Finite Group Databasebeta

← #4467957

Subgroups — #4467957

Order 512.

2856 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2733C211yesmaximal, characteristic, complemented
2C2×#2640C211yesmaximal, characteristic, complemented
3#17086C214yesmaximal, complemented
4#16810C212yesmaximal, complemented
5#16544C214yesmaximal, complemented
6#16509C214yesmaximal, complemented
7#16094C212yesmaximal, complemented
8#15763C212yesmaximal, complemented
9#15761C212yesmaximal, complemented
10#15694C214yesmaximal, complemented
11#15630C212yesmaximal, complemented
12#15120C212yesmaximal, complemented
13#14489C211yesmaximal, characteristic, complemented
14C22×C4⋊Q8C2211yescharacteristic
15C22×#124C2211yescharacteristic
16C22×#124C2212yes—
17C22×#124C2212yes—
18#2733—28no—
19#2640—28no—
20C2×#397C2214yes—
21C2×#397C2214yes—
22C2×#394C2212yes—
23C2×#394C2212yes—
24C2×#393C2214yes—
25C2×#393C2214yes—
26C2×#392C2214yes—
27C2×#392C2214yes—
28C2×#392C2214yes—
29C2×#392C2212yes—
30C2×#392C2214yes—
31C2×#392C2212yes—
32C2×#391C2211yescharacteristic
33C2×#391C2211yescharacteristic
34C2×#390C2211yescharacteristic
35C2×#390C2212yes—
36C2×#390C2211yescharacteristic
37C2×#390C2214yes—
38C2×#388C2214yes—
39C2×#388C2214yes—
40C2×#376C2211yescharacteristic
41#1446C2212yes—
42#1446C2212yes—
43#1444C2212yes—
44#1443C2214yes—
45#1442C2212yes—
46#1440C2214yes—
47#1440C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1438C2214yes—
54#1438C2212yes—
55#1438C2212yes—
56#1438C2214yes—
57#1438C2214yes—
58#1438C2214yes—
59#1438C2214yes—
60#1438C2214yes—
61#1437C2214yes—
62#1437C2212yes—
63#1437C2212yes—
64#1436C2212yes—
65#1436C2212yes—
66#1436C2214yes—
67#1434C2212yes—
68#1434C2214yes—
69C23×Q8C2312yes—
70C23×Q8C2311yescharacteristic
71C23×D4C2311yescharacteristic
72C23×D4C2312yes—
73C23×D4C2312yes—
74C23×D4C2312yes—
75C23×D4C2314yes—
76C23×D4C2314yes—
77C2×C4⋊Q8—24no—
78C2×C4⋊Q8—42no—
79C2×C4⋊Q8—42no—
80C2×#124—24no—
81C2×#124—28no—
82C2×#124—42no—
83C2×#124—42no—
84C2×#124—48no—
85C2×#124—48no—
86C2×#124—44no—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113#397—216no—
114#397—48no—
115#394—28no—
116#394—44no—
117#393—216no—
118#393—48no—
119#392—216no—
120#392—48no—
121#392—48no—
122#392—48no—
123#392—44no—
124#392—44no—
125#391—24no—
126#391D418yescomplemented
127#390—24no—
128#390—24no—
129#390—28no—
130#390—48no—
131#388—216no—
132#388—48no—
133#376—24no—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C25C2412yes—
154C25C2414yes—
155C22×Q8—24no—
156C22×Q8—24no—
157C22×Q8—24no—
158C22×Q8—41no—
159C22×Q8—41no—
160C22×Q8—44no—
161C22×Q8—42no—
162C22×Q8—42no—
163C22×Q8—44no—
164C22×Q8C2×D414yes—
165C22×D4—22no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—28no—
171C22×D4—41no—
172C22×D4—41no—
173C22×D4—44no—
174C22×D4—44no—
175C22×D4—42no—
176C22×D4—42no—
177C22×D4—44no—
178C22×D4—44no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4—42no—
182C22×D4—42no—
183C22×D4—44no—
184C22×D4—44no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—42no—
193C22×D4—42no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4—42no—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2414yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2414yes—
209C23×C4C2411yescharacteristic
210C4⋊Q8—416no—
211#124—416no—
212#124—432no—
213#124—432no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4C2×D414yes—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—28no—
282C2×C42—24no—
283C2×C42—42no—
284C2×C42—44no—
285C2×C42—42no—
286C2×C42C2×D414yes—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—42no—
296#95—44no—
297#95—42no—
298#95—44no—
299#95—44no—
300#95—42no—
301#95—42no—
302#95—44no—
303#95—42no—
304#95—44no—
305#95C2×D4116yes—
306C24—22no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—24no—
311C24—24no—
312C24—22no—
313C24—22no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—24no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24C2511yescharacteristic
331C2×Q8—28no—
332C2×Q8—44no—
333C2×Q8—44no—
334C2×Q8—48no—
335C2×Q8—48no—
336C2×Q8—84no—
337C2×Q8—84no—
338C2×Q8—88no—
339C2×Q8—84no—
340C2×Q8—88no—
341C2×D4—44no—
342C2×D4—48no—
343C2×D4—44no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—416no—
347C2×D4—44no—
348C2×D4—84no—
349C2×D4—84no—
350C2×D4—84no—
351C2×D4—84no—
352C2×D4—84no—
353C2×D4—84no—
354C2×D4—84no—
355C2×D4—88no—
356C2×D4—88no—
357C2×D4—88no—
358C2×D4—88no—
359C2×D4—84no—
360C2×D4—88no—
361C2×D4—88no—
362C2×D4—88no—
363C2×D4—88no—
364C2×D4—88no—
365C2×D4—88no—
366C2×D4—88no—
367C2×D4—88no—
368C2×D4—88no—
369C2×D4—88no—
370C2×D4—88no—
371C2×D4—88no—
372C2×D4—88no—
373C2×D4—88no—
374C2×D4—88no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4C22×D414yes—
427C22×C4C22×D414yes—
428C22×C4C22×D414yes—
429C22×C4C22×D412yes—
430C22×C4H(5,𝔽2)18yes—
431C22×C4H(5,𝔽2)12yes—
432C22×C4H(5,𝔽2)12yes—
433C22×C4H(5,𝔽2)14yes—
434C4⋊C4—44no—
435C4⋊C4—48no—
436C4⋊C4—44no—
437C4⋊C4—84no—
438C4⋊C4—88no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—416no—
442C22⋊C4—416no—
443C22⋊C4—416no—
444C22⋊C4—416no—
445C22⋊C4—416no—
446C22⋊C4—416no—
447C22⋊C4—88no—
448C22⋊C4—88no—
449C22⋊C4—88no—
450C22⋊C4—88no—
451C22⋊C4—88no—
452C22⋊C4—88no—
453C22⋊C4—88no—
454C22⋊C4—88no—
455C42—28no—
456C42—44no—
457C42—48no—
458C42—48no—
459C42—44no—
460C23—44no—
461C23—42no—
462C23—42no—
463C23—44no—
464C23—42no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—42no—
498C23—44no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—42no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23C23×D411yescharacteristic
517C23C2×H(5,𝔽2)12yes—
518C23C2×H(5,𝔽2)11yescharacteristic
519C23C2×H(5,𝔽2)11yescharacteristic
520C23C2×H(5,𝔽2)14yes—
521C23#58512yes—
522C23#58514yes—
523Q8—88no—
524Q8—816no—
525D4—88no—
526D4—816no—
527D4—816no—
528D4—816no—
529D4—832no—
530D4—832no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4#53514yes—
590C22—84no—
591C22—84no—
592C22—84no—
593C22—84no—
594C22—84no—
595C22—84no—
596C22—82no—
597C22—84no—
598C22—84no—
599C22—84no—
600C22—84no—
601C22—82no—
602C22—84no—
603C22—84no—
604C22—84no—
605C22—84no—
606C22—84no—
607C22—82no—
608C22—84no—
609C22—84no—
610C22—84no—
611C22—84no—
612C22—84no—
613C22—84no—
614C22C2×#53511yescharacteristic
615C22#348411yescharacteristic
616C22#348511yescharacteristic
617C22#348712yes—
618C22#349812yes—
619C22#353212yes—
620C22#354014yes—
621C22#354214yes—
622C22#354512yes—
623C22#354614yes—
624C22#354912yes—
625C22#355112yes—
626C22#355812yes—
627C22#355812yes—
628C22#356014yes—
629C4—24no—
630C4—44no—
631C4—44no—
632C4—42no—
633C4—44no—
634C4—42no—
635C4—48no—
636C4—84no—
637C4—84no—
638C4—82no—
639C4—82no—
640C2—162no—
641C2—164no—
642C2#3696711yescharacteristic
643C2#4287612yes—
644C2#4721212yes—
645C2#5190211yescharacteristic
646C2#5208311yescharacteristic
647C2#5552014yes—
648C2#5888414yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
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×#2733C2semidirect
2. C2×#2640C2semidirect
3. #17086C2semidirect
4. #16810C2semidirect
5. #16544C2semidirect
6. #16509C2semidirect
7. #16094C2semidirect
8. #15763C2semidirect
9. #15761C2semidirect
10. #15694C2semidirect
11. #15630C2semidirect
12. #15120C2semidirect
13. #14489C2semidirect
126. #391D4semidirect

Maximal subgroups

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

subgroupindex
C2×#27332
C2×#26402
#170862
#168102
#165442
#165092
#160942
#157632
#157612
#156942
#156302
#151202
#144892