Finite Group Databasebeta

← #4607002

Subgroups — #4607002

Order 512.

2142 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2809C211yesmaximal, characteristic, complemented
2C2×#2547C211yesmaximal, characteristic
3#17130C214yesmaximal, complemented
4#16653C212yesmaximal, complemented
5#16558C212yesmaximal
6#16336C214yesmaximal, complemented
7#16304C214yesmaximal, complemented
8#16094C212yesmaximal
9#15783C212yesmaximal, complemented
10#15393C214yesmaximal, complemented
11#15098C212yesmaximal, complemented
12#14825C212yesmaximal
13#14668C211yesmaximal, characteristic, complemented
14C22×C4⋊Q8C2211yescharacteristic
15C22×#124C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18#2809—28no—
19#2547—28no—
20C2×#400C2212yes—
21C2×#400C2212yes—
22C2×#399C2212yes—
23C2×#398C2214yes—
24C2×#398C2212yes—
25C2×#397C2214yes—
26C2×#397C2214yes—
27C2×#391C2211yescharacteristic
28C2×#391C2212yes—
29C2×#391C2211yescharacteristic
30C2×#389C2212yes—
31C2×#388C2214yes—
32C2×#387C2214yes—
33C2×#380C2214yes—
34C2×#380C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36#1446C2214yes—
37#1446C2212yes—
38#1445C2214yes—
39#1444C2214yes—
40#1444C2212yes—
41#1444C2212yes—
42#1443C2214yes—
43#1443C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1441C2212yes—
48#1441C2214yes—
49#1441C2212yes—
50#1441C2214yes—
51#1441C2212yes—
52#1441C2214yes—
53#1440C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2212yes—
60#1439C2214yes—
61#1438C2214yes—
62#1436C2214yes—
63#1436C2212yes—
64#1436C2212yes—
65#1436C2214yes—
66#1434C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2214yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2311yescharacteristic
71C23×D4C2311yescharacteristic
72C24×C4C2312yes—
73C2×C4⋊Q8—24no—
74C2×C4⋊Q8—42no—
75C2×C4⋊Q8—42no—
76C2×#124—24no—
77C2×#124—24no—
78C2×#124—44no—
79C2×#124—44no—
80C2×C4×D4—24no—
81C2×C4×D4—42no—
82C2×C4×D4—42no—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104C22×C42C2312yes—
105C22×C42C2311yescharacteristic
106#400—28no—
107#400—44no—
108#399—44no—
109#398—216no—
110#398—44no—
111#397—216no—
112#397—48no—
113#391—28no—
114#391—24no—
115#391D418yes—
116#389—28no—
117#388—48no—
118#387—216no—
119#380—216no—
120#380—44no—
121C4×C4⋊C4—24no—
122C2×#95C2312yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C25C2412yes—
149C22×Q8—22no—
150C22×Q8—24no—
151C22×Q8—24no—
152C22×Q8—42no—
153C22×Q8—42no—
154C22×Q8—41no—
155C22×Q8—41no—
156C22×Q8—41no—
157C22×Q8—41no—
158C22×Q8C2×D414yes—
159C22×D4—24no—
160C22×D4—22no—
161C22×D4—24no—
162C22×D4—41no—
163C22×D4—41no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—28no—
167C23×C4—42no—
168C23×C4—44no—
169C23×C4C2414yes—
170C23×C4C2414yes—
171C23×C4C2412yes—
172C23×C4C2412yes—
173C23×C4C2414yes—
174C23×C4C2414yes—
175C23×C4C2412yes—
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2411yescharacteristic
179C23×C4C2411yescharacteristic
180C23×C4C2411yescharacteristic
181C4⋊Q8—416no—
182#124—416no—
183#124—416no—
184C4×D4—416no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
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—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4C2×D414yes—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
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×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—24no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—44no—
247C2×C42C2×D414yes—
248#95—24no—
249#95—28no—
250#95—24no—
251#95—28no—
252#95—28no—
253#95—28no—
254#95—28no—
255#95—28no—
256#95—28no—
257#95—42no—
258#95—42no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—42no—
269#95—44no—
270#95—42no—
271#95—42no—
272#95—44no—
273#95C2×D4116yes—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—24no—
279C24—22no—
280C24—22no—
281C24—24no—
282C24—24no—
283C24—24no—
284C24H(5,𝔽2)14yes—
285C24C2511yescharacteristic
286C2×Q8—28no—
287C2×Q8—44no—
288C2×Q8—44no—
289C2×Q8—44no—
290C2×Q8—44no—
291C2×Q8—44no—
292C2×Q8—84no—
293C2×Q8—84no—
294C2×Q8—84no—
295C2×Q8—84no—
296C2×D4—28no—
297C2×D4—48no—
298C2×D4—48no—
299C2×D4—44no—
300C2×D4—44no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
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—22no—
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—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—48no—
361C22×C4—416no—
362C22×C4—48no—
363C22×C4—84no—
364C22×C4—88no—
365C22×C4C22×D414yes—
366C22×C4C22×D414yes—
367C22×C4C22×D414yes—
368C22×C4C22×D412yes—
369C22×C4H(5,𝔽2)12yes—
370C22×C4H(5,𝔽2)12yes—
371C22×C42−1+414yes—
372C22×C42−1+414yes—
373C4⋊C4—44no—
374C4⋊C4—48no—
375C4⋊C4—48no—
376C4⋊C4—48no—
377C4⋊C4—416no—
378C4⋊C4—48no—
379C4⋊C4—84no—
380C4⋊C4—84no—
381C4⋊C4—84no—
382C4⋊C4—88no—
383C4⋊C4—84no—
384C4⋊C4—88no—
385C22⋊C4—48no—
386C22⋊C4—416no—
387C22⋊C4—416no—
388C22⋊C4—416no—
389C22⋊C4—88no—
390C22⋊C4—88no—
391C22⋊C4—88no—
392C42—28no—
393C42—44no—
394C42—44no—
395C42—84no—
396C23—24no—
397C23—28no—
398C23—24no—
399C23—24no—
400C23—28no—
401C23—42no—
402C23—42no—
403C23—44no—
404C23—44no—
405C23—42no—
406C23—44no—
407C23—42no—
408C23—42no—
409C23—44no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23C23×D411yescharacteristic
420C23C2×H(5,𝔽2)11yescharacteristic
421C23C2×H(5,𝔽2)12yes—
422C23C2×H(5,𝔽2)11yescharacteristic
423C23C2×2−1+412yes—
424C23C2×2−1+412yes—
425C23#58514yes—
426C23#58512yes—
427Q8—88no—
428Q8—88no—
429D4—416no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—28no—
447C2×C4—24no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
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—816no—
496C2×C4#53614yes—
497C2×C4#56314yes—
498C22—44no—
499C22—44no—
500C22—48no—
501C22—48no—
502C22—44no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22—84no—
507C22—84no—
508C22—82no—
509C22C2×#53611yescharacteristic
510C22#348411yescharacteristic
511C22#348411yescharacteristic
512C22#348712yes—
513C22#348912yes—
514C22C2×#56311yescharacteristic
515C22#353314yes—
516C22#353612yes—
517C22#353612yes—
518C22#354511yescharacteristic
519C22#354512yes—
520C22#355112yes—
521C22#355514yes—
522C22#355714yes—
523C22#355914yes—
524C22#356112yes—
525C4—24no—
526C4—24no—
527C4—44no—
528C4—44no—
529C4—42no—
530C4—44no—
531C4—84no—
532C4—82no—
533C4—84no—
534C4—84no—
535C4—82no—
536C4—84no—
537C2—84no—
538C2#3671811yescharacteristic
539C2#4288412yes—
540C2#4613611yescharacteristic
541C2#4783612yes—
542C2#5208311yescharacteristic
543C2#5296112yes—
544C2#5561412yes—
545C2#5726714yes—
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×#2809C2semidirect
3. #17130C2semidirect
4. #16653C2semidirect
6. #16336C2semidirect
7. #16304C2semidirect
9. #15783C2semidirect
10. #15393C2semidirect
11. #15098C2semidirect
13. #14668C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28092
C2×#25472
#171302
#166532
#165582
#163362
#163042
#160942
#157832
#153932
#150982
#148252
#146682