Finite Group Databasebeta

← #2037165

Subgroups — #2037165

Order 512.

2662 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2724C211yesmaximal, characteristic, complemented
2C2×#2850C211yesmaximal, characteristic
3#17078C212yesmaximal, complemented
4#16433C214yesmaximal, complemented
5#15950C212yesmaximal
6#15945C212yesmaximal, complemented
7#15687C214yesmaximal, complemented
8#15430C212yesmaximal, complemented
9#15327C214yesmaximal, complemented
10#15262C212yesmaximal, complemented
11#14697C212yesmaximal, complemented
12#14445C211yesmaximal, characteristic, complemented
13#14405C214yesmaximal, complemented
14C22×#124C2211yescharacteristic
15C22×#123C2212yes—
16C22×C22⋊Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C23×C42C2211yescharacteristic
19#2850—28no—
20#2724—28no—
21C2×#399C2212yes—
22C2×#398C2212yes—
23C2×#397C2214yes—
24C2×#395C2214yes—
25C2×#391C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#388C2214yes—
30C2×#388C2214yes—
31C2×#386C2214yes—
32C2×#386C2212yes—
33C2×#385C2214yes—
34C2×#385C2212yes—
35C2×#385C2214yes—
36C2×#385C2214yes—
37C2×#384C2212yes—
38C2×#382C2211yescharacteristic
39C2×#380C2212yes—
40C2×#380C2212yes—
41#1445C2212yes—
42#1445C2214yes—
43#1444C2214yes—
44#1444C2214yes—
45#1443C2214yes—
46#1443C2214yes—
47#1442C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1440C2212yes—
52#1440C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1438C2212yes—
59#1438C2214yes—
60#1438C2212yes—
61#1437C2212yes—
62#1436C2214yes—
63#1436C2212yes—
64#1436C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67#1434C2214yes—
68C22⋊#95C2214yes—
69C23×Q8C2311yescharacteristic
70C23×D4C2312yes—
71C23×D4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C24×C4C2312yes—
75C2×#124—24no—
76C2×#124—44no—
77C2×#123—28no—
78C2×#123—48no—
79C2×C22⋊Q8—24no—
80C2×C22⋊Q8—42no—
81C2×C22⋊Q8—42no—
82C2×#121—24no—
83C2×#121—42no—
84C2×#121—42no—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C42—24no—
109C22×C42—44no—
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114#399—44no—
115#398—28no—
116#397—216no—
117#395—48no—
118#391—24no—
119#390D418yes—
120#389—28no—
121#389—44no—
122#388—216no—
123#388—216no—
124#386—216no—
125#386—44no—
126#385—216no—
127#385—48no—
128#385—48no—
129#385—44no—
130#384—44no—
131#382—24no—
132#380—28no—
133#380—44no—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C25C2412yes—
155C25C2411yescharacteristic
156C22×Q8—22no—
157C22×Q8—24no—
158C22×Q8—41no—
159C22×Q8—41no—
160C22×Q8—42no—
161C22×D4—24no—
162C22×D4—24no—
163C22×D4—28no—
164C22×D4—24no—
165C22×D4—41no—
166C22×D4—41no—
167C22×D4—42no—
168C22×D4—42no—
169C22×D4—44no—
170C22×D4C2×D414yes—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4C2412yes—
189C23×C4C2414yes—
190C23×C4C2414yes—
191C23×C4C2414yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2414yes—
195C23×C4C2411yescharacteristic
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199#124—416no—
200#123—432no—
201C22⋊Q8—416no—
202#121—416no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4C2×D414yes—
238C2×C4⋊C4#4118yes—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4C2×D4116yes—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—28no—
273C2×C42—24no—
274C2×C42—416no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42C2×D414yes—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—24no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—42no—
299C24—21no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—21no—
304C24—22no—
305C24—22no—
306C24—24no—
307C24—24no—
308C24—22no—
309C24—24no—
310C24—22no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24C22×D414yes—
315C24H(5,𝔽2)12yes—
316C242−1+414yes—
317C24C2511yescharacteristic
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—44no—
321C2×Q8—84no—
322C2×Q8—84no—
323C2×D4—28no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—48no—
327C2×D4—416no—
328C2×D4—48no—
329C2×D4—84no—
330C2×D4—84no—
331C2×D4—84no—
332C2×D4—88no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—28no—
341C22×C4—216no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—44no—
400C22×C4—48no—
401C22×C4—44no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—84no—
405C22×C4—84no—
406C22×C4—88no—
407C22×C4—88no—
408C22×C4C22×D414yes—
409C22×C4C22×D412yes—
410C22×C4C22×D414yes—
411C22×C4C2×#4118yes—
412C22×C4C2×#4114yes—
413C22×C4H(5,𝔽2)14yes—
414C22×C4H(5,𝔽2)12yes—
415C22×C4H(5,𝔽2)14yes—
416C22×C42−1+414yes—
417C4⋊C4—44no—
418C4⋊C4—48no—
419C4⋊C4—48no—
420C4⋊C4—44no—
421C4⋊C4—416no—
422C4⋊C4—48no—
423C4⋊C4—84no—
424C4⋊C4—84no—
425C4⋊C4—84no—
426C4⋊C4—88no—
427C4⋊C4—88no—
428C4⋊C4—88no—
429C22⋊C4—232no—
430C22⋊C4—48no—
431C22⋊C4—416no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—416no—
435C22⋊C4—416no—
436C22⋊C4—48no—
437C22⋊C4—416no—
438C22⋊C4—88no—
439C22⋊C4—84no—
440C42—28no—
441C42—44no—
442C42—48no—
443C42—48no—
444C23—22no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—22no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—28no—
456C23—24no—
457C23—24no—
458C23—28no—
459C23—41no—
460C23—42no—
461C23—42no—
462C23—41no—
463C23—41no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—41no—
469C23—44no—
470C23—42no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—42no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—42no—
481C23—44no—
482C23—42no—
483C23—44no—
484C23—44no—
485C23#56314yes—
486C23C23×D411yescharacteristic
487C23C22×#4112yes—
488C23C2×H(5,𝔽2)12yes—
489C23C2×H(5,𝔽2)12yes—
490C23C2×H(5,𝔽2)11yescharacteristic
491C23C2×H(5,𝔽2)11yescharacteristic
492C23C2×2−1+412yes—
493C23C2×2−1+412yes—
494C23#58512yes—
495Q8—88no—
496D4—88no—
497D4—816no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—28no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—416no—
519C2×C4—432no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—816no—
566C2×C4#53514yes—
567C2×C4#54718yes—
568C22—28no—
569C22—24no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—42no—
574C22—44no—
575C22—42no—
576C22—44no—
577C22—48no—
578C22—44no—
579C22—44no—
580C22—48no—
581C22—44no—
582C22—84no—
583C22—84no—
584C22—84no—
585C22—84no—
586C22C2×#53511yescharacteristic
587C22#345912yes—
588C22C2×#54712yes—
589C22#348111yescharacteristic
590C22#348411yescharacteristic
591C22#348411yescharacteristic
592C22#348912yes—
593C22#349212yes—
594C22#351712yes—
595C22#352012yes—
596C22#352212yes—
597C22C2×#56311yescharacteristic
598C22#353512yes—
599C22#353712yes—
600C22#353712yes—
601C22#353812yes—
602C22#354511yescharacteristic
603C22#354712yes—
604C22#355011yescharacteristic
605C22#355212yes—
606C22#356112yes—
607C4—24no—
608C4—28no—
609C4—48no—
610C4—42no—
611C4—44no—
612C4—44no—
613C4—84no—
614C4—82no—
615C4—84no—
616C4—84no—
617C4—82no—
618C2—44no—
619C2—84no—
620C2#3671711yescharacteristic
621C2#3721912yes—
622C2#3776012yes—
623C2#3876112yes—
624C2#3892012yes—
625C2#4609712yes—
626C2#4613611yescharacteristic
627C2#5319212yes—
628C2#5907011yescharacteristic
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×#2724C2semidirect
3. #17078C2semidirect
4. #16433C2semidirect
6. #15945C2semidirect
7. #15687C2semidirect
8. #15430C2semidirect
9. #15327C2semidirect
10. #15262C2semidirect
11. #14697C2semidirect
12. #14445C2semidirect
13. #14405C2semidirect

Maximal subgroups

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

subgroupindex
C2×#27242
C2×#28502
#170782
#164332
#159502
#159452
#156872
#154302
#153272
#152622
#146972
#144452
#144052