Finite Group Databasebeta

← #1421713

Subgroups — #1421713

Order 512.

2916 conjugacy classes of subgroups.

The subgroup lattice has 622 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#15701C214yesmaximal, complemented
5#15465C212yesmaximal, complemented
6#15296C214yesmaximal, complemented
7#15112C212yesmaximal, complemented
8#14778C214yesmaximal, complemented
9#14501C212yesmaximal, complemented
10#14450C211yesmaximal, characteristic, complemented
11#14240C212yesmaximal, complemented
12#14177C214yesmaximal, complemented
13#13937C212yesmaximal
14C23×#41C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18#2432—28no—
19C2×#400C2212yes—
20C2×#399C2212yes—
21C2×#397C2214yes—
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×#385C2214yes—
35C2×#380C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2214yes—
38C2×C4×C22⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2212yes—
40#1446C2212yes—
41#1445C2212yes—
42#1442C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2212yes—
46#1441C2214yes—
47#1441C2214yes—
48#1440C2214yes—
49#1440C2214yes—
50#1439C2214yes—
51#1439C2212yes—
52#1439C2214yes—
53#1439C2214yes—
54#1438C2214yes—
55#1438C2214yes—
56#1437C2214yes—
57#1437C2214yes—
58#1437C2214yes—
59#1437C2212yes—
60#1436C2214yes—
61#1436C2212yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1434C2212yes—
66#1434C2214yes—
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⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2314yes—
111C22×C42C2312yes—
112#400—44no—
113#399—28no—
114#397—216no—
115#397—44no—
116#394—216no—
117#391—28no—
118#391—28no—
119#391—24no—
120#391—28no—
121#390—24no—
122#390—44no—
123#390—44no—
124#389—216no—
125#388—48no—
126#385—216no—
127#380—44no—
128C4×C4⋊C4—216no—
129C4×C4⋊C4—28no—
130C4×C22⋊C4—24no—
131C4×C22⋊C4—44no—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C25C2411yescharacteristic
155C25C2412yes—
156C2×#41—416no—
157C2×#41—432no—
158C2×#41—416no—
159C2×#41—432no—
160C2×#41—416no—
161C22×Q8—24no—
162C22×Q8—42no—
163C22×Q8—41no—
164C22×Q8—41no—
165C22×Q8—41no—
166C22×Q8—41no—
167C22×Q8—42no—
168C22×Q8C2×D414yes—
169C22×Q8C2×D414yes—
170C22×Q8#4118yes—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—22no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—42no—
180C22×D4—42no—
181C22×D4—41no—
182C22×D4—41no—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2414yes—
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C4⋊Q8—232no—
207#124—416no—
208C22⋊Q8—416no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4C2×D4116yes—
234C2×C4⋊C4#4118yescomplemented
235C2×C22⋊C4—24no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C42—24no—
266C2×C42—22no—
267C2×C42—28no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—42no—
272C2×C42—44no—
273C2×C42—44no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42—44no—
277C2×C42—42no—
278C2×C42—42no—
279C2×C42C2×D414yes—
280#95—24no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—24no—
291#95—28no—
292#95—24no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—42no—
299#95—44no—
300#95—42no—
301#95—44no—
302C24—21no—
303C24—22no—
304C24—24no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—21no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—24no—
319C24H(5,𝔽2)12yes—
320C24H(5,𝔽2)14yes—
321C24C2511yescharacteristic
322#41—4128no—
323C2×Q8—28no—
324C2×Q8—44no—
325C2×Q8—44no—
326C2×Q8—48no—
327C2×Q8—48no—
328C2×Q8—44no—
329C2×Q8—44no—
330C2×Q8—84no—
331C2×Q8—84no—
332C2×Q8#123116yes—
333C2×D4—28no—
334C2×D4—216no—
335C2×D4—48no—
336C2×D4—48no—
337C2×D4—48no—
338C2×D4—48no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—44no—
342C2×D4—48no—
343C2×D4—44no—
344C2×D4—48no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—24no—
354C22×C4—28no—
355C22×C4—216no—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—44no—
408C22×C4—48no—
409C22×C4—44no—
410C22×C4—48no—
411C22×C4C22×D414yes—
412C22×C4C22×D412yes—
413C22×C4C22×D414yes—
414C22×C4C22×D414yes—
415C22×C4C2×#4118yes—
416C22×C4C2×#4114yes—
417C22×C4C2×#4114yes—
418C22×C4C2×#4114yes—
419C22×C4C2×#4114yes—
420C22×C4H(5,𝔽2)12yes—
421C22×C42−1+414yes—
422C4⋊C4—232no—
423C4⋊C4—216no—
424C4⋊C4—48no—
425C4⋊C4—48no—
426C4⋊C4—44no—
427C4⋊C4—84no—
428C4⋊C4—84no—
429C4⋊C4—84no—
430C4⋊C4—88no—
431C22⋊C4—416no—
432C22⋊C4—416no—
433C22⋊C4—48no—
434C22⋊C4—416no—
435C22⋊C4—48no—
436C22⋊C4—416no—
437C22⋊C4—88no—
438C22⋊C4—88no—
439C22⋊C4—88no—
440C22⋊C4—88no—
441C42—28no—
442C42—416no—
443C42—44no—
444C42—84no—
445C42—84no—
446C42—84no—
447C23—22no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—28no—
453C23—24no—
454C23—24no—
455C23—28no—
456C23—41no—
457C23—42no—
458C23—42no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—42no—
464C23—41no—
465C23—42no—
466C23—41no—
467C23—42no—
468C23—44no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—42no—
480C23—41no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—44no—
485C23—44no—
486C23C23×D411yescharacteristic
487C23C22×#4112yes—
488C23C22×#4112yes—
489C23C2×H(5,𝔽2)11yescharacteristic
490C23C2×H(5,𝔽2)12yes—
491C23C2×H(5,𝔽2)11yescharacteristic
492C23C2×2−1+412yes—
493C23#58514yes—
494Q8—416no—
495Q8—88no—
496D4—432no—
497D4—416no—
498C2×C4—24no—
499C2×C4—28no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—28no—
512C2×C4—24no—
513C2×C4—28no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—432no—
521C2×C4—416no—
522C2×C4—41no—
523C2×C4—44no—
524C2×C4—41no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4C2×#12314yes—
563C2×C4C2×#12318yes—
564C2×C4#54818yes—
565C22—44no—
566C22—42no—
567C22—44no—
568C22—48no—
569C22—48no—
570C22—42no—
571C22—44no—
572C22—42no—
573C22—48no—
574C22—82no—
575C22—82no—
576C22—84no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—84no—
583C22—82no—
584C22C22×#12311yescharacteristic
585C22C2×#54812yes—
586C22#348014yes—
587C22#348311yescharacteristic
588C22#348411yescharacteristic
589C22#348411yescharacteristic
590C22#348712yes—
591C22#349012yes—
592C22#349012yes—
593C22#351314yes—
594C22#351614yes—
595C22#352212yes—
596C22#354512yes—
597C22#355011yescharacteristic
598C22#355112yes—
599C22#355914yes—
600C4—28no—
601C4—24no—
602C4—28no—
603C4—42no—
604C4—44no—
605C4—48no—
606C4—84no—
607C4—84no—
608C4—84no—
609C4—82no—
610C4—82no—
611C2—84no—
612C2—82no—
613C2#3535011yescharacteristic
614C2#3657812yes—
615C2#4333014yes—
616C2#4527112yes—
617C2#5198812yes—
618C2#5208311yescharacteristic
619C2#5908411yescharacteristic
620C2#5979812yes—
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. #15701C2semidirect
5. #15465C2semidirect
6. #15296C2semidirect
7. #15112C2semidirect
8. #14778C2semidirect
9. #14501C2semidirect
10. #14450C2semidirect
11. #14240C2semidirect
12. #14177C2semidirect
234. C2×C4⋊C4#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
#160942
#157012
#154652
#152962
#151122
#147782
#145012
#144502
#142402
#141772
#139372