Finite Group Databasebeta

← #4464290

Subgroups — #4464290

Order 512.

1816 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17306C212yesmaximal, complemented
2#17304C212yesmaximal, complemented
3#17235C212yesmaximal, complemented
4#17176C214yesmaximal
5#17153C214yesmaximal
6#17134C212yesmaximal, complemented
7#15231C212yesmaximal, complemented
8#15230C212yesmaximal, complemented
9#15220C212yesmaximal
10#15146C212yesmaximal, complemented
11#15137C212yesmaximal, complemented
12#15135C211yesmaximal, characteristic
13#15125C211yesmaximal, characteristic
14#14340C212yesmaximal
15#13853C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C2×#400C2214yes—
19C2×#399C2212yes—
20C2×#398C2214yes—
21C2×#393C2212yes—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#384C2212yes—
26C2×#384C2211yescharacteristic
27C2×#382C2212yes—
28C2×#382C2212yes—
29C2×#380C2212yes—
30C2×#380C2212yes—
31C2×C4×C22⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33#1446C2214yes—
34#1446C2214yes—
35#1446C2212yes—
36#1445C2211yescharacteristic
37#1445C2211yescharacteristic
38#1445C2211yescharacteristic
39#1445C2211yescharacteristic
40#1445C2212yes—
41#1444C2214yes—
42#1444C2212yes—
43#1443C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1442C2212yes—
48#1442C2214yes—
49#1442C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1440C2212yes—
60#1440C2212yes—
61#1439C2212yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2211yescharacteristic
70C4⋊#95C2212yes—
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2212yes—
73C4×#95C2212yes—
74C4×#95C2212yes—
75C23×Q8C2312yes—
76C24×C4C2311yescharacteristic
77C2×C4⋊Q8—42no—
78C2×C4⋊Q8—44no—
79C2×#117—42no—
80C2×#117—42no—
81C2×#117Q818yescomplemented
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C22⋊C4C2311yescharacteristic
97C22×C22⋊C4C2311yescharacteristic
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104C22×C42C2312yes—
105C22×C42C2312yes—
106#400—48no—
107#399—44no—
108#398—48no—
109#393—44no—
110#389—44no—
111#389—44no—
112#389—44no—
113#384—44no—
114#384—42no—
115#382—44no—
116#382—44no—
117#380—44no—
118#380—44no—
119C4×C22⋊C4—42no—
120C4×C22⋊C4—42no—
121C2×#95C2314yes—
122C2×#95C2312yes—
123C2×#95C2312yes—
124C2×#95C2312yes—
125C2×#95C2312yes—
126C2×#95C2312yes—
127C2×#95C2312yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2311yescharacteristic
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2311yescharacteristic
152C2×#95C2311yescharacteristic
153C2×#95C2311yescharacteristic
154C25C2411yescharacteristic
155C22×Q8—24no—
156C22×Q8—44no—
157C22×Q8—42no—
158C22×Q8—44no—
159C22×Q8—42no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4C2×Q814yes—
165C23×C4C2414yes—
166C23×C4C2414yes—
167C23×C4C2414yes—
168C23×C4C2412yes—
169C23×C4C2414yes—
170C23×C4C2412yes—
171C23×C4C2412yes—
172C23×C4C2412yes—
173C23×C4C2411yescharacteristic
174C23×C4C2411yescharacteristic
175C23×C4C2411yescharacteristic
176C23×C4C2411yescharacteristic
177C23×C4C2411yescharacteristic
178C23×C4C2411yescharacteristic
179C4⋊Q8—416no—
180#117—416no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4C2×Q814yes—
220C2×C4⋊C4C2×Q814yes—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4C2×Q818yes—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—22no—
242C2×C42—24no—
243C2×C42—22no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42C2×Q818yes—
249#95—28no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—42no—
257#95—42no—
258#95—44no—
259#95—44no—
260#95—44no—
261#95—42no—
262#95—42no—
263#95—44no—
264#95—41no—
265#95—41no—
266#95—41no—
267#95—41no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—42no—
278#95—44no—
279#95—42no—
280#95—44no—
281#95—44no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—22no—
287C24—21no—
288C24—21no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24C22×Q812yes—
293C24C2511yescharacteristic
294C2×Q8—48no—
295C2×Q8—84no—
296C2×Q8—88no—
297C2×Q8—88no—
298C2×Q8—84no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—28no—
332C22×C4—216no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—44no—
376C22×C4—48no—
377C22×C4—44no—
378C22×C4C22×Q814yes—
379C22×C4C22×Q814yes—
380C22×C4C22×Q812yes—
381C22×C4C22×Q812yes—
382C22×C4H(5,𝔽2)12yes—
383C22×C4H(5,𝔽2)12yes—
384C22×C4H(5,𝔽2)12yes—
385C22×C4H(5,𝔽2)12yes—
386C22×C4H(5,𝔽2)14yes—
387C22×C42−1+412yes—
388C22×C42−1+412yes—
389C22×C42−1+414yes—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—48no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—84no—
398C4⋊C4—88no—
399C4⋊C4—88no—
400C4⋊C4—88no—
401C4⋊C4—84no—
402C4⋊C4—88no—
403C4⋊C4—88no—
404C22⋊C4—48no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—48no—
408C22⋊C4—88no—
409C22⋊C4—88no—
410C42—216no—
411C42—44no—
412C42—48no—
413C42—44no—
414C23—22no—
415C23—24no—
416C23—22no—
417C23—22no—
418C23—24no—
419C23—41no—
420C23—42no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23C23×Q811yescharacteristic
438C23C2×H(5,𝔽2)11yescharacteristic
439C23C2×H(5,𝔽2)11yescharacteristic
440C23C2×H(5,𝔽2)12yes—
441C23C2×H(5,𝔽2)11yescharacteristic
442C23C2×H(5,𝔽2)11yescharacteristic
443C23C2×2−1+412yes—
444C23C2×2−1+411yescharacteristic
445C23C2×2−1+411yescharacteristic
446C23#58512yes—
447C23#58512yes—
448Q8—816no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—416no—
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—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4#54314yes—
532C2×C4#54418yes—
533C2×C4#56014yes—
534C2×C4#56218yes—
535C2×C4#56314yes—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—44no—
540C22—44no—
541C22—81no—
542C22—81no—
543C22—82no—
544C22—82no—
545C22—81no—
546C22—81no—
547C22C2×#54311yescharacteristic
548C22C2×#54412yes—
549C22#350311yescharacteristic
550C22#350311yescharacteristic
551C22#350412yes—
552C22C2×#56011yescharacteristic
553C22C2×#56212yes—
554C22C2×#56311yescharacteristic
555C22#353611yescharacteristic
556C22#353611yescharacteristic
557C22#354012yes—
558C22#354612yes—
559C22#354712yes—
560C22#354911yescharacteristic
561C22#355011yescharacteristic
562C22#355111yescharacteristic
563C22#355111yescharacteristic
564C22#355412yes—
565C22#355512yes—
566C22#355812yes—
567C22#355912yes—
568C22#356112yes—
569C22#356312yes—
570C4—42no—
571C4—42no—
572C4—44no—
573C4—42no—
574C4—44no—
575C4—42no—
576C4—42no—
577C4—84no—
578C4—82no—
579C4—82no—
580C4—84no—
581C4—84no—
582C4—84no—
583C4#284118yes—
584C2—82no—
585C2C2×#284111yescharacteristic
586C2#3730611yescharacteristic
587C2#3731312yes—
588C2#3731412yes—
589C2#3742611yescharacteristic
590C2#5296011yescharacteristic
591C2#5296111yescharacteristic
592C2#5301711yescharacteristic
593C2#5301811yescharacteristic
594C2#5307012yes—
595C2#5307112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17306C2semidirect
2. #17304C2semidirect
3. #17235C2semidirect
6. #17134C2semidirect
7. #15231C2semidirect
8. #15230C2semidirect
10. #15146C2semidirect
11. #15137C2semidirect
81. C2×#117Q8semidirect

Maximal subgroups

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

subgroupindex
#173062
#173042
#172352
#171762
#171532
#171342
#152312
#152302
#152202
#151462
#151372
#151352
#151252
#143402
#138532