Finite Group Databasebeta

← #4453275

Subgroups — #4453275

Order 512.

2326 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#398C211yesmaximal, characteristic
2C2×#2313C211yesmaximal, characteristic
3#16444C214yesmaximal, complemented
4#16440C214yesmaximal, complemented
5#16313C214yesmaximal, complemented
6#16311C214yesmaximal, complemented
7#15015C211yesmaximal, characteristic
8#15014C211yesmaximal, characteristic
9#14919C212yesmaximal
10#14919C212yesmaximal
11#14822C211yesmaximal, characteristic
12#14821C211yesmaximal, characteristic
13#14339C211yesmaximal, characteristic
14#14130C212yesmaximal
15#13923C212yesmaximal
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C2×#400C2212yes—
21C2×#398—28no—
22C2×#398C2214yes—
23C2×#398C2214yes—
24C2×#398C2212yes—
25#2313—28no—
26C2×#389C2214yes—
27C2×#387C2212yes—
28C2×#387C2212yes—
29C2×#387C2212yes—
30C2×#387C2212yes—
31C2×#385C2212yes—
32C2×#385C2212yes—
33C2×#381C2212yes—
34C2×#381C2212yes—
35C2×#380C2212yes—
36C2×#380C2211yescharacteristic
37C2×#380C2211yescharacteristic
38C2×C4×C22⋊C4C2212yes—
39C22×#95C2211yescharacteristic
40C22×#95C2211yescharacteristic
41C22×#95C2211yescharacteristic
42#1445C2212yes—
43#1445C2212yes—
44#1445C2214yes—
45#1445C2214yes—
46#1444C2214yes—
47#1444C2212yes—
48#1443C2214yes—
49#1443C2214yes—
50#1443C2214yes—
51#1443C2214yes—
52#1443C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2212yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2212yes—
62#1441C2214yes—
63#1441C2212yes—
64#1441C2214yes—
65#1441C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2214yes—
69C22⋊#95C2211yescharacteristic
70C22⋊#95C2211yescharacteristic
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C22×C4⋊C4—28no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—28no—
89C22×C4⋊C4—44no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4D418yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C42C2312yes—
110C22×C42C2312yes—
111#400—44no—
112#398—216no—
113#398—216no—
114#398—44no—
115#389—48no—
116#387—28no—
117#387—28no—
118#387—28no—
119#387—28no—
120#385—44no—
121#385—44no—
122#381—28no—
123#381—44no—
124#380—24no—
125#380—24no—
126#380—28no—
127C4×C22⋊C4—28no—
128C2×#95—24no—
129C2×#95—24no—
130C2×#95—24no—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C25C2411yescharacteristic
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—24no—
170C23×C4—22no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4C2×D414yes—
193C23×C4C2×D414yes—
194C23×C4C2×D414yes—
195C23×C4#4118yes—
196C23×C4C2412yes—
197C23×C4C2414yes—
198C23×C4C2414yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C2×C4⋊C4—232no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—416no—
236C2×C4⋊C4—416no—
237C2×C4⋊C4—416no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4C2×D418yes—
248C2×C4⋊C4C2×D418yes—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—44no—
263C2×C42—44no—
264#95—28no—
265#95—24no—
266#95—24no—
267#95—28no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—28no—
277#95—28no—
278#95—28no—
279#95—28no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—42no—
284#95—42no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—42no—
294#95—44no—
295#95—42no—
296#95—44no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24C22×D412yes—
302C24C2×#4114yes—
303C24H(5,𝔽2)12yes—
304C242−1+414yes—
305C242−1+412yes—
306C24C2511yescharacteristic
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—28no—
310C22×C4—28no—
311C22×C4—28no—
312C22×C4—28no—
313C22×C4—28no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—28no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—416no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—416no—
385C22×C4—48no—
386C22×C4—48no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—88no—
400C22×C4—88no—
401C22×C4—84no—
402C22×C4—84no—
403C22×C4#123116yes—
404C22×C4C22×D414yes—
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C22×D414yes—
408C22×C4C2×#4114yes—
409C22×C4C2×#4114yes—
410C22×C4H(5,𝔽2)14yes—
411C22×C42−1+414yes—
412C4⋊C4—216no—
413C4⋊C4—216no—
414C4⋊C4—416no—
415C4⋊C4—48no—
416C4⋊C4—48no—
417C4⋊C4—416no—
418C4⋊C4—416no—
419C4⋊C4—82no—
420C4⋊C4—82no—
421C4⋊C4—82no—
422C4⋊C4—82no—
423C4⋊C4—88no—
424C4⋊C4—88no—
425C22⋊C4—48no—
426C22⋊C4—48no—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C42—84no—
430C42—84no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—28no—
436C23—28no—
437C23—24no—
438C23—24no—
439C23—28no—
440C23—28no—
441C23—24no—
442C23—24no—
443C23C2×#12314yes—
444C23#53614yes—
445C23#53714yes—
446C23#54118yes—
447C23#54218yes—
448C23C23×D411yescharacteristic
449C23C22×#4112yes—
450C23C2×H(5,𝔽2)11yescharacteristic
451C23C2×H(5,𝔽2)12yes—
452C23C2×2−1+412yes—
453C23C2×2−1+411yescharacteristic
454C23C2×2−1+412yes—
455C23#58512yes—
456C23#58512yes—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—28no—
460C2×C4—28no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—28no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—416no—
476C2×C4—416no—
477C2×C4—416no—
478C2×C4—416no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
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—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—48no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—48no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—88no—
535C2×C4—88no—
536C2×C4—88no—
537C2×C4C2×#12318yes—
538C22—24no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—24no—
543C22—24no—
544C22—216no—
545C22—216no—
546C22#262518yes—
547C22C22×#12311yescharacteristic
548C22C2×#53611yescharacteristic
549C22C2×#53711yescharacteristic
550C22C2×#54112yes—
551C22C2×#54212yes—
552C22#348411yescharacteristic
553C22#348511yescharacteristic
554C22#348611yescharacteristic
555C22#348711yescharacteristic
556C22#349514yes—
557C22#350514yes—
558C22#353412yes—
559C22#353512yes—
560C22#355214yes—
561C22#355314yes—
562C22#355912yes—
563C22#356312yes—
564C4—28no—
565C4—42no—
566C4—42no—
567C4—44no—
568C4—44no—
569C4—44no—
570C4—82no—
571C4—82no—
572C4—82no—
573C4—82no—
574C4—82no—
575C4—82no—
576C4—84no—
577C4—84no—
578C2—28no—
579C2C2×#262511yescharacteristic
580C2#3535111yescharacteristic
581C2#3567211yescharacteristic
582C2#3672211yescharacteristic
583C2#3696811yescharacteristic
584C2#3712011yescharacteristic
585C2#3723311yescharacteristic
586C2#4056414yes—
587C2#4056614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16444C2semidirect
4. #16440C2semidirect
5. #16313C2semidirect
6. #16311C2semidirect

Maximal subgroups

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

subgroupindex
C22×#3982
C2×#23132
#164442
#164402
#163132
#163112
#150152
#150142
#149192
#148222
#148212
#143392
#141302
#139232