Finite Group Databasebeta

← #4389098

Subgroups — #4389098

Order 512.

2566 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2582C211yesmaximal, characteristic, complemented
2C2×#2632C212yesmaximal
3C2×#2326C211yesmaximal, characteristic
4#16629C212yesmaximal, complemented
5#16596C212yesmaximal
6#15832C214yesmaximal, complemented
7#15824C214yesmaximal, complemented
8#15351C214yesmaximal, complemented
9#15294C214yesmaximal, complemented
10#14880C212yesmaximal, complemented
11#14824C212yesmaximal
12#14676C211yesmaximal, characteristic, complemented
13#13961C212yesmaximal, complemented
14C22×#124C2211yescharacteristic
15C22×C4×Q8C2212yes—
16C22×C4×Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18#2632—216no—
19#2582—28no—
20C2×#399C2214yes—
21C2×#398C2212yes—
22C2×#398C2212yes—
23C2×#398C2212yes—
24C2×#397C2212yes—
25C2×#395C2214yes—
26C2×#394C2214yes—
27C2×#393C2214yes—
28#2326—28no—
29C2×#391C2211yescharacteristic
30C2×#391C2214yes—
31C2×#390C2211yescharacteristic
32C2×#389C2214yes—
33C2×#389C2212yes—
34C2×#388C2214yes—
35C2×#388C2214yes—
36C2×#388C2214yes—
37C2×#385C2214yes—
38C2×#384C2211yescharacteristic
39C2×#383C2212yes—
40C2×#380C2212yes—
41C2×#376C2211yescharacteristic
42#1446C2212yes—
43#1446C2212yes—
44#1445C2212yes—
45#1445C2212yes—
46#1444C2214yes—
47#1443C2214yes—
48#1442C2214yes—
49#1441C2212yes—
50#1441C2214yes—
51#1441C2212yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2212yes—
55#1441C2214yes—
56#1440C2214yes—
57#1440C2214yes—
58#1440C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2212yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1434C2212yes—
66#1434C2214yes—
67#1434C2214yes—
68C22⋊#95C2214yes—
69C22⋊#95C2212yes—
70C23×Q8C2312yes—
71C23×Q8C2311yescharacteristic
72C23×D4C2311yescharacteristic
73C24×C4C2312yes—
74C2×#124—24no—
75C2×#124—24no—
76C2×#124D418yes—
77C2×C4×Q8—24no—
78C2×C4×Q8—28no—
79C2×C4×Q8—28no—
80C2×C4×Q8—28no—
81C2×C4×Q8—42no—
82C2×C4×Q8—42no—
83C2×C4×D4—24no—
84C2×C4×D4—42no—
85C2×C4×D4—42no—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C42C2311yescharacteristic
106C22×C42C2314yes—
107C22×C42C2312yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110#399—216no—
111#398—28no—
112#398—28no—
113#398—44no—
114#397—44no—
115#395—216no—
116#394—216no—
117#393—216no—
118#391—24no—
119#391—216no—
120#390—24no—
121#389—216no—
122#389—44no—
123#388—216no—
124#388—216no—
125#388—216no—
126#385—216no—
127#384D418yes—
128#383—28no—
129#380—44no—
130#376—24no—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C25C2412yes—
157C22×Q8—22no—
158C22×Q8—24no—
159C22×Q8—24no—
160C22×Q8—22no—
161C22×Q8—24no—
162C22×Q8—28no—
163C22×Q8—41no—
164C22×Q8—41no—
165C22×Q8—41no—
166C22×Q8—41no—
167C22×Q8—44no—
168C22×Q8C2×D414yes—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4—41no—
172C22×D4—41no—
173C22×D4—41no—
174C22×D4—41no—
175C22×D4C2×D414yes—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—42no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4C2414yes—
182C23×C4C2414yes—
183C23×C4C2414yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2411yescharacteristic
189C23×C4C2411yescharacteristic
190C23×C4C2414yes—
191C23×C4C2412yes—
192C23×C4C2411yescharacteristic
193#124—232no—
194C4×Q8—416no—
195C4×Q8—432no—
196C4×D4—416no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
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—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4C2×D414yes—
229C2×C4⋊C4C2×D414yes—
230C2×C4⋊C4C2×D4116yes—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4C2×D4116yes—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—28no—
257C2×C42—28no—
258C2×C42—22no—
259C2×C42—28no—
260C2×C42—24no—
261C2×C42—42no—
262C2×C42—41no—
263C2×C42—41no—
264C2×C42C2×D414yes—
265C2×C42C2×D414yes—
266C2×C42#4114yes—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—28no—
277#95—28no—
278#95—24no—
279#95—24no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—42no—
291#95—42no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—24no—
297C24—24no—
298C24—24no—
299C24—24no—
300C24—22no—
301C24—22no—
302C24C22×D414yes—
303C24C2511yescharacteristic
304C2×Q8—216no—
305C2×Q8—28no—
306C2×Q8—44no—
307C2×Q8—44no—
308C2×Q8—416no—
309C2×Q8—416no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—44no—
313C2×Q8—48no—
314C2×Q8—44no—
315C2×Q8—48no—
316C2×D4—28no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—44no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—24no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—21no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
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—22no—
354C22×C4—22no—
355C22×C4—22no—
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—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—48no—
382C22×C4—48no—
383C22×C4—88no—
384C22×C4—84no—
385C22×C4—88no—
386C22×C4C22×D414yes—
387C22×C4C22×D414yes—
388C22×C4C22×D414yes—
389C22×C4C22×D414yes—
390C22×C4C22×D412yes—
391C22×C4C22×D414yes—
392C22×C4C22×D412yes—
393C22×C4C2×#4114yes—
394C22×C4C2×#4112yes—
395C22×C4H(5,𝔽2)12yes—
396C22×C4H(5,𝔽2)12yes—
397C22×C42−1+418yes—
398C4⋊C4—416no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—48no—
402C4⋊C4—48no—
403C4⋊C4—416no—
404C4⋊C4—48no—
405C4⋊C4—84no—
406C4⋊C4—84no—
407C4⋊C4—88no—
408C4⋊C4—88no—
409C4⋊C4—84no—
410C22⋊C4—232no—
411C22⋊C4—48no—
412C22⋊C4—416no—
413C22⋊C4—416no—
414C22⋊C4—88no—
415C22⋊C4—88no—
416C22⋊C4—88no—
417C42—28no—
418C42—48no—
419C42—44no—
420C42—416no—
421C42#121116yes—
422C23—24no—
423C23—28no—
424C23—28no—
425C23—24no—
426C23—24no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—44no—
431C23—44no—
432C23—42no—
433C23—44no—
434C23—44no—
435C23—42no—
436C23—42no—
437C23—44no—
438C23—44no—
439C23—42no—
440C23—42no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23C23×D411yescharacteristic
446C23C23×D411yescharacteristic
447C23C22×#4111yescharacteristic
448C23C2×H(5,𝔽2)11yescharacteristic
449C23C2×H(5,𝔽2)11yescharacteristic
450C23C2×2−1+414yes—
451C23#58511yescharacteristic
452C23#58511yescharacteristic
453C23#58514yes—
454Q8—432no—
455Q8—416no—
456D4—416no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—28no—
464C2×C4—28no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—28no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—28no—
476C2×C4—28no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
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—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
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—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—816no—
526C2×C4C2×#12118yes—
527C2×C4C2×#12114yes—
528C22—48no—
529C22—44no—
530C22—44no—
531C22—44no—
532C22—48no—
533C22—82no—
534C22—84no—
535C22—84no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22C22×#12111yescharacteristic
540C22#346011yescharacteristic
541C22#346011yescharacteristic
542C22#348411yescharacteristic
543C22#348512yes—
544C22#348612yes—
545C22#348711yescharacteristic
546C22#348914yes—
547C22#349514yes—
548C22#349611yescharacteristic
549C22#349611yescharacteristic
550C22#353312yes—
551C22#354012yes—
552C22#354412yes—
553C22#355012yes—
554C22#355714yes—
555C22#356514yes—
556C4—28no—
557C4—24no—
558C4—48no—
559C4—44no—
560C4—44no—
561C4—42no—
562C4—44no—
563C4—42no—
564C4—82no—
565C4—84no—
566C4—84no—
567C4—84no—
568C2—84no—
569C2#3516811yescharacteristic
570C2#3563312yes—
571C2#4221112yes—
572C2#4233712yes—
573C2#4778612yes—
574C2#4778812yes—
575C2#5205411yescharacteristic
576C2#5205812yes—
577C2#5207211yescharacteristic
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
1. C2×#2582C2semidirect
4. #16629C2semidirect
6. #15832C2semidirect
7. #15824C2semidirect
8. #15351C2semidirect
9. #15294C2semidirect
10. #14880C2semidirect
12. #14676C2semidirect
13. #13961C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25822
C2×#26322
C2×#23262
#166292
#165962
#158322
#158242
#153512
#152942
#148802
#148242
#146762
#139612