Finite Group Databasebeta

← #876618

Subgroups — #876618

Order 512.

2542 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2579C212yesmaximal
2#17073C212yesmaximal, complemented
3#17036C212yesmaximal, complemented
4#15593C214yesmaximal, complemented
5#15421C214yesmaximal, complemented
6#15206C212yesmaximal, complemented
7#14840C214yesmaximal, complemented
8#14699C212yesmaximal, complemented
9#14486C212yesmaximal
10#14440C211yesmaximal, characteristic
11#14440C211yesmaximal, characteristic
12#14043C214yesmaximal, complemented
13C4×#390C211yesmaximal, characteristic
14C22×C4×D4C2211yescharacteristic
15C23×C4⋊C4C2212yes—
16#2579—216no—
17C2×#398C2214yes—
18C2×#398C2214yes—
19C2×#396C2214yes—
20C2×#390C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#387C2214yes—
26C2×#386C2212yes—
27C2×#386C2212yes—
28C2×#385C2212yes—
29C2×#385C2214yes—
30C2×#385C2212yes—
31C2×#385C2214yes—
32C2×#385C2214yes—
33C2×#380C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2214yes—
36C2×C43C2211yescharacteristic
37#1445C2214yes—
38#1444C2212yes—
39#1444C2214yes—
40#1444C2214yes—
41#1444C2214yes—
42#1442C2212yes—
43#1441C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1441C2212yes—
47#1441C2212yes—
48#1441C2214yes—
49#1441C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1437C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2211yescharacteristic
64C4⋊#95C2211yescharacteristic
65#1434C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C23×D4C2311yescharacteristic
69C24×C4C2312yes—
70C24×C4C2312yes—
71C24×C4C2312yes—
72C2×C4×D4—42no—
73C2×C4×D4—42no—
74C2×C4×D4D418yes—
75C22×C4⋊C4—28no—
76C22×C4⋊C4—44no—
77C22×C4⋊C4—44no—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C42C2312yes—
101C22×C42C2311yescharacteristic
102C22×C42C2312yes—
103C22×C42C2312yes—
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106#398—216no—
107#398—48no—
108#396—216no—
109#390Q818yes—
110#390Q818yes—
111#389—28no—
112#389—44no—
113#389—44no—
114#387—216no—
115#386—28no—
116#386—28no—
117#385—216no—
118#385—216no—
119#385—28no—
120#385—28no—
121#385—48no—
122#380—28no—
123C4×C4⋊C4—42no—
124C4×C22⋊C4—48no—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C43—42no—
147C25C2412yes—
148C22×D4—41no—
149C22×D4—41no—
150C22×D4—41no—
151C22×D4—41no—
152C22×D4C2×D414yes—
153C22×D4C2×Q814yes—
154C22×D4C2×Q814yes—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—28no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—44no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4C2×D418yes—
170C23×C4C2414yes—
171C23×C4C2414yes—
172C23×C4C2414yes—
173C23×C4C2411yescharacteristic
174C23×C4C2412yes—
175C23×C4C2412yes—
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2414yes—
179C23×C4C2411yescharacteristic
180C23×C4C2411yescharacteristic
181C23×C4C2412yes—
182C4×D4—416no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—432no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4C2×D414yes—
224C2×C4⋊C4C2×Q814yes—
225C2×C4⋊C4C2×Q814yes—
226C2×C4⋊C4#4118yes—
227C2×C22⋊C4—28no—
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—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4C2×D418yes—
246C2×C22⋊C4C2×Q8116yes—
247C2×C22⋊C4C2×Q8116yes—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—41no—
255C2×C42—41no—
256C2×C42—42no—
257C2×C42—41no—
258C2×C42—41no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42—44no—
262C2×C42C2×D414yes—
263C2×C42C2×Q814yes—
264C2×C42C2×Q814yes—
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—28no—
274#95—28no—
275#95—44no—
276#95—44no—
277#95—42no—
278#95—42no—
279#95—44no—
280#95—44no—
281#95—42no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286C24—24no—
287C24—24no—
288C24—22no—
289C24—22no—
290C24—24no—
291C24—22no—
292C24—22no—
293C24—24no—
294C24C22×D414yes—
295C24C22×Q814yes—
296C24C22×Q814yes—
297C24C2511yescharacteristic
298C2×D4—44no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—44no—
304C2×D4C4⋊Q8116yes—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—21no—
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—24no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—216no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—48no—
366C22×C4—416no—
367C22×C4—416no—
368C22×C4—416no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—88no—
373C22×C4—84no—
374C22×C4—84no—
375C22×C4—88no—
376C22×C4—88no—
377C22×C4—84no—
378C22×C4—84no—
379C22×C4C22×D412yes—
380C22×C4C22×D414yes—
381C22×C4C22×D412yes—
382C22×C4C22×D412yes—
383C22×C4C22×Q814yes—
384C22×C4C22×Q814yes—
385C22×C4C22×Q814yes—
386C22×C4C22×Q812yes—
387C22×C4C22×Q812yes—
388C22×C4C22×Q814yes—
389C22×C4C2×#4118yes—
390C22×C4C2×#4114yes—
391C22×C4H(5,𝔽2)12yes—
392C22×C4H(5,𝔽2)12yes—
393C22×C4H(5,𝔽2)14yes—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—48no—
398C4⋊C4—416no—
399C4⋊C4—48no—
400C4⋊C4—48no—
401C4⋊C4—416no—
402C4⋊C4—416no—
403C4⋊C4—84no—
404C4⋊C4—84no—
405C4⋊C4—88no—
406C4⋊C4—88no—
407C4⋊C4—84no—
408C4⋊C4—88no—
409C22⋊C4—232no—
410C22⋊C4—48no—
411C22⋊C4—416no—
412C22⋊C4—416no—
413C22⋊C4—48no—
414C22⋊C4—48no—
415C22⋊C4—88no—
416C42—28no—
417C42—44no—
418C42—44no—
419C42—48no—
420C42—48no—
421C42—84no—
422C23—24no—
423C23—24no—
424C23—28no—
425C23—28no—
426C23—28no—
427C23—28no—
428C23—28no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—28no—
434C23—44no—
435C23—42no—
436C23—44no—
437C23—44no—
438C23—42no—
439C23—44no—
440C23—42no—
441C23—42no—
442C23—44no—
443C23—44no—
444C23C2×C4⋊Q818yes—
445C23C23×D411yescharacteristic
446C23C23×Q811yescharacteristic
447C23C23×Q811yescharacteristic
448C23C22×#4112yes—
449C23C2×H(5,𝔽2)11yescharacteristic
450C23C2×H(5,𝔽2)11yescharacteristic
451C23C2×H(5,𝔽2)12yes—
452C23#58511yescharacteristic
453C23#58511yescharacteristic
454C23#58512yes—
455C23#58512yes—
456D4—416no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—28no—
476C2×C4—28no—
477C2×C4—24no—
478C2×C4—28no—
479C2×C4—24no—
480C2×C4—28no—
481C2×C4—28no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—28no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—432no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
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—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—816no—
530C2×C4—816no—
531C2×C4C2×C4⋊Q814yes—
532C2×C4#53414yes—
533C2×C4#55418yes—
534C22—216no—
535C22—28no—
536C22—44no—
537C22—44no—
538C22—48no—
539C22—48no—
540C22—44no—
541C22—44no—
542C22—44no—
543C22—44no—
544C22—48no—
545C22C22×C4⋊Q811yescharacteristic
546C22C2×#53411yescharacteristic
547C22#344911yescharacteristic
548C22#345912yes—
549C22D4×#4112yes—
550C22C2×#55412yes—
551C22Q8×#4112yes—
552C22#348111yescharacteristic
553C22#349312yes—
554C22#349612yes—
555C22#350111yescharacteristic
556C22#350111yescharacteristic
557C22#350112yes—
558C22#350111yescharacteristic
559C22#350111yescharacteristic
560C22#350412yes—
561C22#351712yes—
562C22#352911yescharacteristic
563C22#353412yes—
564C22#354311yescharacteristic
565C22#354511yescharacteristic
566C22#356012yes—
567C22#356712yes—
568C4—24no—
569C4—24no—
570C4—28no—
571C4—44no—
572C4—44no—
573C4—44no—
574C4—42no—
575C4—48no—
576C4—44no—
577C4—84no—
578C4—84no—
579C4—84no—
580C2—48no—
581C2#3501111yescharacteristic
582C2#3626112yes—
583C2#3712911yescharacteristic
584C2#3843312yes—
585C2#3967112yes—
586C2#3971012yes—
587C2#4624011yescharacteristic
588C2#4628211yescharacteristic
589C2#5216311yescharacteristic
590C2#5946611yescharacteristic
591C2#5992611yescharacteristic
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. #17073C2semidirect
3. #17036C2semidirect
4. #15593C2semidirect
5. #15421C2semidirect
6. #15206C2semidirect
7. #14840C2semidirect
8. #14699C2semidirect
12. #14043C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25792
#170732
#170362
#155932
#154212
#152062
#148402
#146992
#144862
#144402
#140432
C4×#3902