Finite Group Databasebeta

← #6434678

Subgroups — #6434678

Order 512.

3220 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×#533C211yesmaximal, characteristic
2C2×#3095C212yesmaximal
3C2×#2855C211yesmaximal, characteristic
4#29645C218yesmaximal, complemented
5#29643C214yesmaximal, complemented
6#29642C212yesmaximal, complemented
7#29641C212yesmaximal, complemented
8#23554C214yesmaximal, complemented
9#23455C214yesmaximal, complemented
10#20702C212yesmaximal
11#20682C211yesmaximal, characteristic
12C2×#533—22no—
13C2×#533—22no—
14C2×#533—216no—
15C2×#533C2214yes—
16C2×#533C2214yes—
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2212yes—
20C22×C22⋊Q8—21no—
21C22×C22⋊Q8—21no—
22C22×C22⋊Q8C2211yescharacteristic
23C22×C4×Q8—21no—
24C22×C4×Q8—21no—
25C22×#117C2212yes—
26C23×C42C2211yescharacteristic
27#3095—216no—
28C2×#484C2214yes—
29C2×#485C2218yes—
30C2×#483C2214yes—
31C2×#428C2214yes—
32C2×#419C2214yes—
33C2×#407C2212yes—
34#2855—21no—
35#2855—21no—
36#2855—22no—
37#2855C2212yes—
38#2855C2212yes—
39#2855C2214yes—
40C2×#391C2212yes—
41C2×#389C2212yes—
42C2×#387—22no—
43C2×#385—21no—
44C2×#381—21no—
45C2×#380C2211yescharacteristic
46#2059C22116yescomplemented
47#2058C2218yescomplemented
48#2057C2218yescomplemented
49#2026C22116yescomplemented
50#2024C2218yescomplemented
51#2024C2218yescomplemented
52#1825C2218yes—
53#1824C2214yes—
54#1823C2214yes—
55#1817C2212yes—
56#1817C2212yes—
57#1817C2214yes—
58#1736C2218yes—
59#1728C2214yes—
60C23×Q8—21no—
61C23×Q8—21no—
62C23×Q8C2311yescharacteristic
63C24×C4D411yescharacteristic
64C24×C4D411yescharacteristic
65C24×C4C2311yescharacteristic
66C22×M16C2314yes—
67#533—232no—
68#533—232no—
69#533—232no—
70#533—232no—
71C2×C4⋊Q8—28no—
72C2×C4⋊Q8—28no—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8—22no—
75C2×C4⋊Q8C2314yes—
76C2×C4⋊Q8C2314yes—
77C2×#125—28no—
78C2×#125—28no—
79C2×#125—24no—
80C2×#125C2318yes—
81C2×C22⋊Q8—216no—
82C2×C22⋊Q8—216no—
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—21no—
88C2×C22⋊Q8—21no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8C2312yes—
91C2×C22⋊Q8C2312yes—
92C2×C4×Q8—216no—
93C2×C4×Q8—216no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8—24no—
97C2×C4×Q8—24no—
98C2×#117—28no—
99C2×#117—22no—
100C2×#117—22no—
101C2×#117D4116yes—
102C2×#117C2314yes—
103C2×#117C2314yes—
104C22×C4⋊C4—22no—
105C22×C4⋊C4—22no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—22no—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4—22no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C42—28no—
118C22×C42—22no—
119C22×C42D411yescharacteristic
120C22×C42D411yescharacteristic
121C22×C42D412yes—
122C22×C42D412yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2312yes—
127#485—232no—
128#484—216no—
129#483—216no—
130#428—216no—
131C2×#107C2318yes—
132C2×#106C2318yes—
133#419—216no—
134C2×#103C2318yes—
135C2×#103C2318yes—
136#407—28no—
137C2×C22⋊C8C2314yes—
138C2×C22⋊C8C2314yes—
139#391—22no—
140#391—22no—
141#391C2314yes—
142#391C2314yes—
143#389—22no—
144#389—22no—
145#389C2314yes—
146#389C2314yes—
147#387—24no—
148#387—24no—
149#387—24no—
150#387—24no—
151#385—22no—
152#385—24no—
153#385—22no—
154#381—22no—
155#381—24no—
156#381—22no—
157#380—21no—
158#380—21no—
159#380C2312yes—
160#380C2312yes—
161C2×#95—21no—
162C2×#95—22no—
163C2×#95—22no—
164C2×#95—22no—
165#340C23116yes—
166#340C23116yes—
167#336C2318yes—
168#336C2318yes—
169C25C2×D411yescharacteristic
170C22×Q8—28no—
171C22×Q8—28no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—21no—
175C22×Q8—24no—
176C22×Q8—28no—
177C22×Q8—44no—
178C22×Q8—48no—
179C22×Q8—44no—
180C22×Q8C2412yes—
181C23×C4—28no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—21no—
188C23×C4—22no—
189C23×C4—21no—
190C23×C4—44no—
191C23×C4—42no—
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D412yes—
195C23×C4C2×D412yes—
196C23×C4C2×D412yes—
197C23×C4C2×D412yes—
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D418yes—
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C2×M16—28no—
208C2×M16—216no—
209C22×C8C2418yes—
210C4⋊Q8—216no—
211C4⋊Q8—216no—
212C4⋊Q8—216no—
213C4⋊Q8—216no—
214#125—232no—
215#125—216no—
216#125—216no—
217C22⋊Q8—232no—
218C22⋊Q8—232no—
219C22⋊Q8—232no—
220C22⋊Q8—232no—
221C22⋊Q8—28no—
222C22⋊Q8—28no—
223C22⋊Q8—44no—
224C22⋊Q8—44no—
225C4×Q8—232no—
226C4×Q8—232no—
227C4×Q8—232no—
228C4×Q8—232no—
229#117—216no—
230#117—216no—
231#117—28no—
232#117—28no—
233#117C2×D4116yes—
234#117C2×D4116yes—
235C2×C4⋊C4—216no—
236C2×C4⋊C4—216no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—48no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4C2×D418yes—
258C2×C4⋊C4C2×D418yes—
259C2×C4⋊C4C2412yes—
260C2×C4⋊C4C2414yes—
261C2×C4⋊C4C2414yes—
262C2×C4⋊C4C2414yes—
263C2×C22⋊C4—216no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C42—216no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—28no—
276C2×C42—28no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—28no—
282C2×C42—28no—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2412yes—
290C2×C42C2412yes—
291#107—216no—
292#106—216no—
293#103—216no—
294#103—216no—
295C22⋊C8—28no—
296C22⋊C8—28no—
297#95—22no—
298#95—22no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305C24—24no—
306C24—22no—
307C24—22no—
308C24—42no—
309C24#12014yes—
310C24C22×D411yescharacteristic
311C24C22×D411yescharacteristic
312C24C22×D411yescharacteristic
313C2×Q8—216no—
314C2×Q8—216no—
315C2×Q8—24no—
316C2×Q8—28no—
317C2×Q8—432no—
318C2×Q8—432no—
319C2×Q8—48no—
320C2×Q8—48no—
321C2×Q8—42no—
322C2×Q8—44no—
323C2×Q8—48no—
324C2×Q8—48no—
325C2×Q8—416no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—232no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—28no—
343C22×C4—28no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—416no—
361C22×C4—416no—
362C22×C4—48no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4#12014yes—
373C22×C4#12014yes—
374C22×C4#12014yes—
375C22×C4#13714yes—
376C22×C4C22×D412yes—
377C22×C4C22×D412yes—
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D414yes—
381C22×C4C22×D414yes—
382C22×C4C22×D412yes—
383C22×C4C22×D412yes—
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D414yes—
386C22×C42−1+418yes—
387C22×C4C2511yescharacteristic
388M16—416no—
389C2×C8—28no—
390C2×C8—216no—
391C4⋊C4—232no—
392C4⋊C4—232no—
393C4⋊C4—216no—
394C4⋊C4—216no—
395C4⋊C4—28no—
396C4⋊C4—28no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—28no—
401C4⋊C4—48no—
402C4⋊C4—42no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—48no—
407C4⋊C4C22×D418yes—
408C4⋊C4C22×D418yes—
409C22⋊C4—216no—
410C22⋊C4—216no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C42—24no—
416C42—24no—
417C42—24no—
418C42—24no—
419C42—24no—
420C42—24no—
421C42—216no—
422C42—216no—
423C42—24no—
424C42—24no—
425C23—22no—
426C23—21no—
427C23—22no—
428C23—22no—
429C23—21no—
430C23—28no—
431C23—22no—
432C23—28no—
433C23—21no—
434C23—21no—
435C23—22no—
436C23—44no—
437C23—48no—
438C23—48no—
439C23C2×#12012yes—
440C23C2×#12012yes—
441C23C2×#12012yes—
442C23#53614yes—
443C23C2×#13712yes—
444C23#57512yes—
445C23C23×D411yescharacteristic
446Q8—432no—
447Q8—432no—
448Q8—48no—
449Q8—84no—
450C2×C4—28no—
451C2×C4—28no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—28no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—28no—
460C2×C4—28no—
461C2×C4—28no—
462C2×C4—28no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—28no—
466C2×C4—28no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—48no—
472C2×C4—48no—
473C2×C4—48no—
474C2×C4—48no—
475C2×C4—44no—
476C2×C4—416no—
477C2×C4—44no—
478C2×C4—416no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—48no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—88no—
490C2×C4—84no—
491C2×C4—84no—
492C2×C4C2×#12012yes—
493C2×C4C2×#12012yes—
494C2×C4C2×#12012yes—
495C2×C4#53614yes—
496C2×C4#536116yes—
497C2×C4C2×#13714yes—
498C2×C4#57514yes—
499C2×C4C23×D412yes—
500C2×C4C2×2−1+414yes—
501C8—48no—
502C22—22no—
503C22—24no—
504C22—24no—
505C22—24no—
506C22—24no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—416no—
514C22—42no—
515C22—42no—
516C22#261414yes—
517C22#300714yes—
518C22C22×#12011yescharacteristic
519C22C2×#53612yes—
520C22C22×#13711yescharacteristic
521C22C2×#57512yes—
522C22C2×#57511yescharacteristic
523C4—22no—
524C4—22no—
525C4—24no—
526C4—24no—
527C4—42no—
528C4—48no—
529C4—42no—
530C4—48no—
531C4—82no—
532C4—84no—
533C4—82no—
534C4#261414yes—
535C4C2×#53614yes—
536C2—24no—
537C2—24no—
538C2—42no—
539C2C2×#261411yescharacteristic
540C2C2×#300711yescharacteristic
541C2#6035911yescharacteristic
542C2#6086512yes—
543C2#6086612yes—
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. #29645C2semidirect
5. #29643C2semidirect
6. #29642C2semidirect
7. #29641C2semidirect
8. #23554C2semidirect
9. #23455C2semidirect
46. #2059C22semidirect
47. #2058C22semidirect
48. #2057C22semidirect
49. #2026C22semidirect
50. #2024C22semidirect
51. #2024C22semidirect

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×#28552
#296452
#296432
#296422
#296412
#235542
#234552
#207022
#206822