Finite Group Databasebeta

← #4464057

Subgroups — #4464057

Order 512.

3038 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2618C212yesmaximal, complemented
2#17221C211yesmaximal, characteristic, complemented
3#17196C212yesmaximal, complemented
4#17187C214yesmaximal, complemented
5#17140C212yesmaximal, complemented
6#17121C212yesmaximal, complemented
7#17093C214yesmaximal, complemented
8#17082C211yesmaximal, characteristic, complemented
9#14926C212yesmaximal, complemented
10#14926C212yesmaximal, complemented
11#14757C214yesmaximal, complemented
12#14735C214yesmaximal, complemented
13C2×#1445C211yesmaximal, characteristic, complemented
14C22×#123C2214yes—
15C22×#121C2212yes—
16C22×#121C2212yes—
17C22×#121C2212yes—
18C23×C4⋊C4C2211yescharacteristic
19#2618—216no—
20C2×#398C2212yes—
21C2×#398C2212yes—
22C2×#397C2214yes—
23C2×#396C2214yes—
24C2×#396C2214yes—
25C2×#394C2212yes—
26C2×#394C2212yes—
27C2×#392C2214yes—
28C2×#386C2214yes—
29C2×#386C2214yes—
30C2×#386C2214yes—
31C2×#380C2212yes—
32C22×#95C2211yescharacteristic
33C22×#95C2211yescharacteristic
34C22×#95C2214yes—
35#1445C2214yes—
36#1445C2211yescharacteristic
37#1445C2211yescharacteristic
38#1445C2211yescharacteristic
39#1445C2211yescharacteristic
40#1445C2214yes—
41#1444C2212yes—
42#1444C2212yes—
43#1444C2214yes—
44#1444C2212yes—
45#1444C2212yes—
46#1443C2214yes—
47#1443C2214yes—
48#1442C2214yes—
49#1440C2214yes—
50#1440C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1438C2212yes—
60#1438C2214yes—
61#1438C2212yes—
62#1438C2212yes—
63#1438C2212yes—
64#1438C2214yes—
65#1438C2214yes—
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C23×D4C2314yes—
69C23×D4C2312yes—
70C23×D4C2312yes—
71C23×D4C2312yes—
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C24×C4C2312yes—
75C24×C4C2312yes—
76C2×#123—216no—
77C2×#123—48no—
78C2×#123—48no—
79C2×#121—28no—
80C2×#121—44no—
81C2×#121—44no—
82C2×#121—44no—
83C2×#121—44no—
84C2×#121—48no—
85C2×#121—48no—
86C22×C4⋊C4—42no—
87C22×C4⋊C4—42no—
88C22×C4⋊C4D418yescomplemented
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110#398—28no—
111#398—28no—
112#397—48no—
113#396—216no—
114#396—216no—
115#394—28no—
116#394—28no—
117#392—48no—
118#386—216no—
119#386—48no—
120#386—48no—
121#380—28no—
122C2×#95—48no—
123C2×#95—42no—
124C2×#95—42no—
125C2×#95C2314yes—
126C2×#95C2312yes—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2311yescharacteristic
138C2×#95C2311yescharacteristic
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2311yescharacteristic
146C2×#95C2311yescharacteristic
147C2×#95C2314yes—
148C2×#95C2314yes—
149C25C2414yes—
150C25C2411yescharacteristic
151C22×D4—28no—
152C22×D4—28no—
153C22×D4—28no—
154C22×D4—28no—
155C22×D4—28no—
156C22×D4—28no—
157C22×D4—28no—
158C22×D4—24no—
159C22×D4—24no—
160C22×D4—24no—
161C22×D4—42no—
162C22×D4—42no—
163C22×D4—44no—
164C22×D4—44no—
165C22×D4—42no—
166C22×D4—42no—
167C22×D4—42no—
168C22×D4—42no—
169C22×D4—44no—
170C22×D4—44no—
171C22×D4—44no—
172C22×D4—44no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—28no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—28no—
181C23×C4—28no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4C2×D414yes—
191C23×C4C2×D418yes—
192C23×C4C2414yes—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2414yes—
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203#123—464no—
204#121—432no—
205#121—432no—
206#121—432no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—416no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4C2×D414yes—
231C2×C4⋊C4C2×D418yes—
232C2×C4⋊C4C2×D414yes—
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—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
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×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—
269#95—24no—
270#95—24no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—24no—
276#95—24no—
277#95—44no—
278#95—44no—
279#95—41no—
280#95—41no—
281#95—42no—
282#95—42no—
283#95—41no—
284#95—41no—
285#95—42no—
286#95—42no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—24no—
298C24—24no—
299C24—24no—
300C24—24no—
301C24—24no—
302C24—24no—
303C24—24no—
304C24—24no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—24no—
311C24—24no—
312C24C22×D412yes—
313C24H(5,𝔽2)12yes—
314C24H(5,𝔽2)12yes—
315C24H(5,𝔽2)18yes—
316C24C2511yescharacteristic
317C2×D4—48no—
318C2×D4—48no—
319C2×D4—48no—
320C2×D4—416no—
321C2×D4—416no—
322C2×D4—416no—
323C2×D4—416no—
324C2×D4—416no—
325C2×D4—416no—
326C2×D4—416no—
327C2×D4—88no—
328C2×D4—88no—
329C2×D4—84no—
330C2×D4—84no—
331C2×D4—88no—
332C2×D4—88no—
333C2×D4—88no—
334C2×D4—84no—
335C2×D4—84no—
336C2×D4—84no—
337C2×D4—84no—
338C2×D4—88no—
339C22×C4—28no—
340C22×C4—216no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—216no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
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—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—416no—
396C22×C4—416no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—416no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—88no—
405C22×C4—88no—
406C22×C4—84no—
407C22×C4—84no—
408C22×C4—84no—
409C22×C4—84no—
410C22×C4—88no—
411C22×C4C22×D414yes—
412C22×C4C22×D414yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4H(5,𝔽2)14yes—
416C22×C42−1+414yes—
417C4⋊C4—44no—
418C4⋊C4—48no—
419C4⋊C4—44no—
420C4⋊C4—48no—
421C4⋊C4—416no—
422C4⋊C4—416no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—88no—
426C22⋊C4—416no—
427C22⋊C4—416no—
428C22⋊C4—416no—
429C22⋊C4—416no—
430C22⋊C4—416no—
431C22⋊C4—416no—
432C22⋊C4—88no—
433C22⋊C4—88no—
434C22⋊C4—88no—
435C22⋊C4—88no—
436C22⋊C4—88no—
437C22⋊C4—88no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—28no—
442C23—24no—
443C23—28no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—28no—
449C23—28no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23#53414yes—
486C23#53518yes—
487C23#56018yes—
488C23#56018yes—
489C23C23×D411yescharacteristic
490C23C2×H(5,𝔽2)11yescharacteristic
491C23C2×H(5,𝔽2)11yescharacteristic
492C23C2×H(5,𝔽2)14yes—
493C23C2×H(5,𝔽2)12yes—
494C23C2×2−1+412yes—
495C23#58512yes—
496C23#58512yes—
497D4—816no—
498D4—816no—
499D4—816no—
500D4—832no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—28no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—416no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—816no—
562C2×C4—816no—
563C2×C4—816no—
564C22—24no—
565C22—28no—
566C22—28no—
567C22—28no—
568C22—24no—
569C22—28no—
570C22—216no—
571C22—84no—
572C22—84no—
573C22—84no—
574C22—84no—
575C22—84no—
576C22—84no—
577C22—84no—
578C22—84no—
579C22—84no—
580C22—84no—
581C22—84no—
582C22—84no—
583C22—84no—
584C22—84no—
585C22—84no—
586C22#281518yes—
587C22C2×#53411yescharacteristic
588C22C2×#53512yes—
589C22#349612yes—
590C22#349712yes—
591C22C2×#56012yes—
592C22C2×#56012yes—
593C22#353412yes—
594C22#353414yes—
595C22#353512yes—
596C22#353614yes—
597C22#354311yescharacteristic
598C22#354411yescharacteristic
599C22#354511yescharacteristic
600C22#354511yescharacteristic
601C22#355914yes—
602C22#356014yes—
603C4—44no—
604C4—44no—
605C4—42no—
606C4—42no—
607C4—44no—
608C4—44no—
609C4—84no—
610C4—82no—
611C4—82no—
612C4—84no—
613C4—84no—
614C2—28no—
615C2—164no—
616C2C2×#281511yescharacteristic
617C2#3674114yes—
618C2#3712911yescharacteristic
619C2#3719011yescharacteristic
620C2#5254812yes—
621C2#5254912yes—
622C2#5258612yes—
623C2#5279912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
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×#2618C2semidirect
2. #17221C2semidirect
3. #17196C2semidirect
4. #17187C2semidirect
5. #17140C2semidirect
6. #17121C2semidirect
7. #17093C2semidirect
8. #17082C2semidirect
9. #14926C2semidirect
10. #14926C2semidirect
11. #14757C2semidirect
12. #14735C2semidirect
13. C2×#1445C2semidirect
88. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26182
#172212
#171962
#171872
#171402
#171212
#170932
#170822
#149262
#147572
#147352
C2×#14452