Finite Group Databasebeta

← #4598548

Subgroups — #4598548

Order 512.

2270 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2844C211yesmaximal, characteristic, complemented
2C2×#2563C211yesmaximal, characteristic
3#17127C214yesmaximal, complemented
4#16692C212yesmaximal
5#16689C212yesmaximal, complemented
6#16511C214yesmaximal, complemented
7#16303C214yesmaximal, complemented
8#15762C212yesmaximal, complemented
9#15662C212yesmaximal
10#15489C214yesmaximal, complemented
11#15086C212yesmaximal, complemented
12#14825C212yesmaximal
13#14476C211yesmaximal, characteristic, complemented
14C22×C4⋊Q8C2211yescharacteristic
15C22×C4⋊Q8C2212yes—
16C22×C4×D4C2211yescharacteristic
17#2844—28no—
18#2563—28no—
19C2×#400C2212yes—
20C2×#400C2214yes—
21C2×#399C2214yes—
22C2×#399C2212yes—
23C2×#397C2214yes—
24C2×#395C2212yes—
25C2×#395C2214yes—
26C2×#395C2214yes—
27C2×#395C2212yes—
28C2×#393C2214yes—
29C2×#393C2214yes—
30C2×#390C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#388C2214yes—
33C2×#382C2211yescharacteristic
34C2×#382C2212yes—
35C2×#382C2211yescharacteristic
36C2×#380C2214yes—
37C2×#380C2212yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39#1446C2212yes—
40#1445C2214yes—
41#1445C2214yes—
42#1444C2214yes—
43#1444C2214yes—
44#1444C2212yes—
45#1444C2214yes—
46#1443C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2212yes—
50#1441C2212yes—
51#1441C2214yes—
52#1441C2212yes—
53#1441C2212yes—
54#1440C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2212yes—
60#1438C2214yes—
61#1436C2212yes—
62#1436C2212yes—
63#1436C2214yes—
64#1434C2214yes—
65#1434C2212yes—
66#1434C2214yes—
67C22⋊#95C2212yes—
68C22⋊#95C2214yes—
69C23×Q8C2312yes—
70C23×Q8C2312yes—
71C23×D4C2311yescharacteristic
72C24×C4C2312yes—
73C2×C4⋊Q8—24no—
74C2×C4⋊Q8—42no—
75C2×C4⋊Q8—48no—
76C2×C4⋊Q8—42no—
77C2×C4⋊Q8—44no—
78C2×C4×D4—24no—
79C2×C4×D4—42no—
80C2×C4×D4—42no—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103C22×C42C2312yes—
104C22×C42C2311yescharacteristic
105#400—28no—
106#400—216no—
107#399—48no—
108#399—44no—
109#397—48no—
110#395—28no—
111#395—216no—
112#395—44no—
113#395—48no—
114#393—216no—
115#393—48no—
116#390—24no—
117#390—24no—
118#388—48no—
119#382—24no—
120#382—28no—
121#382D418yes—
122#380—216no—
123#380—44no—
124C4×C4⋊C4—24no—
125C2×#95C2312yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C25C2412yes—
151C22×Q8—24no—
152C22×Q8—24no—
153C22×Q8—24no—
154C22×Q8—42no—
155C22×Q8—44no—
156C22×Q8—44no—
157C22×Q8—42no—
158C22×Q8—44no—
159C22×Q8—44no—
160C22×Q8—42no—
161C22×D4—24no—
162C22×D4—22no—
163C22×D4—24no—
164C22×D4—41no—
165C22×D4—41no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—28no—
169C23×C4—42no—
170C23×C4—44no—
171C23×C4C2414yes—
172C23×C4C2412yes—
173C23×C4C2414yes—
174C23×C4C2414yes—
175C23×C4C2412yes—
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2412yes—
179C23×C4C2411yescharacteristic
180C23×C4C2414yes—
181C23×C4C2411yescharacteristic
182C23×C4C2411yescharacteristic
183C4⋊Q8—416no—
184C4⋊Q8—432no—
185C4×D4—416no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4C2×D414yes—
221C2×C4⋊C4C2×D414yes—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—24no—
247C2×C42—22no—
248C2×C42—28no—
249C2×C42—24no—
250C2×C42—42no—
251C2×C42C2×D414yes—
252#95—24no—
253#95—28no—
254#95—28no—
255#95—28no—
256#95—24no—
257#95—28no—
258#95—24no—
259#95—28no—
260#95—28no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—28no—
266#95—42no—
267#95—44no—
268#95—42no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—42no—
274#95—44no—
275#95—42no—
276#95C2×D4116yes—
277C24—22no—
278C24—24no—
279C24—24no—
280C24—22no—
281C24—22no—
282C24—24no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—24no—
287C242−1+414yes—
288C24C2511yescharacteristic
289C2×Q8—48no—
290C2×Q8—48no—
291C2×Q8—48no—
292C2×Q8—84no—
293C2×Q8—88no—
294C2×Q8—88no—
295C2×Q8—84no—
296C2×Q8—84no—
297C2×Q8—88no—
298C2×Q8—88no—
299C2×D4—28no—
300C2×D4—48no—
301C2×D4—48no—
302C2×D4—44no—
303C2×D4—44no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—21no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—24no—
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—48no—
363C22×C4—416no—
364C22×C4—48no—
365C22×C4—84no—
366C22×C4—88no—
367C22×C4C22×D414yes—
368C22×C4C22×D414yes—
369C22×C4C22×D414yes—
370C22×C4C22×D412yes—
371C22×C4H(5,𝔽2)18yes—
372C22×C4H(5,𝔽2)12yes—
373C22×C4H(5,𝔽2)12yes—
374C22×C42−1+414yes—
375C22×C42−1+414yes—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—48no—
379C4⋊C4—48no—
380C4⋊C4—416no—
381C4⋊C4—44no—
382C4⋊C4—416no—
383C4⋊C4—84no—
384C4⋊C4—84no—
385C4⋊C4—84no—
386C4⋊C4—88no—
387C4⋊C4—84no—
388C4⋊C4—88no—
389C22⋊C4—48no—
390C22⋊C4—416no—
391C22⋊C4—88no—
392C22⋊C4—88no—
393C22⋊C4—88no—
394C22⋊C4—88no—
395C22⋊C4—88no—
396C42—28no—
397C42—28no—
398C42—48no—
399C42—44no—
400C23—24no—
401C23—28no—
402C23—24no—
403C23—24no—
404C23—28no—
405C23—42no—
406C23—42no—
407C23—44no—
408C23—44no—
409C23—42no—
410C23—44no—
411C23—44no—
412C23—42no—
413C23—42no—
414C23—44no—
415C23—44no—
416C23—42no—
417C23—44no—
418C23—44no—
419C23—42no—
420C23—42no—
421C23—44no—
422C23—44no—
423C23C23×D411yescharacteristic
424C23C2×H(5,𝔽2)11yescharacteristic
425C23C2×H(5,𝔽2)11yescharacteristic
426C23C2×H(5,𝔽2)14yes—
427C23C2×2−1+412yes—
428C23C2×2−1+412yes—
429C23C2×2−1+412yes—
430C23#58512yes—
431Q8—816no—
432Q8—816no—
433D4—416no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—28no—
448C2×C4—24no—
449C2×C4—28no—
450C2×C4—28no—
451C2×C4—24no—
452C2×C4—28no—
453C2×C4—28no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
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—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—816no—
500C2×C4#53614yes—
501C2×C4#56314yes—
502C22—44no—
503C22—44no—
504C22—48no—
505C22—48no—
506C22—44no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22—84no—
511C22—84no—
512C22—82no—
513C22C2×#53611yescharacteristic
514C22#348411yescharacteristic
515C22#348611yescharacteristic
516C22#348912yes—
517C22#349812yes—
518C22C2×#56311yescharacteristic
519C22#353212yes—
520C22#353514yes—
521C22#353812yes—
522C22#353812yes—
523C22#354414yes—
524C22#354511yescharacteristic
525C22#354714yes—
526C22#354912yes—
527C22#355214yes—
528C22#356112yes—
529C4—24no—
530C4—24no—
531C4—44no—
532C4—42no—
533C4—44no—
534C4—48no—
535C4—44no—
536C4—82no—
537C4—84no—
538C4—84no—
539C4—82no—
540C4—84no—
541C2—84no—
542C2#3670811yescharacteristic
543C2#4296812yes—
544C2#4613611yescharacteristic
545C2#5089412yes—
546C2#5190211yescharacteristic
547C2#5300912yes—
548C2#5398012yes—
549C2#5804614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2844C2semidirect
3. #17127C2semidirect
5. #16689C2semidirect
6. #16511C2semidirect
7. #16303C2semidirect
8. #15762C2semidirect
10. #15489C2semidirect
11. #15086C2semidirect
13. #14476C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28442
C2×#25632
#171272
#166922
#166892
#165112
#163032
#157622
#156622
#154892
#150862
#148252
#144762