Finite Group Databasebeta

← #6434677

Subgroups — #6434677

Order 512.

3426 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#533C211yesmaximal, characteristic, complemented
2C2×#3094C212yesmaximal
3C2×#2854C211yesmaximal, characteristic
4#29644C218yesmaximal, complemented
5#29640C214yesmaximal, complemented
6#29639C212yesmaximal, complemented
7#29638C212yesmaximal, complemented
8#23554C214yesmaximal, complemented
9#23454C214yesmaximal, complemented
10#20702C212yesmaximal, complemented
11#20682C211yesmaximal, characteristic, complemented
12C2×#533—22no—
13C2×#533—22no—
14C2×#533—216no—
15C2×#533C2214yes—
16C2×#533C2214yescomplemented
17C22×C4⋊Q8C2211yescharacteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic, complemented
19C22×#125C2212yescomplemented
20C22×#123—21no—
21C22×C22⋊Q8—21no—
22C22×C22⋊Q8—21no—
23C22×#121C2211yescharacteristic
24C22×C4×Q8—21no—
25C22×C4×Q8—21no—
26C22×#117C2212yes—
27C23×C42C2211yescharacteristic
28#3094—216no—
29C2×#481C2214yes—
30C2×#482C2218yes—
31C2×#480C2214yes—
32C2×#428C2214yes—
33C2×#418C2214yes—
34C2×#407C2212yes—
35#2854—21no—
36#2854—21no—
37#2854—22no—
38#2854C2212yes—
39#2854C2212yes—
40#2854C2214yes—
41C2×#390C2212yes—
42C2×#389C2212yes—
43C2×#388—22no—
44C2×#386—21no—
45C2×#380C2211yescharacteristic
46#2059C22116yescomplemented
47#2058C2218yescomplemented
48#2057C2218yescomplemented
49#2025C22116yescomplemented
50#2023C2218yescomplemented
51#2023C2218yescomplemented
52#1825C2218yescomplemented
53#1824C2214yescomplemented
54#1823C2214yescomplemented
55#1817C2212yescomplemented
56#1817C2212yescomplemented
57#1817C2214yescomplemented
58#1736C2218yes—
59#1728C2214yes—
60C23×Q8—21no—
61C23×Q8—21no—
62C23×D4—21no—
63C23×D4C2311yescharacteristic
64C24×C4D411yescharacteristic
65C24×C4D411yescharacteristic
66C24×C4C2311yescharacteristic
67C22×M16C2314yes—
68#533—232no—
69#533—232no—
70#533—232no—
71#533—232no—
72C2×C4⋊Q8—28no—
73C2×C4⋊Q8—28no—
74C2×C4⋊Q8—22no—
75C2×C4⋊Q8—22no—
76C2×C4⋊Q8C2314yes—
77C2×C4⋊Q8C2314yes—
78C2×#125—28no—
79C2×#125—28no—
80C2×#125—24no—
81C2×#125C2318yes—
82C2×#123—22no—
83C2×#123—24no—
84C2×#123—22no—
85C2×#123—48no—
86C2×C22⋊Q8—216no—
87C2×C22⋊Q8—216no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×#121—21no—
93C2×#121—21no—
94C2×#121—28no—
95C2×#121C2312yes—
96C2×#121C2312yes—
97C2×C4×Q8—216no—
98C2×C4×Q8—216no—
99C2×C4×Q8—24no—
100C2×C4×Q8—24no—
101C2×C4×Q8—24no—
102C2×C4×Q8—24no—
103C2×#117—28no—
104C2×#117—22no—
105C2×#117—22no—
106C2×#117D4116yes—
107C2×#117C2314yes—
108C2×#117C2314yes—
109C22×C4⋊C4—22no—
110C22×C4⋊C4—22no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—22no—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C22⋊C4—22no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—22no—
122C22×C22⋊C4—22no—
123C22×C42—28no—
124C22×C42—22no—
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42D412yes—
128C22×C42D412yes—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132C22×C42C2312yes—
133#482—232no—
134#481—216no—
135#480—216no—
136#428—216no—
137C2×#107C2318yes—
138C2×#106C2318yes—
139#418—216no—
140C2×#102C2318yes—
141C2×#102C2318yes—
142#407—28no—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2314yes—
145#390—22no—
146#390—22no—
147#390C2314yes—
148#390C2314yes—
149#389—22no—
150#389—22no—
151#389C2314yes—
152#389C2314yes—
153#388—24no—
154#388—24no—
155#388—24no—
156#388—24no—
157#386—22no—
158#386—24no—
159#386—22no—
160#380—21no—
161#380—21no—
162#380C2312yes—
163#380C2312yes—
164C2×#95—21no—
165C2×#95—22no—
166#340C23116yes—
167#340C23116yes—
168#336C2318yes—
169#336C2318yes—
170C25—21no—
171C25C2×D411yescharacteristic
172C22×Q8—28no—
173C22×Q8—28no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—44no—
177C22×Q8—48no—
178C22×Q8—44no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—24no—
182C22×D4—21no—
183C22×D4—28no—
184C22×D4—44no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4C2412yes—
188C23×C4—28no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—21no—
196C23×C4—44no—
197C23×C4—42no—
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D412yes—
201C23×C4C2×D412yes—
202C23×C4C2×D412yes—
203C23×C4C2×D412yes—
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D418yes—
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C2×M16—28no—
214C2×M16—216no—
215C22×C8C2418yes—
216C4⋊Q8—216no—
217C4⋊Q8—216no—
218C4⋊Q8—216no—
219C4⋊Q8—216no—
220#125—232no—
221#125—216no—
222#125—216no—
223#123—48no—
224#123—416no—
225#123—48no—
226C22⋊Q8—232no—
227C22⋊Q8—232no—
228C22⋊Q8—232no—
229C22⋊Q8—232no—
230#121—28no—
231#121—28no—
232#121—44no—
233#121—44no—
234C4×Q8—232no—
235C4×Q8—232no—
236C4×Q8—232no—
237C4×Q8—232no—
238#117—216no—
239#117—216no—
240#117—28no—
241#117—28no—
242#117C2×D4116yes—
243#117C2×D4116yes—
244C2×C4⋊C4—216no—
245C2×C4⋊C4—216no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—48no—
263C2×C4⋊C4C2×D418yes—
264C2×C4⋊C4C2×D418yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2414yes—
267C2×C4⋊C4C2414yes—
268C2×C4⋊C4C2414yes—
269C2×C22⋊C4—216no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—48no—
281C2×C22⋊C4—48no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C42—216no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—28no—
288C2×C42—28no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—28no—
294C2×C42—28no—
295C2×C42C2×D412yes—
296C2×C42C2×D412yes—
297C2×C42C2×D412yes—
298C2×C42C2×D412yes—
299C2×C42C2×D412yes—
300C2×C42C2×D412yes—
301C2×C42C2412yes—
302C2×C42C2412yes—
303#107—216no—
304#106—216no—
305#102—216no—
306#102—216no—
307C22⋊C8—28no—
308C22⋊C8—28no—
309#95—22no—
310#95—22no—
311#95—24no—
312#95—24no—
313C24—24no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—42no—
318C24—42no—
319C24—44no—
320C24—42no—
321C24—42no—
322C24—44no—
323C24#12014yes—
324C24C22×D411yescharacteristic
325C24C22×D411yescharacteristic
326C24C22×D411yescharacteristic
327C2×Q8—216no—
328C2×Q8—216no—
329C2×Q8—432no—
330C2×Q8—432no—
331C2×Q8—48no—
332C2×Q8—48no—
333C2×Q8—48no—
334C2×Q8—48no—
335C2×D4—24no—
336C2×D4—28no—
337C2×D4—44no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—42no—
344C2×D4—416no—
345C2×D4—44no—
346C2×D4—816no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—232no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—416no—
381C22×C4—416no—
382C22×C4—48no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4#12014yes—
392C22×C4#12014yes—
393C22×C4#12014yes—
394C22×C4AGL(1,ℤ/8)14yes—
395C22×C4C22×D412yes—
396C22×C4C22×D412yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D414yes—
400C22×C4C22×D414yes—
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D414yes—
405C22×C42−1+418yes—
406C22×C4C2511yescharacteristic
407M16—416no—
408C2×C8—28no—
409C2×C8—216no—
410C4⋊C4—232no—
411C4⋊C4—232no—
412C4⋊C4—216no—
413C4⋊C4—216no—
414C4⋊C4—28no—
415C4⋊C4—28no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—28no—
420C4⋊C4—48no—
421C4⋊C4—42no—
422C4⋊C4—44no—
423C4⋊C4—48no—
424C4⋊C4C22×D418yes—
425C4⋊C4C22×D418yes—
426C22⋊C4—216no—
427C22⋊C4—216no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—48no—
435C22⋊C4—48no—
436C42—24no—
437C42—24no—
438C42—24no—
439C42—24no—
440C42—24no—
441C42—24no—
442C42—216no—
443C42—216no—
444C42—24no—
445C42—24no—
446C23—22no—
447C23—21no—
448C23—22no—
449C23—22no—
450C23—21no—
451C23—28no—
452C23—22no—
453C23—28no—
454C23—21no—
455C23—22no—
456C23—21no—
457C23—44no—
458C23—48no—
459C23—48no—
460C23—42no—
461C23—44no—
462C23—42no—
463C23—44no—
464C23—42no—
465C23—84no—
466C23—88no—
467C23—88no—
468C23—84no—
469C23—84no—
470C23C2×#12012yes—
471C23C2×#12012yes—
472C23C2×#12012yes—
473C23#53614yes—
474C23C2×AGL(1,ℤ/8)12yes—
475C23#57512yes—
476C23C23×D411yescharacteristic
477Q8—432no—
478Q8—432no—
479D4—48no—
480D4—88no—
481D4—88no—
482D4—84no—
483C2×C4—28no—
484C2×C4—28no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—28no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—28no—
493C2×C4—28no—
494C2×C4—28no—
495C2×C4—28no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—48no—
505C2×C4—48no—
506C2×C4—48no—
507C2×C4—48no—
508C2×C4—44no—
509C2×C4—416no—
510C2×C4—44no—
511C2×C4—416no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—48no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—88no—
522C2×C4—84no—
523C2×C4C2×#12012yes—
524C2×C4C2×#12012yes—
525C2×C4C2×#12012yes—
526C2×C4#53614yes—
527C2×C4#536116yes—
528C2×C4C2×AGL(1,ℤ/8)14yes—
529C2×C4#57514yes—
530C2×C4C23×D412yes—
531C2×C4C2×2−1+414yes—
532C8—48no—
533C22—22no—
534C22—24no—
535C22—24no—
536C22—24no—
537C22—24no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—416no—
545C22—42no—
546C22—42no—
547C22—82no—
548C22—84no—
549C22—84no—
550C22—82no—
551C22—82no—
552C22—168no—
553C22#261414yes—
554C22#300614yes—
555C22C22×#12011yescharacteristic
556C22C2×#53612yes—
557C22C22×AGL(1,ℤ/8)11yescharacteristic
558C22C2×#57512yes—
559C22C2×#57511yescharacteristic
560C4—22no—
561C4—22no—
562C4—24no—
563C4—24no—
564C4—42no—
565C4—48no—
566C4—42no—
567C4—48no—
568C4—84no—
569C4—82no—
570C4#261414yes—
571C4C2×#53614yes—
572C2—24no—
573C2—24no—
574C2—42no—
575C2—162no—
576C2C2×#261411yescharacteristic
577C2C2×#300611yescharacteristic
578C2#6035911yescharacteristic
579C2#6086412yes—
580C2#6086612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
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
1. C22×#533C2semidirect
4. #29644C2semidirect
5. #29640C2semidirect
6. #29639C2semidirect
7. #29638C2semidirect
8. #23554C2semidirect
9. #23454C2semidirect
10. #20702C2semidirect
11. #20682C2semidirect
16. C2×#533C22semidirect
17. C22×C4⋊Q8C22semidirect
18. C22×C4⋊Q8C22semidirect
19. C22×#125C22semidirect
46. #2059C22semidirect
47. #2058C22semidirect
48. #2057C22semidirect
49. #2025C22semidirect
50. #2023C22semidirect
51. #2023C22semidirect
52. #1825C22semidirect
53. #1824C22semidirect
54. #1823C22semidirect
55. #1817C22semidirect
56. #1817C22semidirect
57. #1817C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5332
C2×#30942
C2×#28542
#296442
#296402
#296392
#296382
#235542
#234542
#207022
#206822