Finite Group Databasebeta

← #4386451

Subgroups — #4386451

Order 512.

2806 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2815C211yesmaximal, characteristic
2C2×#2607C211yesmaximal, characteristic, complemented
3#16357C212yesmaximal, complemented
4#16338C212yesmaximal, complemented
5#15759C214yesmaximal, complemented
6#15697C214yesmaximal, complemented
7#15428C214yesmaximal, complemented
8#15317C214yesmaximal, complemented
9#14977C212yesmaximal, complemented
10#14915C212yesmaximal, complemented
11#14290C212yesmaximal, complemented
12#14137C212yesmaximal, complemented
13#13776C211yesmaximal, characteristic, complemented
14C22×#124C2212yes—
15C22×#121C2212yes—
16C23×C4⋊C4C2211yescharacteristic
17#2815—28no—
18#2607—28no—
19C2×#399C2212yes—
20C2×#397C2214yes—
21C2×#394C2214yes—
22C2×#393C2212yes—
23C2×#393C2214yes—
24C2×#392C2212yes—
25C2×#392C2214yes—
26C2×#392C2212yes—
27C2×#391C2214yes—
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#388C2214yes—
32C2×#387C2214yes—
33C2×#386C2214yes—
34C2×#386C2214yes—
35C2×#381C2211yescharacteristic
36C2×#381C2211yescharacteristic
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2211yescharacteristic
40#1446C2212yes—
41#1445C2214yes—
42#1445C2212yes—
43#1443C2214yes—
44#1442C2214yes—
45#1440C2214yes—
46#1440C2214yes—
47#1440C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1438C2212yes—
53#1438C2214yes—
54#1438C2214yes—
55#1438C2214yes—
56#1438C2214yes—
57#1438C2212yes—
58#1437C2214yes—
59#1437C2214yes—
60#1437C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65#1434C2212yes—
66#1434C2212yes—
67C22⋊#95C2211yescharacteristic
68C22⋊#95C2211yescharacteristic
69C23×Q8C2312yes—
70C23×D4C2314yes—
71C23×D4C2312yes—
72C23×D4C2312yes—
73C23×D4C2312yes—
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C2×#124—44no—
78C2×#124—48no—
79C2×#121—28no—
80C2×#121—28no—
81C2×#121—28no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4—42no—
84C22×C4⋊C4—42no—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2311yescharacteristic
99C22×C22⋊C4C2311yescharacteristic
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2311yescharacteristic
106C22×C42C2312yes—
107C22×C42C2312yes—
108C22×C42C2312yes—
109C22×C42C2312yes—
110#399—28no—
111#397—216no—
112#394—48no—
113#393—28no—
114#393—48no—
115#392—28no—
116#392—28no—
117#392—216no—
118#391—216no—
119#390—44no—
120#390—44no—
121#390—44no—
122#388—48no—
123#387—216no—
124#386—48no—
125#386—48no—
126#381—24no—
127#381D418yes—
128C4×C22⋊C4—28no—
129C4×C22⋊C4—42no—
130C4×C22⋊C4—42no—
131C2×#95C2312yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C25C2414yes—
152C25C2411yescharacteristic
153C22×Q8—24no—
154C22×Q8—28no—
155C22×Q8—42no—
156C22×Q8—44no—
157C22×Q8—42no—
158C22×D4—28no—
159C22×D4—28no—
160C22×D4—28no—
161C22×D4—24no—
162C22×D4—24no—
163C22×D4—24no—
164C22×D4—24no—
165C22×D4—24no—
166C22×D4—42no—
167C22×D4—44no—
168C22×D4—42no—
169C22×D4—42no—
170C22×D4—44no—
171C22×D4—44no—
172C22×D4—44no—
173C22×D4—44no—
174C22×D4—44no—
175C22×D4—42no—
176C22×D4—42no—
177C22×D4—42no—
178C22×D4—42no—
179C22×D4—44no—
180C22×D4—42no—
181C22×D4—44no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—41no—
187C23×C4—41no—
188C23×C4—41no—
189C23×C4—41no—
190C23×C4—41no—
191C23×C4—41no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4C2×D414yes—
195C23×C4#4114yes—
196C23×C4#4114yes—
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208#124—432no—
209#121—432no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—416no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
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×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⋊C4C2×D414yes—
273C2×C22⋊C4C2×D414yes—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—44no—
279C2×C42—44no—
280C2×C42—42no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42—42no—
284#95—28no—
285#95—24no—
286#95—28no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—44no—
294#95—44no—
295#95—42no—
296#95—42no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95C2×D4116yes—
304C24—24no—
305C24—21no—
306C24—21no—
307C24—21no—
308C24—21no—
309C24—24no—
310C24—24no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—24no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24C22×D412yes—
326C24C2×#4112yes—
327C24C2×#4112yes—
328C24C2511yescharacteristic
329C2×Q8—48no—
330C2×Q8—416no—
331C2×Q8—84no—
332C2×Q8—88no—
333C2×Q8—84no—
334C2×D4—48no—
335C2×D4—48no—
336C2×D4—416no—
337C2×D4—416no—
338C2×D4—48no—
339C2×D4—48no—
340C2×D4—416no—
341C2×D4—48no—
342C2×D4—84no—
343C2×D4—84no—
344C2×D4—84no—
345C2×D4—84no—
346C2×D4—88no—
347C2×D4—88no—
348C2×D4—84no—
349C2×D4—84no—
350C2×D4—88no—
351C2×D4—88no—
352C2×D4—88no—
353C2×D4—88no—
354C2×D4—88no—
355C2×D4—84no—
356C2×D4—84no—
357C2×D4—88no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—28no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—84no—
420C22×C4—84no—
421C22×C4—84no—
422C22×C4—84no—
423C22×C4—84no—
424C22×C4—84no—
425C22×C4#124116yescomplemented
426C22×C4C22×D414yes—
427C22×C4C22×D414yes—
428C22×C4C22×D414yes—
429C22×C4C2×#4114yes—
430C22×C4C2×#4114yes—
431C22×C4H(5,𝔽2)18yes—
432C22×C4H(5,𝔽2)12yes—
433C22×C4H(5,𝔽2)12yes—
434C4⋊C4—416no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—84no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—416no—
442C22⋊C4—416no—
443C22⋊C4—416no—
444C22⋊C4—416no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—416no—
448C22⋊C4—88no—
449C22⋊C4—88no—
450C22⋊C4—88no—
451C22⋊C4—88no—
452C22⋊C4—88no—
453C22⋊C4—88no—
454C22⋊C4—88no—
455C42—48no—
456C42—84no—
457C42—84no—
458C42—84no—
459C23—22no—
460C23—22no—
461C23—28no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—28no—
467C23—24no—
468C23—24no—
469C23—42no—
470C23—44no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23C2×#12414yes—
511C23C23×D411yescharacteristic
512C23C22×#4111yescharacteristic
513C23C22×#4111yescharacteristic
514C23C2×H(5,𝔽2)11yescharacteristic
515C23C2×H(5,𝔽2)11yescharacteristic
516C23C2×H(5,𝔽2)14yes—
517C23#58511yescharacteristic
518C23#58511yescharacteristic
519C23#58514yes—
520Q8—816no—
521D4—816no—
522D4—816no—
523D4—816no—
524D4—832no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—416no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—88no—
587C2×C4—88no—
588C2×C4C2×#12418yes—
589C22—24no—
590C22—28no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—42no—
596C22—42no—
597C22—44no—
598C22—84no—
599C22—84no—
600C22—84no—
601C22—84no—
602C22—84no—
603C22—84no—
604C22—84no—
605C22—84no—
606C22—84no—
607C22—84no—
608C22—84no—
609C22—84no—
610C22—84no—
611C22—84no—
612C22—84no—
613C22C22×#12411yescharacteristic
614C22#345911yescharacteristic
615C22#345911yescharacteristic
616C22#348814yes—
617C22#349011yescharacteristic
618C22#349111yescharacteristic
619C22#349611yescharacteristic
620C22#349611yescharacteristic
621C22#350614yes—
622C22#350812yes—
623C22#351012yes—
624C22#353212yes—
625C22#353314yes—
626C22#354012yes—
627C22#354312yes—
628C22#355112yes—
629C22#356214yes—
630C4—28no—
631C4—44no—
632C4—48no—
633C4—42no—
634C4—42no—
635C4—44no—
636C4—82no—
637C4—82no—
638C4—82no—
639C4—84no—
640C4—84no—
641C2—44no—
642C2—164no—
643C2#3499311yescharacteristic
644C2#3555612yes—
645C2#4100112yes—
646C2#4101612yes—
647C2#4703111yescharacteristic
648C2#4703611yescharacteristic
649C2#4722312yes—
650C2#5068412yes—
651C2#5106012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2815C2
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×#2607C2semidirect
3. #16357C2semidirect
4. #16338C2semidirect
5. #15759C2semidirect
6. #15697C2semidirect
7. #15428C2semidirect
8. #15317C2semidirect
9. #14977C2semidirect
10. #14915C2semidirect
11. #14290C2semidirect
12. #14137C2semidirect
13. #13776C2semidirect
425. C22×C4#124semidirect

Maximal subgroups

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

subgroupindex
C2×#28152
C2×#26072
#163572
#163382
#157592
#156972
#154282
#153172
#149772
#149152
#142902
#141372
#137762