Finite Group Databasebeta

← #764242

Subgroups — #764242

Order 512.

4238 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#544C211yesmaximal, characteristic, complemented
2C2×#2561C212yesmaximal
3C22×#396C211yesmaximal, characteristic
4#15880C218yesmaximal, complemented
5#15322C218yesmaximal, complemented
6#14934C212yesmaximal, complemented
7#14717C212yesmaximal
8#14139C212yesmaximal, complemented
9#13870C212yesmaximal, complemented
10#13868C212yesmaximal
11#13776C211yesmaximal, characteristic, complemented
12C2×#544—216no—
13C2×#544—28no—
14C22×C4⋊Q8C2214yescomplemented
15C22×#125C2214yescomplemented
16C22×C22⋊Q8C2212yes—
17C22×C22⋊Q8C2212yes—
18C22×#117C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24C23×C22⋊C4C2211yescharacteristic
25#2561—216no—
26C2×#397C2214yes—
27C2×#396—28no—
28C2×#396C2218yes—
29C2×#393C2214yes—
30C2×#393C2214yes—
31C2×#391C2214yescomplemented
32C2×#388C2218yes—
33C2×#381C2214yes—
34C2×#381C2212yes—
35C2×#379C2214yes—
36C2×C4×C22⋊C4C2212yes—
37C22×#95C2211yescharacteristic
38#1446C2214yescomplemented
39#1442C2214yescomplemented
40#1442C2218yescomplemented
41#1441C2214yescomplemented
42#1441C2218yescomplemented
43#1441C2214yescomplemented
44#1439C2214yes—
45#1439C2214yes—
46#1439C2218yes—
47#1439C2214yes—
48#1437C2218yes—
49#1436C2218yescomplemented
50#1436C2218yescomplemented
51C4⋊#95C2214yescomplemented
52C22⋊#95C2214yes—
53C22⋊#95C2214yes—
54C22⋊#95C2212yes—
55C26C2311yescharacteristic
56C23×Q8C2312yes—
57C23×Q8C2312yes—
58C24×C4C2311yescharacteristic
59C24×C4C2311yescharacteristic
60C24×C4C2311yescharacteristic
61C24×C4C2311yescharacteristic
62C24×C4C2311yescharacteristic
63C24×C4C2311yescharacteristic
64#544—2128no—
65C2×C4⋊Q8—216no—
66C2×C4⋊Q8—216no—
67C2×C4⋊Q8—216no—
68C2×#125—216no—
69C2×#125—216no—
70C2×#125—216no—
71C2×C22⋊Q8—28no—
72C2×C22⋊Q8—28no—
73C2×C22⋊Q8—28no—
74C2×C22⋊Q8—216no—
75C2×C22⋊Q8D4116yes—
76C2×#117—28no—
77C2×#117—216no—
78C22×C4⋊C4—28no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4—24no—
81C22×C4⋊C4—28no—
82C22×C4⋊C4—44no—
83C22×C4⋊C4D418yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C22⋊C4—28no—
94C22×C22⋊C4—24no—
95C22×C22⋊C4—24no—
96C22×C22⋊C4—24no—
97C22×C22⋊C4—28no—
98C22×C22⋊C4—44no—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2318yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2318yes—
106C22×C42C2314yes—
107C22×C42C2314yes—
108#397—216no—
109#396—232no—
110#393—216no—
111#393—48no—
112#391—48no—
113#388—232no—
114#381—28no—
115#381—216no—
116#379—216no—
117C4×C22⋊C4—28no—
118C2×#95—24no—
119C2×#95C2314yes—
120C2×#95C2318yes—
121C2×#95C2318yes—
122C2×#95C2314yes—
123C2×#95C2318yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2318yes—
128C2×#95C2318yes—
129C2×#95C2314yes—
130C2×#95C2318yes—
131C25—22no—
132C25—24no—
133C25—24no—
134C25—24no—
135C25—24no—
136C25—42no—
137C25—42no—
138C25—42no—
139C25C2412yes—
140C25C2411yescharacteristic
141C22×Q8—24no—
142C22×Q8—24no—
143C22×Q8—24no—
144C22×Q8—28no—
145C22×Q8—44no—
146C22×Q8—44no—
147C22×Q8—44no—
148C22×Q8—44no—
149C22×Q8C2×D418yes—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4C2×D414yes—
170C23×C4C2×D414yes—
171C23×C4C2×D414yes—
172C23×C4C2×D418yes—
173C23×C4#4118yes—
174C23×C4C2418yes—
175C23×C4C2412yes—
176C23×C4C2412yes—
177C23×C4C2414yes—
178C23×C4C2414yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C4⋊Q8—2128no—
184#125—2128no—
185C22⋊Q8—264no—
186C22⋊Q8—264no—
187#117—264no—
188C2×C4⋊C4—232no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—416no—
211C2×C4⋊C4—416no—
212C2×C4⋊C4C2×D4116yes—
213C2×C4⋊C4C2×D418yes—
214C2×C4⋊C4C2×D4116yes—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—216no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—216no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—216no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—48no—
233C2×C22⋊C4—416no—
234C2×C22⋊C4—416no—
235C2×C22⋊C4—416no—
236C2×C22⋊C4—48no—
237C2×C22⋊C4—48no—
238C2×C22⋊C4C2×D4116yes—
239C2×C42—28no—
240C2×C42—28no—
241C2×C42—28no—
242C2×C42—28no—
243C2×C42—44no—
244C2×C42—44no—
245#95—28no—
246#95—28no—
247#95—216no—
248#95—216no—
249#95—28no—
250#95—28no—
251#95—216no—
252#95—48no—
253#95—48no—
254#95—44no—
255#95—48no—
256#95—44no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—28no—
262C24—22no—
263C24—24no—
264C24—24no—
265C24—24no—
266C24—22no—
267C24—24no—
268C24—24no—
269C24—24no—
270C24—416no—
271C24—416no—
272C24—416no—
273C24—44no—
274C24—44no—
275C24—44no—
276C24—44no—
277C24—44no—
278C24—44no—
279C24—416no—
280C24—88no—
281C24—88no—
282C24—88no—
283C24C22×D412yes—
284C24C22×D414yes—
285C24C2×#4114yes—
286C242−1+414yes—
287C242−1+414yes—
288C24C2511yescharacteristic
289C2×Q8—216no—
290C2×Q8—216no—
291C2×Q8—416no—
292C2×Q8—84no—
293C2×Q8—84no—
294C2×Q8—88no—
295C2×Q8—84no—
296C2×Q8—84no—
297C2×Q8—816no—
298C22×C4—28no—
299C22×C4—28no—
300C22×C4—232no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—28no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—28no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—28no—
317C22×C4—28no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—28no—
321C22×C4—28no—
322C22×C4—24no—
323C22×C4—28no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—28no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—28no—
330C22×C4—24no—
331C22×C4—28no—
332C22×C4—28no—
333C22×C4—24no—
334C22×C4—28no—
335C22×C4—24no—
336C22×C4—28no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—416no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—416no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—416no—
352C22×C4—88no—
353C22×C4—88no—
354C22×C4#120116yes—
355C22×C4#124116yes—
356C22×C4C22×D418yes—
357C22×C4C22×D418yes—
358C22×C4C22×D414yes—
359C22×C4C22×D414yes—
360C22×C4C22×D414yes—
361C22×C4C22×D414yes—
362C22×C4C22×D414yes—
363C22×C4C2×#4114yes—
364C22×C4C2×#4114yes—
365C22×C4H(5,𝔽2)14yes—
366C4⋊C4—216no—
367C4⋊C4—216no—
368C4⋊C4—216no—
369C4⋊C4—232no—
370C4⋊C4—232no—
371C4⋊C4—216no—
372C4⋊C4—232no—
373C4⋊C4—416no—
374C4⋊C4—416no—
375C22⋊C4—232no—
376C22⋊C4—416no—
377C22⋊C4—432no—
378C22⋊C4—416no—
379C22⋊C4—416no—
380C22⋊C4—416no—
381C22⋊C4—816no—
382C42—232no—
383C42—88no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—28no—
389C23—28no—
390C23—28no—
391C23—28no—
392C23—28no—
393C23—28no—
394C23—24no—
395C23—28no—
396C23—28no—
397C23—28no—
398C23—416no—
399C23—416no—
400C23—416no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—48no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—44no—
409C23—48no—
410C23—44no—
411C23—48no—
412C23—44no—
413C23—48no—
414C23—864no—
415C23—88no—
416C23—88no—
417C23—88no—
418C23—88no—
419C23C2×#12014yes—
420C23C2×#12414yes—
421C23#53614yes—
422C23#548116yes—
423C23C23×D411yescharacteristic
424C23C23×D412yes—
425C23C22×#4112yes—
426C23C2×H(5,𝔽2)12yes—
427C23C2×2−1+412yes—
428C23C2×2−1+412yes—
429C23#58512yes—
430C23#58512yes—
431Q8—816no—
432Q8—816no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—232no—
436C2×C4—28no—
437C2×C4—28no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—28no—
441C2×C4—28no—
442C2×C4—28no—
443C2×C4—28no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—216no—
449C2×C4—28no—
450C2×C4—216no—
451C2×C4—416no—
452C2×C4—416no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—48no—
460C2×C4—44no—
461C2×C4—48no—
462C2×C4—48no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—48no—
468C2×C4—48no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—48no—
472C2×C4—48no—
473C2×C4—48no—
474C2×C4—48no—
475C2×C4—48no—
476C2×C4—48no—
477C2×C4—48no—
478C2×C4—88no—
479C2×C4—88no—
480C2×C4—88no—
481C2×C4C2×#12018yes—
482C2×C4C2×#12418yes—
483C22—24no—
484C22—24no—
485C22—24no—
486C22—24no—
487C22—24no—
488C22—24no—
489C22—232no—
490C22—44no—
491C22—44no—
492C22—44no—
493C22—416no—
494C22—832no—
495C22—82no—
496C22—82no—
497C22—82no—
498C22—88no—
499C22—82no—
500C22#243618yes—
501C22C22×#12011yescharacteristic
502C22C22×#12411yescharacteristic
503C22C2×#53611yescharacteristic
504C22C2×#54814yes—
505C22#348411yescharacteristic
506C22#348511yescharacteristic
507C22#348611yescharacteristic
508C22#348711yescharacteristic
509C22#348914yes—
510C22#349414yes—
511C22#349514yes—
512C22#349614yes—
513C22#353814yes—
514C22#356414yes—
515C4—28no—
516C4—28no—
517C4—44no—
518C4—44no—
519C4—48no—
520C4—48no—
521C4—82no—
522C4—88no—
523C4—82no—
524C2—28no—
525C2—84no—
526C2C2×#243611yescharacteristic
527C2#3534711yescharacteristic
528C2#3558411yescharacteristic
529C2#3564411yescharacteristic
530C2#3565911yescharacteristic
531C2#3670811yescharacteristic
532C2#3696211yescharacteristic
533C2#4390818yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#544C2semidirect
4. #15880C2semidirect
5. #15322C2semidirect
6. #14934C2semidirect
8. #14139C2semidirect
9. #13870C2semidirect
11. #13776C2semidirect
14. C22×C4⋊Q8C22semidirect
15. C22×#125C22semidirect
31. C2×#391C22semidirect
38. #1446C22semidirect
39. #1442C22semidirect
40. #1442C22semidirect
41. #1441C22semidirect
42. #1441C22semidirect
43. #1441C22semidirect
49. #1436C22semidirect
50. #1436C22semidirect
51. C4⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
C2×#25612
C22×#3962
#158802
#153222
#149342
#147172
#141392
#138702
#138682
#137762