Finite Group Databasebeta

← #2923601

Subgroups — #2923601

Order 512.

2376 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2724C211yesmaximal, characteristic, complemented
2C2×#2500C211yesmaximal, characteristic
3#16688C212yesmaximal, complemented
4#16468C214yesmaximal, complemented
5#15983C212yesmaximal, complemented
6#15950C212yesmaximal
7#15716C214yesmaximal, complemented
8#15554C214yesmaximal, complemented
9#15197C212yesmaximal, complemented
10#14907C212yesmaximal, complemented
11#14442C214yesmaximal, complemented
12#14230C212yesmaximal, complemented
13#14157C211yesmaximal, characteristic, complemented
14C22×#123C2212yes—
15C22×#121C2211yescharacteristic
16C22×#117C2211yescharacteristic
17#2724—28no—
18#2500—28no—
19C2×#399C2212yes—
20C2×#398C2212yes—
21C2×#397C2214yes—
22C2×#397C2212yes—
23C2×#394C2214yes—
24C2×#390C2211yescharacteristic
25C2×#390C2211yescharacteristic
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#388C2214yes—
29C2×#388C2214yes—
30C2×#387C2214yes—
31C2×#386C2212yes—
32C2×#385C2214yes—
33C2×#385C2214yes—
34C2×#385C2212yes—
35C2×#384C2211yescharacteristic
36C2×#382C2211yescharacteristic
37C2×C4×C22⋊C4C2211yescharacteristic
38#1445C2212yes—
39#1444C2212yes—
40#1443C2214yes—
41#1442C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1441C2212yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1440C2214yes—
50#1440C2212yes—
51#1440C2214yes—
52#1439C2212yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2212yes—
57#1438C2212yes—
58#1438C2212yes—
59#1438C2214yes—
60#1437C2214yes—
61#1437C2214yes—
62#1436C2214yes—
63C4⋊#95C2214yes—
64C4⋊#95C2214yes—
65C4⋊#95C2214yes—
66#1434C2212yes—
67C22⋊#95C2214yes—
68C4×#95C2212yes—
69C23×D4C2312yes—
70C23×D4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2312yes—
73C2×#123—28no—
74C2×#123—48no—
75C2×#121—24no—
76C2×#121—42no—
77C2×#121—42no—
78C2×#117—24no—
79C2×#117—42no—
80C2×#117—42no—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C42C2311yescharacteristic
105C22×C42C2312yes—
106C22×C42C2314yes—
107C22×C42C2311yescharacteristic
108#399—28no—
109#398—28no—
110#397—216no—
111#397—44no—
112#394—216no—
113#390—24no—
114#390D418yes—
115#389—28no—
116#389—44no—
117#388—216no—
118#388—216no—
119#387—48no—
120#386—44no—
121#385—216no—
122#385—48no—
123#385—44no—
124#384—24no—
125#382—24no—
126C4×C22⋊C4—24no—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
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×#95C2312yes—
149C25C2412yes—
150C25C2411yescharacteristic
151C22×D4—24no—
152C22×D4—24no—
153C22×D4—24no—
154C22×D4—41no—
155C22×D4—41no—
156C22×D4—42no—
157C22×D4—42no—
158C22×D4—44no—
159C22×D4—44no—
160C22×D4C2×D414yes—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—42no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4C2414yes—
172C23×C4C2412yes—
173C23×C4C2412yes—
174C23×C4C2414yes—
175C23×C4C2414yes—
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2414yes—
179C23×C4C2411yescharacteristic
180C23×C4C2412yes—
181C23×C4C2411yescharacteristic
182#123—432no—
183#121—416no—
184#117—416no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4C2×D414yes—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—42no—
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—42no—
248C2×C22⋊C4C2×D4116yes—
249C2×C42—24no—
250C2×C42—22no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—28no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—44no—
257C2×C42—44no—
258C2×C42C2×D414yes—
259#95—24no—
260#95—28no—
261#95—28no—
262#95—28no—
263#95—24no—
264#95—28no—
265#95—28no—
266#95—24no—
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—42no—
279#95—44no—
280#95#4118yes—
281C24—21no—
282C24—24no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—24no—
287C24—21no—
288C24—22no—
289C24—24no—
290C24—22no—
291C24—22no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24—24no—
296C24C22×D414yes—
297C24H(5,𝔽2)12yes—
298C24C2511yescharacteristic
299C2×D4—28no—
300C2×D4—44no—
301C2×D4—44no—
302C2×D4—48no—
303C2×D4—48no—
304C2×D4—84no—
305C2×D4—84no—
306C2×D4—88no—
307C2×D4—84no—
308C2×D4—88no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—28no—
315C22×C4—21no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
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—24no—
365C22×C4—48no—
366C22×C4—48no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—48no—
370C22×C4—84no—
371C22×C4—84no—
372C22×C4—88no—
373C22×C4—88no—
374C22×C4C22×D414yes—
375C22×C4C22×D412yes—
376C22×C4C22×D414yes—
377C22×C4C2×#4118yes—
378C22×C4C2×#4114yes—
379C22×C4H(5,𝔽2)12yes—
380C22×C4H(5,𝔽2)14yes—
381C22×C42−1+414yes—
382C4⋊C4—44no—
383C4⋊C4—48no—
384C4⋊C4—416no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—416no—
388C4⋊C4—48no—
389C4⋊C4—84no—
390C4⋊C4—88no—
391C4⋊C4—84no—
392C4⋊C4—88no—
393C22⋊C4—232no—
394C22⋊C4—48no—
395C22⋊C4—48no—
396C22⋊C4—48no—
397C22⋊C4—416no—
398C22⋊C4—48no—
399C22⋊C4—416no—
400C22⋊C4—48no—
401C22⋊C4—88no—
402C22⋊C4—88no—
403C22⋊C4—88no—
404C22⋊C4—84no—
405C42—44no—
406C42—48no—
407C42—44no—
408C42—88no—
409C23—24no—
410C23—28no—
411C23—22no—
412C23—24no—
413C23—24no—
414C23—28no—
415C23—24no—
416C23—24no—
417C23—24no—
418C23—41no—
419C23—42no—
420C23—42no—
421C23—41no—
422C23—41no—
423C23—42no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—42no—
440C23—41no—
441C23—42no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23C23×D411yescharacteristic
447C23C22×#4112yes—
448C23C2×H(5,𝔽2)12yes—
449C23C2×H(5,𝔽2)11yescharacteristic
450C23C2×H(5,𝔽2)11yescharacteristic
451C23C2×2−1+412yes—
452C23#58512yes—
453C23#58514yes—
454D4—88no—
455D4—816no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—28no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—28no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—28no—
469C2×C4—28no—
470C2×C4—24no—
471C2×C4—416no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
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—44no—
503C2×C4—42no—
504C2×C4—42no—
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—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—816no—
519C2×C4#53514yes—
520C2×C4#54718yes—
521C22—42no—
522C22—44no—
523C22—48no—
524C22—42no—
525C22—44no—
526C22—44no—
527C22—42no—
528C22—48no—
529C22—48no—
530C22—82no—
531C22—82no—
532C22—84no—
533C22—84no—
534C22—82no—
535C22—82no—
536C22—84no—
537C22—84no—
538C22C2×#53511yescharacteristic
539C22C2×#54712yes—
540C22#348111yescharacteristic
541C22#348411yescharacteristic
542C22#348511yescharacteristic
543C22#350012yes—
544C22#350614yes—
545C22#352114yes—
546C22#352212yes—
547C22#353312yes—
548C22#353412yes—
549C22#353412yes—
550C22#354714yes—
551C22#355011yescharacteristic
552C22#355112yes—
553C22#355314yes—
554C4—24no—
555C4—28no—
556C4—42no—
557C4—48no—
558C4—44no—
559C4—82no—
560C4—82no—
561C4—82no—
562C4—84no—
563C4—84no—
564C4—84no—
565C2—82no—
566C2—84no—
567C2#3672111yescharacteristic
568C2#3949112yes—
569C2#4344212yes—
570C2#4349512yes—
571C2#5015114yes—
572C2#5197811yescharacteristic
573C2#5873112yes—
574C2#5907011yescharacteristic
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×#2724C2semidirect
3. #16688C2semidirect
4. #16468C2semidirect
5. #15983C2semidirect
7. #15716C2semidirect
8. #15554C2semidirect
9. #15197C2semidirect
10. #14907C2semidirect
11. #14442C2semidirect
12. #14230C2semidirect
13. #14157C2semidirect

Maximal subgroups

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

subgroupindex
C2×#27242
C2×#25002
#166882
#164682
#159832
#159502
#157162
#155542
#151972
#149072
#144422
#142302
#141572