—232no—
217C22⋊Q8—232no—
218C22⋊Q8—232no—
219C22⋊Q8—216no—
220C22⋊Q8—216no—
221C22⋊Q8—28no—
222C22⋊Q8—28no—
223C22⋊Q8C2×D4116yes—
224C22⋊Q8C2×D4116yes—
225C4×Q8—232no—
226C4×Q8—232no—
227C4×Q8—232no—
228C4×Q8—216no—
229C4×Q8—216no—
230#117—28no—
231#117—28no—
232#117—232no—
233#117—232no—
234#117—24no—
235#117—28no—
236#117—28no—
237#117—28no—
238#117—44no—
239#117—44no—
240#117#4118yes—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—216no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—216no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
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—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—48no—
265C2×C4⋊C4—48no—
266C2×C4⋊C4C2×D418yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2414yes—
270C2×C4⋊C4C2414yes—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—216no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—44no—
280C2×C42—216no—
281C2×C42—28no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—28no—
285C2×C42—28no—
286C2×C42—28no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—28no—
292C2×C42—28no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—44no—
296C2×C42C2×D412yes—
297C2×C42C2×D412yes—
298C2×C42C2×D412yes—
299C2×C42C2×D412yes—
300C2×C42C2×D412yes—
301C2×C42C2×D412yes—
302C2×C42C2412yes—
303C2×C42C2412yes—
304#107—216no—
305#106—216no—
306#103—216no—
307#103—216no—
308#103—232no—
309C22⋊C8—28no—
310C22⋊C8—28no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315C24—22no—
316C24—22no—
317C24—24no—
318C24—42no—
319C24#12014yes—
320C24C22×D411yescharacteristic
321C24C22×D411yescharacteristic
322C24C22×D411yescharacteristic
323C2×Q8—216no—
324C2×Q8—28no—
325C2×Q8—24no—
326C2×Q8—232no—
327C2×Q8—28no—
328C2×Q8—28no—
329C2×Q8—432no—
330C2×Q8—48no—
331C2×Q8—44no—
332C2×Q8—44no—
333C2×Q8—416no—
334C2×Q8—48no—
335C2×Q8C22×D418yes—
336C22×C4—232no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—232no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—28no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—416no—
376C22×C4—44no—
377C22×C4—48no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—44no—
385C22×C4#12014yes—
386C22×C4#12014yes—
387C22×C4#12014yes—
388C22×C4#13718yes—
389C22×C4C22×D412yes—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D414yes—
393C22×C4C22×D414yes—
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C22×D412yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D414yes—
399C22×C4C2×#4114yes—
400C22×C42−1+414yes—
401C22×C4C2511yescharacteristic
402M16—416no—
403C2×C8—28no—
404C2×C8—28no—
405C2×C8—28no—
406C4⋊C4—28no—
407C4⋊C4—216no—
408C4⋊C4—216no—
409C4⋊C4—232no—
410C4⋊C4—216no—
411C4⋊C4—216no—
412C4⋊C4—216no—
413C4⋊C4—28no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—48no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—48no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4C22×D418yes—
425C4⋊C4C2×#4118yes—
426C22⋊C4—28no—
427C22⋊C4—216no—
428C22⋊C4—216no—
429C22⋊C4—28no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C42—28no—
433C42—232no—
434C42—24no—
435C42—24no—
436C42—24no—
437C42—24no—
438C42—24no—
439C42—24no—
440C42—28no—
441C42—24no—
442C42—24no—
443C42—44no—
444C42—44no—
445C23—21no—
446C23—21no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—28no—
451C23—21no—
452C23—21no—
453C23—28no—
454C23—22no—
455C23—22no—
456C23—44no—
457C23—48no—
458C23—48no—
459C23C2×#12012yes—
460C23C2×#12012yes—
461C23C2×#12012yes—
462C23#52512yes—
463C23#53612yes—
464C23C2×#13714yes—
465C23C23×D411yescharacteristic
466Q8—216no—
467Q8—432no—
468Q8—416no—
469Q8—48no—
470C2×C4—28no—
471C2×C4—28no—
472C2×C4—216no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—216no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—28no—
489C2×C4—28no—
490C2×C4—22no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—28no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—48no—
497C2×C4—48no—
498C2×C4—416no—
499C2×C4—44no—
500C2×C4—416no—
501C2×C4—416no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—48no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—84no—
512C2×C4—84no—
513C2×C4C2×#12012yes—
514C2×C4C2×#12012yes—
515C2×C4C2×#12012yes—
516C2×C4#52512yes—
517C2×C4#53612yes—
518C2×C4#547116yes—
519C2×C4C2×#13714yes—
520C2×C4C2×#13714yes—
521C2×C4#57514yes—
522C2×C4C23×D412yes—
523C2×C4C22×#4112yes—
524C2×C4C2×2−1+412yes—
525C8—48no—
526C22—24no—
527C22—24no—
528C22—22no—
529C22—24no—
530C22—24no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—416no—
538C22—42no—
539C22—42no—
540C22#240614yes—
541C22#300414yes—
542C22C22×#12011yescharacteristic
543C22C2×#52511yescharacteristic
544C22C2×#53611yescharacteristic
545C22C22×#13712yes—
546C22C2×#57512yes—
547C4—22no—
548C4—22no—
549C4—24no—
550C4—24no—
551C4—24no—
552C4—48no—
553C4—48no—
554C4—42no—
555C4—48no—
556C4—82no—
557C4—82no—
558C4#240614yes—
559C4C2×#54714yes—
560C2—24no—
561C2—24no—
562C2—42no—
563C2C2×#240611yescharacteristic
564C2C2×#300411yescharacteristic
565C2#6035911yescharacteristic
566C2#6056212yes—
567C2#6086512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #24065C2semidirect
5. #24064C2semidirect
6. #24063C2semidirect
7. #23552C2semidirect
8. #23507C2semidirect
10. #20874C2semidirect
47. #2058C22semidirect
48. #2057C22semidirect
49. #2043C22semidirect
50. #2043C22semidirect
53. #1865C22semidirect
54. #1863C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5332
C2×#30952
C2×#22962
#240652
#240642
#240632
#235522
#235072
#209112
#208742
#206822