Finite Group Databasebeta

← #951618

Subgroups — #951618

Order 512.

2198 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2494C212yesmaximal, complemented
2#17230C212yesmaximal, complemented
3#17109C212yesmaximal
4#16585C214yesmaximal, complemented
5#16342C214yesmaximal, complemented
6#16088C212yesmaximal, complemented
7#15780C212yesmaximal
8#15223C212yesmaximal, complemented
9#15131C212yesmaximal, complemented
10#15129C212yesmaximal
11#14277C211yesmaximal, characteristic, complemented
12#14277C211yesmaximal, characteristic, complemented
13#14045C212yesmaximal, complemented
14#13876C212yesmaximal, complemented
15#13847C211yesmaximal, characteristic
16C22×C4×D4C2211yescharacteristic
17#2494—216no—
18C2×#401C2214yes—
19C2×#397C2214yes—
20C2×#396C2214yes—
21C2×#394C2214yes—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#388C2214yes—
27C2×#388C2214yes—
28C2×#387C2214yes—
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2212yes—
41C2×#376C2211yescharacteristic
42#1445C2211yescharacteristic
43#1445C2211yescharacteristic
44#1445C2214yes—
45#1445C2214yes—
46#1444C2212yes—
47#1444C2212yes—
48#1444C2212yes—
49#1444C2212yes—
50#1444C2214yes—
51#1442C2214yes—
52#1442C2212yes—
53#1442C2212yes—
54#1442C2212yes—
55#1442C2212yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1440C2214yes—
61#1440C2214yes—
62#1439C2212yes—
63#1439C2212yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1438C2211yescharacteristic
69#1438C2211yescharacteristic
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2214yes—
76C23×D4C2311yescharacteristic
77C24×C4C2312yes—
78C2×C4×D4—24no—
79C2×C4×D4—42no—
80C2×C4×D4—42no—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C42C2312yes—
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2314yes—
109#401—216no—
110#397—216no—
111#396—48no—
112#394—48no—
113#389—28no—
114#389—28no—
115#389—28no—
116#389—44no—
117#388—216no—
118#388—48no—
119#387—216no—
120#384—42no—
121#384—42no—
122#382—24no—
123#382D418yescomplemented
124#380—28no—
125#380—28no—
126#380—44no—
127#380—44no—
128C4×C4⋊C4—28no—
129C4×C4⋊C4—42no—
130C4×C22⋊C4—28no—
131C4×C22⋊C4—28no—
132#376—42no—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2311yescharacteristic
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2311yescharacteristic
155C2×#95C2311yescharacteristic
156C2×#95C2311yescharacteristic
157C2×#95C2311yescharacteristic
158C2×#95C2311yescharacteristic
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C25C2412yes—
163C22×D4—24no—
164C22×D4—22no—
165C22×D4—41no—
166C22×D4—41no—
167C22×D4—42no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4C2414yes—
176C23×C4C2414yes—
177C23×C4C2414yes—
178C23×C4C2412yes—
179C23×C4C2412yes—
180C23×C4C2411yescharacteristic
181C23×C4C2414yes—
182C23×C4C2411yescharacteristic
183C23×C4C2411yescharacteristic
184C23×C4C2411yescharacteristic
185C23×C4C2411yescharacteristic
186C23×C4C2411yescharacteristic
187C23×C4C2412yes—
188C23×C4C2411yescharacteristic
189C4×D4—416no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4C2×D418yes—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—28no—
265C2×C42—44no—
266C2×C42—44no—
267C2×C42C2×D414yes—
268C2×C42#4114yes—
269C2×C42#4114yes—
270#95—24no—
271#95—24no—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—28no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—28no—
282#95—44no—
283#95—44no—
284#95—41no—
285#95—41no—
286#95—41no—
287#95—41no—
288#95—41no—
289#95—41no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95C2×D418yes—
298#95C2×D418yes—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24H(5,𝔽2)14yes—
314C24C2511yescharacteristic
315C2×D4—48no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—48no—
319C2×D4—44no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4C22×D412yes—
399C22×C4C22×D418yes—
400C22×C4C22×D412yes—
401C22×C4C22×D412yes—
402C22×C4C2×#4112yes—
403C22×C4C2×#4112yes—
404C22×C4C2×#4112yes—
405C22×C4C2×#4112yes—
406C22×C4C2×#4112yes—
407C22×C4C2×#4112yes—
408C22×C4H(5,𝔽2)12yes—
409C22×C4H(5,𝔽2)12yes—
410C22×C42−1+414yes—
411C22×C42−1+412yes—
412C22×C42−1+412yes—
413C4⋊C4—216no—
414C4⋊C4—416no—
415C4⋊C4—48no—
416C4⋊C4—44no—
417C4⋊C4—48no—
418C4⋊C4—48no—
419C4⋊C4—48no—
420C4⋊C4—88no—
421C4⋊C4—88no—
422C4⋊C4—84no—
423C4⋊C4—84no—
424C4⋊C4—84no—
425C4⋊C4—84no—
426C4⋊C4—88no—
427C22⋊C4—216no—
428C22⋊C4—48no—
429C22⋊C4—416no—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C22⋊C4—48no—
433C22⋊C4—88no—
434C22⋊C4—88no—
435C22⋊C4—88no—
436C42—28no—
437C42—28no—
438C42—28no—
439C42—48no—
440C42—88no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23C23×D411yescharacteristic
477C23C22×#4111yescharacteristic
478C23C22×#4111yescharacteristic
479C23C2×H(5,𝔽2)11yescharacteristic
480C23C2×H(5,𝔽2)11yescharacteristic
481C23C2×H(5,𝔽2)12yes—
482C23C2×2−1+411yescharacteristic
483C23C2×2−1+411yescharacteristic
484C23C2×2−1+412yes—
485C23#58512yes—
486C23#58512yes—
487D4—416no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—28no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—816no—
568C2×C4#53714yes—
569C2×C4#53814yes—
570C2×C4#53914yes—
571C2×C4#54714yes—
572C2×C4#55014yes—
573C2×C4#55614yes—
574C2×C4#56314yes—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22C2×#53711yescharacteristic
591C22C2×#53811yescharacteristic
592C22C2×#53911yescharacteristic
593C22#346112yes—
594C22C2×#54711yescharacteristic
595C22C2×#55011yescharacteristic
596C22C2×#55611yescharacteristic
597C22#348812yes—
598C22#348912yes—
599C22#349212yes—
600C22#349312yes—
601C22#349412yes—
602C22C2×#56311yescharacteristic
603C22#353711yescharacteristic
604C22#354011yescharacteristic
605C22#354112yes—
606C22#354212yes—
607C22#354412yes—
608C22#355011yescharacteristic
609C22#355111yescharacteristic
610C22#355512yes—
611C22#355912yes—
612C22#356512yes—
613C4—42no—
614C4—48no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—44no—
619C4—42no—
620C4—42no—
621C4—84no—
622C4—82no—
623C4—84no—
624C4—82no—
625C4—84no—
626C4#272718yes—
627C2—84no—
628C2#3728411yescharacteristic
629C2#3751411yescharacteristic
630C2#3776612yes—
631C2#3796012yes—
632C2#4250611yescharacteristic
633C2#4268411yescharacteristic
634C2#4292911yescharacteristic
635C2#4295211yescharacteristic
636C2#4498012yes—
637C2C2×#272711yescharacteristic
638C2#5313312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2494C2semidirect
2. #17230C2semidirect
4. #16585C2semidirect
5. #16342C2semidirect
6. #16088C2semidirect
8. #15223C2semidirect
9. #15131C2semidirect
11. #14277C2semidirect
12. #14277C2semidirect
13. #14045C2semidirect
14. #13876C2semidirect
123. #382D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24942
#172302
#171092
#165852
#163422
#160882
#157802
#152232
#151312
#151292
#142772
#140452
#138762
#138472