Finite Group Databasebeta

← #2810880

Subgroups — #2810880

Order 512.

2424 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2571C211yesmaximal, characteristic, complemented
2C2×#2501C211yesmaximal, characteristic
3#16599C212yesmaximal, complemented
4#16532C214yesmaximal, complemented
5#15886C214yesmaximal, complemented
6#15798C212yesmaximal, complemented
7#15674C212yesmaximal
8#15669C214yesmaximal, complemented
9#15112C212yesmaximal, complemented
10#14822C212yesmaximal, complemented
11#14424C211yesmaximal, characteristic, complemented
12#14360C214yesmaximal, complemented
13#14240C212yesmaximal, complemented
14C22×#123C2214yes—
15C22×#121C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17#2571—28no—
18#2501—28no—
19C2×#400C2212yes—
20C2×#400C2212yes—
21C2×#400C2212yes—
22C2×#399C2214yes—
23C2×#399C2212yes—
24C2×#397C2214yes—
25C2×#397C2212yes—
26C2×#390C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#390C2212yes—
29C2×#389C2214yes—
30C2×#387C2214yes—
31C2×#384C2211yescharacteristic
32C2×#384C2211yescharacteristic
33C2×#382C2214yes—
34C2×#380C2212yes—
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C22⋊C4C2211yescharacteristic
37#1445C2212yes—
38#1445C2214yes—
39#1442C2214yes—
40#1442C2214yes—
41#1442C2212yes—
42#1441C2212yes—
43#1441C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1441C2212yes—
47#1440C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2212yes—
52#1439C2214yes—
53#1439C2214yes—
54#1438C2214yes—
55#1438C2214yes—
56#1438C2212yes—
57#1437C2214yes—
58#1437C2214yes—
59#1437C2212yes—
60#1437C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2212yes—
64#1436C2214yes—
65#1434C2212yes—
66C22⋊#95C2214yes—
67C22⋊#95C2212yes—
68C22⋊#95C2214yes—
69C23×D4C2312yes—
70C23×D4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2312yes—
73C2×#123—216no—
74C2×#123—216no—
75C2×#123—48no—
76C2×#121—24no—
77C2×#121—42no—
78C2×#121—42no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4—42no—
81C22×C4⋊C4—42no—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2314yes—
107C22×C42C2312yes—
108#400—28no—
109#400—28no—
110#400—44no—
111#399—216no—
112#399—28no—
113#397—44no—
114#397—48no—
115#390—24no—
116#390—24no—
117#390—44no—
118#389—216no—
119#387—48no—
120#384—24no—
121#384D418yes—
122#382—216no—
123#380—44no—
124C4×C4⋊C4—28no—
125C4×C22⋊C4—24no—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2312yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C25C2412yes—
150C25C2411yescharacteristic
151C22×D4—28no—
152C22×D4—24no—
153C22×D4—24no—
154C22×D4—22no—
155C22×D4—24no—
156C22×D4—41no—
157C22×D4—41no—
158C22×D4—42no—
159C22×D4—44no—
160C22×D4—42no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—28no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—44no—
171C23×C4C2414yes—
172C23×C4C2414yes—
173C23×C4C2414yes—
174C23×C4C2414yes—
175C23×C4C2412yes—
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2412yes—
179C23×C4C2411yescharacteristic
180C23×C4C2412yes—
181C23×C4C2411yescharacteristic
182#123—464no—
183#121—416no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—416no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4C2×D4116yes—
217C2×C4⋊C4C2×D414yes—
218C2×C4⋊C4C2×D414yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—42no—
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×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4#4118yes—
249C2×C42—24no—
250C2×C42—22no—
251C2×C42—28no—
252C2×C42—44no—
253C2×C42—44no—
254C2×C42—42no—
255C2×C42—44no—
256C2×C42—42no—
257C2×C42C2×D414yes—
258#95—24no—
259#95—28no—
260#95—24no—
261#95—24no—
262#95—28no—
263#95—28no—
264#95—24no—
265#95—24no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—42no—
277#95—44no—
278#95—44no—
279#95—42no—
280#95—44no—
281C24—21no—
282C24—22no—
283C24—21no—
284C24—22no—
285C24—24no—
286C24—22no—
287C24—22no—
288C24—24no—
289C24—24no—
290C24—22no—
291C24—22no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24—24no—
296C24C2×#4114yes—
297C24H(5,𝔽2)12yes—
298C24C2511yescharacteristic
299C2×D4—48no—
300C2×D4—416no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—48no—
304C2×D4—44no—
305C2×D4—84no—
306C2×D4—84no—
307C2×D4—88no—
308C2×D4—84no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—28no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—24no—
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—22no—
345C22×C4—22no—
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—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—48no—
365C22×C4—48no—
366C22×C4—44no—
367C22×C4—48no—
368C22×C4—416no—
369C22×C4—44no—
370C22×C4—48no—
371C22×C4—88no—
372C22×C4—84no—
373C22×C4C22×D414yes—
374C22×C4C22×D414yes—
375C22×C4C22×D412yes—
376C22×C4C22×D414yes—
377C22×C4C2×#4114yes—
378C22×C4C2×#4114yes—
379C22×C4H(5,𝔽2)12yes—
380C22×C4H(5,𝔽2)14yes—
381C22×C42−1+414yes—
382C4⋊C4—232no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—48no—
386C4⋊C4—416no—
387C4⋊C4—48no—
388C4⋊C4—44no—
389C4⋊C4—48no—
390C4⋊C4—84no—
391C4⋊C4—84no—
392C4⋊C4—88no—
393C4⋊C4—84no—
394C22⋊C4—216no—
395C22⋊C4—48no—
396C22⋊C4—416no—
397C22⋊C4—416no—
398C22⋊C4—88no—
399C22⋊C4—88no—
400C22⋊C4—88no—
401C22⋊C4—88no—
402C22⋊C4—88no—
403C22⋊C4—88no—
404C42—28no—
405C42—416no—
406C42—44no—
407C42—84no—
408C23—24no—
409C23—22no—
410C23—24no—
411C23—28no—
412C23—28no—
413C23—24no—
414C23—22no—
415C23—24no—
416C23—41no—
417C23—41no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—44no—
422C23—44no—
423C23—42no—
424C23—44no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—42no—
437C23—44no—
438C23—42no—
439C23—42no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23#55914yes—
445C23C23×D411yescharacteristic
446C23C22×#4112yes—
447C23C2×H(5,𝔽2)12yes—
448C23C2×H(5,𝔽2)11yescharacteristic
449C23C2×H(5,𝔽2)11yescharacteristic
450C23C2×2−1+412yes—
451C23#58514yes—
452C23#58512yes—
453D4—88no—
454D4—816no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—28no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—28no—
463C2×C4—24no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—28no—
470C2×C4—24no—
471C2×C4—416no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
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—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—816no—
518C2×C4#53614yes—
519C2×C4#54618yes—
520C22—24no—
521C22—28no—
522C22—42no—
523C22—42no—
524C22—48no—
525C22—44no—
526C22—44no—
527C22—44no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—84no—
532C22—84no—
533C22—84no—
534C22—84no—
535C22—84no—
536C22—84no—
537C22C2×#53611yescharacteristic
538C22C2×#54612yes—
539C22C2×#55911yescharacteristic
540C22#348411yescharacteristic
541C22#348711yescharacteristic
542C22#348712yes—
543C22#350812yes—
544C22#351514yes—
545C22#352114yes—
546C22#353512yes—
547C22#353512yes—
548C22#354311yescharacteristic
549C22#354512yes—
550C22#355112yes—
551C22#355914yes—
552C22#356214yes—
553C4—24no—
554C4—28no—
555C4—44no—
556C4—44no—
557C4—42no—
558C4—44no—
559C4—82no—
560C4—84no—
561C4—84no—
562C4—84no—
563C4—84no—
564C2—44no—
565C2—162no—
566C2#3671111yescharacteristic
567C2#3890712yes—
568C2#4371012yes—
569C2#4526912yes—
570C2#5085314yes—
571C2#5208311yescharacteristic
572C2#5348812yes—
573C2#5914311yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2571C2semidirect
3. #16599C2semidirect
4. #16532C2semidirect
5. #15886C2semidirect
6. #15798C2semidirect
8. #15669C2semidirect
9. #15112C2semidirect
10. #14822C2semidirect
11. #14424C2semidirect
12. #14360C2semidirect
13. #14240C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25712
C2×#25012
#165992
#165322
#158862
#157982
#156742
#156692
#151122
#148222
#144242
#143602
#142402