Finite Group Databasebeta

← #518154

Subgroups — #518154

Order 512.

3126 conjugacy classes of subgroups.

The subgroup lattice has 754 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2C2×#2352C211yesmaximal, characteristic
3C2×#2274C211yesmaximal, characteristic, complemented
4#14308C212yesmaximal, complemented
5#14264C211yesmaximal, characteristic, complemented
6#14244C212yesmaximal, complemented
7#14225C212yesmaximal, complemented
8#14104C214yesmaximal, complemented
9#14096C214yesmaximal
10#13982C212yesmaximal, complemented
11#13974C212yesmaximal, complemented
12#13924C211yesmaximal, characteristic, complemented
13C4×#397C212yesmaximal, complemented
14#13839C211yesmaximal, characteristic
15#13818C212yesmaximal, complemented
16#13801C211yesmaximal, characteristic, complemented
17C22⋊C4×C4⋊C4C212yesmaximal, complemented
18C22×#123C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24#2486C4116yescomplemented
25C2×#398C2212yes—
26C2×#397C2212yes—
27#2352—28no—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#388C2212yes—
31C2×#388C2214yes—
32C2×#387C2212yes—
33C2×#386C2212yes—
34C2×#382C2211yescharacteristic
35C2×#381C2212yes—
36C2×#380C2211yescharacteristic
37C2×#380C2211yescharacteristic
38C2×#380C2212yes—
39C2×#380C2211yescharacteristic
40#2274—28no—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2214yes—
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2214yes—
52#1445C2214yes—
53#1445C2212yes—
54#1445C2212yes—
55#1444C2212yes—
56#1443C2214yes—
57#1443C2214yes—
58#1442C2212yes—
59#1441C2211yescharacteristic
60#1441C2211yescharacteristic
61#1440C2212yes—
62#1440C2214yes—
63#1439C2211yescharacteristic
64#1439C2211yescharacteristic
65#1439C2214yes—
66#1438C2212yes—
67#1437C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74C4⋊#95C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79#1434C2214yes—
80#1434C2212yes—
81C22⋊#95C2214yes—
82C22⋊#95C2214yes—
83C4×#95C2212yes—
84C4×#95C2214yes—
85C4×#95C2212yes—
86C23×D4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C2×#123—44no—
92C2×#123—44no—
93C2×#123C2×C4116yes—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C2×C4×D4C2×C418yes—
97C2×#117—24no—
98C2×#117—28no—
99C2×#117—28no—
100C2×#117D418yes—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—28no—
105C22×C4⋊C4C2×C418yescomplemented
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2314yes—
135C22×C42C2312yes—
136C22×C42C2314yes—
137C22×C42C2314yes—
138#398C2×C4116yescomplemented
139#397C2×C4116yes—
140#389—28no—
141#389C2×C4116yescomplemented
142#388—28no—
143#388—48no—
144#387—28no—
145#386—28no—
146#382—24no—
147#381C2×C4116yescomplemented
148#380—24no—
149#380—24no—
150#380C2×C4116yes—
151#380C2×C418yescomplemented
152C4×C4⋊C4—24no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—28no—
155C4×C4⋊C4—28no—
156C4×C4⋊C4—28no—
157C4×C4⋊C4—28no—
158C4×C22⋊C4—24no—
159C4×C22⋊C4—216no—
160C4×C22⋊C4—28no—
161C4×C22⋊C4—28no—
162C4×C22⋊C4—48no—
163C2×#95C2312yes—
164C2×#95C2311yescharacteristic
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2311yescharacteristic
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C25C2411yescharacteristic
192C25C2411yescharacteristic
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—41no—
198C22×D4—41no—
199C22×D4C22×C414yes—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—24no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—28no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—44no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4C22×C414yes—
218C23×C4C22×C414yes—
219C23×C4C22×C418yes—
220C23×C4C2×D414yes—
221C23×C4#4114yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2414yes—
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237#123—432no—
238C4×D4—232no—
239#117—232no—
240#117—232no—
241C2×C4⋊C4—232no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—416no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4C22×C418yes—
268C2×C4⋊C4C22×C418yes—
269C2×C4⋊C4C22×C414yes—
270C2×C4⋊C4C22×C418yes—
271C2×C4⋊C4C22×C4116yes—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4#4118yes—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4C22×C414yes—
302C2×C22⋊C4C22×C414yes—
303C2×C22⋊C4C22×C4116yes—
304C2×C22⋊C4C22×C418yes—
305C2×C22⋊C4C2×D418yes—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—28no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—28no—
315C2×C42—28no—
316C2×C42—24no—
317C2×C42—28no—
318C2×C42—44no—
319C2×C42—44no—
320C2×C42—44no—
321C2×C42—44no—
322C2×C42—44no—
323C2×C42—44no—
324C2×C42—44no—
325C2×C42C22×C414yes—
326C2×C42C2×D418yes—
327C2×C42#4118yes—
328#95—28no—
329#95—28no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—22no—
338#95—22no—
339#95—28no—
340#95—28no—
341#95—24no—
342#95—28no—
343#95—42no—
344#95—42no—
345#95—44no—
346#95—44no—
347#95—42no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95C22×C4116yes—
354#95C22×C418yes—
355#95C22×C4116yes—
356C24—21no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—21no—
361C24—22no—
362C24—21no—
363C24—21no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—21no—
368C24—22no—
369C24—21no—
370C24—21no—
371C24—21no—
372C24C23×C412yes—
373C24C23×C412yes—
374C24C22×D412yes—
375C24C2×#4112yes—
376C24H(5,𝔽2)12yes—
377C24H(5,𝔽2)12yes—
378C24C2511yescharacteristic
379C2×D4—28no—
380C2×D4—28no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—44no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—216no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—216no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—216no—
444C22×C4—28no—
445C22×C4—216no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—416no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—84no—
472C22×C4—88no—
473C22×C4—84no—
474C22×C4—84no—
475C22×C4—84no—
476C22×C4—84no—
477C22×C4C4×D4116yes—
478C22×C4C23×C412yes—
479C22×C4C23×C412yes—
480C22×C4C23×C418yes—
481C22×C4C23×C418yes—
482C22×C4C23×C412yes—
483C22×C4C23×C412yes—
484C22×C4C23×C412yes—
485C22×C4C22×D414yes—
486C22×C4C22×D412yes—
487C22×C4C22×D414yes—
488C22×C4C22×D412yes—
489C22×C4C2×#4114yes—
490C22×C4C2×#4114yes—
491C22×C4C2×#4112yes—
492C22×C4C2×#4118yes—
493C22×C4C2×#4112yes—
494C22×C4C2×#4114yes—
495C22×C4C2×#4114yes—
496C22×C42−1+414yes—
497C4⋊C4—28no—
498C4⋊C4—28no—
499C4⋊C4—216no—
500C4⋊C4—216no—
501C4⋊C4—28no—
502C4⋊C4—216no—
503C4⋊C4—28no—
504C4⋊C4—416no—
505C4⋊C4—48no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—88no—
509C4⋊C4#123116yescomplemented
510C22⋊C4—216no—
511C22⋊C4—216no—
512C22⋊C4—28no—
513C22⋊C4—232no—
514C22⋊C4—28no—
515C22⋊C4—216no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—84no—
521C42—28no—
522C42—216no—
523C42—216no—
524C42—48no—
525C42—88no—
526C42—84no—
527C42—88no—
528C42—88no—
529C23—22no—
530C23—24no—
531C23—24no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—24no—
536C23—24no—
537C23—22no—
538C23—22no—
539C23—24no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—24no—
545C23—24no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—24no—
551C23—22no—
552C23—22no—
553C23—24no—
554C23—24no—
555C23—41no—
556C23—41no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—41no—
562C23—41no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23C2×C4×D414yes—
574C23#51814yes—
575C23C24×C411yescharacteristic
576C23C23×D411yescharacteristic
577C23C22×#4111yescharacteristic
578C23C22×#4112yes—
579C23C22×#4112yes—
580C23C2×H(5,𝔽2)11yescharacteristic
581C23C2×H(5,𝔽2)11yescharacteristic
582C23C2×2−1+412yes—
583C23#58511yescharacteristic
584C23#58511yescharacteristic
585C23#58512yes—
586D4—416no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—232no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—22no—
605C2×C4—28no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—28no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—416no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—816no—
672C2×C4—816no—
673C2×C4C2×C4×D414yes—
674C2×C4C2×C4×D414yes—
675C2×C4C4×#4118yes—
676C2×C4C2×#12318yes—
677C2×C4C2×#12314yes—
678C2×C4#52514yes—
679C2×C4#52918yes—
680C22—24no—
681C22—24no—
682C22—24no—
683C22—24no—
684C22—28no—
685C22—44no—
686C22—44no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—44no—
692C22—44no—
693C22—42no—
694C22—42no—
695C22—42no—
696C22—42no—
697C22—42no—
698C22—42no—
699C22—44no—
700C22—44no—
701C22—44no—
702C22C22×C4×D411yescharacteristic
703C22C2×C4×#4112yes—
704C22C2×#51811yescharacteristic
705C22#342811yescharacteristic
706C22C4×2−1+412yes—
707C22C22×#12311yescharacteristic
708C22C2×#52511yescharacteristic
709C22C2×#52912yes—
710C22#348811yescharacteristic
711C22#348811yescharacteristic
712C22#349212yes—
713C22#349411yescharacteristic
714C22#349411yescharacteristic
715C22#349412yes—
716C22#349512yes—
717C22#349911yescharacteristic
718C22#350011yescharacteristic
719C22#350712yes—
720C22#351212yes—
721C22#351312yes—
722C22#351612yes—
723C22#352112yes—
724C22#352212yes—
725C4—28no—
726C4—24no—
727C4—44no—
728C4—44no—
729C4—48no—
730C4—42no—
731C4—44no—
732C4—42no—
733C4—42no—
734C4—44no—
735C4—48no—
736C4—82no—
737C4—82no—
738C4—82no—
739C4C4×#12318yes—
740C2—44no—
741C2—44no—
742C2C2×C4×#12311yescharacteristic
743C2#3420411yescharacteristic
744C2#3427611yescharacteristic
745C2#3429812yes—
746C2#3431812yes—
747C2#3568611yescharacteristic
748C2#3573412yes—
749C2#3583312yes—
750C2#3597611yescharacteristic
751C2#3599611yescharacteristic
752C2#3608311yescharacteristic
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
3. C2×#2274C2semidirect
4. #14308C2semidirect
5. #14264C2semidirect
6. #14244C2semidirect
7. #14225C2semidirect
8. #14104C2semidirect
10. #13982C2semidirect
11. #13974C2semidirect
12. #13924C2semidirect
13. C4×#397C2semidirect
15. #13818C2semidirect
16. #13801C2semidirect
17. C22⋊C4×C4⋊C4C2semidirect
24. #2486C4semidirect
105. C22×C4⋊C4C2×C4semidirect
138. #398C2×C4semidirect
141. #389C2×C4semidirect
147. #381C2×C4semidirect
151. #380C2×C4semidirect
509. C4⋊C4#123semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
C2×#23522
C2×#22742
#143082
#142642
#142442
#142252
#141042
#140962
#139822
#139742
#139242
C4×#3972
#138392
#138182
#138012
C22⋊C4×C4⋊C42