Finite Group Databasebeta

← #4121486

Subgroups — #4121486

Order 512.

2242 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17298C212yesmaximal, complemented
2#17137C211yesmaximal, characteristic, complemented
3#16159C214yesmaximal, complemented
4#16110C212yesmaximal, complemented
5#15966C211yesmaximal, characteristic, complemented
6#15269C214yesmaximal, complemented
7#15260C212yesmaximal
8#15050C212yesmaximal, complemented
9#14817C214yesmaximal, complemented
10#14575C214yesmaximal, complemented
11#14516C212yesmaximal, complemented
12#14152C211yesmaximal, characteristic
13#13966C212yesmaximal, complemented
14C22×#123C2212yes—
15C22×#121C2211yescharacteristic
16C2×#397C2212yes—
17C2×#394C2214yes—
18C2×#393C2214yes—
19C2×#393C2212yes—
20C2×#391C2212yes—
21C2×#391C2214yes—
22C2×#386C2212yes—
23C2×#386C2214yes—
24C2×#385C2214yes—
25C2×#385C2212yes—
26C2×#380C2212yes—
27C2×C4×C22⋊C4C2212yes—
28C2×C4×C22⋊C4C2211yescharacteristic
29C22×#95C2211yescharacteristic
30#1446C2212yes—
31#1445C2211yescharacteristic
32#1445C2211yescharacteristic
33#1445C2212yes—
34#1444C2214yes—
35#1444C2214yes—
36#1444C2214yes—
37#1443C2214yes—
38#1443C2214yes—
39#1442C2212yes—
40#1442C2214yes—
41#1442C2212yes—
42#1442C2212yes—
43#1442C2212yes—
44#1442C2214yes—
45#1441C2214yes—
46#1440C2212yes—
47#1440C2214yes—
48#1440C2214yes—
49#1440C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1438C2211yescharacteristic
53#1438C2211yescharacteristic
54#1438C2212yes—
55#1438C2212yes—
56#1437C2214yes—
57#1437C2214yes—
58#1437C2214yes—
59#1434C2212yes—
60#1434C2214yes—
61#1434C2214yes—
62#1434C2214yes—
63#1434C2214yes—
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C23×Q8C2312yes—
68C23×D4C2312yes—
69C23×D4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C24×C4C2312yes—
72C24×C4C2312yes—
73C2×#123—44no—
74C2×#123—48no—
75C2×#121—24no—
76C2×#121—42no—
77C2×#121—42no—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2312yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2312yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2312yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C42C2314yes—
96C22×C42C2312yes—
97#397—44no—
98#394—48no—
99#393—216no—
100#393—44no—
101#391—44no—
102#391—48no—
103#386—28no—
104#386—48no—
105#385—44no—
106#385—48no—
107#380—44no—
108C4×C22⋊C4—28no—
109C4×C22⋊C4—42no—
110C2×#95—42no—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2312yes—
119C2×#95C2312yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2312yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2311yescharacteristic
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2311yescharacteristic
136C2×#95C2311yescharacteristic
137C2×#95C2311yescharacteristic
138C2×#95C2312yes—
139C25C2412yes—
140C25C2411yescharacteristic
141C22×Q8—28no—
142C22×Q8—24no—
143C22×Q8—44no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×D4—24no—
147C22×D4—24no—
148C22×D4—28no—
149C22×D4—41no—
150C22×D4—41no—
151C22×D4—42no—
152C22×D4—44no—
153C22×D4—44no—
154C22×D4#4114yes—
155C23×C4—22no—
156C23×C4—24no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—22no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—42no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4—42no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4C2414yes—
170C23×C4C2414yes—
171C23×C4C2412yes—
172C23×C4C2414yes—
173C23×C4C2412yes—
174C23×C4C2414yes—
175C23×C4C2412yes—
176C23×C4C2411yescharacteristic
177C23×C4C2412yes—
178C23×C4C2411yescharacteristic
179C23×C4C2412yes—
180#123—432no—
181#121—416no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—42no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C42—28no—
229C2×C42—24no—
230C2×C42—44no—
231C2×C42—44no—
232C2×C42—44no—
233#95—24no—
234#95—28no—
235#95—28no—
236#95—28no—
237#95—28no—
238#95—28no—
239#95—28no—
240#95—28no—
241#95—41no—
242#95—42no—
243#95—41no—
244#95—44no—
245#95—44no—
246#95—44no—
247#95—41no—
248#95—41no—
249#95—44no—
250#95—44no—
251#95—44no—
252#95—44no—
253#95—44no—
254#95—44no—
255#95—42no—
256#95—44no—
257#95—44no—
258#95—42no—
259#95—44no—
260#95#4118yes—
261C24—22no—
262C24—21no—
263C24—22no—
264C24—21no—
265C24—22no—
266C24—24no—
267C24—24no—
268C24—24no—
269C24—22no—
270C24—24no—
271C24—22no—
272C24—24no—
273C24—24no—
274C24—24no—
275C24C2×#4114yes—
276C24H(5,𝔽2)14yes—
277C24H(5,𝔽2)12yes—
278C24C2511yescharacteristic
279C2×Q8—48no—
280C2×Q8—416no—
281C2×Q8—88no—
282C2×Q8—84no—
283C2×Q8—84no—
284C2×D4—44no—
285C2×D4—48no—
286C2×D4—48no—
287C2×D4—416no—
288C2×D4—48no—
289C2×D4—88no—
290C2×D4—84no—
291C2×D4—84no—
292C2×D4—84no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—28no—
318C22×C4—216no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
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—48no—
351C22×C4—84no—
352C22×C4—88no—
353C22×C4—84no—
354C22×C4—88no—
355C22×C4—84no—
356C22×C4—84no—
357C22×C4—84no—
358C22×C4—84no—
359C22×C4—88no—
360C22×C4—88no—
361C22×C4—84no—
362C22×C4C2×#4112yes—
363C22×C4C2×#4114yes—
364C22×C4C2×#4118yes—
365C22×C4H(5,𝔽2)12yes—
366C22×C4H(5,𝔽2)14yes—
367C22×C42−1+414yes—
368C4⋊C4—44no—
369C4⋊C4—416no—
370C4⋊C4—84no—
371C4⋊C4—88no—
372C4⋊C4—84no—
373C4⋊C4—84no—
374C22⋊C4—48no—
375C22⋊C4—416no—
376C22⋊C4—416no—
377C22⋊C4—48no—
378C22⋊C4—48no—
379C22⋊C4—48no—
380C22⋊C4—416no—
381C22⋊C4—88no—
382C22⋊C4—88no—
383C22⋊C4—84no—
384C22⋊C4—88no—
385C42—88no—
386C42—84no—
387C23—22no—
388C23—24no—
389C23—28no—
390C23—24no—
391C23—24no—
392C23—22no—
393C23—24no—
394C23—28no—
395C23—28no—
396C23—24no—
397C23—24no—
398C23—24no—
399C23—28no—
400C23—41no—
401C23—42no—
402C23—42no—
403C23—44no—
404C23—42no—
405C23—41no—
406C23—44no—
407C23—42no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23—44no—
412C23—44no—
413C23—42no—
414C23—44no—
415C23—42no—
416C23—44no—
417C23—44no—
418C23—42no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23H(3,𝔽4)14yes—
424C23C22×#4112yes—
425C23C22×#4111yescharacteristic
426C23C2×H(5,𝔽2)11yescharacteristic
427C23C2×H(5,𝔽2)11yescharacteristic
428C23C2×H(5,𝔽2)12yes—
429C23C2×H(5,𝔽2)12yes—
430C23C2×2−1+412yes—
431C23#58514yes—
432Q8—816no—
433D4—88no—
434D4—816no—
435C2×C4—28no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—28no—
442C2×C4—28no—
443C2×C4—28no—
444C2×C4—28no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—28no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—816no—
497C2×C4—88no—
498C2×C4—816no—
499C22—24no—
500C22—28no—
501C22—44no—
502C22—44no—
503C22—44no—
504C22—42no—
505C22—48no—
506C22—42no—
507C22—44no—
508C22—44no—
509C22—48no—
510C22—48no—
511C22—44no—
512C22—84no—
513C22—84no—
514C22—84no—
515C22—84no—
516C22#348111yescharacteristic
517C22#348212yes—
518C22#349011yescharacteristic
519C22#349112yes—
520C22#349212yes—
521C22#349414yes—
522C22#351314yes—
523C22#351912yes—
524C22C2×H(3,𝔽4)11yescharacteristic
525C22#353712yes—
526C22#354311yescharacteristic
527C22#354512yes—
528C22#354512yes—
529C22#355011yescharacteristic
530C22#355914yes—
531C22#356114yes—
532C4—48no—
533C4—44no—
534C4—44no—
535C4—44no—
536C4—42no—
537C4—42no—
538C4—84no—
539C4—82no—
540C4—82no—
541C4—84no—
542C4—84no—
543C2—44no—
544C2—84no—
545C2#4346514yes—
546C2#4481312yes—
547C2#4599112yes—
548C2#4608911yescharacteristic
549C2#4647512yes—
550C2#4683711yescharacteristic
551C2#5364212yes—
552C2#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. #17298C2semidirect
2. #17137C2semidirect
3. #16159C2semidirect
4. #16110C2semidirect
5. #15966C2semidirect
6. #15269C2semidirect
8. #15050C2semidirect
9. #14817C2semidirect
10. #14575C2semidirect
11. #14516C2semidirect
13. #13966C2semidirect

Maximal subgroups

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

subgroupindex
#172982
#171372
#161592
#161102
#159662
#152692
#152602
#150502
#148172
#145752
#145162
#141522
#139662