Finite Group Databasebeta

← #4121485

Subgroups — #4121485

Order 512.

2346 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17243C212yesmaximal, complemented
2#17135C211yesmaximal, characteristic, complemented
3#16088C212yesmaximal, complemented
4#15950C212yesmaximal
5#15949C211yesmaximal, characteristic, complemented
6#15894C214yesmaximal, complemented
7#15324C214yesmaximal, complemented
8#14969C212yesmaximal, complemented
9#14756C214yesmaximal, complemented
10#14549C214yesmaximal, complemented
11#14506C212yesmaximal, complemented
12#14152C211yesmaximal, characteristic
13#13965C212yesmaximal, complemented
14C22×#123C2212yes—
15C22×#121C2211yescharacteristic
16C2×#396C2214yes—
17C2×#396C2214yes—
18C2×#389C2212yes—
19C2×#388C2214yes—
20C2×#387C2214yes—
21C2×#387C2214yes—
22C2×#386C2214yes—
23C2×#386C2212yes—
24C2×#385C2214yes—
25C2×#385C2214yes—
26C2×#385C2212yes—
27C2×#385C2214yes—
28C2×#380C2212yes—
29C2×C4×C22⋊C4C2212yes—
30C2×C4×C22⋊C4C2211yescharacteristic
31C22×#95C2211yescharacteristic
32#1445C2214yes—
33#1445C2211yescharacteristic
34#1445C2211yescharacteristic
35#1445C2214yes—
36#1444C2212yes—
37#1444C2212yes—
38#1444C2212yes—
39#1444C2212yes—
40#1444C2212yes—
41#1444C2212yes—
42#1443C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1440C2212yes—
49#1440C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2212yes—
53#1439C2214yes—
54#1439C2212yes—
55#1438C2211yescharacteristic
56#1438C2211yescharacteristic
57#1438C2212yes—
58#1438C2214yes—
59#1438C2212yes—
60#1437C2214yes—
61#1437C2214yes—
62#1436C2214yes—
63C4⋊#95C2214yes—
64C4⋊#95C2214yes—
65#1434C2212yes—
66C22⋊#95C2214yes—
67C23×D4C2312yes—
68C23×D4C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C24×C4C2312yes—
71C24×C4C2312yes—
72C2×#123—28no—
73C2×#123—48no—
74C2×#121—24no—
75C2×#121—42no—
76C2×#121—42no—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C22⋊C4C2312yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2312yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C42C2312yes—
99C22×C42C2314yes—
100#396—216no—
101#396—48no—
102#389—44no—
103#388—48no—
104#387—48no—
105#387—48no—
106#386—48no—
107#386—44no—
108#385—48no—
109#385—48no—
110#385—48no—
111#385—44no—
112#380—44no—
113C4×C22⋊C4—28no—
114C4×C22⋊C4—42no—
115C2×#95—42no—
116C2×#95C2312yes—
117C2×#95C2314yes—
118C2×#95C2312yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2312yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2311yescharacteristic
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2311yescharacteristic
137C2×#95C2311yescharacteristic
138C2×#95C2311yescharacteristic
139C2×#95C2314yes—
140C2×#95C2312yes—
141C25C2412yes—
142C25C2411yescharacteristic
143C22×D4—24no—
144C22×D4—24no—
145C22×D4—28no—
146C22×D4—41no—
147C22×D4—41no—
148C22×D4—42no—
149C22×D4—44no—
150C22×D4—44no—
151C22×D4#4114yes—
152C23×C4—22no—
153C23×C4—24no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—42no—
164C23×C4—44no—
165C23×C4—42no—
166C23×C4C2414yes—
167C23×C4C2412yes—
168C23×C4C2414yes—
169C23×C4C2412yes—
170C23×C4C2414yes—
171C23×C4C2412yes—
172C23×C4C2411yescharacteristic
173C23×C4C2412yes—
174C23×C4C2414yes—
175C23×C4C2411yescharacteristic
176C23×C4C2412yes—
177#123—432no—
178#121—416no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C42—28no—
236C2×C42—24no—
237C2×C42—44no—
238C2×C42—44no—
239C2×C42—44no—
240#95—24no—
241#95—28no—
242#95—28no—
243#95—28no—
244#95—28no—
245#95—24no—
246#95—28no—
247#95—24no—
248#95—41no—
249#95—42no—
250#95—41no—
251#95—41no—
252#95—41no—
253#95—44no—
254#95—44no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95#4118yes—
265C24—22no—
266C24—21no—
267C24—24no—
268C24—22no—
269C24—21no—
270C24—22no—
271C24—24no—
272C24—24no—
273C24—24no—
274C24—24no—
275C24—22no—
276C24—22no—
277C24—24no—
278C24—24no—
279C24C2×#4114yes—
280C24H(5,𝔽2)12yes—
281C242−1+414yes—
282C24C2511yescharacteristic
283C2×D4—44no—
284C2×D4—48no—
285C2×D4—48no—
286C2×D4—416no—
287C2×D4—48no—
288C2×D4—88no—
289C2×D4—84no—
290C2×D4—84no—
291C2×D4—84no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—21no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—21no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—28no—
323C22×C4—216no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—48no—
349C22×C4—84no—
350C22×C4—88no—
351C22×C4—88no—
352C22×C4—84no—
353C22×C4—84no—
354C22×C4—84no—
355C22×C4—84no—
356C22×C4—84no—
357C22×C4—84no—
358C22×C4—88no—
359C22×C4—88no—
360C22×C4C2×#4114yes—
361C22×C4C2×#4118yes—
362C22×C4C2×#4112yes—
363C22×C4H(5,𝔽2)18yes—
364C22×C4H(5,𝔽2)14yes—
365C22×C4H(5,𝔽2)12yes—
366C22×C42−1+414yes—
367C4⋊C4—48no—
368C4⋊C4—416no—
369C4⋊C4—416no—
370C4⋊C4—416no—
371C4⋊C4—44no—
372C4⋊C4—88no—
373C4⋊C4—88no—
374C4⋊C4—84no—
375C4⋊C4—84no—
376C4⋊C4—84no—
377C22⋊C4—232no—
378C22⋊C4—216no—
379C22⋊C4—48no—
380C22⋊C4—416no—
381C22⋊C4—48no—
382C22⋊C4—48no—
383C22⋊C4—416no—
384C22⋊C4—88no—
385C22⋊C4—88no—
386C22⋊C4—88no—
387C22⋊C4—84no—
388C42—88no—
389C42—84no—
390C23—22no—
391C23—28no—
392C23—24no—
393C23—24no—
394C23—24no—
395C23—24no—
396C23—22no—
397C23—28no—
398C23—28no—
399C23—24no—
400C23—24no—
401C23—24no—
402C23—28no—
403C23—41no—
404C23—42no—
405C23—42no—
406C23—44no—
407C23—44no—
408C23—42no—
409C23—41no—
410C23—42no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23—42no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—42no—
421C23—44no—
422C23—44no—
423C23—42no—
424C23—44no—
425C23—44no—
426C23#56314yes—
427C23C22×#4112yes—
428C23C22×#4111yescharacteristic
429C23C2×H(5,𝔽2)11yescharacteristic
430C23C2×H(5,𝔽2)11yescharacteristic
431C23C2×H(5,𝔽2)14yes—
432C23C2×H(5,𝔽2)12yes—
433C23C2×2−1+412yes—
434C23C2×2−1+412yes—
435D4—88no—
436D4—816no—
437C2×C4—28no—
438C2×C4—28no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—28no—
443C2×C4—28no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—28no—
447C2×C4—28no—
448C2×C4—28no—
449C2×C4—24no—
450C2×C4—28no—
451C2×C4—28no—
452C2×C4—24no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—88no—
499C2×C4—816no—
500C2×C4—816no—
501C22—24no—
502C22—28no—
503C22—42no—
504C22—48no—
505C22—44no—
506C22—42no—
507C22—44no—
508C22—44no—
509C22—44no—
510C22—48no—
511C22—48no—
512C22—44no—
513C22—44no—
514C22—84no—
515C22—84no—
516C22—84no—
517C22—84no—
518C22#348111yescharacteristic
519C22#348212yes—
520C22#349012yes—
521C22#349011yescharacteristic
522C22#349214yes—
523C22#349212yes—
524C22#351914yes—
525C22#352012yes—
526C22C2×#56311yescharacteristic
527C22#353614yes—
528C22#353612yes—
529C22#354312yes—
530C22#354312yes—
531C22#354311yescharacteristic
532C22#355011yescharacteristic
533C22#355214yes—
534C4—42no—
535C4—42no—
536C4—48no—
537C4—48no—
538C4—44no—
539C4—44no—
540C4—44no—
541C4—84no—
542C4—82no—
543C4—84no—
544C4—82no—
545C2—44no—
546C2—84no—
547C2#4336614yes—
548C2#4340412yes—
549C2#4619911yescharacteristic
550C2#4632612yes—
551C2#4638711yescharacteristic
552C2#4650712yes—
553C2#5349212yes—
554C2#6004611yescharacteristic
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. #17243C2semidirect
2. #17135C2semidirect
3. #16088C2semidirect
5. #15949C2semidirect
6. #15894C2semidirect
7. #15324C2semidirect
8. #14969C2semidirect
9. #14756C2semidirect
10. #14549C2semidirect
11. #14506C2semidirect
13. #13965C2semidirect

Maximal subgroups

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

subgroupindex
#172432
#171352
#160882
#159502
#159492
#158942
#153242
#149692
#147562
#145492
#145062
#141522
#139652