Finite Group Databasebeta

← #5035180

Subgroups — #5035180

Order 512.

2288 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2620C211yesmaximal, characteristic
2C2×#2379C211yesmaximal, characteristic, complemented
3#16112C212yesmaximal, complemented
4#16074C212yesmaximal, complemented
5#15984C212yesmaximal
6#15710C214yesmaximal, complemented
7#15705C214yesmaximal, complemented
8#15554C214yesmaximal, complemented
9#15470C214yesmaximal, complemented
10#15465C212yesmaximal, complemented
11#14348C212yesmaximal, complemented
12#14169C211yesmaximal, characteristic, complemented
13#13957C212yesmaximal, complemented
14C22×#124C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic
16C22×#121C2211yescharacteristic
17C22×#117C2211yescharacteristic
18#2620—28no—
19C2×#400C2212yes—
20C2×#397C2212yes—
21C2×#396C2214yes—
22C2×#394C2214yes—
23#2379—28no—
24C2×#391C2212yes—
25C2×#391C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#388C2214yes—
28C2×#386C2212yes—
29C2×#386C2212yes—
30C2×#385C2214yes—
31C2×#384C2211yescharacteristic
32C2×#382C2212yes—
33C2×#381C2212yes—
34C2×#376C2212yes—
35#1446C2212yes—
36#1443C2214yes—
37#1443C2214yes—
38#1442C2214yes—
39#1442C2214yes—
40#1442C2214yes—
41#1442C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1441C2212yes—
45#1441C2212yes—
46#1440C2214yes—
47#1440C2214yes—
48#1440C2214yes—
49#1440C2212yes—
50#1440C2214yes—
51#1440C2214yes—
52#1439C2212yes—
53#1439C2212yes—
54#1438C2212yes—
55#1438C2212yes—
56#1438C2214yes—
57#1437C2214yes—
58#1437C2214yes—
59#1437C2212yes—
60#1437C2214yes—
61#1436C2212yes—
62#1436C2214yes—
63C4⋊#95C2214yes—
64C4⋊#95C2214yes—
65#1434C2214yes—
66#1434C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C23×Q8C2311yescharacteristic
70C23×D4C2312yes—
71C23×D4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C2×#124—28no—
75C2×#124D418yes—
76C2×C22⋊Q8—24no—
77C2×C22⋊Q8—42no—
78C2×C22⋊Q8—42no—
79C2×#121—24no—
80C2×#121—42no—
81C2×#121—42no—
82C2×#117—24no—
83C2×#117—28no—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C42C2311yescharacteristic
105C22×C42C2312yes—
106C22×C42C2314yes—
107C22×C42C2311yescharacteristic
108#400—44no—
109#397—44no—
110#396—216no—
111#394—216no—
112#391—24no—
113#391—28no—
114#390—24no—
115#388—216no—
116#386—28no—
117#386—44no—
118#385—216no—
119#384—24no—
120#382—28no—
121#381—44no—
122#376—28no—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2312yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C25C2412yes—
148C25C2411yescharacteristic
149C22×Q8—24no—
150C22×Q8—41no—
151C22×Q8—41no—
152C22×Q8—42no—
153C22×Q8C2×D414yes—
154C22×D4—24no—
155C22×D4—24no—
156C22×D4—24no—
157C22×D4—41no—
158C22×D4—41no—
159C22×D4—42no—
160C22×D4—42no—
161C22×D4—44no—
162C22×D4—44no—
163C22×D4C2×D414yes—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—42no—
170C23×C4—44no—
171C23×C4—42no—
172C23×C4—44no—
173C23×C4—42no—
174C23×C4C2412yes—
175C23×C4C2414yes—
176C23×C4C2412yes—
177C23×C4C2414yes—
178C23×C4C2414yes—
179C23×C4C2414yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2411yescharacteristic
183C23×C4C2412yes—
184C23×C4C2411yescharacteristic
185#124—232no—
186C22⋊Q8—416no—
187#121—416no—
188#117—416no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
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×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4C2×D4116yes—
244C2×C42—22no—
245C2×C42—24no—
246C2×C42—28no—
247C2×C42—24no—
248C2×C42—44no—
249C2×C42—44no—
250C2×C42—44no—
251C2×C42C2×D414yes—
252C2×C42#4118yes—
253#95—28no—
254#95—28no—
255#95—28no—
256#95—28no—
257#95—28no—
258#95—24no—
259#95—24no—
260#95—28no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—42no—
274#95—44no—
275#95—42no—
276#95—44no—
277C24—21no—
278C24—22no—
279C24—24no—
280C24—22no—
281C24—21no—
282C24—22no—
283C24—24no—
284C24—22no—
285C24—24no—
286C24—24no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24C22×D414yes—
292C24H(5,𝔽2)12yes—
293C24C2511yescharacteristic
294C2×Q8—28no—
295C2×Q8—44no—
296C2×Q8—44no—
297C2×Q8—84no—
298C2×Q8—84no—
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—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
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—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—24no—
365C22×C4—24no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—48no—
371C22×C4—84no—
372C22×C4—88no—
373C22×C4—84no—
374C22×C4—88no—
375C22×C4C22×D414yes—
376C22×C4C22×D412yes—
377C22×C4C22×D414yes—
378C22×C4C2×#4114yes—
379C22×C4C2×#4114yes—
380C22×C4C2×#4114yes—
381C22×C4H(5,𝔽2)12yes—
382C4⋊C4—48no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—48no—
386C4⋊C4—84no—
387C4⋊C4—88no—
388C4⋊C4—84no—
389C4⋊C4—84no—
390C22⋊C4—232no—
391C22⋊C4—48no—
392C22⋊C4—48no—
393C22⋊C4—48no—
394C22⋊C4—84no—
395C22⋊C4—88no—
396C22⋊C4—88no—
397C22⋊C4—88no—
398C22⋊C4—84no—
399C22⋊C4—84no—
400C22⋊C4—88no—
401C22⋊C4—88no—
402C42—416no—
403C42—44no—
404C42—84no—
405C42#123116yes—
406C23—24no—
407C23—22no—
408C23—24no—
409C23—24no—
410C23—24no—
411C23—28no—
412C23—28no—
413C23—24no—
414C23—24no—
415C23—41no—
416C23—41no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—42no—
421C23—44no—
422C23—44no—
423C23—42no—
424C23—42no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—44no—
434C23—42no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23C23×D411yescharacteristic
443C23C22×#4112yes—
444C23C2×H(5,𝔽2)11yescharacteristic
445C23C2×H(5,𝔽2)11yescharacteristic
446C23#58512yes—
447C23#58514yes—
448C23#58514yes—
449Q8—88no—
450D4—88no—
451D4—816no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—28no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—28no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—416no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
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—816no—
515C2×C4C2×#12314yes—
516C2×C4C2×#12318yes—
517C22—42no—
518C22—44no—
519C22—48no—
520C22—44no—
521C22—42no—
522C22—48no—
523C22—48no—
524C22—42no—
525C22—44no—
526C22—82no—
527C22—82no—
528C22—84no—
529C22—84no—
530C22—84no—
531C22—84no—
532C22—84no—
533C22C22×#12311yescharacteristic
534C22#348411yescharacteristic
535C22#348511yescharacteristic
536C22#349314yes—
537C22#349412yes—
538C22#349412yes—
539C22#350012yes—
540C22#350012yes—
541C22#351214yes—
542C22#353312yes—
543C22#354012yes—
544C22#354512yes—
545C22#355112yes—
546C22#356614yes—
547C22#356614yes—
548C4—28no—
549C4—24no—
550C4—42no—
551C4—44no—
552C4—82no—
553C4—82no—
554C4—82no—
555C4—84no—
556C4—84no—
557C4—84no—
558C4—84no—
559C2—82no—
560C2—84no—
561C2#3554011yescharacteristic
562C2#3561812yes—
563C2#5174911yescharacteristic
564C2#5175312yes—
565C2#5176514yes—
566C2#5176614yes—
567C2#5197811yescharacteristic
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
2. C2×#2379C2semidirect
3. #16112C2semidirect
4. #16074C2semidirect
6. #15710C2semidirect
7. #15705C2semidirect
8. #15554C2semidirect
9. #15470C2semidirect
10. #15465C2semidirect
11. #14348C2semidirect
12. #14169C2semidirect
13. #13957C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26202
C2×#23792
#161122
#160742
#159842
#157102
#157052
#155542
#154702
#154652
#143482
#141692
#139572