Finite Group Databasebeta

← #521076

Subgroups — #521076

Order 512.

3338 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2392C211yesmaximal, characteristic
2C2×#2293C212yesmaximal, complemented
3#14589C212yesmaximal, complemented
4#14587C212yesmaximal, complemented
5#14581C212yesmaximal, complemented
6#14573C214yesmaximal, complemented
7#14565C214yesmaximal, complemented
8#14521C212yesmaximal
9#13852C211yesmaximal, characteristic, complemented
10#13844C212yesmaximal, complemented
11#13839C212yesmaximal, complemented
12C4×#387C212yesmaximal, complemented
13C4×#388C212yesmaximal, complemented
14C4×#386C212yesmaximal, complemented
15#13756C211yesmaximal, characteristic, complemented
16C22×C4×D4C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×#117C2212yes—
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2392C4116yescomplemented
23C2×#398C2211yescharacteristic
24C2×#397C2211yescharacteristic
25C2×#396C2212yes—
26C2×#394C2212yes—
27C2×#394C2211yescharacteristic
28C2×#392C2211yescharacteristic
29#2293—216no—
30C2×#390C2214yes—
31C2×#389C2214yes—
32C2×#388C2214yes—
33C2×#388C2214yes—
34C2×#388C2214yes—
35C2×#387C2214yes—
36C2×#386C2214yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2212yes—
42C2×C4×C22⋊C4C2212yes—
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C2×C43C2212yes—
47#1444C2214yes—
48#1442C2214yes—
49#1441C2214yes—
50#1441C2212yes—
51#1441C2212yes—
52#1440C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2212yes—
57#1439C2212yes—
58#1438C2214yes—
59#1437C2214yes—
60#1437C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67#1434C2214yes—
68#1434C2214yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2214yes—
72C4×#95C2212yes—
73C4×#95C2212yes—
74C4×#95C2212yes—
75C4×#95C2212yes—
76C4×#95C2212yes—
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C2×C4×D4—28no—
83C2×C4×D4—28no—
84C2×C4×D4—42no—
85C2×C4×D4—42no—
86C2×C4×D4C2×C4116yes—
87C2×C4×D4C2×C418yescomplemented
88C2×#117—28no—
89C2×#117—42no—
90C2×#117—42no—
91C2×#117—44no—
92C2×#117—44no—
93C2×#117—42no—
94C2×#117—42no—
95C2×#117C2×C418yes—
96C2×#117D418yescomplemented
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C42—24no—
121C22×C42—42no—
122C22×C42—42no—
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2314yes—
127C22×C42C2314yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2312yes—
133#398C2×C418yescomplemented
134#397C2×C418yes—
135#396C2×C4116yescomplemented
136#394C2×C4116yes—
137#394C2×C418yescomplemented
138#392C2×C418yes—
139#390—216no—
140#389—216no—
141#388—216no—
142#388—216no—
143#388—216no—
144#387—216no—
145#386—216no—
146C4×C4⋊C4—28no—
147C4×C4⋊C4—42no—
148C4×C4⋊C4—42no—
149C4×C4⋊C4C2×C418yescomplemented
150C4×C22⋊C4—28no—
151C4×C22⋊C4—28no—
152C4×C22⋊C4—44no—
153C4×C22⋊C4—44no—
154C4×C22⋊C4C2×C4116yescomplemented
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2311yescharacteristic
172C2×#95C2311yescharacteristic
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C43—28no—
177C25C2412yes—
178C25C2411yescharacteristic
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—28no—
182C22×D4—41no—
183C22×D4—41no—
184C22×D4—44no—
185C22×D4—44no—
186C22×D4C22×C418yes—
187C22×D4C22×C414yes—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—28no—
200C23×C4—44no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4C22×C418yes—
204C23×C4C22×C414yes—
205C23×C4C2×D414yes—
206C23×C4#4114yes—
207C23×C4C2414yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C4×D4—264no—
221C4×D4—416no—
222#117—416no—
223#117—416no—
224#117—432no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4C22×C418yes—
247C2×C4⋊C4C22×C418yes—
248C2×C4⋊C4C22×C414yes—
249C2×C4⋊C4C22×C414yes—
250C2×C4⋊C4C22×C414yes—
251C2×C4⋊C4C2×D418yes—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4C22×C418yes—
282C2×C22⋊C4C22×C418yes—
283C2×C22⋊C4C22×C418yes—
284C2×C22⋊C4C22×C414yes—
285C2×C22⋊C4C22×C414yes—
286C2×C22⋊C4C22×C418yes—
287C2×C22⋊C4C22×C418yes—
288C2×C22⋊C4C2×D418yes—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—28no—
300C2×C42—28no—
301C2×C42—28no—
302C2×C42—28no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—42no—
312C2×C42—42no—
313C2×C42—44no—
314C2×C42—44no—
315C2×C42—416no—
316C2×C42C22×C414yes—
317C2×C42C22×C414yes—
318C2×C42C22×C418yes—
319C2×C42C2×D418yes—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—42no—
335#95—42no—
336#95C22×C418yes—
337#95C22×C418yes—
338#95C22×C414yes—
339#95C22×C414yes—
340#95#41116yes—
341C24—24no—
342C24—21no—
343C24—21no—
344C24—21no—
345C24—21no—
346C24—24no—
347C24—22no—
348C24—24no—
349C24—24no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—24no—
357C24C23×C414yes—
358C24C23×C412yes—
359C24C22×D412yes—
360C24C2×#4112yes—
361C24C2511yescharacteristic
362C2×D4—216no—
363C2×D4—216no—
364C2×D4—44no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—416no—
368C2×D4—416no—
369C2×D4—48no—
370C2×D4—48no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—24no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—216no—
425C22×C4—216no—
426C22×C4—216no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—48no—
435C22×C4—416no—
436C22×C4—416no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4C4×D4116yes—
444C22×C4C23×C414yes—
445C22×C4C23×C412yes—
446C22×C4C23×C412yes—
447C22×C4C23×C412yes—
448C22×C4C23×C414yes—
449C22×C4C23×C412yes—
450C22×C4C23×C412yes—
451C22×C4C23×C414yes—
452C22×C4C23×C412yes—
453C22×C4C22×D418yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C2×#4118yes—
457C22×C4C2×#4118yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4118yes—
460C22×C4C2×#4112yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C42−1+412yes—
465C4⋊C4—216no—
466C4⋊C4—216no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C4⋊C4—416no—
472C4⋊C4—48no—
473C4⋊C4—416no—
474C4⋊C4—48no—
475C22⋊C4—216no—
476C22⋊C4—216no—
477C22⋊C4—216no—
478C22⋊C4—48no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C22⋊C4—416no—
484C22⋊C4—416no—
485C22⋊C4—416no—
486C22⋊C4—416no—
487C22⋊C4—416no—
488C42—216no—
489C42—28no—
490C42—28no—
491C42—216no—
492C42—416no—
493C42—416no—
494C42—48no—
495C42—48no—
496C42—48no—
497C42—48no—
498C23—24no—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—28no—
507C23—28no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—44no—
516C23—44no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23C2×C4×D414yes—
538C23C24×C411yescharacteristic
539C23C23×D411yescharacteristic
540C23C22×#4111yescharacteristic
541C23C22×#4114yes—
542C23C2×H(5,𝔽2)11yescharacteristic
543C23C2×H(5,𝔽2)11yescharacteristic
544C23C2×H(5,𝔽2)11yescharacteristic
545C23C2×2−1+411yescharacteristic
546C23#58514yes—
547D4—432no—
548D4—416no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—232no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—432no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—432no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4C2×C4×D414yes—
619C2×C4C2×C4×D414yes—
620C2×C4C4×#4118yes—
621C2×C4#51814yes—
622C2×C4#52014yes—
623C2×C4#53414yes—
624C2×C4#53714yes—
625C2×C4#53814yes—
626C2×C4#53914yes—
627C22—24no—
628C22—28no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—42no—
633C22—42no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—84no—
644C22—84no—
645C22—84no—
646C22—84no—
647C22C22×C4×D411yescharacteristic
648C22C2×C4×#4112yes—
649C22C2×#51811yescharacteristic
650C22C2×#52011yescharacteristic
651C22#342812yes—
652C22C2×#53411yescharacteristic
653C22C2×#53711yescharacteristic
654C22C2×#53811yescharacteristic
655C22C2×#53911yescharacteristic
656C22#346014yes—
657C22#346114yes—
658C22D4×#4114yes—
659C22#347514yes—
660C22#348112yes—
661C22#348312yes—
662C22#352912yes—
663C22#352912yes—
664C4—24no—
665C4—28no—
666C4—44no—
667C4—48no—
668C4—42no—
669C4—42no—
670C4—44no—
671C4—42no—
672C4—42no—
673C4—48no—
674C4—48no—
675C4—84no—
676C4#232518yes—
677C2—44no—
678C2—84no—
679C2C2×#232511yescharacteristic
680C2#3408412yes—
681C2#3413612yes—
682C2#3415511yescharacteristic
683C2#3434211yescharacteristic
684C2#5216712yes—
685C2#5216812yes—
686C2#5224512yes—
687C2#5224812yes—
subgroupG/H
centerC23×C4C24
commutatorC23C24×C4
Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2293C2semidirect
3. #14589C2semidirect
4. #14587C2semidirect
5. #14581C2semidirect
6. #14573C2semidirect
7. #14565C2semidirect
9. #13852C2semidirect
10. #13844C2semidirect
11. #13839C2semidirect
12. C4×#387C2semidirect
13. C4×#388C2semidirect
14. C4×#386C2semidirect
15. #13756C2semidirect
22. #2392C4semidirect
87. C2×C4×D4C2×C4semidirect
96. C2×#117D4semidirect
133. #398C2×C4semidirect
135. #396C2×C4semidirect
137. #394C2×C4semidirect
149. C4×C4⋊C4C2×C4semidirect
154. C4×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23922
C2×#22932
#145892
#145872
#145812
#145732
#145652
#145212
#138522
#138442
#138392
C4×#3872
C4×#3882
C4×#3862
#137562