Finite Group Databasebeta

← #674685

Subgroups — #674685

Order 512.

3258 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×#2537C211yesmaximal, characteristic
2C2×#2500C212yesmaximal, complemented
3C2×#2354C211yesmaximal, characteristic, complemented
4#15647C218yesmaximal, complemented
5#15233C212yesmaximal, complemented
6#15219C212yesmaximal, complemented
7#15126C214yesmaximal, complemented
8#14687C211yesmaximal, characteristic, complemented
9#13974C214yesmaximal, complemented
10#13910C211yesmaximal, characteristic, complemented
11#13906C211yesmaximal, characteristic, complemented
12#13879C212yesmaximal, complemented
13C4×#390C211yesmaximal, characteristic, complemented
14C4×#390C211yesmaximal, characteristic, complemented
15C22×#121C2212yes—
16C22×#121C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117C2212yes—
20#2537—28no—
21#2500—216no—
22C2×#394C2218yes—
23C2×#394C2214yes—
24#2354—28no—
25C2×#390C2212yes—
26C2×#390C2212yes—
27C2×#390C2212yes—
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#389C2214yes—
31C2×#388C2214yes—
32C2×#388C2218yes—
33C2×#384C2212yes—
34C2×#384C2211yescharacteristic
35C2×#384C2211yescharacteristic
36C2×#382C2212yes—
37C2×#382C2212yes—
38C2×#380C2214yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2214yes—
48C2×C43C2211yescharacteristic
49#1444C2214yes—
50#1443C2218yes—
51#1440C2218yes—
52#1440C2218yes—
53#1439C2218yes—
54#1439C2214yes—
55#1438C2218yes—
56#1437C2214yes—
57#1437C2214yes—
58#1436C2218yes—
59C4⋊#95C2214yes—
60C4⋊#95C2214yes—
61C4⋊#95C2211yescharacteristic
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2212yes—
66C23×D4C2311yescharacteristic
67C23×D4C2311yescharacteristic
68C23×D4C2314yes—
69C24×C4C2312yes—
70C24×C4C2312yes—
71C2×#121—28no—
72C2×#121—28no—
73C2×#121—28no—
74C2×#121—44no—
75C2×#121—48no—
76C2×C4×D4—24no—
77C2×C4×D4—24no—
78C2×C4×D4—24no—
79C2×C4×D4—24no—
80C2×C4×D4—24no—
81C2×C4×D4D418yes—
82C2×#117—28no—
83C2×#117—28no—
84C2×#117—44no—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2318yes—
96C22×C4⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2318yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2312yes—
111C22×C42C2314yes—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116#394—232no—
117#394—216no—
118#390—28no—
119#390—28no—
120#390—28no—
121#390—24no—
122#390D418yescomplemented
123#389—48no—
124#388—216no—
125#388—232no—
126#384—28no—
127#384—24no—
128#384—24no—
129#382—28no—
130#382—44no—
131#380—28no—
132#380—28no—
133#380—28no—
134#380—216no—
135C4×C4⋊C4—28no—
136C4×C4⋊C4—24no—
137C4×C4⋊C4—24no—
138C4×C22⋊C4—28no—
139C4×C22⋊C4—28no—
140C4×C22⋊C4—48no—
141C2×#95C2318yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2318yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2318yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2318yes—
153C43—24no—
154C25C2412yes—
155C25C2412yes—
156C22×D4—28no—
157C22×D4—28no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—24no—
165C22×D4—24no—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—44no—
169C22×D4—48no—
170C22×D4—48no—
171C22×D4C2×D414yes—
172C22×D4C2×D414yes—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—28no—
183C23×C4—28no—
184C23×C4C2×D418yes—
185C23×C4C2414yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2414yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2414yes—
193C23×C4C2412yes—
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197#121—432no—
198#121—432no—
199C4×D4—232no—
200C4×D4—416no—
201#117—432no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—216no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—48no—
230C2×C4⋊C4—48no—
231C2×C4⋊C4C2×D414yes—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4#4118yes—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—216no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—216no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—48no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—48no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—48no—
263C2×C22⋊C4C2×D418yes—
264C2×C22⋊C4C2×D4116yes—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—28no—
270C2×C42—28no—
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—44no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42C2×D414yes—
284C2×C42C2×D414yes—
285C2×C42#4114yes—
286C2×C42#4114yes—
287C2×C42#4114yes—
288#95—28no—
289#95—28no—
290#95—24no—
291#95—28no—
292#95—216no—
293#95—28no—
294#95—24no—
295#95—216no—
296#95—44no—
297#95—44no—
298#95—48no—
299#95—48no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—24no—
304C24—22no—
305C24—24no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—24no—
312C24—24no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—24no—
317C24—24no—
318C24—24no—
319C24—24no—
320C24C22×D414yes—
321C24C22×D414yes—
322C24C2511yescharacteristic
323C2×D4—28no—
324C2×D4—28no—
325C2×D4—28no—
326C2×D4—28no—
327C2×D4—28no—
328C2×D4—28no—
329C2×D4—416no—
330C2×D4—416no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—48no—
336C2×D4—48no—
337C2×D4—816no—
338C2×D4—88no—
339C2×D4—816no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—216no—
371C22×C4—216no—
372C22×C4—216no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—28no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—28no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—416no—
401C22×C4—416no—
402C22×C4C22×D414yes—
403C22×C4C22×D414yes—
404C22×C4C22×D414yes—
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C2×#4114yes—
410C22×C4C2×#4118yes—
411C22×C4C2×#4112yes—
412C22×C4C2×#4112yes—
413C22×C4C2×#4112yes—
414C22×C4C2×#4112yes—
415C22×C4C2×#4112yes—
416C22×C4C2×#4112yes—
417C22×C4C2×#4112yes—
418C22×C4C2×#4114yes—
419C22×C4H(5,𝔽2)14yes—
420C22×C4H(5,𝔽2)12yes—
421C22×C42−1+412yes—
422C4⋊C4—28no—
423C4⋊C4—28no—
424C4⋊C4—216no—
425C4⋊C4—44no—
426C4⋊C4—48no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—48no—
430C4⋊C4—44no—
431C4⋊C4—84no—
432C4⋊C4—84no—
433C4⋊C4—816no—
434C22⋊C4—216no—
435C22⋊C4—416no—
436C22⋊C4—416no—
437C22⋊C4—416no—
438C22⋊C4—432no—
439C22⋊C4—416no—
440C22⋊C4—416no—
441C22⋊C4—416no—
442C22⋊C4—416no—
443C22⋊C4—48no—
444C22⋊C4—88no—
445C42—28no—
446C42—28no—
447C42—416no—
448C42—48no—
449C42—48no—
450C42—48no—
451C42—48no—
452C42#123116yes—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—28no—
462C23—28no—
463C23—24no—
464C23—24no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—44no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—42no—
486C23—42no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23C23×D411yescharacteristic
500C23C23×D411yescharacteristic
501C23C22×#4111yescharacteristic
502C23C22×#4111yescharacteristic
503C23C22×#4111yescharacteristic
504C23C22×#4112yes—
505C23C2×H(5,𝔽2)12yes—
506C23C2×H(5,𝔽2)11yescharacteristic
507C23C2×2−1+411yescharacteristic
508C23#58512yes—
509C23#58512yes—
510D4—416no—
511D4—416no—
512D4—832no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—432no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—48no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—48no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—44no—
571C2×C4—816no—
572C2×C4C2×#12314yes—
573C2×C4C2×#12314yes—
574C2×C4C2×#12314yes—
575C2×C4#53214yes—
576C2×C4#53214yes—
577C2×C4#53514yes—
578C2×C4#54214yes—
579C2×C4D4214yes—
580C2×C4#54714yes—
581C2×C4#54714yes—
582C2×C4#55314yes—
583C2×C4#55318yes—
584C2×C4#55518yes—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—48no—
591C22—48no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—84no—
598C22—84no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—84no—
603C22—84no—
604C22—84no—
605C22—84no—
606C22—82no—
607C22C22×#12311yescharacteristic
608C22C2×#53211yescharacteristic
609C22C2×#53211yescharacteristic
610C22C2×#53511yescharacteristic
611C22C2×#54211yescharacteristic
612C22C2×D4211yescharacteristic
613C22C2×#54711yescharacteristic
614C22C2×#54711yescharacteristic
615C22D4×#4112yes—
616C22C2×#55312yes—
617C22C2×#55311yescharacteristic
618C22C2×#55512yes—
619C22#347512yes—
620C22#349312yes—
621C22#349612yes—
622C22#350011yescharacteristic
623C22#350011yescharacteristic
624C22#350612yes—
625C22#351212yes—
626C22#351712yes—
627C22#351812yes—
628C22#354612yes—
629C22#354712yes—
630C4—24no—
631C4—44no—
632C4—44no—
633C4—48no—
634C4—44no—
635C4—44no—
636C4—84no—
637C4—82no—
638C4—84no—
639C4—82no—
640C4#242218yes—
641C4#250018yescomplemented
642C2—84no—
643C2—84no—
644C2C2×#242211yescharacteristic
645C2C2×#250011yescharacteristic
646C2#3561811yescharacteristic
647C2#3612511yescharacteristic
648C2#3618511yescharacteristic
649C2#3646812yes—
650C2#3919412yes—
651C2#3936211yescharacteristic
652C2#3949111yescharacteristic
653C2#5922812yes—
654C2#5922912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2537C2
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×#2500C2semidirect
3. C2×#2354C2semidirect
4. #15647C2semidirect
5. #15233C2semidirect
6. #15219C2semidirect
7. #15126C2semidirect
8. #14687C2semidirect
9. #13974C2semidirect
10. #13910C2semidirect
11. #13906C2semidirect
12. #13879C2semidirect
13. C4×#390C2semidirect
14. C4×#390C2semidirect
122. #390D4semidirect
641. C4#2500semidirect

Maximal subgroups

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

subgroupindex
C2×#25372
C2×#25002
C2×#23542
#156472
#152332
#152192
#151262
#146872
#139742
#139102
#139062
#138792
C4×#3902