Finite Group Databasebeta

← #2926568

Subgroups — #2926568

Order 512.

2526 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2683C211yesmaximal, characteristic
2C2×#2545C211yesmaximal, characteristic
3#16630C212yesmaximal, complemented
4#16518C214yesmaximal, complemented
5#16032C212yesmaximal
6#15784C212yesmaximal
7#15706C214yesmaximal, complemented
8#15614C214yesmaximal, complemented
9#15230C212yesmaximal, complemented
10#14881C212yesmaximal, complemented
11#14364C214yesmaximal, complemented
12#14226C212yesmaximal, complemented
13#14157C211yesmaximal, characteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×#124C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×C22⋊Q8C2214yes—
18C22×C4×Q8C2211yescharacteristic
19#2683—28no—
20#2545—28no—
21C2×#398C2212yes—
22C2×#397C2214yes—
23C2×#396C2214yes—
24C2×#396C2214yes—
25C2×#395C2212yes—
26C2×#395C2212yes—
27C2×#391C2211yescharacteristic
28C2×#391C2212yes—
29C2×#390C2211yescharacteristic
30C2×#389C2212yes—
31C2×#389C2214yes—
32C2×#388C2214yes—
33C2×#388C2214yes—
34C2×#386C2212yes—
35C2×#385C2212yes—
36C2×#382C2214yes—
37C2×#381C2212yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39#1446C2214yes—
40#1446C2214yes—
41#1446C2214yes—
42#1445C2212yes—
43#1445C2214yes—
44#1444C2214yes—
45#1444C2212yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2212yes—
49#1441C2212yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2212yes—
53#1440C2214yes—
54#1440C2214yes—
55#1440C2212yes—
56#1440C2212yes—
57#1439C2214yes—
58#1439C2214yes—
59#1438C2212yes—
60#1437C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65C4⋊#95C2214yes—
66#1434C2212yes—
67C22⋊#95C2214yes—
68C4×#95C2212yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2312yes—
71C23×Q8C2311yescharacteristic
72C23×D4C2311yescharacteristic
73C24×C4C2312yes—
74C2×C4⋊Q8—24no—
75C2×C4⋊Q8—44no—
76C2×#124—24no—
77C2×#124—24no—
78C2×#124—42no—
79C2×#124—42no—
80C2×#124—44no—
81C2×C22⋊Q8—216no—
82C2×C22⋊Q8—216no—
83C2×C22⋊Q8—48no—
84C2×C4×Q8—24no—
85C2×C4×Q8—42no—
86C2×C4×Q8—42no—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109C22×C42C2314yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112#398—28no—
113#397—48no—
114#396—216no—
115#396—216no—
116#395—28no—
117#395—44no—
118#391—24no—
119#391—44no—
120#390D418yes—
121#389—28no—
122#389—48no—
123#388—216no—
124#388—48no—
125#386—28no—
126#385—44no—
127#382—48no—
128#381—44no—
129C4×C4⋊C4—24no—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C25C2412yes—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—24no—
157C22×Q8—28no—
158C22×Q8—24no—
159C22×Q8—24no—
160C22×Q8—24no—
161C22×Q8—41no—
162C22×Q8—41no—
163C22×Q8—44no—
164C22×Q8—41no—
165C22×Q8—41no—
166C22×Q8—42no—
167C22×Q8—42no—
168C22×Q8—42no—
169C22×D4—24no—
170C22×D4—41no—
171C22×D4—41no—
172C22×D4—42no—
173C22×D4C2×D414yes—
174C23×C4—24no—
175C23×C4—28no—
176C23×C4—44no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4C2414yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2414yes—
183C23×C4C2412yes—
184C23×C4C2414yes—
185C23×C4C2411yescharacteristic
186C23×C4C2412yes—
187C23×C4C2411yescharacteristic
188C23×C4C2414yes—
189C23×C4C2412yes—
190C23×C4C2411yescharacteristic
191C4⋊Q8—416no—
192#124—416no—
193#124—416no—
194C22⋊Q8—464no—
195C4×Q8—416no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4C2×D414yes—
226C2×C4⋊C4#4118yes—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4C2×D4116yes—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—24no—
257C2×C42—28no—
258C2×C42—28no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42—44no—
263C2×C42C2×D414yes—
264#95—24no—
265#95—28no—
266#95—28no—
267#95—24no—
268#95—24no—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—24no—
274#95—28no—
275#95—28no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—42no—
285#95—44no—
286#95—44no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—22no—
291C24—22no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24C22×D414yes—
296C24C2511yescharacteristic
297C2×Q8—28no—
298C2×Q8—48no—
299C2×Q8—48no—
300C2×Q8—416no—
301C2×Q8—48no—
302C2×Q8—44no—
303C2×Q8—44no—
304C2×Q8—44no—
305C2×Q8—44no—
306C2×Q8—44no—
307C2×Q8—84no—
308C2×Q8—84no—
309C2×Q8—84no—
310C2×Q8—88no—
311C2×Q8—84no—
312C2×D4—28no—
313C2×D4—44no—
314C2×D4—44no—
315C2×D4—84no—
316C2×D4—84no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—21no—
323C22×C4—24no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—48no—
376C22×C4—416no—
377C22×C4—88no—
378C22×C4—84no—
379C22×C4—84no—
380C22×C4C22×D414yes—
381C22×C4C22×D412yes—
382C22×C4C22×D414yes—
383C22×C4C2×#4118yes—
384C22×C4C2×#4114yes—
385C22×C4H(5,𝔽2)12yes—
386C22×C4H(5,𝔽2)14yes—
387C22×C4H(5,𝔽2)12yes—
388C22×C4H(5,𝔽2)14yes—
389C22×C42−1+418yes—
390C22×C42−1+414yes—
391C4⋊C4—216no—
392C4⋊C4—44no—
393C4⋊C4—416no—
394C4⋊C4—48no—
395C4⋊C4—416no—
396C4⋊C4—416no—
397C4⋊C4—48no—
398C4⋊C4—416no—
399C4⋊C4—84no—
400C4⋊C4—88no—
401C4⋊C4—84no—
402C22⋊C4—232no—
403C22⋊C4—48no—
404C22⋊C4—416no—
405C22⋊C4—48no—
406C22⋊C4—48no—
407C22⋊C4—416no—
408C22⋊C4—88no—
409C22⋊C4—84no—
410C22⋊C4—84no—
411C42—28no—
412C42—44no—
413C42—416no—
414C42—48no—
415C42—44no—
416C23—24no—
417C23—28no—
418C23—24no—
419C23—24no—
420C23—28no—
421C23—42no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—42no—
432C23—42no—
433C23—44no—
434C23—42no—
435C23—44no—
436C23—44no—
437C23C23×D411yescharacteristic
438C23C22×#4112yes—
439C23C2×H(5,𝔽2)12yes—
440C23C2×H(5,𝔽2)11yescharacteristic
441C23C2×H(5,𝔽2)12yes—
442C23C2×H(5,𝔽2)11yescharacteristic
443C23C2×2−1+412yes—
444C23C2×2−1+414yes—
445Q8—416no—
446Q8—88no—
447Q8—816no—
448D4—88no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—28no—
454C2×C4—28no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—28no—
458C2×C4—28no—
459C2×C4—28no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—28no—
466C2×C4—28no—
467C2×C4—28no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—28no—
471C2×C4—24no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
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—816no—
517C2×C4#53614yes—
518C2×C4#54618yes—
519C22—48no—
520C22—44no—
521C22—44no—
522C22—48no—
523C22—44no—
524C22—84no—
525C22—84no—
526C22—84no—
527C22—84no—
528C22C2×#53611yescharacteristic
529C22C2×#54612yes—
530C22#348111yescharacteristic
531C22#348411yescharacteristic
532C22#348411yescharacteristic
533C22#349912yes—
534C22#351114yes—
535C22#351912yes—
536C22#352314yes—
537C22#353612yes—
538C22#353614yes—
539C22#353612yes—
540C22#353812yes—
541C22#354412yes—
542C22#354911yescharacteristic
543C22#355014yes—
544C4—24no—
545C4—28no—
546C4—42no—
547C4—48no—
548C4—44no—
549C4—42no—
550C4—48no—
551C4—44no—
552C4—84no—
553C4—82no—
554C4—84no—
555C4—82no—
556C2—84no—
557C2#3671811yescharacteristic
558C2#3895412yes—
559C2#4388812yes—
560C2#4486112yes—
561C2#4976414yes—
562C2#5154211yescharacteristic
563C2#5895912yes—
564C2#5904111yescharacteristic
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
3. #16630C2semidirect
4. #16518C2semidirect
7. #15706C2semidirect
8. #15614C2semidirect
9. #15230C2semidirect
10. #14881C2semidirect
11. #14364C2semidirect
12. #14226C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26832
C2×#25452
#166302
#165182
#160322
#157842
#157062
#156142
#152302
#148812
#143642
#142262
#141572