Finite Group Databasebeta

← #1424399

Subgroups — #1424399

Order 512.

3044 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2432C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic, complemented
3#16094C212yesmaximal, complemented
4#15696C214yesmaximal, complemented
5#15466C212yesmaximal, complemented
6#15294C214yesmaximal, complemented
7#15084C212yesmaximal, complemented
8#14799C214yesmaximal, complemented
9#14467C212yesmaximal, complemented
10#14450C211yesmaximal, characteristic, complemented
11#14225C212yesmaximal, complemented
12#14118C214yesmaximal, complemented
13#13937C212yesmaximal
14C23×#41C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18#2432—28no—
19C2×#399C2212yes—
20C2×#399C2214yes—
21C2×#398C2212yes—
22C2×#397C2212yes—
23C2×#394C2214yes—
24#2369—28no—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#389C2214yes—
33C2×#388C2214yes—
34C2×#388C2214yes—
35C2×#388C2214yes—
36C2×#388C2214yes—
37C2×#385C2214yes—
38C2×#380C2212yes—
39C2×C4×C4⋊C4C2214yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2212yes—
43#1446C2212yes—
44#1445C2212yes—
45#1444C2212yes—
46#1444C2214yes—
47#1444C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1440C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2212yes—
57#1438C2214yes—
58#1438C2214yes—
59#1437C2212yes—
60#1436C2212yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64C4⋊#95C2214yes—
65#1434C2214yes—
66#1434C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C22×#41—24no—
70C22×#41—28no—
71C22×#41—42no—
72C22×#41—42no—
73C22×#41—44no—
74C23×Q8C2311yescharacteristic
75C23×Q8C2311yescharacteristic
76C23×D4C2312yes—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8D418yes—
82C2×#124—24no—
83C2×#124—44no—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8—42no—
86C2×C22⋊Q8—42no—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C42C2314yes—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2312yes—
114#399—48no—
115#399—44no—
116#398—44no—
117#397—28no—
118#394—216no—
119#391—28no—
120#391—28no—
121#391—24no—
122#391—28no—
123#390—24no—
124#390—44no—
125#390—44no—
126#389—216no—
127#388—216no—
128#388—48no—
129#388—48no—
130#388—48no—
131#385—216no—
132#380—44no—
133C4×C4⋊C4—216no—
134C4×C4⋊C4—28no—
135C4×C22⋊C4—24no—
136C4×C22⋊C4—44no—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C25C2411yescharacteristic
158C25C2412yes—
159C2×#41—416no—
160C2×#41—432no—
161C2×#41—416no—
162C2×#41—432no—
163C2×#41—416no—
164C22×Q8—24no—
165C22×Q8—42no—
166C22×Q8—41no—
167C22×Q8—41no—
168C22×Q8—41no—
169C22×Q8—41no—
170C22×Q8—42no—
171C22×Q8C2×D414yes—
172C22×Q8C2×D414yes—
173C22×Q8#4118yes—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—22no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—42no—
183C22×D4—42no—
184C22×D4—41no—
185C22×D4—41no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2411yescharacteristic
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C4⋊Q8—232no—
210#124—416no—
211C22⋊Q8—416no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4C2×D4116yes—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42—28no—
277C2×C42—28no—
278C2×C42—28no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—42no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42—44no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42C2×D414yes—
289#95—24no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—24no—
300#95—28no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—42no—
307#95—42no—
308#95#4118yescomplemented
309C24—21no—
310C24—22no—
311C24—24no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—24no—
318C24—21no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24H(5,𝔽2)12yes—
327C24H(5,𝔽2)14yes—
328C24C2511yescharacteristic
329#41—4128no—
330C2×Q8—28no—
331C2×Q8—44no—
332C2×Q8—44no—
333C2×Q8—48no—
334C2×Q8—48no—
335C2×Q8—44no—
336C2×Q8—44no—
337C2×Q8—84no—
338C2×Q8—84no—
339C2×Q8#123116yes—
340C2×D4—28no—
341C2×D4—216no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—48no—
347C2×D4—48no—
348C2×D4—44no—
349C2×D4—48no—
350C2×D4—44no—
351C2×D4—48no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—24no—
361C22×C4—28no—
362C22×C4—216no—
363C22×C4—24no—
364C22×C4—21no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—21no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
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—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—44no—
414C22×C4—48no—
415C22×C4—44no—
416C22×C4—48no—
417C22×C4C22×D414yes—
418C22×C4C22×D412yes—
419C22×C4C22×D414yes—
420C22×C4C22×D414yes—
421C22×C4C2×#4118yes—
422C22×C4C2×#4114yes—
423C22×C4C2×#4114yes—
424C22×C4C2×#4114yes—
425C22×C4C2×#4114yes—
426C22×C4H(5,𝔽2)12yes—
427C22×C4H(5,𝔽2)14yes—
428C22×C42−1+418yes—
429C4⋊C4—216no—
430C4⋊C4—232no—
431C4⋊C4—48no—
432C4⋊C4—44no—
433C4⋊C4—48no—
434C4⋊C4—84no—
435C4⋊C4—84no—
436C4⋊C4—84no—
437C4⋊C4—88no—
438C4⋊C4—88no—
439C4⋊C4—88no—
440C22⋊C4—232no—
441C22⋊C4—416no—
442C22⋊C4—416no—
443C22⋊C4—48no—
444C22⋊C4—416no—
445C22⋊C4—48no—
446C22⋊C4—416no—
447C22⋊C4—88no—
448C22⋊C4—88no—
449C22⋊C4—88no—
450C42—28no—
451C42—44no—
452C42—416no—
453C42—84no—
454C42—84no—
455C42—84no—
456C23—22no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—28no—
462C23—24no—
463C23—28no—
464C23—24no—
465C23—41no—
466C23—42no—
467C23—44no—
468C23—42no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—42no—
473C23—41no—
474C23—41no—
475C23—42no—
476C23—44no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—42no—
491C23—41no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23C23×D411yescharacteristic
496C23C22×#4112yes—
497C23C22×#4112yes—
498C23C2×H(5,𝔽2)12yes—
499C23C2×H(5,𝔽2)11yescharacteristic
500C23C2×H(5,𝔽2)12yes—
501C23C2×H(5,𝔽2)11yescharacteristic
502C23C2×2−1+414yes—
503Q8—416no—
504Q8—88no—
505D4—432no—
506D4—416no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—432no—
533C2×C4—416no—
534C2×C4—41no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—41no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4C2×#12314yes—
572C2×C4C2×#12318yes—
573C2×C4#55018yes—
574C22—42no—
575C22—44no—
576C22—48no—
577C22—42no—
578C22—48no—
579C22—48no—
580C22—44no—
581C22—42no—
582C22—44no—
583C22—82no—
584C22—82no—
585C22—84no—
586C22—82no—
587C22—84no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22C22×#12311yescharacteristic
594C22C2×#55012yes—
595C22#348214yes—
596C22#348311yescharacteristic
597C22#348411yescharacteristic
598C22#348411yescharacteristic
599C22#348612yes—
600C22#349012yes—
601C22#349012yes—
602C22#349214yes—
603C22#351912yes—
604C22#352214yes—
605C22#353614yes—
606C22#354412yes—
607C22#354911yescharacteristic
608C22#355012yes—
609C4—28no—
610C4—24no—
611C4—28no—
612C4—42no—
613C4—44no—
614C4—48no—
615C4—48no—
616C4—82no—
617C4—84no—
618C4—82no—
619C4—84no—
620C2—84no—
621C2—82no—
622C2#3535011yescharacteristic
623C2#3648312yes—
624C2#4340614yes—
625C2#4535612yes—
626C2#5192312yes—
627C2#5207211yescharacteristic
628C2#5905611yescharacteristic
629C2#6004712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2369C2semidirect
3. #16094C2semidirect
4. #15696C2semidirect
5. #15466C2semidirect
6. #15294C2semidirect
7. #15084C2semidirect
8. #14799C2semidirect
9. #14467C2semidirect
10. #14450C2semidirect
11. #14225C2semidirect
12. #14118C2semidirect
308. #95#41semidirect

Maximal subgroups

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

subgroupindex
C2×#24322
C2×#23692
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