Finite Group Databasebeta

← #1495450

Subgroups — #1495450

Order 512.

2630 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2724C211yesmaximal, characteristic
2C2×#2542C211yesmaximal, characteristic
3#15790C212yesmaximal, complemented
4#15783C212yesmaximal, complemented
5#15670C214yesmaximal, complemented
6#15469C214yesmaximal, complemented
7#15242C212yesmaximal, complemented
8#15016C214yesmaximal, complemented
9#14505C212yesmaximal
10#14242C212yesmaximal, complemented
11#14052C214yesmaximal, complemented
12#13968C212yesmaximal
13#13824C211yesmaximal, characteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×#123C2212yes—
16C22×C22⋊Q8C2214yes—
17C22×C4×D4C2211yescharacteristic
18#2724—28no—
19#2542—28no—
20C2×#400C2212yes—
21C2×#400C2212yes—
22C2×#399C2214yes—
23C2×#398C2212yes—
24C2×#397C2214yes—
25C2×#391C2212yes—
26C2×#390C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#389C2212yes—
29C2×#386C2212yes—
30C2×#385C2214yes—
31C2×#384C2211yescharacteristic
32C2×#384C2211yescharacteristic
33C2×#382C2214yes—
34C2×#381C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4C2214yes—
37C22×#95C2212yes—
38#1446C2212yes—
39#1446C2214yes—
40#1445C2214yes—
41#1445C2214yes—
42#1445C2212yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2212yes—
46#1442C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2212yes—
51#1440C2214yes—
52#1440C2214yes—
53#1440C2214yes—
54#1437C2214yes—
55#1436C2212yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2212yes—
60#1436C2214yes—
61C4⋊#95C2214yes—
62#1434C2214yes—
63C22⋊#95C2212yes—
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2212yes—
67C22⋊#95C2214yes—
68C4×#95C2212yes—
69C23×Q8C2312yes—
70C23×D4C2311yescharacteristic
71C24×C4C2312yes—
72C24×C4C2312yes—
73C24×C4C2312yes—
74C2×C4⋊Q8—24no—
75C2×C4⋊Q8—42no—
76C2×C4⋊Q8—42no—
77C2×#123—28no—
78C2×#123—28no—
79C2×#123—28no—
80C2×C22⋊Q8—216no—
81C2×C22⋊Q8—216no—
82C2×C22⋊Q8—48no—
83C2×C4×D4—24no—
84C2×C4×D4—42no—
85C2×C4×D4—42no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C42C2311yescharacteristic
105C22×C42C2314yes—
106C22×C42C2311yescharacteristic
107C22×C42C2312yes—
108C22×C42C2311yescharacteristic
109#400—28no—
110#400—44no—
111#399—216no—
112#398—28no—
113#397—216no—
114#391—44no—
115#390—24no—
116#390D418yes—
117#389—28no—
118#386—28no—
119#385—216no—
120#384—24no—
121#384—24no—
122#382—48no—
123#381—44no—
124C4×C4⋊C4—24no—
125C4×C22⋊C4—48no—
126C2×#95—44no—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C25C2412yes—
153C22×Q8—24no—
154C22×Q8—28no—
155C22×Q8—24no—
156C22×Q8—44no—
157C22×Q8—42no—
158C22×D4—41no—
159C22×D4—41no—
160C22×D4—42no—
161C22×D4C2×D414yes—
162C22×D4#4118yes—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—28no—
169C23×C4—24no—
170C23×C4—28no—
171C23×C4—42no—
172C23×C4—44no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—44no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4C2412yes—
181C23×C4C2414yes—
182C23×C4C2414yes—
183C23×C4C2412yes—
184C23×C4C2414yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2411yescharacteristic
188C23×C4C2414yes—
189C23×C4C2411yescharacteristic
190C23×C4C2412yes—
191C23×C4C2411yescharacteristic
192C4⋊Q8—416no—
193#123—432no—
194C22⋊Q8—464no—
195C4×D4—416no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4C2×D414yes—
227C2×C4⋊C4#4118yes—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4C2×D4116yes—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—28no—
254C2×C42—28no—
255C2×C42—42no—
256C2×C42—44no—
257C2×C42—42no—
258C2×C42—44no—
259C2×C42—42no—
260C2×C42C2×D414yes—
261#95—24no—
262#95—28no—
263#95—28no—
264#95—24no—
265#95—28no—
266#95—24no—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—24no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—44no—
275#95—44no—
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—
286C24—24no—
287C24—24no—
288C24—22no—
289C24—22no—
290C24—24no—
291C24—24no—
292C24—22no—
293C24—22no—
294C24C22×D414yes—
295C24C2×#4114yes—
296C24C2×#4114yes—
297C24C2511yescharacteristic
298C2×Q8—416no—
299C2×Q8—48no—
300C2×Q8—48no—
301C2×Q8—88no—
302C2×Q8—84no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—48no—
306C2×D4—48no—
307C2×D4#124116yes—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
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—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
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—416no—
367C22×C4—416no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—88no—
373C22×C4—84no—
374C22×C4—88no—
375C22×C4—84no—
376C22×C4—84no—
377C22×C4—84no—
378C22×C4—84no—
379C22×C4—84no—
380C22×C4—84no—
381C22×C4C22×D414yes—
382C22×C4C22×D412yes—
383C22×C4C22×D414yes—
384C22×C4C2×#4114yes—
385C22×C4C2×#4114yes—
386C22×C4C2×#4114yes—
387C22×C4C2×#4114yes—
388C22×C4H(5,𝔽2)12yes—
389C22×C4H(5,𝔽2)14yes—
390C22×C4H(5,𝔽2)12yes—
391C22×C42−1+418yes—
392C4⋊C4—216no—
393C4⋊C4—44no—
394C4⋊C4—416no—
395C4⋊C4—48no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—416no—
399C4⋊C4—48no—
400C4⋊C4—416no—
401C4⋊C4—84no—
402C4⋊C4—84no—
403C22⋊C4—232no—
404C22⋊C4—416no—
405C22⋊C4—48no—
406C22⋊C4—416no—
407C22⋊C4—48no—
408C22⋊C4—416no—
409C22⋊C4—84no—
410C42—28no—
411C42—44no—
412C42—416no—
413C42—44no—
414C42—84no—
415C23—28no—
416C23—28no—
417C23—28no—
418C23—24no—
419C23—24no—
420C23—28no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—24no—
425C23—28no—
426C23—28no—
427C23—42no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—44no—
437C23C2×#12418yes—
438C23C23×D411yescharacteristic
439C23C22×#4112yes—
440C23C22×#4112yes—
441C23C2×H(5,𝔽2)12yes—
442C23C2×H(5,𝔽2)11yescharacteristic
443C23C2×H(5,𝔽2)11yescharacteristic
444C23C2×2−1+414yes—
445C23#58512yes—
446Q8—816no—
447D4—416no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—28no—
458C2×C4—28no—
459C2×C4—22no—
460C2×C4—28no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—28no—
467C2×C4—28no—
468C2×C4—24no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—44no—
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—42no—
492C2×C4—42no—
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—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—816no—
513C2×C4—816no—
514C2×C4—816no—
515C2×C4C2×#12414yes—
516C2×C4#54618yes—
517C2×C4#55914yes—
518C22—28no—
519C22—216no—
520C22—44no—
521C22—48no—
522C22—48no—
523C22—44no—
524C22—44no—
525C22—44no—
526C22—48no—
527C22—44no—
528C22—44no—
529C22C22×#12411yescharacteristic
530C22C2×#54612yes—
531C22C2×#55911yescharacteristic
532C22#348214yes—
533C22#348411yescharacteristic
534C22#348511yescharacteristic
535C22#349912yes—
536C22#350512yes—
537C22#350512yes—
538C22#350812yes—
539C22#351514yes—
540C22#352014yes—
541C22#353514yes—
542C22#353812yes—
543C22#354311yescharacteristic
544C22#354412yes—
545C4—24no—
546C4—24no—
547C4—28no—
548C4—44no—
549C4—48no—
550C4—42no—
551C4—44no—
552C4—82no—
553C4—82no—
554C4—84no—
555C4—84no—
556C4—84no—
557C2—48no—
558C2#3537711yescharacteristic
559C2#3630312yes—
560C2#4349612yes—
561C2#4383414yes—
562C2#4821412yes—
563C2#5154211yescharacteristic
564C2#5914311yescharacteristic
565C2#5963112yes—
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. #15790C2semidirect
4. #15783C2semidirect
5. #15670C2semidirect
6. #15469C2semidirect
7. #15242C2semidirect
8. #15016C2semidirect
10. #14242C2semidirect
11. #14052C2semidirect

Maximal subgroups

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

subgroupindex
C2×#27242
C2×#25422
#157902
#157832
#156702
#154692
#152422
#150162
#145052
#142422
#140522
#139682
#138242