Finite Group Databasebeta

← #577204

Subgroups — #577204

Order 512.

3318 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2815C211yesmaximal, characteristic
2C2×#2333C212yesmaximal, complemented
3#14572C212yesmaximal, complemented
4#14544C214yesmaximal, complemented
5#14521C212yesmaximal, complemented
6#14521C214yesmaximal, complemented
7#14228C212yesmaximal, complemented
8#14209C214yesmaximal, complemented
9#14187C214yesmaximal, complemented
10#14175C212yesmaximal, complemented
11#14165C211yesmaximal, characteristic, complemented
12#14137C212yesmaximal, complemented
13#13776C211yesmaximal, characteristic, complemented
14C22×#121C2212yes—
15C22×C4×D4C2212yes—
16C23×C4⋊C4C2211yescharacteristic
17#2815C4116yescomplemented
18C2×#397C2214yes—
19C2×#392C2212yes—
20C2×#392C2214yes—
21C2×#392C2212yes—
22#2333—216no—
23C2×#390C2214yes—
24C2×#390C2212yes—
25C2×#390C2212yes—
26C2×#390C2212yes—
27C2×#388C2214yes—
28C2×#388C2214yes—
29C2×#388C2214yes—
30C2×#387C2214yes—
31C2×#386C2214yes—
32C2×#386C2214yes—
33C2×#386C2214yes—
34C2×#386C2214yes—
35C2×#386C2214yes—
36C2×#381C2211yescharacteristic
37C2×#381C2211yescharacteristic
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2214yes—
40C2×C4×C22⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2211yescharacteristic
42#1444C2212yes—
43#1443C2214yes—
44#1442C2212yes—
45#1439C2214yes—
46#1439C2214yes—
47#1439C2212yes—
48#1439C2212yes—
49#1439C2214yes—
50#1439C2214yes—
51#1438C2212yes—
52#1438C2212yes—
53#1438C2214yes—
54#1437C2214yes—
55#1437C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63#1434C2212yes—
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2211yescharacteristic
67C22⋊#95C2211yescharacteristic
68C4×#95C2212yes—
69C23×D4C2314yes—
70C23×D4C2312yes—
71C23×D4C2312yes—
72C23×D4C2312yes—
73C24×C4C2311yescharacteristic
74C24×C4C2314yes—
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C2×#121—28no—
78C2×#121—28no—
79C2×#121C2×C4116yescomplemented
80C2×C4×D4—44no—
81C2×C4×D4—48no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4—42no—
84C22×C4⋊C4—42no—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2311yescharacteristic
106C22×C42C2312yes—
107C22×C42C2312yes—
108C22×C42C2314yes—
109C22×C42C2312yes—
110C22×C42C2312yes—
111#397C2×C4132yescomplemented
112#392C2×C4132yescomplemented
113#392C2×C4116yescomplemented
114#392C2×C4116yescomplemented
115#390—216no—
116#390—44no—
117#390—44no—
118#390—44no—
119#388—216no—
120#388—48no—
121#388—48no—
122#387—216no—
123#386—216no—
124#386—216no—
125#386—48no—
126#386—48no—
127#386—48no—
128#381C2×C418yescomplemented
129#381D418yescomplemented
130C4×C22⋊C4—28no—
131C4×C22⋊C4—216no—
132C4×C22⋊C4—42no—
133C4×C22⋊C4—42no—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C25C2414yes—
154C25C2411yescharacteristic
155C22×D4—28no—
156C22×D4—28no—
157C22×D4—24no—
158C22×D4—24no—
159C22×D4—42no—
160C22×D4—44no—
161C22×D4—42no—
162C22×D4—44no—
163C22×D4—42no—
164C22×D4—42no—
165C22×D4—42no—
166C22×D4—44no—
167C22×D4—44no—
168C22×D4—44no—
169C22×D4—44no—
170C22×D4—44no—
171C22×D4—44no—
172C22×D4—42no—
173C22×D4—42no—
174C22×D4—42no—
175C22×D4C22×C4116yes—
176C22×D4C22×C418yes—
177C22×D4C22×C418yes—
178C22×D4C22×C418yes—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—28no—
184C23×C4—28no—
185C23×C4—41no—
186C23×C4—41no—
187C23×C4—41no—
188C23×C4—41no—
189C23×C4—41no—
190C23×C4—41no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4C22×C414yes—
199C23×C4C2×D414yes—
200C23×C4#4114yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2414yes—
211C23×C4C2412yes—
212#121—264no—
213C4×D4—432no—
214C2×C4⋊C4—24no—
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—28no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—416no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4C22×C418yes—
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—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—41no—
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⋊C4C22×C4116yes—
270C2×C22⋊C4C22×C414yes—
271C2×C22⋊C4C22×C414yes—
272C2×C22⋊C4C22×C4116yes—
273C2×C22⋊C4C22×C4116yes—
274C2×C22⋊C4C2×D414yes—
275C2×C22⋊C4C2×D414yes—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—28no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—44no—
282C2×C42—44no—
283C2×C42—44no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—44no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—42no—
296#95—44no—
297#95—42no—
298#95—42no—
299#95—42no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95C22×C4116yes—
305#95C22×C418yes—
306#95C22×C418yes—
307#95C2×D4116yes—
308C24—24no—
309C24—21no—
310C24—21no—
311C24—21no—
312C24—21no—
313C24—24no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—24no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24C23×C418yes—
329C24C23×C412yes—
330C24C22×D412yes—
331C24C2×#4112yes—
332C24C2511yescharacteristic
333C2×D4—232no—
334C2×D4—216no—
335C2×D4—48no—
336C2×D4—416no—
337C2×D4—48no—
338C2×D4—48no—
339C2×D4—416no—
340C2×D4—416no—
341C2×D4—416no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—416no—
345C2×D4—48no—
346C2×D4—84no—
347C2×D4—84no—
348C2×D4—84no—
349C2×D4—84no—
350C2×D4—88no—
351C2×D4—88no—
352C2×D4—88no—
353C2×D4—88no—
354C2×D4—84no—
355C2×D4—84no—
356C2×D4—88no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
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—22no—
395C22×C4—28no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—416no—
415C22×C4—416no—
416C22×C4—44no—
417C22×C4—88no—
418C22×C4—88no—
419C22×C4—84no—
420C22×C4—84no—
421C22×C4—88no—
422C22×C4—84no—
423C22×C4—84no—
424C22×C4—88no—
425C22×C4—88no—
426C22×C4—84no—
427C22×C4—84no—
428C22×C4C4×D4116yescomplemented
429C22×C4C23×C414yes—
430C22×C4C23×C414yes—
431C22×C4C23×C418yes—
432C22×C4C23×C414yes—
433C22×C4C22×D414yes—
434C22×C4C22×D414yes—
435C22×C4C22×D414yes—
436C22×C4C2×#4114yes—
437C22×C4H(5,𝔽2)18yes—
438C22×C4H(5,𝔽2)12yes—
439C22×C4H(5,𝔽2)12yes—
440C4⋊C4—216no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—88no—
445C4⋊C4—84no—
446C22⋊C4—232no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—416no—
450C22⋊C4—416no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—416no—
454C22⋊C4—416no—
455C22⋊C4—416no—
456C22⋊C4—88no—
457C22⋊C4—88no—
458C22⋊C4—88no—
459C22⋊C4—88no—
460C22⋊C4—88no—
461C42—48no—
462C42—84no—
463C42—84no—
464C42—88no—
465C42—84no—
466C23—22no—
467C23—22no—
468C23—28no—
469C23—28no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—24no—
475C23—24no—
476C23—28no—
477C23—28no—
478C23—28no—
479C23—28no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—42no—
484C23—44no—
485C23—42no—
486C23—42no—
487C23—42no—
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—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23C2×C4×D414yes—
518C23C24×C411yescharacteristic
519C23C23×D411yescharacteristic
520C23C22×#4111yescharacteristic
521C23C2×H(5,𝔽2)14yes—
522C23C2×H(5,𝔽2)11yescharacteristic
523C23C2×H(5,𝔽2)11yescharacteristic
524C23#58514yes—
525C23#58511yescharacteristic
526C23#58511yescharacteristic
527D4—432no—
528D4—816no—
529D4—832no—
530D4—816no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—416no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—88no—
593C2×C4—832no—
594C2×C4—88no—
595C2×C4C2×C4×D418yes—
596C22—28no—
597C22—24no—
598C22—44no—
599C22—44no—
600C22—42no—
601C22—42no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—48no—
606C22—48no—
607C22—48no—
608C22—48no—
609C22—48no—
610C22—48no—
611C22—48no—
612C22—84no—
613C22—84no—
614C22—84no—
615C22—84no—
616C22—84no—
617C22—84no—
618C22—84no—
619C22—84no—
620C22C22×C4×D411yescharacteristic
621C22C4×H(5,𝔽2)11yescharacteristic
622C22C4×H(5,𝔽2)14yes—
623C22C4×H(5,𝔽2)11yescharacteristic
624C22#348814yes—
625C22#349011yescharacteristic
626C22#349111yescharacteristic
627C22#349611yescharacteristic
628C22#349611yescharacteristic
629C22#350812yes—
630C22#351012yes—
631C22#353212yes—
632C22#353914yes—
633C22#354012yes—
634C22#354312yes—
635C22#355112yes—
636C22#356014yes—
637C4—28no—
638C4—44no—
639C4—42no—
640C4—42no—
641C4—44no—
642C4—44no—
643C4—48no—
644C4—48no—
645C4—44no—
646C4—84no—
647C4—82no—
648C2—44no—
649C2—88no—
650C2#3426912yes—
651C2#3427311yescharacteristic
652C2#3451112yes—
653C2#3452012yes—
654C2#4683612yes—
655C2#4685612yes—
656C2#4703111yescharacteristic
657C2#4703611yescharacteristic
658C2#4745712yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC2×#2815C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2333C2semidirect
3. #14572C2semidirect
4. #14544C2semidirect
5. #14521C2semidirect
6. #14521C2semidirect
7. #14228C2semidirect
8. #14209C2semidirect
9. #14187C2semidirect
10. #14175C2semidirect
11. #14165C2semidirect
12. #14137C2semidirect
13. #13776C2semidirect
17. #2815C4semidirect
79. C2×#121C2×C4semidirect
111. #397C2×C4semidirect
112. #392C2×C4semidirect
113. #392C2×C4semidirect
114. #392C2×C4semidirect
128. #381C2×C4semidirect
129. #381D4semidirect
428. C22×C4C4×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28152
C2×#23332
#145722
#145442
#145212
#142282
#142092
#141872
#141752
#141652
#141372
#137762