Finite Group Databasebeta

← #4100382

Subgroups — #4100382

Order 512.

2970 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2712C214yesmaximal, complemented
2#17118C212yesmaximal, complemented
3#17082C211yesmaximal, characteristic
4#15964C211yesmaximal, characteristic, complemented
5#15824C214yesmaximal, complemented
6#15761C212yesmaximal, complemented
7#15632C214yesmaximal, complemented
8#15506C212yesmaximal, complemented
9#15070C212yesmaximal, complemented
10#14806C214yesmaximal, complemented
11#14571C212yesmaximal, complemented
12#14176C212yesmaximal, complemented
13#14155C211yesmaximal, characteristic, complemented
14C22×C4⋊Q8C2211yescharacteristic
15C22×#124C2212yes—
16C22×#124C2212yes—
17C22×C4×D4C2212yes—
18#2712—232no—
19C2×#397C2214yes—
20C2×#395C2212yes—
21C2×#394C2212yes—
22C2×#394C2214yes—
23C2×#394C2212yes—
24C2×#394C2212yes—
25C2×#394C2212yes—
26C2×#393C2214yes—
27C2×#393C2214yes—
28C2×#392C2214yes—
29C2×#391C2214yes—
30C2×#391C2214yes—
31C2×#388C2214yes—
32C2×#388C2214yes—
33C2×#388C2214yes—
34C2×#386C2214yes—
35C2×#386C2214yes—
36C2×#385C2214yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39#1446C2212yes—
40#1446C2212yes—
41#1445C2211yescharacteristic
42#1445C2211yescharacteristic
43#1445C2211yescharacteristic
44#1445C2211yescharacteristic
45#1444C2212yes—
46#1443C2214yes—
47#1442C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2212yes—
51#1440C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1438C2212yes—
58#1438C2212yes—
59#1437C2214yes—
60#1436C2212yes—
61#1436C2212yes—
62C4⋊#95C2214yes—
63#1434C2212yes—
64#1434C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C23×Q8C2312yes—
69C23×Q8C2312yes—
70C23×D4C2312yes—
71C23×D4C2312yes—
72C23×D4C2312yes—
73C24×C4C2314yes—
74C2×C4⋊Q8—24no—
75C2×C4⋊Q8—42no—
76C2×C4⋊Q8—42no—
77C2×#124—216no—
78C2×#124—216no—
79C2×#124—44no—
80C2×#124—44no—
81C2×C4×D4—44no—
82C2×C4×D4—48no—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
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×C42C2312yes—
103C22×C42C2312yes—
104C22×C42C2311yescharacteristic
105C22×C42C2312yes—
106#397—216no—
107#395—44no—
108#394—216no—
109#394—28no—
110#394—44no—
111#394—44no—
112#394D4116yescomplemented
113#393—216no—
114#393—216no—
115#392—216no—
116#391—216no—
117#391—216no—
118#388—216no—
119#388—216no—
120#388—48no—
121#386—216no—
122#386—216no—
123#385—216no—
124C4×C4⋊C4—42no—
125C4×C4⋊C4—42no—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2311yescharacteristic
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2311yescharacteristic
147C2×#95C2311yescharacteristic
148C2×#95C2311yescharacteristic
149C2×#95C2312yes—
150C25C2414yes—
151C22×Q8—28no—
152C22×Q8—28no—
153C22×Q8—24no—
154C22×Q8—24no—
155C22×Q8—24no—
156C22×Q8—42no—
157C22×Q8—42no—
158C22×Q8—44no—
159C22×Q8—44no—
160C22×Q8—42no—
161C22×D4—28no—
162C22×D4—28no—
163C22×D4—24no—
164C22×D4—24no—
165C22×D4—42no—
166C22×D4—44no—
167C22×D4—42no—
168C22×D4—44no—
169C22×D4—42no—
170C22×D4—42no—
171C22×D4—44no—
172C22×D4—44no—
173C22×D4—42no—
174C22×D4—42no—
175C22×D4C2×D418yes—
176C23×C4—28no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4—44no—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2414yes—
186C23×C4C2414yes—
187C23×C4C2414yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2411yescharacteristic
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C4⋊Q8—416no—
196#124—432no—
197#124—432no—
198C4×D4—432no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4C2×D418yes—
221C2×C4⋊C4#4114yes—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
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—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
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⋊C4C2×D4116yes—
254C2×C22⋊C4C2×D4116yes—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—28no—
259C2×C42—28no—
260C2×C42—24no—
261C2×C42—28no—
262C2×C42—24no—
263C2×C42—42no—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—24no—
274#95—28no—
275#95—44no—
276#95—42no—
277#95—44no—
278#95—44no—
279#95—41no—
280#95—41no—
281#95—41no—
282#95—41no—
283#95—44no—
284#95—42no—
285#95—42no—
286#95—42no—
287#95C2×D418yes—
288C24—24no—
289C24—24no—
290C24—24no—
291C24—24no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24—24no—
296C24—24no—
297C24—24no—
298C24—24no—
299C24—24no—
300C24—24no—
301C24—24no—
302C24C22×D418yes—
303C24C2511yescharacteristic
304C2×Q8—48no—
305C2×Q8—48no—
306C2×Q8—416no—
307C2×Q8—416no—
308C2×Q8—48no—
309C2×Q8—88no—
310C2×Q8—84no—
311C2×Q8—88no—
312C2×Q8—84no—
313C2×Q8—84no—
314C2×D4—416no—
315C2×D4—48no—
316C2×D4—416no—
317C2×D4—416no—
318C2×D4—48no—
319C2×D4—416no—
320C2×D4—48no—
321C2×D4—48no—
322C2×D4—48no—
323C2×D4—84no—
324C2×D4—84no—
325C2×D4—88no—
326C2×D4—88no—
327C2×D4—84no—
328C2×D4—84no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
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—416no—
385C22×C4—88no—
386C22×C4—88no—
387C22×C4—88no—
388C22×C4—88no—
389C22×C4—88no—
390C22×C4—88no—
391C22×C4C22×D414yes—
392C22×C4C22×D414yes—
393C22×C4C22×D418yes—
394C22×C4C22×D414yes—
395C22×C4C2×#4114yes—
396C22×C4C2×#4112yes—
397C22×C4H(5,𝔽2)12yes—
398C22×C4H(5,𝔽2)14yes—
399C22×C4H(5,𝔽2)12yes—
400C22×C42−1+414yes—
401C4⋊C4—48no—
402C4⋊C4—44no—
403C4⋊C4—48no—
404C4⋊C4—48no—
405C4⋊C4—44no—
406C4⋊C4—48no—
407C4⋊C4—88no—
408C4⋊C4—88no—
409C22⋊C4—232no—
410C22⋊C4—416no—
411C22⋊C4—416no—
412C22⋊C4—416no—
413C22⋊C4—416no—
414C22⋊C4—416no—
415C22⋊C4—416no—
416C22⋊C4—88no—
417C22⋊C4—88no—
418C22⋊C4—88no—
419C22⋊C4—88no—
420C42—28no—
421C42—48no—
422C42—48no—
423C42—48no—
424C23—28no—
425C23—28no—
426C23—28no—
427C23—28no—
428C23—28no—
429C23—28no—
430C23—28no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23C23×D412yes—
460C23C22×#4111yescharacteristic
461C23C2×H(5,𝔽2)11yescharacteristic
462C23C2×H(5,𝔽2)11yescharacteristic
463C23C2×H(5,𝔽2)12yes—
464C23C2×2−1+412yes—
465C23#58512yes—
466C23#58514yes—
467Q8—816no—
468Q8—816no—
469D4—432no—
470D4—816no—
471D4—816no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—28no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—28no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—832no—
536C2×C4D4214yes—
537C2×C4H(3,𝔽4)14yes—
538C22—48no—
539C22—48no—
540C22—48no—
541C22—48no—
542C22—48no—
543C22—48no—
544C22—48no—
545C22—84no—
546C22—84no—
547C22—84no—
548C22—84no—
549C22—84no—
550C22—84no—
551C22—84no—
552C22—84no—
553C22C2×D4211yescharacteristic
554C22D4×#4112yes—
555C22#348914yes—
556C22#349011yescharacteristic
557C22#349614yes—
558C22#349612yes—
559C22#351012yes—
560C22#352212yes—
561C22C2×H(3,𝔽4)11yescharacteristic
562C22#353612yes—
563C22#354014yes—
564C22#354012yes—
565C22#354411yescharacteristic
566C22#354511yescharacteristic
567C22#355112yes—
568C22#355714yes—
569C4—24no—
570C4—24no—
571C4—44no—
572C4—48no—
573C4—44no—
574C4—44no—
575C4—44no—
576C4—42no—
577C4—82no—
578C4—84no—
579C4—82no—
580C4—84no—
581C2—88no—
582C2#4391714yes—
583C2#4510312yes—
584C2#4547912yes—
585C2#4609911yescharacteristic
586C2#4703611yescharacteristic
587C2#4778812yes—
588C2#5688512yes—
589C2#5981411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#17082C2
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×#2712C2semidirect
2. #17118C2semidirect
4. #15964C2semidirect
5. #15824C2semidirect
6. #15761C2semidirect
7. #15632C2semidirect
8. #15506C2semidirect
9. #15070C2semidirect
10. #14806C2semidirect
11. #14571C2semidirect
12. #14176C2semidirect
13. #14155C2semidirect
112. #394D4semidirect

Maximal subgroups

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

subgroupindex
C2×#27122
#171182
#170822
#159642
#158242
#157612
#156322
#155062
#150702
#148062
#145712
#141762
#141552