Finite Group Databasebeta

← #5486630

Subgroups — #5486630

Order 512.

2124 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17273C211yesmaximal, characteristic, complemented
2#17232C214yesmaximal, complemented
3#16928C214yesmaximal, complemented
4#16533C214yesmaximal, complemented
5#16114C214yesmaximal, complemented
6#15949C211yesmaximal, characteristic, complemented
7#15452C214yesmaximal, complemented
8#15261C211yesmaximal, characteristic
9#15261C211yesmaximal, characteristic
10#14756C214yesmaximal, complemented
11#14661C211yesmaximal, characteristic
12#14658C211yesmaximal, characteristic, complemented
13#14167C211yesmaximal, characteristic, complemented
14C22×#127C2211yescharacteristic
15C22×#123C2211yescharacteristic
16C22×#123C2211yescharacteristic
17C2×#396C2214yes—
18C2×#394C2212yes—
19C2×#394C2212yes—
20C2×#388C2214yes—
21C2×#387C2214yes—
22C2×#386C2212yes—
23C2×#386C2212yes—
24C2×#385C2212yes—
25C2×#385C2212yes—
26C2×#383C2212yes—
27C2×#380C2212yes—
28C2×C4×C22⋊C4C2212yes—
29#1445C2214yes—
30#1444C2214yes—
31#1443C2214yes—
32#1443C2214yes—
33#1443C2214yes—
34#1442C2214yes—
35#1442C2214yes—
36#1442C2214yes—
37#1441C2214yes—
38#1440C2212yes—
39#1440C2212yes—
40#1440C2214yes—
41#1440C2214yes—
42#1440C2212yes—
43#1440C2212yes—
44#1440C2214yes—
45#1439C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1438C2212yes—
51#1438C2214yes—
52#1438C2214yes—
53#1438C2211yescharacteristic
54#1438C2211yescharacteristic
55#1438C2211yescharacteristic
56#1438C2214yes—
57#1438C2212yes—
58#1438C2212yes—
59#1438C2214yes—
60#1438C2211yescharacteristic
61#1437C2214yes—
62#1437C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2212yes—
66#1436C2212yes—
67C23×D4C2312yes—
68C23×D4C2312yes—
69C23×D4C2311yescharacteristic
70C23×D4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C2×#127—24no—
74C2×#127—44no—
75C2×#123—24no—
76C2×#123—24no—
77C2×#123—44no—
78C2×#123—44no—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2314yes—
84C22×C22⋊C4C2312yes—
85C22×C22⋊C4C2312yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2311yescharacteristic
89C22×C22⋊C4C2311yescharacteristic
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C42C2311yescharacteristic
103C22×C42C2314yes—
104#396—48no—
105#394—44no—
106#394—44no—
107#388—48no—
108#387—48no—
109#386—44no—
110#386—44no—
111#385—44no—
112#385—44no—
113#383—44no—
114#380—44no—
115C4×C22⋊C4—44no—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2312yes—
122C2×#95C2312yes—
123C2×#95C2312yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C25C2412yes—
139C25C2411yescharacteristic
140C25C2411yescharacteristic
141C22×D4—24no—
142C22×D4—24no—
143C22×D4—22no—
144C22×D4—24no—
145C22×D4—24no—
146C22×D4—42no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4—42no—
152C22×D4—44no—
153C22×D4—44no—
154C22×D4—44no—
155C22×D4—44no—
156C22×D4—42no—
157C22×D4—42no—
158C22×D4#4114yes—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4C2414yes—
168C23×C4C2414yes—
169C23×C4C2412yes—
170C23×C4C2412yes—
171C23×C4C2414yes—
172C23×C4C2414yes—
173C23×C4C2411yescharacteristic
174C23×C4C2412yes—
175C23×C4C2412yes—
176C23×C4C2412yes—
177#127—416no—
178#123—416no—
179#123—416no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
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×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C42—22no—
240C2×C42—24no—
241C2×C42—44no—
242C2×C42—44no—
243C2×C42—44no—
244#95—28no—
245#95—28no—
246#95—28no—
247#95—28no—
248#95—28no—
249#95—44no—
250#95—44no—
251#95—44no—
252#95—44no—
253#95—44no—
254#95—44no—
255#95—44no—
256#95—44no—
257#95—42no—
258#95—44no—
259#95—42no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—42no—
265#95—42no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—24no—
273C24—24no—
274C24—24no—
275C24—24no—
276C24—22no—
277C24—24no—
278C24—24no—
279C24—22no—
280C24—22no—
281C24—24no—
282C24—24no—
283C24—24no—
284C24—24no—
285C24C2×#4112yes—
286C24C2×#4112yes—
287C24C2511yescharacteristic
288C2×D4—44no—
289C2×D4—44no—
290C2×D4—44no—
291C2×D4—48no—
292C2×D4—48no—
293C2×D4—44no—
294C2×D4—48no—
295C2×D4—48no—
296C2×D4—82no—
297C2×D4—82no—
298C2×D4—84no—
299C2×D4—84no—
300C2×D4—88no—
301C2×D4—88no—
302C2×D4—88no—
303C2×D4—88no—
304C2×D4—88no—
305C2×D4—88no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—28no—
322C22×C4—28no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
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—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—88no—
362C22×C4—88no—
363C22×C4—82no—
364C22×C4—82no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4C2×#4112yes—
368C22×C4H(5,𝔽2)14yes—
369C22×C4H(5,𝔽2)14yes—
370C22×C4H(5,𝔽2)14yes—
371C4⋊C4—416no—
372C4⋊C4—48no—
373C4⋊C4—48no—
374C4⋊C4—88no—
375C4⋊C4—88no—
376C22⋊C4—48no—
377C22⋊C4—48no—
378C22⋊C4—44no—
379C22⋊C4—44no—
380C22⋊C4—48no—
381C22⋊C4—48no—
382C22⋊C4—416no—
383C22⋊C4—48no—
384C22⋊C4—48no—
385C22⋊C4—88no—
386C22⋊C4—88no—
387C22⋊C4—84no—
388C22⋊C4—84no—
389C22⋊C4—88no—
390C22⋊C4—88no—
391C22⋊C4—84no—
392C22⋊C4—84no—
393C22⋊C4—88no—
394C42—28no—
395C42—88no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—28no—
403C23—28no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—44no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23C22×#4111yescharacteristic
446C23C2×H(5,𝔽2)12yes—
447C23C2×H(5,𝔽2)12yes—
448C23C2×H(5,𝔽2)12yes—
449C23#58514yes—
450C23#58514yes—
451D4—88no—
452D4—816no—
453D4—816no—
454D4—88no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—28no—
464C2×C4—28no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
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—88no—
513C2×C4—88no—
514C22—44no—
515C22—44no—
516C22—44no—
517C22—44no—
518C22—42no—
519C22—42no—
520C22—44no—
521C22—44no—
522C22—84no—
523C22—84no—
524C22—84no—
525C22—84no—
526C22—84no—
527C22—84no—
528C22—84no—
529C22—84no—
530C22—82no—
531C22—82no—
532C22—84no—
533C22—82no—
534C22—84no—
535C22#349011yescharacteristic
536C22#349011yescharacteristic
537C22#349011yescharacteristic
538C22#352114yes—
539C22#352912yes—
540C22#353312yes—
541C22#353312yes—
542C22#353714yes—
543C22#353914yes—
544C22#354012yes—
545C22#354012yes—
546C22#355112yes—
547C22#356214yes—
548C22#356214yes—
549C4—44no—
550C4—44no—
551C4—44no—
552C4—42no—
553C4—84no—
554C4—84no—
555C4—84no—
556C4—84no—
557C4—82no—
558C4—82no—
559C2—82no—
560C2—82no—
561C2—162no—
562C2#4573012yes—
563C2#4638411yescharacteristic
564C2#5021912yes—
565C2#5199212yes—
566C2#5248814yes—
567C2#5506014yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #17273C2semidirect
2. #17232C2semidirect
3. #16928C2semidirect
4. #16533C2semidirect
5. #16114C2semidirect
6. #15949C2semidirect
7. #15452C2semidirect
10. #14756C2semidirect
12. #14658C2semidirect
13. #14167C2semidirect

Maximal subgroups

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

subgroupindex
#172732
#172322
#169282
#165332
#161142
#159492
#154522
#152612
#147562
#146612
#146582
#141672