Finite Group Databasebeta

← #4929008

Subgroups — #4929008

Order 512.

5078 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#520C211yesmaximal, characteristic
2C2×#2484C211yesmaximal, characteristic, complemented
3C2×#2406C211yesmaximal, characteristic
4C22×#398C211yesmaximal, characteristic, complemented
5#14906C214yesmaximal, complemented
6#14879C214yesmaximal, complemented
7#14801C214yesmaximal, complemented
8#14786C214yesmaximal, complemented
9#14342C211yesmaximal, characteristic, complemented
10#14015C214yesmaximal, complemented
11#13968C214yesmaximal, complemented
12#13958C211yesmaximal, characteristic, complemented
13#13956C211yesmaximal, characteristic
14C23×#41C2211yescharacteristic
15C22×#123C2212yes—
16C2×#520—28no—
17C2×#520—28no—
18C2×#520—28no—
19C22×C4×Q8C2212yes—
20C22×C4×D4C2214yes—
21C22×C4×D4C2212yes—
22C22×#117C2212yes—
23C22×#117C2211yescharacteristic
24C23×C4⋊C4C2211yescharacteristic
25C23×C4⋊C4C2212yes—
26C23×C4⋊C4C2211yescharacteristic
27C23×C4⋊C4C2212yes—
28#2484—28no—
29#2406—28no—
30C2×#400C2212yes—
31C2×#398—28no—
32C2×#398C2214yes—
33C2×#398C2214yes—
34C2×#398C2212yes—
35C2×#395C2212yes—
36C2×#394C2212yes—
37C2×#387C2214yes—
38C2×#386C2214yes—
39C2×#386C2212yes—
40C2×#385C2214yes—
41C2×#385C2212yes—
42C2×#385C2218yes—
43C2×#381C2211yescharacteristic
44C2×#381C2211yescharacteristic
45C2×#381C2212yes—
46C22×#95C2211yescharacteristic
47C22×#95C2212yes—
48#1446C2212yes—
49#1445C2214yes—
50#1445C2214yes—
51#1445C2212yes—
52#1445C2212yes—
53#1444C2218yes—
54#1441C2218yes—
55#1440C2218yes—
56#1439C2218yes—
57#1438C2212yes—
58#1436C2218yes—
59#1436C2218yes—
60#1434C2214yes—
61C22⋊#95C2214yes—
62C22⋊#95C2214yes—
63C22⋊#95C2214yes—
64C22⋊#95C2218yes—
65C22×#41—216no—
66C22×#41—24no—
67C22×#41—24no—
68C22×#41D418yes—
69C23×Q8C2311yescharacteristic
70C23×D4C2311yescharacteristic
71C23×D4C2312yes—
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C24×C4C2311yescharacteristic
75C24×C4C2312yes—
76C24×C4C2311yescharacteristic
77C24×C4C2314yes—
78C24×C4C2312yes—
79C2×#123—28no—
80C2×#123—216no—
81#520—2128no—
82C2×C4×Q8—28no—
83C2×C4×Q8—28no—
84C2×C4×Q8—28no—
85C2×C4×D4—28no—
86C2×C4×D4—216no—
87C2×C4×D4—216no—
88C2×C4×D4—28no—
89C2×C4×D4—216no—
90C2×C4×D4—28no—
91C2×#117—28no—
92C2×#117—24no—
93C2×#117—24no—
94C2×#117—28no—
95C2×#117—28no—
96C2×#117D418yes—
97C22×C4⋊C4—28no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—28no—
101C22×C4⋊C4—28no—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—48no—
107C22×C4⋊C4D418yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2318yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2318yes—
124C22×C22⋊C4C2312yes—
125C22×C42C2314yes—
126C22×C42C2312yes—
127#400—28no—
128#398—216no—
129#398—216no—
130#398—28no—
131#395—28no—
132#394—28no—
133#387—216no—
134#386—28no—
135#386—216no—
136#385—216no—
137#385—28no—
138#385—416no—
139#381—24no—
140#381—24no—
141#381—28no—
142C2×#95—24no—
143C2×#95—28no—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2318yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2318yes—
154C2×#95C2318yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2318yes—
158C2×#95C2318yes—
159C2×#95C2318yes—
160C25C2412yes—
161C25C2411yescharacteristic
162C2×#41—264no—
163C2×#41—264no—
164C2×#41—264no—
165C2×#41—232no—
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8—28no—
169C22×Q8C2×D414yes—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—28no—
173C22×D4—28no—
174C22×D4—22no—
175C22×D4—22no—
176C22×D4C2×D414yes—
177C22×D4C2×D418yes—
178C22×D4#41116yes—
179C23×C4—28no—
180C23×C4—28no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—28no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—28no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—28no—
191C23×C4—28no—
192C23×C4—28no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—28no—
199C23×C4—28no—
200C23×C4—44no—
201C23×C4—42no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—48no—
205C23×C4—48no—
206C23×C4C2×D418yes—
207C23×C4C2×D414yes—
208C23×C4C2×D414yes—
209C23×C4C2×D414yes—
210C23×C4C2×D418yes—
211C23×C4C2414yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222#123—432no—
223C4×Q8—264no—
224C4×D4—2128no—
225C4×D4—264no—
226#117—264no—
227#117—232no—
228C2×C4⋊C4—264no—
229C2×C4⋊C4—232no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—216no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—216no—
251C2×C4⋊C4—216no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—48no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—432no—
256C2×C4⋊C4—416no—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2×D418yes—
259C2×C4⋊C4C2×D418yes—
260C2×C4⋊C4C2×D414yes—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—216no—
272C2×C22⋊C4—216no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—216no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—48no—
278C2×C22⋊C4C2×D418yes—
279C2×C42—28no—
280C2×C42—28no—
281C2×C42—28no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42C2×D418yes—
285#95—24no—
286#95—24no—
287#95—216no—
288#95—24no—
289#95—24no—
290#95—28no—
291#95—28no—
292#95—24no—
293#95—24no—
294#95—28no—
295#95—28no—
296#95—216no—
297#95—216no—
298#95—48no—
299#95—48no—
300#95—48no—
301C24—22no—
302C24—24no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—24no—
307C24—24no—
308C24—28no—
309C24—24no—
310C24C22×D414yes—
311C24C22×D412yes—
312C24C22×D412yes—
313C24C22×D412yes—
314C24C2×#4118yes—
315C24C2×#4118yes—
316C24C2511yescharacteristic
317#41—2256no—
318C2×Q8—216no—
319C2×Q8—216no—
320C2×Q8—28no—
321C2×Q8—216no—
322C2×D4—216no—
323C2×D4—216no—
324C2×D4—216no—
325C2×D4—216no—
326C2×D4—216no—
327C2×D4—232no—
328C2×D4—232no—
329C2×D4—28no—
330C2×D4#123132yes—
331C22×C4—232no—
332C22×C4—232no—
333C22×C4—216no—
334C22×C4—216no—
335C22×C4—216no—
336C22×C4—216no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—216no—
343C22×C4—216no—
344C22×C4—28no—
345C22×C4—216no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—28no—
357C22×C4—24no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—22no—
361C22×C4—28no—
362C22×C4—24no—
363C22×C4—28no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—232no—
368C22×C4—216no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—48no—
384C22×C4—48no—
385C22×C4—48no—
386C22×C4—48no—
387C22×C4—432no—
388C22×C4—48no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—48no—
392C22×C4—44no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—88no—
397C22×C4—88no—
398C22×C4—816no—
399C22×C4—88no—
400C22×C4—88no—
401C22×C4—816no—
402C22×C4#120116yes—
403C22×C4C22×D414yes—
404C22×C4C22×D418yes—
405C22×C4C22×D414yes—
406C22×C4C22×D412yes—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D414yes—
413C22×C4C2×#4118yes—
414C22×C4H(5,𝔽2)14yes—
415C22×C42−1+418yes—
416C4⋊C4—232no—
417C4⋊C4—28no—
418C4⋊C4—216no—
419C4⋊C4—216no—
420C4⋊C4—28no—
421C4⋊C4—416no—
422C4⋊C4—432no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—416no—
426C22⋊C4—232no—
427C22⋊C4—216no—
428C22⋊C4—416no—
429C22⋊C4—432no—
430C22⋊C4—48no—
431C22⋊C4—416no—
432C22⋊C4—48no—
433C42—232no—
434C42—216no—
435C23—22no—
436C23—22no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—216no—
443C23—216no—
444C23—24no—
445C23—24no—
446C23—28no—
447C23—28no—
448C23—28no—
449C23—28no—
450C23—216no—
451C23—216no—
452C23—42no—
453C23—42no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—48no—
458C23—48no—
459C23C2×#12014yes—
460C23C2×#12318yes—
461C23C2×#12318yes—
462C23#52518yes—
463C23#52518yes—
464C23C23×D411yescharacteristic
465C23C23×D411yescharacteristic
466C23C23×D411yescharacteristic
467C23C22×#4114yes—
468C23C2×H(5,𝔽2)12yes—
469C23C2×2−1+414yes—
470C23#58512yes—
471Q8—232no—
472D4—232no—
473D4—264no—
474C2×C4—264no—
475C2×C4—232no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—28no—
485C2×C4—28no—
486C2×C4—24no—
487C2×C4—28no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—216no—
498C2×C4—432no—
499C2×C4—416no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—48no—
511C2×C4—48no—
512C2×C4—44no—
513C2×C4—48no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—48no—
525C2×C4—48no—
526C2×C4—48no—
527C2×C4—48no—
528C2×C4—816no—
529C2×C4—832no—
530C2×C4—88no—
531C2×C4C2×#12014yes—
532C2×C4C2×#12014yes—
533C2×C4C2×#12318yes—
534C22—28no—
535C22—24no—
536C22—28no—
537C22—216no—
538C22—216no—
539C22—24no—
540C22—28no—
541C22—216no—
542C22—48no—
543C22—44no—
544C22—48no—
545C22—48no—
546C22—416no—
547C22—44no—
548C22#284918yes—
549C22C22×#12011yescharacteristic
550C22C22×#12312yes—
551C22C2×#52512yes—
552C22C2×#52512yes—
553C22#348411yescharacteristic
554C22#348511yescharacteristic
555C22#348612yes—
556C22#348812yes—
557C22#348812yes—
558C22#349912yes—
559C22#349912yes—
560C22#351518yes—
561C22#352018yes—
562C4—24no—
563C4—24no—
564C4—28no—
565C4—48no—
566C4—44no—
567C4—44no—
568C4—48no—
569C4—44no—
570C4—82no—
571C4—84no—
572C4—84no—
573C2—28no—
574C2—48no—
575C2#3535712yes—
576C2#3545312yes—
577C2#3551211yescharacteristic
578C2#3566611yescharacteristic
579C2#3599814yes—
580C2#3600114yes—
581C2C2×#284911yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2484C2semidirect
4. C22×#398C2semidirect
5. #14906C2semidirect
6. #14879C2semidirect
7. #14801C2semidirect
8. #14786C2semidirect
9. #14342C2semidirect
10. #14015C2semidirect
11. #13968C2semidirect
12. #13958C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5202
C2×#24842
C2×#24062
C22×#3982
#149062
#148792
#148012
#147862
#143422
#140152
#139682
#139582
#139562