Finite Group Databasebeta

← #4463918

Subgroups — #4463918

Order 512.

2058 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17296C214yesmaximal, complemented
2#17287C214yesmaximal, complemented
3#17283C214yesmaximal, complemented
4#17232C214yesmaximal
5#15236C212yesmaximal, complemented
6#15235C211yesmaximal, characteristic, complemented
7#15183C212yesmaximal, complemented
8#15180C211yesmaximal, characteristic, complemented
9#15179C212yesmaximal, complemented
10#15178C212yesmaximal, complemented
11#15177C212yesmaximal
12#14261C212yesmaximal, complemented
13#13847C211yesmaximal, characteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C22×C22⋊Q8C2212yes—
17C22×C4×D4C2211yescharacteristic
18C2×#391C2212yes—
19C2×#391C2212yes—
20C2×#391C2212yes—
21C2×#391C2212yes—
22C2×#389C2212yes—
23C2×#382C2211yescharacteristic
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C22⋊C4C2212yes—
28C2×C4×C22⋊C4C2212yes—
29C2×#376C2211yescharacteristic
30#1445C2212yes—
31#1445C2214yes—
32#1444C2214yes—
33#1444C2214yes—
34#1444C2214yes—
35#1443C2214yes—
36#1443C2214yes—
37#1443C2214yes—
38#1443C2214yes—
39#1442C2214yes—
40#1442C2214yes—
41#1442C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1440C2214yes—
49#1440C2214yes—
50#1440C2214yes—
51#1440C2214yes—
52#1439C2212yes—
53#1439C2212yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1438C2212yes—
61#1438C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2211yescharacteristic
66C4⋊#95C2212yes—
67C23×Q8C2312yes—
68C23×D4C2311yescharacteristic
69C24×C4C2312yes—
70C2×C4⋊Q8—44no—
71C2×C4⋊Q8—42no—
72C2×C22⋊Q8—44no—
73C2×C22⋊Q8—44no—
74C2×C22⋊Q8—44no—
75C2×C22⋊Q8—44no—
76C2×C22⋊Q8—44no—
77C2×C22⋊Q8—44no—
78C2×C4×D4—24no—
79C2×C4×D4—42no—
80C2×C4×D4—42no—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C22⋊C4C2312yes—
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C42C2311yescharacteristic
100C22×C42C2312yes—
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103#391—28no—
104#391—44no—
105#391—44no—
106#391—44no—
107#389—44no—
108#382—42no—
109#380—44no—
110#380—44no—
111C4×C4⋊C4—42no—
112C4×C22⋊C4—44no—
113C4×C22⋊C4—44no—
114#376—42no—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2312yes—
126C2×#95C2312yes—
127C2×#95C2314yes—
128C2×#95C2311yescharacteristic
129C2×#95C2311yescharacteristic
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C25C2412yes—
144C22×Q8—24no—
145C22×Q8—24no—
146C22×Q8—24no—
147C22×Q8—24no—
148C22×Q8—24no—
149C22×Q8—24no—
150C22×Q8—24no—
151C22×D4—24no—
152C22×D4—22no—
153C22×D4—41no—
154C22×D4—41no—
155C22×D4—42no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4C2414yes—
164C23×C4C2414yes—
165C23×C4C2414yes—
166C23×C4C2414yes—
167C23×C4C2412yes—
168C23×C4C2412yes—
169C23×C4C2412yes—
170C23×C4C2412yes—
171C23×C4C2411yescharacteristic
172C23×C4C2411yescharacteristic
173C23×C4C2412yes—
174C23×C4C2411yescharacteristic
175C4⋊Q8—416no—
176C22⋊Q8—432no—
177C22⋊Q8—432no—
178C4×D4—416no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
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—42no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
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×C4⋊C4—44no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
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—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—22no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—22no—
241C2×C42—24no—
242C2×C42#4114yes—
243#95—28no—
244#95—28no—
245#95—28no—
246#95—44no—
247#95—44no—
248#95—44no—
249#95—44no—
250#95—44no—
251#95—44no—
252#95—44no—
253#95—41no—
254#95—41no—
255#95—42no—
256#95—42no—
257#95—42no—
258#95—42no—
259#95—42no—
260#95—42no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C242−1+414yes—
286C24C2511yescharacteristic
287C2×Q8—48no—
288C2×Q8—48no—
289C2×Q8—48no—
290C2×Q8—48no—
291C2×Q8—48no—
292C2×Q8—48no—
293C2×Q8—48no—
294C2×D4—48no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—48no—
298C2×D4—44no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
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—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
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—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4C2×#4112yes—
377C22×C4C2×#4112yes—
378C22×C4C2×#4112yes—
379C22×C4H(5,𝔽2)12yes—
380C22×C4H(5,𝔽2)12yes—
381C22×C4H(5,𝔽2)14yes—
382C22×C4H(5,𝔽2)14yes—
383C22×C42−1+414yes—
384C4⋊C4—48no—
385C4⋊C4—44no—
386C4⋊C4—416no—
387C4⋊C4—416no—
388C4⋊C4—416no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—48no—
392C4⋊C4—88no—
393C22⋊C4—48no—
394C22⋊C4—48no—
395C22⋊C4—48no—
396C22⋊C4—48no—
397C22⋊C4—48no—
398C22⋊C4—416no—
399C22⋊C4—416no—
400C22⋊C4—88no—
401C22⋊C4—88no—
402C42—28no—
403C42—44no—
404C42—48no—
405C42—44no—
406C23—24no—
407C23—24no—
408C23—24no—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—24no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23C22×#4111yescharacteristic
442C23C2×H(5,𝔽2)11yescharacteristic
443C23C2×H(5,𝔽2)11yescharacteristic
444C23C2×H(5,𝔽2)12yes—
445C23C2×H(5,𝔽2)12yes—
446C23C2×2−1+412yes—
447C23C2×2−1+412yes—
448C23#58512yes—
449C23#58512yes—
450Q8—816no—
451D4—416no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—816no—
526C2×C4#53814yes—
527C2×C4#54018yes—
528C2×C4#56014yes—
529C2×C4#56018yes—
530C2×C4H(3,𝔽4)14yes—
531C22—44no—
532C22—44no—
533C22—44no—
534C22—44no—
535C22—44no—
536C22—44no—
537C22—44no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22C2×#53811yescharacteristic
547C22C2×#54012yes—
548C22#349512yes—
549C22#350911yescharacteristic
550C22#351211yescharacteristic
551C22C2×#56012yes—
552C22C2×#56011yescharacteristic
553C22C2×H(3,𝔽4)11yescharacteristic
554C22#353811yescharacteristic
555C22#353812yes—
556C22#353911yescharacteristic
557C22#354012yes—
558C22#354112yes—
559C22#354112yes—
560C22#354112yes—
561C22#354112yes—
562C22#354712yes—
563C22#355212yes—
564C22#355412yes—
565C22#356112yes—
566C22#356212yes—
567C4—42no—
568C4—42no—
569C4—44no—
570C4—82no—
571C4—84no—
572C4—82no—
573C4—84no—
574C4—82no—
575C4—84no—
576C4—82no—
577C4—84no—
578C4#282818yes—
579C2—84no—
580C2C2×#282811yescharacteristic
581C2#3738011yescharacteristic
582C2#3739311yescharacteristic
583C2#3739912yes—
584C2#3741712yes—
585C2#5301112yes—
586C2#5305112yes—
587C2#5313712yes—
588C2#5314412yes—
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. #17296C2semidirect
2. #17287C2semidirect
3. #17283C2semidirect
5. #15236C2semidirect
6. #15235C2semidirect
7. #15183C2semidirect
8. #15180C2semidirect
9. #15179C2semidirect
10. #15178C2semidirect
12. #14261C2semidirect

Maximal subgroups

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

subgroupindex
#172962
#172872
#172832
#172322
#152362
#152352
#151832
#151802
#151792
#151782
#151772
#142612
#138472