Finite Group Databasebeta

← #524636

Subgroups — #524636

Order 512.

2154 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2369C211yesmaximal, characteristic
2#14714C211yesmaximal, characteristic
3#14432C214yesmaximal
4#14380C214yesmaximal
5#14266C212yesmaximal
6#14161C211yesmaximal, characteristic
7#14133C212yesmaximal
8#14042C212yesmaximal
9#13982C212yesmaximal
10#13955C212yesmaximal
11#13921C212yesmaximal
12#13921C212yesmaximal
13#13911C211yesmaximal, characteristic
14C4⋊C42C211yesmaximal, characteristic
15#13797C212yesmaximal
16C4×#389C211yesmaximal, characteristic
17#13755C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19#2369C4116yescomplemented
20C2×#391C2212yes—
21C2×#391C2212yes—
22C2×#391C2214yes—
23C2×#389C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#387C2214yes—
28C2×#384C2211yescharacteristic
29C2×#383C2214yes—
30C2×#382C2212yes—
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×#376C2212yes—
41C2×C43C2211yescharacteristic
42#1446C2214yes—
43#1445C2214yes—
44#1443C2214yes—
45#1443C2214yes—
46#1443C2214yes—
47#1443C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1441C2214yes—
52#1436C2212yes—
53#1436C2214yes—
54#1436C2214yes—
55#1436C2212yes—
56C4⋊#95C2212yes—
57C4⋊#95C2212yes—
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2211yescharacteristic
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2211yescharacteristic
74C4⋊#95C2211yescharacteristic
75#1434C2212yes—
76#1434C2214yes—
77#1434C2214yes—
78#1434C2212yes—
79#1434C2214yes—
80#1434C2214yes—
81#1434C2214yes—
82C4×#95C2212yes—
83C4×#95C2212yes—
84C23×Q8C2312yes—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8C2×C418yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2312yes—
107C22×C42C2312yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2314yes—
115C22×C42C2314yes—
116C22×C42C2311yescharacteristic
117#391—44no—
118#391—44no—
119#391C2×C4132yes—
120#389—24no—
121#389—24no—
122#389C2×C4116yes—
123#389C2×C4116yes—
124#387—48no—
125#384—24no—
126#383—48no—
127#382—44no—
128C4×C4⋊C4—28no—
129C4×C4⋊C4—24no—
130C4×C4⋊C4—24no—
131C4×C4⋊C4—44no—
132C4×C4⋊C4—44no—
133C4×C4⋊C4C2×C4116yes—
134C4×C4⋊C4C2×C4116yes—
135C4×C4⋊C4C2×C418yes—
136C4×C4⋊C4C2×C418yes—
137#376—28no—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C43—24no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—42no—
177C22×Q8—42no—
178C22×Q8—42no—
179C22×Q8—42no—
180C22×Q8C22×C418yes—
181C23×C4C2414yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2414yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2411yescharacteristic
192C23×C4C2411yescharacteristic
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C4⋊Q8—232no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
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—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4C22×C418yes—
240C2×C4⋊C4C22×C418yes—
241C2×C4⋊C4C22×C414yes—
242C2×C4⋊C4C22×C414yes—
243C2×C4⋊C4C22×C414yes—
244C2×C4⋊C4C22×C414yes—
245C2×C4⋊C4C22×C4116yes—
246C2×C4⋊C4#4114yes—
247C2×C42—22no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—24no—
272C2×C42—28no—
273C2×C42—44no—
274C2×C42—44no—
275C2×C42—41no—
276C2×C42—41no—
277C2×C42—44no—
278C2×C42—44no—
279C2×C42C22×C418yes—
280C2×C42C22×C418yes—
281C2×C42C22×C4116yes—
282C2×C42C22×C414yes—
283C2×C42#4114yes—
284C2×C42#4114yes—
285C2×C42#4114yes—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—28no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—28no—
298#95—24no—
299#95—28no—
300#95—42no—
301#95—44no—
302#95—42no—
303#95—42no—
304#95—44no—
305#95—42no—
306#95—42no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95C22×C4116yes—
318#95C22×C4116yes—
319#95C22×C418yes—
320#95C22×C418yes—
321C24C2511yescharacteristic
322C2×Q8—216no—
323C2×Q8—48no—
324C2×Q8—48no—
325C2×Q8—84no—
326C2×Q8—84no—
327C2×Q8—84no—
328C2×Q8—84no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4C23×C414yes—
422C22×C4C23×C414yes—
423C22×C4C23×C414yes—
424C22×C4C23×C414yes—
425C22×C4C23×C418yes—
426C22×C4C23×C412yes—
427C22×C4C23×C412yes—
428C22×C4C23×C412yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4114yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4112yes—
440C22×C4H(5,𝔽2)12yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C4H(5,𝔽2)12yes—
443C22×C42−1+412yes—
444C22×C42−1+412yes—
445C22×C42−1+412yes—
446C22×C42−1+412yes—
447C22×C42−1+412yes—
448C4⋊C4—216no—
449C4⋊C4—216no—
450C4⋊C4—28no—
451C4⋊C4—28no—
452C4⋊C4—28no—
453C4⋊C4—28no—
454C4⋊C4—28no—
455C4⋊C4—416no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—84no—
464C42—216no—
465C42—216no—
466C42—28no—
467C42—416no—
468C42—44no—
469C42—44no—
470C42—44no—
471C42—44no—
472C42—48no—
473C42—44no—
474C42—48no—
475C42—88no—
476C42#117116yes—
477C23C24×C411yescharacteristic
478C23C22×#4111yescharacteristic
479C23C22×#4111yescharacteristic
480C23C22×#4111yescharacteristic
481C23C22×#4111yescharacteristic
482C23C2×H(5,𝔽2)11yescharacteristic
483C23C2×H(5,𝔽2)11yescharacteristic
484C23C2×H(5,𝔽2)11yescharacteristic
485C23C2×2−1+411yescharacteristic
486C23C2×2−1+411yescharacteristic
487C23C2×2−1+411yescharacteristic
488C23C2×2−1+411yescharacteristic
489C23C2×2−1+411yescharacteristic
490C23#58511yescharacteristic
491C23#58511yescharacteristic
492Q8—816no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—28no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—28no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4C2×#11714yes—
598C2×C4C2×#11714yes—
599C2×C4C2×#11714yes—
600C2×C4C4×#4114yes—
601C2×C4#51914yes—
602C2×C4#51914yes—
603C2×C4#52014yes—
604C2×C4#54114yes—
605C2×C4#54114yes—
606C2×C4#54214yes—
607C2×C4#54214yes—
608C2×C4#55214yes—
609C2×C4#55214yes—
610C2×C4#55614yes—
611C2×C4#55614yes—
612C22C22×#11711yescharacteristic
613C22C2×C4×#4111yescharacteristic
614C22C2×#51911yescharacteristic
615C22C2×#51911yescharacteristic
616C22C2×#52011yescharacteristic
617C22C4×H(5,𝔽2)11yescharacteristic
618C22C4×H(5,𝔽2)11yescharacteristic
619C22C2×#54111yescharacteristic
620C22C2×#54111yescharacteristic
621C22C2×#54211yescharacteristic
622C22C2×#54211yescharacteristic
623C22C2×#55211yescharacteristic
624C22C2×#55211yescharacteristic
625C22C2×#55611yescharacteristic
626C22C2×#55611yescharacteristic
627C22#349411yescharacteristic
628C22#349411yescharacteristic
629C22#350611yescharacteristic
630C22#350711yescharacteristic
631C22#350811yescharacteristic
632C22#351011yescharacteristic
633C22#351111yescharacteristic
634C22#351111yescharacteristic
635C22#351511yescharacteristic
636C22#351811yescharacteristic
637C22#351911yescharacteristic
638C22#352211yescharacteristic
639C22#354411yescharacteristic
640C22#354411yescharacteristic
641C22#354711yescharacteristic
642C22#354811yescharacteristic
643C22#355011yescharacteristic
644C22#355511yescharacteristic
645C22#355511yescharacteristic
646C22#355511yescharacteristic
647C4—24no—
648C4—44no—
649C4—44no—
650C4—44no—
651C4—42no—
652C4—44no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—44no—
657C4—48no—
658C4—82no—
659C4—84no—
660C4—82no—
661C4—82no—
662C4#232018yes—
663C4#235618yes—
664C2C2×#232011yescharacteristic
665C2C2×#235611yescharacteristic
666C2#3425511yescharacteristic
667C2#3431711yescharacteristic
668C2#3432111yescharacteristic
669C2#3448411yescharacteristic
670C2#3449111yescharacteristic
671C2#3893111yescharacteristic
672C2#3893211yescharacteristic
673C2#3898511yescharacteristic
674C2#3898611yescharacteristic
675C2#3947911yescharacteristic
676C2#3948011yescharacteristic
677C2#3951411yescharacteristic
678C2#3951511yescharacteristic
subgroupG/H
center, Frattini, socle, prime radical, cosocleC24C25
commutatorC23C24×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
19. #2369C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23692
#147142
#144322
#143802
#142662
#141612
#141332
#140422
#139822
#139552
#139212
#139112
C4⋊C422
#137972
C4×#3892
#137552