Finite Group Databasebeta

← #6434683

Subgroups — #6434683

Order 512.

3018 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#544C211yesmaximal, characteristic
2C2×#3095C211yesmaximal, characteristic
3C22×#483C211yesmaximal, characteristic
4#35311C211yesmaximal, characteristic
5#29648C218yesmaximal, complemented
6#29643C214yesmaximal, complemented
7#29641C214yesmaximal, complemented
8#23556C214yesmaximal, complemented
9#23457C214yesmaximal, complemented
10#20700C211yesmaximal, characteristic
11#20699C211yesmaximal, characteristic
12#20685C211yesmaximal, characteristic
13C2×#544—22no—
14C2×#544—22no—
15C2×#544—28no—
16C2×#544—28no—
17C2×#544C2214yes—
18C2×#544C2214yes—
19C22×C4⋊Q8—21no—
20C22×C4⋊Q8C2212yes—
21C22×#125—21no—
22C22×#125C2212yes—
23C22×C22⋊Q8—22no—
24C22×C22⋊Q8C2211yescharacteristic
25C22×#117C2211yescharacteristic
26C22×#117C2211yescharacteristic
27C23×C4⋊C4C2211yescharacteristic
28#3095—28no—
29C2×#484C2212yes—
30C2×#485C2214yes—
31C2×#483—28no—
32C2×#483C2214yes—
33C2×#483C2214yes—
34C2×#483C2212yes—
35C2×#428C2212yes—
36C22×#106C2212yes—
37C2×#419C2212yes—
38C22×#103C2212yes—
39C2×#407C2211yescharacteristic
40C22×C22⋊C8C2211yescharacteristic
41#2855C2214yes—
42#2625C2214yes—
43#2503—21no—
44#2502—21no—
45#2499—22no—
46#2484—22no—
47#2483—22no—
48#2317C2212yes—
49#2310C2212yes—
50#2280C2211yescharacteristic
51#2059C22116yescomplemented
52#2058C22116yescomplemented
53#2026C22116yescomplemented
54#2024C22116yescomplemented
55#1825C2214yes—
56#1824C2212yes—
57#1824C2214yes—
58#1823C2212yes—
59#1823C2214yes—
60#1818C2214yes—
61#1817C2214yes—
62#1736C2214yes—
63#1735C2214yes—
64#1729C2214yes—
65C23×Q8—22no—
66C23×Q8C2311yescharacteristic
67C24×C4—21no—
68C24×C4C2311yescharacteristic
69C22×M16C2312yes—
70C23×C8C2312yes—
71#544—232no—
72#544—232no—
73#544—216no—
74#544—216no—
75#544—216no—
76#544—216no—
77C2×C4⋊Q8—216no—
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8—28no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8C2318yes—
84C2×#125—216no—
85C2×#125—24no—
86C2×#125—24no—
87C2×#125—28no—
88C2×#125—28no—
89C2×#125—24no—
90C2×#125C2318yes—
91C2×C22⋊Q8—232no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2312yes—
100C2×#117—24no—
101C2×#117—21no—
102C2×#117—21no—
103C2×#117—24no—
104C2×#117—24no—
105C2×#117D418yes—
106C2×#117D412yes—
107C2×#117D412yes—
108C2×#117C2312yes—
109C2×#117C2312yes—
110C2×#117C2312yes—
111C2×#117C2312yes—
112C2×#117C2312yes—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—24no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4D418yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4D412yes—
133C22×C42—21no—
134C22×C42C2312yes—
135#485—216no—
136#484—28no—
137#483—216no—
138#483—216no—
139#483—28no—
140#428—28no—
141C2×#107C2314yes—
142C2×#106—28no—
143C2×#106C2314yes—
144C2×#106C2318yes—
145#419—28no—
146C2×#103—28no—
147C2×#103C2318yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150#407—24no—
151C2×C22⋊C8—24no—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2312yes—
156#400—24no—
157#398—24no—
158#397—28no—
159#391—22no—
160#391—22no—
161#391C2318yes—
162#389C2318yes—
163#387—28no—
164#385—24no—
165#385—24no—
166#384—22no—
167#382—22no—
168#381—24no—
169#381—21no—
170#381—24no—
171#380C2312yes—
172#380C2312yes—
173#379—21no—
174C4×C22⋊C4—21no—
175C4×C22⋊C4—21no—
176#340C2318yes—
177#340C23116yes—
178#340C2318yes—
179#336C23116yes—
180C25C2×D411yescharacteristic
181C22×Q8—216no—
182C22×Q8—28no—
183C22×Q8—21no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—28no—
187C22×Q8—88no—
188C22×Q8C2412yes—
189C23×C4—28no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—22no—
193C23×C4—28no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4—84no—
199C23×C4C2×D412yes—
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D414yes—
204C23×C4C2×D414yes—
205C23×C4C2×D412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C2×M16—24no—
210C2×M16—28no—
211C22×C8—24no—
212C22×C8—28no—
213C22×C8C2414yes—
214C22×C8C2414yes—
215C4⋊Q8—232no—
216C4⋊Q8—232no—
217C4⋊Q8—232no—
218C4⋊Q8—216no—
219C4⋊Q8—216no—
220#125—232no—
221#125—232no—
222#125—232no—
223#125—216no—
224#125—216no—
225C22⋊Q8—264no—
226C22⋊Q8—264no—
227C22⋊Q8—24no—
228C22⋊Q8—24no—
229C22⋊Q8—24no—
230C22⋊Q8—24no—
231C22⋊Q8—44no—
232C22⋊Q8—44no—
233#117—28no—
234#117—28no—
235#117—28no—
236#117—28no—
237#117—24no—
238#117—24no—
239#117—24no—
240#117—24no—
241#117—24no—
242#117—24no—
243#117—28no—
244#117C2×D418yes—
245#117C2×D418yes—
246C2×C4⋊C4—216no—
247C2×C4⋊C4—216no—
248C2×C4⋊C4—216no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D418yes—
279C2×C4⋊C4C2×D418yes—
280C2×C4⋊C4C2×D418yes—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2414yes—
285C2×C4⋊C4C2414yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4C2×D414yes—
299C2×C22⋊C4C2×D414yes—
300C2×C42—28no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42C2×D414yes—
307C2×C42C2414yes—
308#107—28no—
309#106—216no—
310#106—28no—
311#103—216no—
312#103—28no—
313#103—28no—
314C22⋊C8—24no—
315C22⋊C8—24no—
316C22⋊C8—24no—
317C22⋊C8—24no—
318#95—22no—
319#95—22no—
320#95—28no—
321#95—28no—
322#95—28no—
323C24—24no—
324C24—24no—
325C24—42no—
326C24#12014yes—
327C24C22×D411yescharacteristic
328C24C22×D411yescharacteristic
329C24C22×D411yescharacteristic
330C2×Q8—232no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—28no—
334C2×Q8—42no—
335C2×Q8—44no—
336C2×Q8—48no—
337C2×Q8—48no—
338C2×Q8—816no—
339C2×Q8—832no—
340C22×C4—24no—
341C22×C4—216no—
342C22×C4—216no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—48no—
364C22×C4—44no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4—82no—
369C22×C4—816no—
370C22×C4—84no—
371C22×C4—84no—
372C22×C4#12014yes—
373C22×C4#12014yes—
374C22×C4#12014yes—
375C22×C4C22×D414yes—
376C22×C4C22×D414yes—
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D412yes—
380C22×C4C22×D412yes—
381C22×C4C22×D412yes—
382C22×C4C22×D412yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D412yes—
385C22×C4C22×D412yes—
386C22×C42−1+418yes—
387C22×C4C2511yescharacteristic
388M16—48no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—28no—
392C2×C8—28no—
393C2×C8—48no—
394C4⋊C4—232no—
395C4⋊C4—232no—
396C4⋊C4—232no—
397C4⋊C4—28no—
398C4⋊C4—28no—
399C4⋊C4—28no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—28no—
412C4⋊C4—42no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—88no—
416C4⋊C4C22×D418yes—
417C4⋊C4C22×D414yes—
418C4⋊C4C22×D414yes—
419C22⋊C4—28no—
420C22⋊C4—28no—
421C22⋊C4—28no—
422C22⋊C4—28no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—82no—
428C42—28no—
429C42—28no—
430C42—28no—
431C42—24no—
432C42—24no—
433C42—44no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—28no—
440C23—22no—
441C23—28no—
442C23—22no—
443C23—44no—
444C23—416no—
445C23C2×#12012yes—
446C23C2×#12012yes—
447C23C2×#12012yes—
448C23#53614yes—
449C23#57714yes—
450C23C23×D411yescharacteristic
451Q8—44no—
452Q8—44no—
453Q8—832no—
454Q8—84no—
455C2×C4—28no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—28no—
459C2×C4—28no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—28no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—28no—
470C2×C4—28no—
471C2×C4—48no—
472C2×C4—48no—
473C2×C4—44no—
474C2×C4—48no—
475C2×C4—416no—
476C2×C4—48no—
477C2×C4—48no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—88no—
487C2×C4—88no—
488C2×C4—84no—
489C2×C4—84no—
490C2×C4C2×#12012yes—
491C2×C4C2×#12012yes—
492C2×C4C2×#12012yes—
493C2×C4#53614yes—
494C2×C4#53618yes—
495C2×C4#53618yes—
496C2×C4C23×D411yescharacteristic
497C2×C4C23×D411yescharacteristic
498C2×C4C2×2−1+414yes—
499C8—44no—
500C8—44no—
501C22—22no—
502C22—28no—
503C22—28no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—416no—
511C22—44no—
512C22#261414yes—
513C22#300914yes—
514C22C22×#12011yescharacteristic
515C22C2×#53612yes—
516C22C2×#57712yes—
517C22#358612yes—
518C4—22no—
519C4—22no—
520C4—44no—
521C4—48no—
522C4—44no—
523C4—44no—
524C4—44no—
525C4—82no—
526C4—84no—
527C4—82no—
528C4#261414yes—
529C4C2×#53612yes—
530C4C2×#53612yes—
531C2—24no—
532C2—24no—
533C2—42no—
534C2C2×#261411yescharacteristic
535C2C2×#300911yescharacteristic
536C2#6036811yescharacteristic
537C2#6087214yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #29648C2semidirect
6. #29643C2semidirect
7. #29641C2semidirect
8. #23556C2semidirect
9. #23457C2semidirect
51. #2059C22semidirect
52. #2058C22semidirect
53. #2026C22semidirect
54. #2024C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
C2×#30952
C22×#4832
#353112
#296482
#296432
#296412
#235562
#234572
#207002
#206992
#206852