Finite Group Databasebeta

← #6489547

Subgroups — #6489547

Order 512.

3202 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#125C211yesmaximal, characteristic
2C2×#3236C212yesmaximal, complemented
3#36625C211yesmaximal, characteristic
4#32983C212yesmaximal
5#32982C212yesmaximal
6#29643C214yesmaximal, complemented
7#25912C214yesmaximal, complemented
8#25819C214yesmaximal, complemented
9#24307C218yesmaximal, complemented
10#21838C212yesmaximal, complemented
11#20715C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C22×#125—22no—
14C22×#125—21no—
15C22×#125—21no—
16C22×#125—22no—
17C22×#125—216no—
18C22×#125C2214yes—
19C22×#125C2212yes—
20C22×#125C2214yes—
21C22×#125C2212yes—
22C23×C4⋊C4—21no—
23C23×C4⋊C4—21no—
24C23×C4⋊C4C2211yescharacteristic
25C23×C4⋊C4C2211yescharacteristic
26C23×C42C2211yescharacteristic
27#3236—216no—
28C2×#501C2214yescomplemented
29C2×#491C2214yescomplemented
30C2×#484C2214yes—
31C2×#483C2214yes—
32C2×#479C2218yescomplemented
33C2×#432C2212yes—
34#2857C2212yes—
35#2857C2212yes—
36#2855—21no—
37#2855C2212yes—
38#2851—22no—
39#2848—21no—
40#2848C2212yes—
41C4×C22⋊Q8—22no—
42C4×C22⋊Q8C2214yes—
43#2296—21no—
44#2285C2211yescharacteristic
45#2280—21no—
46#2059C22116yescomplemented
47#2026C22116yescomplemented
48#1992C2218yescomplemented
49#1989C2218yescomplemented
50#1872C2218yes—
51#1872C2218yes—
52#1853C2218yes—
53#1852C2218yes—
54#1824C2214yes—
55#1823C2214yes—
56#1817C2212yes—
57#1797C2214yes—
58#1779C2212yes—
59#1749C2214yes—
60C24×C4—21no—
61C24×C4—21no—
62C24×C4D411yescharacteristic
63C24×C4D411yescharacteristic
64C24×C4C2311yescharacteristic
65#533—28no—
66C2×C4⋊Q8C2311yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×#125—216no—
69C2×#125—216no—
70C2×#125—232no—
71C2×#125—232no—
72C2×#125—232no—
73C2×#125—232no—
74C2×#125—28no—
75C2×#125—28no—
76C2×#125—24no—
77C2×#125—24no—
78C2×#125—24no—
79C2×#125—24no—
80C2×#125—24no—
81C2×#125—22no—
82C2×#125—22no—
83C2×#125—28no—
84C2×#125—21no—
85C2×#125D4116yes—
86C2×#125C2318yes—
87C2×#125C2314yes—
88C2×#125C2314yes—
89C2×C22⋊Q8—21no—
90C2×C22⋊Q8—22no—
91C2×C22⋊Q8C2312yes—
92C2×C22⋊Q8C2314yes—
93C2×C4×Q8—21no—
94C2×C4×Q8C2312yes—
95C2×#117—21no—
96C2×#117—21no—
97C22×C4⋊C4—216no—
98C22×C4⋊C4—216no—
99C22×C4⋊C4—22no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—22no—
105C22×C4⋊C4—22no—
106C22×C4⋊C4—22no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—22no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—22no—
111C22×C4⋊C4—28no—
112C22×C4⋊C4—28no—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C42—28no—
118C22×C42D412yes—
119C22×C42D411yescharacteristic
120C22×C42D411yescharacteristic
121C22×C42D412yes—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126#501—216no—
127#491—216no—
128#484—216no—
129#483—216no—
130#479—232no—
131#432—28no—
132C2×#107C2318yes—
133C2×#106C2318yes—
134C2×C4⋊C8C2314yes—
135C2×#103C2318yes—
136C2×#103C2318yes—
137C2×C22⋊C8C2318yes—
138C2×C22⋊C8C2314yes—
139C2×#97C2314yes—
140#391—22no—
141#391C2314yes—
142#389—22no—
143#389—22no—
144#389—22no—
145#389C2314yes—
146#387—24no—
147#387—24no—
148#385—24no—
149#385—22no—
150#385—24no—
151#385—22no—
152#384C2312yes—
153#382—21no—
154#381—22no—
155#380—21no—
156#380—22no—
157#380—22no—
158#380—22no—
159#380C2312yes—
160#380C2314yes—
161#379—22no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4—24no—
164C4×C4⋊C4C2312yes—
165C4×C22⋊C4—24no—
166#340C23116yes—
167#340C23116yes—
168#336C2318yes—
169C25C2×D411yescharacteristic
170C22×Q8—21no—
171C22×Q8C2412yes—
172C23×C4—24no—
173C23×C4—28no—
174C23×C4—28no—
175C23×C4—28no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—42no—
184C23×C4—44no—
185C23×C4—48no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D412yes—
191C23×C4C2×D412yes—
192C23×C4C2×D412yes—
193C23×C4C2×D412yes—
194C23×C4C2×D412yes—
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C22×C8C2418yes—
201C4⋊Q8—24no—
202C4⋊Q8—24no—
203#125—232no—
204#125—232no—
205#125—232no—
206#125—232no—
207#125—232no—
208#125—216no—
209#125—216no—
210#125—216no—
211#125—216no—
212#125—28no—
213#125—28no—
214#125—44no—
215#125C2×D4116yes—
216#125C2×D4116yes—
217C22⋊Q8—28no—
218C22⋊Q8—216no—
219C22⋊Q8—48no—
220C22⋊Q8—44no—
221C4×Q8—28no—
222C4×Q8—44no—
223#117—44no—
224#117—44no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—216no—
228C2×C4⋊C4—216no—
229C2×C4⋊C4—232no—
230C2×C4⋊C4—232no—
231C2×C4⋊C4—232no—
232C2×C4⋊C4—232no—
233C2×C4⋊C4—216no—
234C2×C4⋊C4—216no—
235C2×C4⋊C4—216no—
236C2×C4⋊C4—216no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—216no—
250C2×C4⋊C4—216no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4C2×D418yes—
261C2×C4⋊C4C2×D418yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2414yes—
264C2×C4⋊C4C2414yes—
265C2×C4⋊C4C2414yes—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—28no—
274C2×C42—28no—
275C2×C42—28no—
276C2×C42—28no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D412yes—
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D418yes—
287C2×C42C2412yes—
288C2×C42C2411yescharacteristic
289C2×C42C2411yescharacteristic
290#107—216no—
291#106—216no—
292C4⋊C8—28no—
293#103—216no—
294#103—216no—
295C22⋊C8—28no—
296C22⋊C8—216no—
297#97—28no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—22no—
302#95—24no—
303#95—22no—
304C24—22no—
305C24—24no—
306C24—24no—
307C24—22no—
308C24#13714yes—
309C24C22×D411yescharacteristic
310C24C22×D412yes—
311C2×Q8—24no—
312C2×Q8—28no—
313C2×Q8—44no—
314C2×Q8—42no—
315C22×C4—232no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—216no—
327C22×C4—216no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—28no—
331C22×C4—28no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—416no—
354C22×C4—416no—
355C22×C4—432no—
356C22×C4—432no—
357C22×C4—48no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—44no—
362C22×C4—48no—
363C22×C4—44no—
364C22×C4#12014yes—
365C22×C4#12014yes—
366C22×C4C22×D412yes—
367C22×C4C22×D412yes—
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D414yes—
374C22×C4C22×D412yes—
375C22×C42−1+418yes—
376C22×C4C2511yescharacteristic
377C2×C8—28no—
378C2×C8—216no—
379C4⋊C4—216no—
380C4⋊C4—216no—
381C4⋊C4—232no—
382C4⋊C4—232no—
383C4⋊C4—216no—
384C4⋊C4—216no—
385C4⋊C4—216no—
386C4⋊C4—216no—
387C4⋊C4—28no—
388C4⋊C4—28no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—28no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—48no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—42no—
401C4⋊C4—48no—
402C4⋊C4C22×D418yes—
403C4⋊C4C22×D418yes—
404C22⋊C4—44no—
405C22⋊C4—48no—
406C22⋊C4—44no—
407C22⋊C4—48no—
408C42—24no—
409C42—24no—
410C42—24no—
411C42—24no—
412C42—24no—
413C42—24no—
414C42—44no—
415C42—44no—
416C42#12014yes—
417C42#12014yes—
418C42C22×D414yes—
419C42C22×D414yes—
420C23—28no—
421C23—28no—
422C23—216no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—216no—
427C23—28no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23C2×#12012yes—
432C23#53614yes—
433C23C2×#13714yes—
434C23C2×#13712yes—
435C23#57514yes—
436C23#57512yes—
437C23C23×D411yescharacteristic
438Q8—48no—
439Q8—84no—
440C2×C4—216no—
441C2×C4—28no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—28no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—28no—
453C2×C4—216no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—28no—
457C2×C4—28no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—44no—
463C2×C4—48no—
464C2×C4—48no—
465C2×C4—48no—
466C2×C4—48no—
467C2×C4—432no—
468C2×C4—44no—
469C2×C4—416no—
470C2×C4—432no—
471C2×C4—44no—
472C2×C4—416no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—48no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4C2×#12012yes—
484C2×C4C2×#12012yes—
485C2×C4C2×#12012yes—
486C2×C4C2×#12012yes—
487C2×C4C2×#12012yes—
488C2×C4#53614yes—
489C2×C4C23×D412yes—
490C2×C4C2×2−1+414yes—
491C8—48no—
492C22—232no—
493C22—22no—
494C22—24no—
495C22—24no—
496C22—24no—
497C22—24no—
498C22—24no—
499C22—24no—
500C22—24no—
501C22—22no—
502C22—24no—
503C22#300714yes—
504C22#323618yescomplemented
505C22C22×#12011yescharacteristic
506C22C2×#53612yes—
507C22C22×#13711yescharacteristic
508C22C2×#57512yes—
509C22C2×#57511yescharacteristic
510C4—22no—
511C4—24no—
512C4—22no—
513C4—22no—
514C4—22no—
515C4—22no—
516C4—42no—
517C4—48no—
518C4—42no—
519C4—48no—
520C4—82no—
521C4—84no—
522C4—82no—
523C4#348614yes—
524C2—28no—
525C2—24no—
526C2#3524211yescharacteristic
527C2C2×#300711yescharacteristic
528C2#6035911yescharacteristic
529C2C2×#323612yes—
530C2#6086112yes—
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
2. C2×#3236C2semidirect
6. #29643C2semidirect
7. #25912C2semidirect
8. #25819C2semidirect
9. #24307C2semidirect
10. #21838C2semidirect
28. C2×#501C22semidirect
29. C2×#491C22semidirect
32. C2×#479C22semidirect
46. #2059C22semidirect
47. #2026C22semidirect
48. #1992C22semidirect
49. #1989C22semidirect
504. C22#3236semidirect

Maximal subgroups

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

subgroupindex
C23×#1252
C2×#32362
#366252
#329832
#329822
#296432
#259122
#258192
#243072
#218382
#207152