Finite Group Databasebeta

← #760863

Subgroups — #760863

Order 512.

3126 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2549C212yesmaximal
2C2×#2440C212yesmaximal
3#16827C214yesmaximal, complemented
4#15133C212yesmaximal, complemented
5#15128C212yesmaximal, complemented
6#15011C212yesmaximal, complemented
7#15008C212yesmaximal, complemented
8#14914C214yesmaximal
9#14149C212yesmaximal
10#14130C212yesmaximal
11#14053C212yesmaximal, complemented
12#14005C212yesmaximal, complemented
13C4×#380C211yesmaximal, characteristic
14C4×#380C211yesmaximal, characteristic
15C2×C4⋊#95C211yesmaximal, characteristic
16C22×C4⋊Q8C2212yes—
17C22×C22⋊Q8C2214yes—
18C23×C4⋊C4C2212yes—
19C23×C42C2211yescharacteristic
20#2549—216no—
21#2440—216no—
22C2×#398C2214yes—
23C2×#398C2214yes—
24C2×#395C2214yes—
25C2×#395C2214yes—
26C2×#391C2214yes—
27C2×#389C2214yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#385C2214yes—
33C2×#385C2214yes—
34C2×#384C2212yes—
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×#380C2211yescharacteristic
38C2×#380C2211yescharacteristic
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C22⋊C4C2212yes—
42C2×C4×C22⋊C4C2212yes—
43C22×#95C2212yes—
44C22×#95C2212yes—
45#1446C2214yes—
46#1446C2214yes—
47#1444C2214yes—
48#1444C2214yes—
49#1444C2214yes—
50#1442C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59C4⋊#95C2211yescharacteristic
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2211yescharacteristic
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68#1434C2211yescharacteristic
69#1434C2211yescharacteristic
70#1434C2212yes—
71C22⋊#95C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C4×#95C2212yes—
75C4×#95C2211yescharacteristic
76C4×#95C2211yescharacteristic
77C23×Q8C2314yes—
78C24×C4C2314yes—
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C2×C4⋊Q8—28no—
83C2×C4⋊Q8—28no—
84C2×C4⋊Q8—44no—
85C2×C22⋊Q8—216no—
86C2×C22⋊Q8—216no—
87C2×C22⋊Q8—48no—
88C22×C4⋊C4—28no—
89C22×C4⋊C4—28no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C42—44no—
110C22×C42Q818yescomplemented
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2314yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118#398—216no—
119#398—216no—
120#395—216no—
121#395—216no—
122#391—216no—
123#389—28no—
124#389—28no—
125#389—28no—
126#389—216no—
127#389—44no—
128#385—216no—
129#385—216no—
130#384—28no—
131#380D418yes—
132#380D418yes—
133#380Q818yescomplemented
134#380Q818yescomplemented
135C4×C4⋊C4—44no—
136C4×C4⋊C4—44no—
137C4×C22⋊C4—28no—
138C4×C22⋊C4—28no—
139C2×#95—28no—
140C2×#95—44no—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2311yescharacteristic
167C2×#95C2311yescharacteristic
168C2×#95C2311yescharacteristic
169C2×#95C2312yes—
170C25C2411yescharacteristic
171C22×Q8—28no—
172C22×Q8—28no—
173C22×Q8—28no—
174C22×Q8—44no—
175C22×Q8—44no—
176C22×Q8—44no—
177C22×Q8—44no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—28no—
183C23×C4—28no—
184C23×C4—28no—
185C23×C4—28no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4C2×D414yes—
194C23×C4C2×D414yes—
195C23×C4C2×Q814yes—
196C23×C4C2×Q814yes—
197C23×C4C2×Q814yes—
198C23×C4C2×Q814yes—
199C23×C4C2×Q814yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2414yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C4⋊Q8—432no—
215C22⋊Q8—464no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—432no—
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—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4C2×D418yes—
255C2×C4⋊C4C2×D418yes—
256C2×C4⋊C4C2×Q818yes—
257C2×C4⋊C4C2×Q818yes—
258C2×C4⋊C4#4118yes—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4C2×D418yes—
268C2×C22⋊C4C2×D418yes—
269C2×C22⋊C4C2×Q818yes—
270C2×C22⋊C4C2×Q818yes—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—28no—
280C2×C42—416no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42—44no—
284C2×C42—44no—
285C2×C42C2×Q814yes—
286C2×C42C2×Q814yes—
287C2×C42C2×Q814yes—
288C2×C42C2×Q814yes—
289#95—28no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—28no—
294#95—24no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—42no—
301#95—44no—
302#95—42no—
303#95—44no—
304#95—42no—
305#95—44no—
306#95—44no—
307#95—42no—
308#95—42no—
309#95—44no—
310#95—42no—
311#95—44no—
312#95C2×D414yes—
313#95C2×D414yes—
314#95C2×D418yes—
315#95C2×Q814yes—
316#95C2×Q814yes—
317#95C2×Q818yes—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24C22×D412yes—
323C24C22×D412yes—
324C24C22×Q812yes—
325C24C22×Q812yes—
326C24C22×Q812yes—
327C24H(5,𝔽2)12yes—
328C242−1+412yes—
329C24C2511yescharacteristic
330C2×Q8—416no—
331C2×Q8—416no—
332C2×Q8—416no—
333C2×Q8—88no—
334C2×Q8—88no—
335C2×Q8—88no—
336C2×Q8—88no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—28no—
344C22×C4—28no—
345C22×C4—216no—
346C22×C4—216no—
347C22×C4—28no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—48no—
408C22×C4—44no—
409C22×C4—48no—
410C22×C4—44no—
411C22×C4—416no—
412C22×C4—416no—
413C22×C4—416no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—416no—
417C22×C4—88no—
418C22×C4—88no—
419C22×C4—88no—
420C22×C4C4⋊Q8116yes—
421C22×C4C4⋊Q8116yescomplemented
422C22×C4C22×D414yes—
423C22×C4C22×D418yes—
424C22×C4C22×D414yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×Q818yes—
430C22×C4C22×Q814yes—
431C22×C4C22×Q814yes—
432C22×C4C22×Q812yes—
433C22×C4C22×Q812yes—
434C22×C4C22×Q812yes—
435C22×C4C22×Q812yes—
436C22×C4C22×Q812yes—
437C22×C4C22×Q812yes—
438C22×C4C22×Q812yes—
439C22×C4C22×Q812yes—
440C22×C4C22×Q812yes—
441C22×C4C22×Q812yes—
442C22×C4C2×#4118yes—
443C22×C4C2×#4114yes—
444C22×C4H(5,𝔽2)14yes—
445C4⋊C4—216no—
446C4⋊C4—216no—
447C4⋊C4—216no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—416no—
451C4⋊C4—416no—
452C4⋊C4—416no—
453C4⋊C4—416no—
454C4⋊C4—416no—
455C4⋊C4—416no—
456C4⋊C4—48no—
457C4⋊C4—88no—
458C4⋊C4—88no—
459C22⋊C4—216no—
460C22⋊C4—216no—
461C22⋊C4—216no—
462C22⋊C4—216no—
463C42—28no—
464C42—28no—
465C42—44no—
466C42—44no—
467C42—48no—
468C42—48no—
469C42—88no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23C2×C4⋊Q814yes—
487C23C2×C4⋊Q814yes—
488C23#54414yes—
489C23D4214yes—
490C23D4×Q814yes—
491C23D4×Q814yes—
492C23Q8214yes—
493C23C23×D411yescharacteristic
494C23C23×D411yescharacteristic
495C23C23×Q811yescharacteristic
496C23C23×Q811yescharacteristic
497C23C23×Q811yescharacteristic
498C23C22×#4112yes—
499C23C2×H(5,𝔽2)12yes—
500C23C2×H(5,𝔽2)11yescharacteristic
501C23C2×2−1+411yescharacteristic
502C23#58512yes—
503C23#58512yes—
504Q8—832no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—232no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—416no—
546C2×C4—416no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—832no—
587C2×C4C2×C4⋊Q814yes—
588C2×C4C2×C4⋊Q814yes—
589C2×C4C2×C4⋊Q814yes—
590C2×C4C2×C4⋊Q814yes—
591C2×C4#53314yes—
592C2×C4#54314yes—
593C2×C4#54618yes—
594C2×C4#55418yes—
595C22—24no—
596C22—24no—
597C22—28no—
598C22—24no—
599C22—24no—
600C22—28no—
601C22—28no—
602C22—24no—
603C22—28no—
604C22—24no—
605C22#255118yes—
606C22C22×C4⋊Q811yescharacteristic
607C22C22×C4⋊Q811yescharacteristic
608C22C2×#53311yescharacteristic
609C22C2×#54311yescharacteristic
610C22C2×#54411yescharacteristic
611C22#345811yescharacteristic
612C22#345811yescharacteristic
613C22C2×D4211yescharacteristic
614C22C2×#54612yes—
615C22C2×D4×Q811yescharacteristic
616C22C2×D4×Q811yescharacteristic
617C22D4×#4112yes—
618C22C2×#55412yes—
619C22C2×Q8211yescharacteristic
620C22Q8×#4112yes—
621C22#349612yes—
622C22#350012yes—
623C22#350112yes—
624C22#350412yes—
625C22#351312yes—
626C22#351612yes—
627C22#354112yes—
628C22#354212yes—
629C4—24no—
630C4—24no—
631C4—24no—
632C4—42no—
633C4—42no—
634C4—44no—
635C4—48no—
636C4—48no—
637C4—44no—
638C4—44no—
639C4—44no—
640C4—48no—
641C4—84no—
642C4#254918yescomplemented
643C2—28no—
644C2C2×#254911yescharacteristic
645C2C2×#255111yescharacteristic
646C2#3493811yescharacteristic
647C2#3500211yescharacteristic
648C2#3523511yescharacteristic
649C2#3533011yescharacteristic
650C2#3747611yescharacteristic
651C2#5980812yes—
652C2#5980912yes—
653C2#5983112yes—
654C2#5991812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16827C2semidirect
4. #15133C2semidirect
5. #15128C2semidirect
6. #15011C2semidirect
7. #15008C2semidirect
11. #14053C2semidirect
12. #14005C2semidirect
110. C22×C42Q8semidirect
133. #380Q8semidirect
134. #380Q8semidirect
421. C22×C4C4⋊Q8semidirect
642. C4#2549semidirect

Maximal subgroups

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

subgroupindex
C2×#25492
C2×#24402
#168272
#151332
#151282
#150112
#150082
#149142
#141492
#141302
#140532
#140052
C4×#3802
C2×C4⋊#952