Finite Group Databasebeta

← #2926324

Subgroups — #2926324

Order 512.

2510 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2804C211yesmaximal, characteristic, complemented
2C2×#2501C211yesmaximal, characteristic
3#16803C214yesmaximal, complemented
4#16570C212yesmaximal, complemented
5#16074C212yesmaximal, complemented
6#15984C212yesmaximal
7#15668C214yesmaximal, complemented
8#15469C214yesmaximal, complemented
9#15185C212yesmaximal, complemented
10#14822C212yesmaximal, complemented
11#14359C214yesmaximal, complemented
12#14243C212yesmaximal, complemented
13#14157C211yesmaximal, characteristic, complemented
14C22×#123C2214yes—
15C22×#121C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17#2804—28no—
18#2501—28no—
19C2×#400C2212yes—
20C2×#400C2212yes—
21C2×#399C2214yes—
22C2×#393C2214yes—
23C2×#393C2212yes—
24C2×#391C2212yes—
25C2×#391C2214yes—
26C2×#390C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#388C2214yes—
29C2×#386C2212yes—
30C2×#386C2212yes—
31C2×#385C2214yes—
32C2×#384C2211yescharacteristic
33C2×#384C2211yescharacteristic
34C2×#381C2212yes—
35C2×#380C2212yes—
36C2×C4×C22⋊C4C2211yescharacteristic
37#1446C2212yes—
38#1445C2214yes—
39#1445C2214yes—
40#1442C2214yes—
41#1442C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1442C2212yes—
45#1442C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2212yes—
49#1441C2212yes—
50#1440C2214yes—
51#1440C2214yes—
52#1440C2214yes—
53#1440C2214yes—
54#1440C2212yes—
55#1439C2212yes—
56#1438C2214yes—
57#1438C2212yes—
58#1438C2212yes—
59#1437C2214yes—
60#1437C2214yes—
61C4⋊#95C2214yes—
62#1434C2214yes—
63#1434C2214yes—
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2212yes—
67C22⋊#95C2214yes—
68C4×#95C2212yes—
69C23×Q8C2312yes—
70C23×D4C2312yes—
71C23×D4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C24×C4C2312yes—
75C2×#123—216no—
76C2×#123—216no—
77C2×#123—48no—
78C2×#121—24no—
79C2×#121—42no—
80C2×#121—42no—
81C22×C4⋊C4—24no—
82C22×C4⋊C4—42no—
83C22×C4⋊C4—42no—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C42C2311yescharacteristic
105C22×C42C2314yes—
106C22×C42C2311yescharacteristic
107#400—28no—
108#400—44no—
109#399—216no—
110#393—216no—
111#393—28no—
112#391—28no—
113#391—48no—
114#390—24no—
115#390D418yes—
116#388—48no—
117#386—28no—
118#386—44no—
119#385—216no—
120#384—24no—
121#384—24no—
122#381—44no—
123#380—44no—
124C4×C22⋊C4—24no—
125C2×#95C2312yes—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C25C2412yes—
150C25C2411yescharacteristic
151C22×Q8—24no—
152C22×Q8—24no—
153C22×Q8—24no—
154C22×Q8—44no—
155C22×Q8—44no—
156C22×D4—24no—
157C22×D4—28no—
158C22×D4—24no—
159C22×D4—24no—
160C22×D4—41no—
161C22×D4—41no—
162C22×D4—42no—
163C22×D4—44no—
164C22×D4—42no—
165C22×D4C2×D414yes—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—28no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—42no—
179C23×C4—44no—
180C23×C4—42no—
181C23×C4C2414yes—
182C23×C4C2414yes—
183C23×C4C2414yes—
184C23×C4C2412yes—
185C23×C4C2414yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2411yescharacteristic
190C23×C4C2412yes—
191C23×C4C2411yescharacteristic
192#123—464no—
193#121—416no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—416no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4C2×D414yes—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—42no—
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—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4C2×D4116yes—
249C2×C22⋊C4#4118yes—
250C2×C42—22no—
251C2×C42—24no—
252C2×C42—28no—
253C2×C42—42no—
254C2×C42—44no—
255C2×C42—44no—
256C2×C42C2×D414yes—
257#95—24no—
258#95—28no—
259#95—28no—
260#95—28no—
261#95—24no—
262#95—24no—
263#95—28no—
264#95—28no—
265#95—24no—
266#95—28no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
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—24no—
288C24—24no—
289C24—22no—
290C24—22no—
291C24—24no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24C22×D414yes—
296C24C2×#4114yes—
297C24H(5,𝔽2)12yes—
298C24C2511yescharacteristic
299C2×Q8—416no—
300C2×Q8—48no—
301C2×Q8—84no—
302C2×Q8—84no—
303C2×Q8—88no—
304C2×D4—28no—
305C2×D4—44no—
306C2×D4—44no—
307C2×D4—48no—
308C2×D4—416no—
309C2×D4—48no—
310C2×D4—84no—
311C2×D4—84no—
312C2×D4—88no—
313C2×D4—84no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—28no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
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—22no—
352C22×C4—24no—
353C22×C4—22no—
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—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—44no—
373C22×C4—48no—
374C22×C4—416no—
375C22×C4—44no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—88no—
379C22×C4—84no—
380C22×C4—84no—
381C22×C4—88no—
382C22×C4—84no—
383C22×C4—88no—
384C22×C4C22×D414yes—
385C22×C4C22×D412yes—
386C22×C4C22×D414yes—
387C22×C4C2×#4114yes—
388C22×C4C2×#4114yes—
389C22×C4H(5,𝔽2)12yes—
390C22×C4H(5,𝔽2)14yes—
391C22×C42−1+414yes—
392C4⋊C4—44no—
393C4⋊C4—48no—
394C4⋊C4—416no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—416no—
398C4⋊C4—84no—
399C4⋊C4—84no—
400C4⋊C4—84no—
401C22⋊C4—232no—
402C22⋊C4—216no—
403C22⋊C4—48no—
404C22⋊C4—416no—
405C22⋊C4—84no—
406C22⋊C4—88no—
407C22⋊C4—88no—
408C22⋊C4—88no—
409C22⋊C4—88no—
410C22⋊C4—84no—
411C22⋊C4—84no—
412C42—44no—
413C42—44no—
414C42—88no—
415C23—24no—
416C23—22no—
417C23—24no—
418C23—28no—
419C23—28no—
420C23—24no—
421C23—22no—
422C23—24no—
423C23—24no—
424C23—28no—
425C23—24no—
426C23—24no—
427C23—28no—
428C23—41no—
429C23—41no—
430C23—42no—
431C23—44no—
432C23—42no—
433C23—42no—
434C23—44no—
435C23—42no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23#55914yes—
452C23C23×D411yescharacteristic
453C23C22×#4112yes—
454C23C2×H(5,𝔽2)12yes—
455C23C2×H(5,𝔽2)11yescharacteristic
456C23C2×H(5,𝔽2)11yescharacteristic
457C23C2×2−1+412yes—
458C23#58512yes—
459C23#58514yes—
460Q8—816no—
461D4—88no—
462D4—816no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—28no—
468C2×C4—28no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—28no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—28no—
476C2×C4—24no—
477C2×C4—416no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
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—44no—
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—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—816no—
526C2×C4—816no—
527C2×C4#53614yes—
528C2×C4#54618yes—
529C22—24no—
530C22—28no—
531C22—48no—
532C22—42no—
533C22—44no—
534C22—44no—
535C22—42no—
536C22—44no—
537C22—44no—
538C22—44no—
539C22—48no—
540C22—44no—
541C22—48no—
542C22—84no—
543C22—84no—
544C22—84no—
545C22—84no—
546C22C2×#53611yescharacteristic
547C22C2×#54612yes—
548C22C2×#55911yescharacteristic
549C22#348411yescharacteristic
550C22#348511yescharacteristic
551C22#350012yes—
552C22#350812yes—
553C22#351514yes—
554C22#352114yes—
555C22#353314yes—
556C22#353512yes—
557C22#353512yes—
558C22#354012yes—
559C22#354311yescharacteristic
560C22#354512yes—
561C22#354714yes—
562C4—24no—
563C4—28no—
564C4—44no—
565C4—44no—
566C4—42no—
567C4—84no—
568C4—82no—
569C4—82no—
570C4—84no—
571C4—84no—
572C4—84no—
573C2—44no—
574C2—84no—
575C2#3672011yescharacteristic
576C2#3895512yes—
577C2#4371012yes—
578C2#4445012yes—
579C2#5043314yes—
580C2#5174911yescharacteristic
581C2#5484812yes—
582C2#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×#2804C2semidirect
3. #16803C2semidirect
4. #16570C2semidirect
5. #16074C2semidirect
7. #15668C2semidirect
8. #15469C2semidirect
9. #15185C2semidirect
10. #14822C2semidirect
11. #14359C2semidirect
12. #14243C2semidirect
13. #14157C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28042
C2×#25012
#168032
#165702
#160742
#159842
#156682
#154692
#151852
#148222
#143592
#142432
#141572