Finite Group Databasebeta

← #538362

Subgroups — #538362

Order 512.

2564 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2#14613C212yesmaximal, complemented
3#14593C214yesmaximal, complemented
4#14506C212yesmaximal
5#14253C212yesmaximal, complemented
6#14167C212yesmaximal
7#14104C214yesmaximal, complemented
8#13994C212yesmaximal, complemented
9#13993C212yesmaximal, complemented
10#13982C212yesmaximal, complemented
11#13905C211yesmaximal, characteristic
12#13839C211yesmaximal, characteristic
13#13818C212yesmaximal, complemented
14C22⋊C4×C4⋊C4C212yesmaximal, complemented
15#13777C212yesmaximal, complemented
16C22×#123C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2486C4116yescomplemented
21C2×#398C2212yes—
22C2×#397C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#388C2214yes—
26C2×#382C2211yescharacteristic
27C2×#381C2212yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×#380C2212yes—
31C2×#380C2211yescharacteristic
32C2×C4×C4⋊C4C2212yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C22⋊C4C2212yes—
37C2×C4×C22⋊C4C2211yescharacteristic
38C2×C4×C22⋊C4C2214yes—
39#1445C2212yes—
40#1445C2214yes—
41#1443C2214yes—
42#1443C2214yes—
43#1442C2214yes—
44#1441C2212yes—
45#1440C2211yescharacteristic
46#1440C2211yescharacteristic
47#1440C2214yes—
48#1439C2212yes—
49#1439C2214yes—
50#1438C2211yescharacteristic
51#1438C2211yescharacteristic
52#1438C2212yes—
53#1437C2212yes—
54#1437C2214yes—
55#1437C2212yes—
56#1437C2214yes—
57#1437C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66C4⋊#95C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71#1434C2214yes—
72#1434C2214yes—
73C22⋊#95C2214yes—
74C4×#95C2212yes—
75C4×#95C2214yes—
76C23×D4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C2×#123—48no—
81C2×#123C2×C4116yes—
82C2×C4×D4—42no—
83C2×C4×D4—42no—
84C2×C4×D4C2×C418yes—
85C2×#117—28no—
86C2×#117—42no—
87C22×C4⋊C4—42no—
88C22×C4⋊C4—42no—
89C22×C4⋊C4C2×C418yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2311yescharacteristic
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2314yes—
115C22×C42C2312yes—
116C22×C42C2314yes—
117C22×C42C2314yes—
118#398C2×C4116yes—
119#397C2×C4116yes—
120#389—28no—
121#389C2×C4116yes—
122#388—48no—
123#382—42no—
124#381C2×C4116yes—
125#380—28no—
126#380—28no—
127#380C2×C4116yes—
128#380C2×C418yes—
129C4×C4⋊C4—28no—
130C4×C4⋊C4—44no—
131C4×C4⋊C4—42no—
132C4×C4⋊C4—44no—
133C4×C22⋊C4—28no—
134C4×C22⋊C4—42no—
135C4×C22⋊C4—48no—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C25C2411yescharacteristic
164C25C2411yescharacteristic
165C22×D4—24no—
166C22×D4—41no—
167C22×D4—41no—
168C22×D4—42no—
169C22×D4C22×C414yes—
170C23×C4—28no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—44no—
182C23×C4C22×C414yes—
183C23×C4C22×C414yes—
184C23×C4C22×C418yes—
185C23×C4C2412yes—
186C23×C4C2414yes—
187C23×C4C2414yes—
188C23×C4C2411yescharacteristic
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C23×C4C2414yes—
192C23×C4C2411yescharacteristic
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2411yescharacteristic
198#123—432no—
199C4×D4—416no—
200#117—416no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—416no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4C22×C418yes—
224C2×C4⋊C4C22×C418yes—
225C2×C4⋊C4C22×C418yes—
226C2×C4⋊C4C22×C414yes—
227C2×C4⋊C4C22×C4116yes—
228C2×C4⋊C4#4118yes—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4C22×C418yes—
257C2×C22⋊C4C22×C414yes—
258C2×C22⋊C4C22×C414yes—
259C2×C22⋊C4C22×C4116yes—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—28no—
268C2×C42—44no—
269C2×C42—44no—
270C2×C42—42no—
271C2×C42—44no—
272C2×C42—44no—
273C2×C42—44no—
274C2×C42—44no—
275C2×C42—44no—
276C2×C42—44no—
277C2×C42—44no—
278C2×C42C22×C414yes—
279C2×C42#4118yes—
280#95—24no—
281#95—24no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—24no—
287#95—28no—
288#95—42no—
289#95—42no—
290#95—44no—
291#95—44no—
292#95—42no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—42no—
297#95—44no—
298#95—42no—
299#95—44no—
300#95—42no—
301#95—42no—
302#95—44no—
303#95—44no—
304#95C22×C4116yes—
305#95C22×C418yes—
306#95C22×C4116yes—
307C24—22no—
308C24—22no—
309C24—21no—
310C24—21no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—21no—
318C24—21no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24C23×C412yes—
323C24C23×C412yes—
324C24H(5,𝔽2)14yes—
325C24C2511yescharacteristic
326C2×D4—28no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—48no—
330C2×D4—48no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—21no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—216no—
384C22×C4—216no—
385C22×C4—216no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—88no—
411C22×C4—88no—
412C22×C4C23×C412yes—
413C22×C4C23×C412yes—
414C22×C4C23×C418yes—
415C22×C4C23×C412yes—
416C22×C4C23×C418yes—
417C22×C4C23×C412yes—
418C22×C4C23×C412yes—
419C22×C4C2×#4114yes—
420C22×C4C2×#4118yes—
421C22×C4C2×#4114yes—
422C22×C4C2×#4114yes—
423C22×C4C2×#4114yes—
424C22×C4H(5,𝔽2)12yes—
425C22×C4H(5,𝔽2)12yes—
426C22×C42−1+414yes—
427C4⋊C4—216no—
428C4⋊C4—216no—
429C4⋊C4—216no—
430C4⋊C4—48no—
431C4⋊C4—44no—
432C4⋊C4—416no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—88no—
436C22⋊C4—216no—
437C22⋊C4—216no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—416no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—48no—
445C22⋊C4—416no—
446C22⋊C4—44no—
447C22⋊C4—88no—
448C42—28no—
449C42—216no—
450C42—48no—
451C42—88no—
452C42—84no—
453C42—88no—
454C42—88no—
455C23—24no—
456C23—22no—
457C23—22no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—41no—
471C23—42no—
472C23—41no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—41no—
478C23—41no—
479C23—42no—
480C23—41no—
481C23—42no—
482C23—41no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—41no—
492C23—41no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23#51814yes—
497C23C24×C411yescharacteristic
498C23C22×#4112yes—
499C23C22×#4112yes—
500C23C2×H(5,𝔽2)11yescharacteristic
501C23C2×H(5,𝔽2)11yescharacteristic
502C23C2×H(5,𝔽2)12yes—
503C23C2×2−1+412yes—
504C23#58512yes—
505C23#58512yes—
506D4—416no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—28no—
515C2×C4—22no—
516C2×C4—28no—
517C2×C4—22no—
518C2×C4—28no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—416no—
570C2×C4—416no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—816no—
586C2×C4C4×#4118yes—
587C2×C4#51814yes—
588C2×C4#55114yes—
589C2×C4#55318yes—
590C2×C4#55914yes—
591C22—24no—
592C22—28no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—42no—
597C22—42no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—44no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22C2×C4×#4112yes—
610C22C2×#51811yescharacteristic
611C22C2×#51811yescharacteristic
612C22#342811yescharacteristic
613C22C4×2−1+412yes—
614C22C2×#55111yescharacteristic
615C22C2×#55312yes—
616C22C2×#55911yescharacteristic
617C22#349212yes—
618C22#349412yes—
619C22#350712yes—
620C22#351212yes—
621C22#351312yes—
622C22#351612yes—
623C22#352112yes—
624C22#352212yes—
625C22#353811yescharacteristic
626C22#354011yescharacteristic
627C22#354212yes—
628C22#354212yes—
629C22#354712yes—
630C4—48no—
631C4—48no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—42no—
636C4—44no—
637C4—44no—
638C4—44no—
639C4—48no—
640C4—82no—
641C4—84no—
642C4#237418yes—
643C2—44no—
644C2—82no—
645C2C2×#237411yescharacteristic
646C2#3429812yes—
647C2#3431812yes—
648C2#3436311yescharacteristic
649C2#3441211yescharacteristic
650C2#4491612yes—
651C2#4492612yes—
652C2#4506612yes—
653C2#4507512yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14613C2semidirect
3. #14593C2semidirect
5. #14253C2semidirect
7. #14104C2semidirect
8. #13994C2semidirect
9. #13993C2semidirect
10. #13982C2semidirect
13. #13818C2semidirect
14. C22⋊C4×C4⋊C4C2semidirect
15. #13777C2semidirect
20. #2486C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
#146132
#145932
#145062
#142532
#141672
#141042
#139942
#139932
#139822
#139052
#138392
#138182
C22⋊C4×C4⋊C42
#137772