Finite Group Databasebeta

← #803520

Subgroups — #803520

Order 512.

2166 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2517C212yesmaximal
2#16473C212yesmaximal
3#16115C212yesmaximal, complemented
4#16096C212yesmaximal, complemented
5#16080C214yesmaximal, complemented
6#15135C211yesmaximal, characteristic
7#15132C212yesmaximal
8#14829C214yesmaximal
9#14133C212yesmaximal, complemented
10#14130C211yesmaximal, characteristic
11#14042C212yesmaximal, complemented
12#13981C212yesmaximal, complemented
13#13918C212yesmaximal, complemented
14C4×#397C211yesmaximal, characteristic
15C22⋊C4×C4⋊C4C212yesmaximal
16C23×C4⋊C4C2211yescharacteristic
17#2517—216no—
18C2×#399C2212yes—
19C2×#397C2214yes—
20C2×#397C2211yescharacteristic
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#387C2214yes—
25C2×#387C2214yes—
26C2×#387C2214yes—
27C2×#385C2212yes—
28C2×#384C2211yescharacteristic
29C2×#382C2214yes—
30C2×#380C2211yescharacteristic
31C2×#380C2212yes—
32C2×C4×C4⋊C4C2212yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C22⋊C4C2212yes—
36C2×C4×C22⋊C4C2214yes—
37C2×C4×C22⋊C4C2211yescharacteristic
38#1445C2212yes—
39#1445C2211yescharacteristic
40#1444C2212yes—
41#1444C2212yes—
42#1443C2214yes—
43#1443C2212yes—
44#1443C2214yes—
45#1443C2214yes—
46#1442C2214yes—
47#1442C2211yescharacteristic
48#1442C2212yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1440C2214yes—
56#1440C2212yes—
57#1439C2212yes—
58#1439C2211yescharacteristic
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2211yescharacteristic
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70#1434C2214yes—
71#1434C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C4×#95C2212yes—
75C4×#95C2211yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2312yes—
78C22×C4⋊C4—28no—
79C22×C4⋊C4—42no—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2311yescharacteristic
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2311yescharacteristic
98C22×C42C2312yes—
99C22×C42C2311yescharacteristic
100C22×C42C2312yes—
101C22×C42C2311yescharacteristic
102C22×C42C2314yes—
103C22×C42C2314yes—
104#399—28no—
105#397—216no—
106#397D418yes—
107#389—28no—
108#389—28no—
109#389—28no—
110#387—216no—
111#387—216no—
112#387—216no—
113#385—28no—
114#384—42no—
115#382—216no—
116#380—24no—
117#380—28no—
118C4×C4⋊C4—28no—
119C4×C4⋊C4—42no—
120C4×C4⋊C4—44no—
121C4×C22⋊C4—28no—
122C4×C22⋊C4—216no—
123C4×C22⋊C4—42no—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2311yescharacteristic
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2311yescharacteristic
154C2×#95C2311yescharacteristic
155C2×#95C2311yescharacteristic
156C2×#95C2312yes—
157C25C2411yescharacteristic
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4—28no—
164C23×C4—42no—
165C23×C4—44no—
166C23×C4—42no—
167C23×C4#4114yes—
168C23×C4C2414yes—
169C23×C4C2414yes—
170C23×C4C2412yes—
171C23×C4C2412yes—
172C23×C4C2411yescharacteristic
173C23×C4C2411yescharacteristic
174C23×C4C2411yescharacteristic
175C23×C4C2412yes—
176C23×C4C2412yes—
177C23×C4C2414yes—
178C23×C4C2414yes—
179C23×C4C2411yescharacteristic
180C23×C4C2411yescharacteristic
181C23×C4C2411yescharacteristic
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—416no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4#4118yes—
215C2×C4⋊C4#4118yes—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4C2×D418yes—
233C2×C22⋊C4C2×D414yes—
234C2×C42—28no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—28no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—24no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—28no—
247C2×C42—44no—
248C2×C42—44no—
249C2×C42—44no—
250#95—28no—
251#95—28no—
252#95—28no—
253#95—28no—
254#95—28no—
255#95—28no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—28no—
260#95—28no—
261#95—24no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—24no—
266#95—24no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—42no—
271#95—44no—
272#95—41no—
273#95—42no—
274#95—42no—
275#95—44no—
276#95—44no—
277#95—41no—
278#95—41no—
279#95—42no—
280#95—44no—
281#95C2×D414yes—
282#95C2×D418yes—
283C24—22no—
284C24—22no—
285C24—21no—
286C24—21no—
287C24—21no—
288C24—22no—
289C24—22no—
290C24—21no—
291C24C22×D412yes—
292C24C2×#4112yes—
293C242−1+412yes—
294C24C2511yescharacteristic
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—24no—
321C22×C4—21no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—216no—
336C22×C4—28no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—48no—
371C22×C4—416no—
372C22×C4—44no—
373C22×C4—48no—
374C22×C4—44no—
375C22×C4—88no—
376C22×C4—84no—
377C22×C4—84no—
378C22×C4C22×D414yes—
379C22×C4C22×D414yes—
380C22×C4C22×D412yes—
381C22×C4C22×D412yes—
382C22×C4C2×#4112yes—
383C22×C4C2×#4118yes—
384C22×C4C2×#4118yes—
385C22×C4C2×#4112yes—
386C22×C4C2×#4114yes—
387C22×C4C2×#4114yes—
388C22×C4H(5,𝔽2)12yes—
389C22×C42−1+412yes—
390C22×C42−1+412yes—
391C4⋊C4—216no—
392C4⋊C4—216no—
393C4⋊C4—44no—
394C4⋊C4—416no—
395C4⋊C4—416no—
396C4⋊C4—416no—
397C4⋊C4—48no—
398C4⋊C4—44no—
399C4⋊C4—48no—
400C4⋊C4—48no—
401C4⋊C4—84no—
402C4⋊C4—84no—
403C22⋊C4—216no—
404C22⋊C4—44no—
405C22⋊C4—416no—
406C22⋊C4—44no—
407C22⋊C4—48no—
408C22⋊C4—48no—
409C42—416no—
410C42—48no—
411C42—48no—
412C42—44no—
413C42—44no—
414C42—88no—
415C23—24no—
416C23—22no—
417C23—24no—
418C23—24no—
419C23—24no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—24no—
424C23—24no—
425C23—22no—
426C23—22no—
427C23—41no—
428C23—42no—
429C23—41no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—41no—
435C23—41no—
436C23—42no—
437C23#54714yes—
438C23C23×D411yescharacteristic
439C23C22×#4111yescharacteristic
440C23C22×#4112yes—
441C23C22×#4112yes—
442C23C2×H(5,𝔽2)11yescharacteristic
443C23C2×2−1+411yescharacteristic
444C23C2×2−1+411yescharacteristic
445C23C2×2−1+411yescharacteristic
446C23#58512yes—
447C23#58512yes—
448C23#58511yescharacteristic
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—28no—
455C2×C4—28no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—28no—
460C2×C4—28no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—28no—
466C2×C4—24no—
467C2×C4—28no—
468C2×C4—22no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—416no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—41no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—816no—
532C2×C4#52918yes—
533C2×C4#53914yes—
534C2×C4#55114yes—
535C2×C4#55218yes—
536C2×C4#55914yes—
537C22—28no—
538C22—24no—
539C22—44no—
540C22—42no—
541C22—42no—
542C22—44no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—44no—
547C22—42no—
548C22C2×#52912yes—
549C22C2×#53911yescharacteristic
550C22C2×#54711yescharacteristic
551C22D4×#4112yes—
552C22D4×#4112yes—
553C22C2×#55111yescharacteristic
554C22C2×#55212yes—
555C22C2×#55911yescharacteristic
556C22#349312yes—
557C22#349611yescharacteristic
558C22#349911yescharacteristic
559C22#350711yescharacteristic
560C22#351512yes—
561C22#351612yes—
562C22#351612yes—
563C22#351612yes—
564C22#352112yes—
565C22#352412yes—
566C22#353811yescharacteristic
567C22#354011yescharacteristic
568C22#355111yescharacteristic
569C22#356311yescharacteristic
570C22#356512yes—
571C4—44no—
572C4—48no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—48no—
577C4—44no—
578C4—42no—
579C4—84no—
580C4—81no—
581C4—82no—
582C4—82no—
583C4—84no—
584C4#251718yescomplemented
585C2—44no—
586C2C2×#251711yescharacteristic
587C2#3580312yes—
588C2#3611812yes—
589C2#3937611yescharacteristic
590C2#4268411yescharacteristic
591C2#4522611yescharacteristic
592C2#4522912yes—
593C2#4542011yescharacteristic
594C2#5949812yes—
595C2#5962811yescharacteristic
596C2#5984311yescharacteristic
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. #16115C2semidirect
4. #16096C2semidirect
5. #16080C2semidirect
9. #14133C2semidirect
11. #14042C2semidirect
12. #13981C2semidirect
13. #13918C2semidirect
584. C4#2517semidirect

Maximal subgroups

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

subgroupindex
C2×#25172
#164732
#161152
#160962
#160802
#151352
#151322
#148292
#141332
#141302
#140422
#139812
#139182
C4×#3972
C22⋊C4×C4⋊C42