Finite Group Databasebeta

← #2923264

Subgroups — #2923264

Order 512.

3294 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2683C211yesmaximal, characteristic, complemented
2C2×#2680C214yesmaximal, complemented
3C2×#2501C211yesmaximal, characteristic
4#16522C212yesmaximal, complemented
5#16469C214yesmaximal, complemented
6#15612C214yesmaximal, complemented
7#15430C214yesmaximal, complemented
8#15342C212yesmaximal, complemented
9#15260C212yesmaximal
10#15145C212yesmaximal, complemented
11#14720C212yesmaximal, complemented
12#14229C212yesmaximal, complemented
13#14157C211yesmaximal, characteristic, complemented
14C22×#123C2212yes—
15C22×#123C2214yes—
16C22×C22⋊Q8C2214yes—
17C22×C22⋊Q8C2212yes—
18C22×#121C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2683—28no—
21#2680—232no—
22#2501—28no—
23C2×#399C2214yes—
24C2×#398C2214yes—
25C2×#398C2214yes—
26C2×#398C2214yes—
27C2×#397C2214yes—
28C2×#397C2212yes—
29C2×#396C2214yes—
30C2×#395C2214yes—
31C2×#395C2212yes—
32C2×#395C2212yes—
33C2×#394C2214yes—
34C2×#390C2211yescharacteristic
35C2×#390C2211yescharacteristic
36C2×#389C2212yes—
37C2×#388C2214yes—
38C2×#388C2214yes—
39C2×#386C2212yes—
40C2×#385C2214yes—
41C2×#385C2214yes—
42C2×#385C2212yes—
43C2×#384C2211yescharacteristic
44C2×#384C2214yes—
45C2×#384C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47#1445C2212yes—
48#1444C2214yes—
49#1444C2214yes—
50#1444C2214yes—
51#1444C2214yes—
52#1442C2212yes—
53#1442C2214yes—
54#1441C2212yes—
55#1441C2212yes—
56#1440C2212yes—
57#1440C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2212yes—
61#1438C2212yes—
62#1438C2212yes—
63#1437C2214yes—
64#1436C2214yes—
65C4⋊#95C2214yes—
66#1434C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2212yes—
69C4×#95C2212yes—
70C23×Q8C2312yes—
71C23×D4C2312yes—
72C23×D4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2312yes—
75C24×C4C2312yes—
76C2×#123—216no—
77C2×#123—216no—
78C2×#123—216no—
79C2×#123—48no—
80C2×#123—44no—
81C2×C22⋊Q8—216no—
82C2×C22⋊Q8—216no—
83C2×C22⋊Q8—216no—
84C2×C22⋊Q8—44no—
85C2×C22⋊Q8—48no—
86C2×#121—24no—
87C2×#121—42no—
88C2×#121—42no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—42no—
91C22×C4⋊C4—42no—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2314yes—
120C22×C42C2311yescharacteristic
121#399—216no—
122#398—216no—
123#398—216no—
124#398—48no—
125#397—216no—
126#397D4116yescomplemented
127#396—216no—
128#395—28no—
129#395—216no—
130#395—44no—
131#394—216no—
132#390—24no—
133#390D418yes—
134#389—28no—
135#388—216no—
136#388—216no—
137#386—28no—
138#385—216no—
139#385—216no—
140#385—44no—
141#384—216no—
142#384—24no—
143#384—24no—
144C4×C22⋊C4—24no—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C25C2412yes—
164C25C2411yescharacteristic
165C22×Q8—24no—
166C22×Q8—28no—
167C22×Q8—42no—
168C22×Q8—44no—
169C22×Q8—42no—
170C22×D4—24no—
171C22×D4—28no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—41no—
175C22×D4—41no—
176C22×D4—42no—
177C22×D4—44no—
178C22×D4—42no—
179C22×D4C2×D414yes—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—28no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—28no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—44no—
192C23×C4—42no—
193C23×C4—44no—
194C23×C4—42no—
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2414yes—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206#123—432no—
207#123—464no—
208C22⋊Q8—432no—
209C22⋊Q8—464no—
210#121—416no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—416no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4C2×D418yes—
254C2×C4⋊C4C2×D414yes—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4C2×D4116yes—
283C2×C22⋊C4C2×D4116yes—
284C2×C22⋊C4C2×D418yes—
285C2×C42—22no—
286C2×C42—28no—
287C2×C42—28no—
288C2×C42—24no—
289C2×C42—42no—
290C2×C42—44no—
291C2×C42C2×D414yes—
292#95—24no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—24no—
300#95—28no—
301#95—24no—
302#95—28no—
303#95—28no—
304#95—44no—
305#95—42no—
306#95—44no—
307#95—44no—
308#95C2×D4116yes—
309#95C2×D418yes—
310C24—21no—
311C24—22no—
312C24—21no—
313C24—24no—
314C24—22no—
315C24—22no—
316C24—24no—
317C24—24no—
318C24—22no—
319C24—22no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24C22×D414yes—
325C24C22×D414yes—
326C24H(5,𝔽2)12yes—
327C24C2511yescharacteristic
328C2×Q8—48no—
329C2×Q8—416no—
330C2×Q8—84no—
331C2×Q8—88no—
332C2×Q8—84no—
333C2×D4—28no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—48no—
337C2×D4—416no—
338C2×D4—48no—
339C2×D4—84no—
340C2×D4—84no—
341C2×D4—88no—
342C2×D4—84no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—28no—
349C22×C4—24no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—21no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—44no—
400C22×C4—48no—
401C22×C4—416no—
402C22×C4—416no—
403C22×C4—44no—
404C22×C4—48no—
405C22×C4—88no—
406C22×C4—84no—
407C22×C4—88no—
408C22×C4—84no—
409C22×C4—84no—
410C22×C4C22×D414yes—
411C22×C4C22×D418yes—
412C22×C4C22×D414yes—
413C22×C4C22×D418yes—
414C22×C4C22×D412yes—
415C22×C4C22×D414yes—
416C22×C4C22×D414yes—
417C22×C4H(5,𝔽2)12yes—
418C22×C4H(5,𝔽2)14yes—
419C22×C42−1+418yes—
420C22×C42−1+414yes—
421C4⋊C4—216no—
422C4⋊C4—44no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—416no—
426C4⋊C4—416no—
427C4⋊C4—416no—
428C4⋊C4—416no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—416no—
432C4⋊C4—416no—
433C4⋊C4—84no—
434C4⋊C4—88no—
435C4⋊C4—88no—
436C22⋊C4—232no—
437C22⋊C4—232no—
438C22⋊C4—216no—
439C22⋊C4—416no—
440C22⋊C4—48no—
441C22⋊C4—48no—
442C22⋊C4—416no—
443C22⋊C4—48no—
444C22⋊C4—416no—
445C22⋊C4—416no—
446C22⋊C4—84no—
447C42—44no—
448C42—416no—
449C42—44no—
450C23—24no—
451C23—22no—
452C23—28no—
453C23—24no—
454C23—28no—
455C23—24no—
456C23—22no—
457C23—24no—
458C23—24no—
459C23—28no—
460C23—24no—
461C23—24no—
462C23—28no—
463C23—41no—
464C23—41no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—44no—
469C23—44no—
470C23—42no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23D4214yes—
487C23C23×D411yescharacteristic
488C23C23×D412yes—
489C23C2×H(5,𝔽2)11yescharacteristic
490C23C2×H(5,𝔽2)12yes—
491C23C2×H(5,𝔽2)11yescharacteristic
492C23C2×2−1+412yes—
493C23C2×2−1+414yes—
494C23#58512yes—
495Q8—816no—
496D4—88no—
497D4—816no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—28no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—28no—
506C2×C4—24no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—416no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—41no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—816no—
561C2×C4—816no—
562C2×C4#53614yes—
563C2×C4D4218yes—
564C22—28no—
565C22—24no—
566C22—42no—
567C22—42no—
568C22—44no—
569C22—44no—
570C22—48no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—48no—
575C22—48no—
576C22—44no—
577C22—84no—
578C22—84no—
579C22—84no—
580C22—84no—
581C22C2×#53611yescharacteristic
582C22C2×D4212yes—
583C22C2×D4211yescharacteristic
584C22#348411yescharacteristic
585C22#348511yescharacteristic
586C22#348914yes—
587C22#349812yes—
588C22#349912yes—
589C22#349914yes—
590C22#353512yes—
591C22#353512yes—
592C22#353812yes—
593C22#354311yescharacteristic
594C22#354412yes—
595C22#355214yes—
596C22#355914yes—
597C4—24no—
598C4—28no—
599C4—42no—
600C4—44no—
601C4—48no—
602C4—48no—
603C4—44no—
604C4—48no—
605C4—84no—
606C4—82no—
607C4—82no—
608C2—44no—
609C2—84no—
610C2#3672011yescharacteristic
611C2#3936012yes—
612C2#4444712yes—
613C2#4475812yes—
614C2#5065114yes—
615C2#5154211yescharacteristic
616C2#5348712yes—
617C2#5912611yescharacteristic
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. C2×#2683C2semidirect
2. C2×#2680C2semidirect
4. #16522C2semidirect
5. #16469C2semidirect
6. #15612C2semidirect
7. #15430C2semidirect
8. #15342C2semidirect
10. #15145C2semidirect
11. #14720C2semidirect
12. #14229C2semidirect
13. #14157C2semidirect
126. #397D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26832
C2×#26802
C2×#25012
#165222
#164692
#156122
#154302
#153422
#152602
#151452
#147202
#142292
#141572