Finite Group Databasebeta

← #4457122

Subgroups — #4457122

Order 512.

2656 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2501C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic, complemented
3#16532C214yesmaximal, complemented
4#16531C214yesmaximal, complemented
5#16360C214yesmaximal, complemented
6#16359C214yesmaximal, complemented
7#15148C212yesmaximal, complemented
8#15146C212yesmaximal, complemented
9#15138C212yesmaximal, complemented
10#15133C212yesmaximal, complemented
11#14257C211yesmaximal, characteristic
12#14136C211yesmaximal, characteristic
13#14129C211yesmaximal, characteristic, complemented
14#13924C211yesmaximal, characteristic, complemented
15#13844C211yesmaximal, characteristic, complemented
16C22×#123C2214yes—
17C22×#121C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21#2501—28no—
22C2×#400C2214yes—
23C2×#399C2214yes—
24C2×#399C2214yes—
25C2×#398C2214yes—
26C2×#397C2214yes—
27#2316—28no—
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2214yes—
35C2×#389C2212yes—
36C2×#384C2211yescharacteristic
37C2×#384C2211yescharacteristic
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2211yescharacteristic
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C22⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2211yescharacteristic
44#1445C2212yes—
45#1445C2212yes—
46#1444C2212yes—
47#1442C2212yes—
48#1442C2214yes—
49#1442C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1440C2214yes—
55#1440C2214yes—
56#1439C2212yes—
57#1439C2212yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1438C2214yes—
65#1438C2212yes—
66#1438C2212yes—
67#1437C2212yes—
68#1437C2212yes—
69#1436C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C22⋊#95C2214yes—
76C4×#95C2212yes—
77C23×D4C2311yescharacteristic
78C23×D4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C2×#123—216no—
82C2×#123—48no—
83C2×#123—48no—
84C2×#121—24no—
85C2×#121—24no—
86C2×#121—24no—
87C2×#117—24no—
88C2×#117—24no—
89C2×#117—44no—
90C2×#117—44no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4D418yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126#400—48no—
127#399—216no—
128#399—48no—
129#398—48no—
130#397—48no—
131#390—24no—
132#390—24no—
133#390—44no—
134#390—44no—
135#389—28no—
136#389—28no—
137#389—216no—
138#389—44no—
139#384—24no—
140#384—24no—
141#380—28no—
142#380—28no—
143#380—24no—
144C4×C4⋊C4—44no—
145C4×C22⋊C4—24no—
146C4×C22⋊C4—24no—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2311yescharacteristic
154C2×#95C2311yescharacteristic
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C25C2412yes—
171C25C2411yescharacteristic
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4—22no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—44no—
181C22×D4—44no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4C2×D414yes—
189C23×C4C2×D418yes—
190C23×C4#4114yes—
191C23×C4#4114yes—
192C23×C4C2412yes—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205#123—464no—
206#121—416no—
207#117—416no—
208#117—416no—
209C2×C4⋊C4—232no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4C2×D414yes—
250C2×C4⋊C4C2×D414yes—
251C2×C4⋊C4C2×D418yes—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
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—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—44no—
299#95—24no—
300#95—28no—
301#95—28no—
302#95—22no—
303#95—22no—
304#95—24no—
305#95—24no—
306#95—28no—
307#95—24no—
308#95—24no—
309#95—42no—
310#95—44no—
311#95—42no—
312#95—42no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24C22×D412yes—
338C24C2×#4112yes—
339C24C2×#4112yes—
340C24C2511yescharacteristic
341C2×D4—416no—
342C2×D4—48no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—44no—
346C2×D4—48no—
347C2×D4—48no—
348C2×D4—88no—
349C2×D4—84no—
350C2×D4—84no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—216no—
356C22×C4—216no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—216no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—416no—
425C22×C4—88no—
426C22×C4—84no—
427C22×C4—84no—
428C22×C4#123116yes—
429C22×C4C22×D414yes—
430C22×C4C22×D414yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4C2×#4112yes—
437C22×C4H(5,𝔽2)14yes—
438C22×C4H(5,𝔽2)14yes—
439C22×C4H(5,𝔽2)14yes—
440C22×C42−1+414yes—
441C22×C42−1+414yes—
442C4⋊C4—216no—
443C4⋊C4—28no—
444C4⋊C4—28no—
445C4⋊C4—48no—
446C4⋊C4—416no—
447C4⋊C4—416no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—44no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—88no—
454C4⋊C4—88no—
455C4⋊C4—88no—
456C22⋊C4—216no—
457C22⋊C4—416no—
458C22⋊C4—416no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—48no—
464C22⋊C4—88no—
465C22⋊C4—84no—
466C22⋊C4—88no—
467C22⋊C4—88no—
468C22⋊C4—88no—
469C42—48no—
470C42—44no—
471C42—48no—
472C42—44no—
473C42—48no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23C2×#12314yes—
516C23C23×D411yescharacteristic
517C23C22×#4111yescharacteristic
518C23C22×#4111yescharacteristic
519C23C2×H(5,𝔽2)12yes—
520C23C2×H(5,𝔽2)12yes—
521C23C2×H(5,𝔽2)12yes—
522C23C2×2−1+412yes—
523C23C2×2−1+412yes—
524C23#58512yes—
525D4—816no—
526D4—88no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—232no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—816no—
604C2×C4C2×#12314yes—
605C2×C4C2×#12314yes—
606C2×C4#53514yes—
607C2×C4#53614yes—
608C2×C4#53918yes—
609C2×C4#53918yes—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—82no—
620C22—82no—
621C22—84no—
622C22—84no—
623C22—84no—
624C22—84no—
625C22—84no—
626C22—84no—
627C22—82no—
628C22C22×#12311yescharacteristic
629C22C2×#53511yescharacteristic
630C22C2×#53611yescharacteristic
631C22C2×#53912yes—
632C22C2×#53912yes—
633C22#348412yes—
634C22#349512yes—
635C22#349712yes—
636C22#350812yes—
637C22#350811yescharacteristic
638C22#350811yescharacteristic
639C22#350911yescharacteristic
640C22#351011yescharacteristic
641C22#353612yes—
642C22#354212yes—
643C22#354412yes—
644C22#354612yes—
645C22#354712yes—
646C22#354812yes—
647C22#355212yes—
648C22#355912yes—
649C4—24no—
650C4—44no—
651C4—44no—
652C4—44no—
653C4—44no—
654C4—44no—
655C4—84no—
656C4—82no—
657C4—82no—
658C4—84no—
659C4—82no—
660C4—82no—
661C4#262318yes—
662C2—44no—
663C2—162no—
664C2C2×#262311yescharacteristic
665C2#3555411yescharacteristic
666C2#3567311yescharacteristic
667C2#3733012yes—
668C2#3733512yes—
669C2#4242112yes—
670C2#4242312yes—
671C2#4293912yes—
672C2#4296012yes—
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
2. C2×#2316C2semidirect
3. #16532C2semidirect
4. #16531C2semidirect
5. #16360C2semidirect
6. #16359C2semidirect
7. #15148C2semidirect
8. #15146C2semidirect
9. #15138C2semidirect
10. #15133C2semidirect
13. #14129C2semidirect
14. #13924C2semidirect
15. #13844C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25012
C2×#23162
#165322
#165312
#163602
#163592
#151482
#151462
#151382
#151332
#142572
#141362
#141292
#139242
#138442