Finite Group Databasebeta

← #4453541

Subgroups — #4453541

Order 512.

2294 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2398C211yesmaximal, characteristic
2C2×#2330C211yesmaximal, characteristic, complemented
3#16452C214yesmaximal, complemented
4#16408C214yesmaximal, complemented
5#16335C214yesmaximal, complemented
6#16312C214yesmaximal
7#15216C212yesmaximal, complemented
8#15202C212yesmaximal, complemented
9#15135C212yesmaximal, complemented
10#15129C212yesmaximal
11#14350C211yesmaximal, characteristic, complemented
12#13994C212yesmaximal, complemented
13#13993C212yesmaximal, complemented
14C22×#125C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17#2398—28no—
18C2×#400C2212yes—
19C2×#399C2214yes—
20C2×#397C2212yes—
21C2×#394C2212yes—
22C2×#394C2214yes—
23C2×#393C2212yes—
24#2330—28no—
25C2×#389C2214yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#388C2214yes—
30C2×#387C2214yes—
31C2×#384C2211yescharacteristic
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C2×#376C2211yescharacteristic
40#1446C2214yes—
41#1445C2214yes—
42#1444C2214yes—
43#1444C2214yes—
44#1444C2214yes—
45#1444C2214yes—
46#1443C2214yes—
47#1443C2214yes—
48#1443C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2214yes—
53#1440C2214yes—
54#1440C2214yes—
55#1440C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1437C2214yes—
61#1436C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2211yescharacteristic
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68#1434C2214yes—
69C23×Q8C2311yescharacteristic
70C23×D4C2311yescharacteristic
71C24×C4C2312yes—
72C2×#125—28no—
73C2×#125—24no—
74C2×C4×Q8—24no—
75C2×C4×Q8—42no—
76C2×C4×Q8—42no—
77C2×C4×D4—24no—
78C2×C4×D4—42no—
79C2×C4×D4—42no—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C42C2312yes—
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2314yes—
106#400—28no—
107#399—48no—
108#397—28no—
109#394—28no—
110#394—48no—
111#393—28no—
112#389—28no—
113#389—216no—
114#389—44no—
115#389—44no—
116#388—48no—
117#387—216no—
118#384—24no—
119#380—28no—
120#380—28no—
121#380—44no—
122#380—44no—
123C4×C4⋊C4—28no—
124C4×C22⋊C4—28no—
125C4×C22⋊C4—28no—
126#376D418yes—
127C2×#95C2312yes—
128C2×#95C2312yes—
129C2×#95C2312yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C25C2412yes—
152C22×Q8—22no—
153C22×Q8—24no—
154C22×Q8—41no—
155C22×Q8—41no—
156C22×Q8—42no—
157C22×D4—24no—
158C22×D4—22no—
159C22×D4—41no—
160C22×D4—41no—
161C22×D4—42no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4C2412yes—
170C23×C4C2414yes—
171C23×C4C2414yes—
172C23×C4C2414yes—
173C23×C4C2412yes—
174C23×C4C2412yes—
175C23×C4C2411yescharacteristic
176C23×C4C2412yes—
177C23×C4C2411yescharacteristic
178C23×C4C2414yes—
179C23×C4C2412yes—
180C23×C4C2411yescharacteristic
181#125—232no—
182C4×Q8—416no—
183C4×D4—416no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4#4118yes—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
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×C42—24no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—28no—
255C2×C42—44no—
256C2×C42—44no—
257C2×C42C2×D414yes—
258C2×C42C2×D414yes—
259C2×C42C2×D414yes—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95C2×D418yes—
283#95C2×D418yes—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24H(5,𝔽2)14yes—
299C24C2511yescharacteristic
300C2×Q8—48no—
301C2×Q8—44no—
302C2×Q8—44no—
303C2×Q8—48no—
304C2×Q8—44no—
305C2×D4—48no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—48no—
309C2×D4—44no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—48no—
383C22×C4—48no—
384C22×C4C22×D414yes—
385C22×C4C22×D412yes—
386C22×C4C22×D414yes—
387C22×C4C22×D412yes—
388C22×C4C22×D412yes—
389C22×C4C2×#4118yes—
390C22×C4C2×#4114yes—
391C22×C4H(5,𝔽2)12yes—
392C22×C4H(5,𝔽2)12yes—
393C22×C42−1+414yes—
394C22×C42−1+414yes—
395C4⋊C4—232no—
396C4⋊C4—216no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—48no—
400C4⋊C4—48no—
401C4⋊C4—48no—
402C4⋊C4—48no—
403C4⋊C4—88no—
404C4⋊C4—88no—
405C4⋊C4—88no—
406C4⋊C4—88no—
407C22⋊C4—48no—
408C22⋊C4—48no—
409C22⋊C4—48no—
410C22⋊C4—48no—
411C22⋊C4—416no—
412C22⋊C4—48no—
413C22⋊C4—88no—
414C22⋊C4—88no—
415C22⋊C4—88no—
416C42—28no—
417C42—28no—
418C42—28no—
419C42—48no—
420C42—88no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—24no—
425C23—24no—
426C23—24no—
427C23—24no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23C23×D411yescharacteristic
457C23C22×#4112yes—
458C23C2×H(5,𝔽2)11yescharacteristic
459C23C2×H(5,𝔽2)11yescharacteristic
460C23C2×H(5,𝔽2)12yes—
461C23C2×2−1+412yes—
462C23C2×2−1+412yes—
463C23#58512yes—
464C23#58512yes—
465Q8—416no—
466D4—416no—
467C2×C4—24no—
468C2×C4—28no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—28no—
480C2×C4—28no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—816no—
541C2×C4#52514yes—
542C2×C4#53514yes—
543C2×C4#53614yes—
544C2×C4#54218yes—
545C2×C4#54218yes—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—44no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22C2×#52511yescharacteristic
562C22C2×#53511yescharacteristic
563C22C2×#53611yescharacteristic
564C22C2×#54212yes—
565C22C2×#54212yes—
566C22#348812yes—
567C22#348912yes—
568C22#351312yes—
569C22#351512yes—
570C22#352212yes—
571C22#352412yes—
572C22#353711yescharacteristic
573C22#353811yescharacteristic
574C22#353911yescharacteristic
575C22#353911yescharacteristic
576C22#354112yes—
577C22#354212yes—
578C22#354412yes—
579C22#355512yes—
580C22#355912yes—
581C22#356512yes—
582C4—42no—
583C4—44no—
584C4—44no—
585C4—48no—
586C4—42no—
587C4—44no—
588C4—82no—
589C4—84no—
590C4—84no—
591C4—82no—
592C4—84no—
593C4#263218yes—
594C2—84no—
595C2C2×#263211yescharacteristic
596C2#3552012yes—
597C2#3727011yescharacteristic
598C2#3728112yes—
599C2#3737511yescharacteristic
600C2#4253712yes—
601C2#4262512yes—
602C2#4271112yes—
603C2#4277512yes—
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
2. C2×#2330C2semidirect
3. #16452C2semidirect
4. #16408C2semidirect
5. #16335C2semidirect
7. #15216C2semidirect
8. #15202C2semidirect
9. #15135C2semidirect
11. #14350C2semidirect
12. #13994C2semidirect
13. #13993C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23982
C2×#23302
#164522
#164082
#163352
#163122
#152162
#152022
#151352
#151292
#143502
#139942
#139932