Finite Group Databasebeta

← #770346

Subgroups — #770346

Order 512.

2262 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2399C211yesmaximal, characteristic
2C2×#2330C211yesmaximal, characteristic
3#15682C214yesmaximal
4#15625C214yesmaximal
5#15592C214yesmaximal
6#15438C214yesmaximal
7#15202C212yesmaximal
8#15158C212yesmaximal
9#14160C212yesmaximal
10#14151C212yesmaximal
11#14150C212yesmaximal
12#13910C212yesmaximal
13#13779C211yesmaximal, characteristic
14C22×#125C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C22×C4×Q8C2211yescharacteristic
17#2399—28no—
18C2×#398C2212yes—
19C2×#398C2212yes—
20C2×#395C2212yes—
21C2×#395C2212yes—
22C2×#395C2214yes—
23#2330—28no—
24C2×#389C2214yes—
25C2×#389C2212yes—
26C2×#389C2214yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#387C2214yes—
32C2×#384C2211yescharacteristic
33C2×#384C2214yes—
34C2×#384C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2212yes—
40C2×#376C2211yescharacteristic
41#1446C2214yes—
42#1444C2214yes—
43#1444C2214yes—
44#1444C2214yes—
45#1444C2214yes—
46#1444C2214yes—
47#1444C2214yes—
48#1442C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2214yes—
66C4⋊#95C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2311yescharacteristic
71C2×#125—24no—
72C2×#125—24no—
73C2×#125Q818yescomplemented
74C2×C4×Q8—24no—
75C2×C4×Q8—24no—
76C2×C4×Q8—42no—
77C2×C4×Q8—42no—
78C2×C4×Q8—42no—
79C2×C4×Q8—42no—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C42C2312yes—
101C22×C42C2312yes—
102C22×C42C2311yescharacteristic
103C22×C42C2314yes—
104C22×C42C2314yes—
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107#398—28no—
108#398—28no—
109#395—28no—
110#395—28no—
111#395—48no—
112#389—28no—
113#389—216no—
114#389—48no—
115#389—44no—
116#389—44no—
117#389—44no—
118#389—44no—
119#387—216no—
120#384—24no—
121#384—48no—
122#384Q818yescomplemented
123#382Q818yescomplemented
124C4×C4⋊C4—28no—
125C4×C4⋊C4—28no—
126C4×C4⋊C4—28no—
127C4×C4⋊C4—28no—
128#376D418yes—
129C2×#95C2312yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
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×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—24no—
156C22×Q8—24no—
157C22×Q8—41no—
158C22×Q8—41no—
159C22×Q8—41no—
160C22×Q8—41no—
161C22×Q8—42no—
162C22×Q8—42no—
163C23×C4C2414yes—
164C23×C4C2412yes—
165C23×C4C2412yes—
166C23×C4C2411yescharacteristic
167C23×C4C2411yescharacteristic
168C23×C4C2414yes—
169C23×C4C2414yes—
170C23×C4C2414yes—
171C23×C4C2412yes—
172C23×C4C2412yes—
173C23×C4C2412yes—
174C23×C4C2412yes—
175C23×C4C2411yescharacteristic
176#125—232no—
177C4×Q8—416no—
178C4×Q8—416no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
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—24no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4C2×Q8116yes—
235C2×C4⋊C4C2×Q818yes—
236C2×C4⋊C4C2×Q818yes—
237C2×C4⋊C4C2×Q818yes—
238C2×C4⋊C4C2×Q8116yes—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—28no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—28no—
249C2×C42—44no—
250C2×C42—44no—
251C2×C42—44no—
252C2×C42—44no—
253C2×C42C2×D414yes—
254C2×C42C2×D414yes—
255C2×C42C2×D414yes—
256C2×C42C2×Q814yes—
257C2×C42C2×Q814yes—
258C2×C42C2×Q814yes—
259C2×C42#4114yes—
260#95—28no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
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#95C2×D418yes—
282#95C2×D418yes—
283#95C2×Q8116yes—
284C24C2511yescharacteristic
285C2×Q8—48no—
286C2×Q8—48no—
287C2×Q8—44no—
288C2×Q8—44no—
289C2×Q8—44no—
290C2×Q8—44no—
291C2×Q8—48no—
292C2×Q8—48no—
293C2×Q8—44no—
294C2×Q8—44no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4C22×D412yes—
363C22×C4C22×D412yes—
364C22×C4C22×D414yes—
365C22×C4C22×D414yes—
366C22×C4C22×D412yes—
367C22×C4C22×Q818yes—
368C22×C4C22×Q818yes—
369C22×C4C22×Q818yes—
370C22×C4C22×Q814yes—
371C22×C4C22×Q814yes—
372C22×C4C22×Q814yes—
373C22×C4C22×Q812yes—
374C22×C4C22×Q812yes—
375C22×C4C22×Q812yes—
376C22×C4C2×#4114yes—
377C22×C4C2×#4112yes—
378C22×C4H(5,𝔽2)14yes—
379C22×C4H(5,𝔽2)12yes—
380C22×C4H(5,𝔽2)12yes—
381C22×C42−1+414yes—
382C4⋊C4—232no—
383C4⋊C4—216no—
384C4⋊C4—216no—
385C4⋊C4—216no—
386C4⋊C4—416no—
387C4⋊C4—48no—
388C4⋊C4—48no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—48no—
392C4⋊C4—48no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—48no—
396C4⋊C4—88no—
397C4⋊C4—88no—
398C4⋊C4—88no—
399C4⋊C4—88no—
400C4⋊C4—88no—
401C4⋊C4—88no—
402C42—28no—
403C42—416no—
404C42—48no—
405C42—48no—
406C42—88no—
407C42C22⋊Q8116yes—
408C42C4⋊Q8116yes—
409C23C23×D411yescharacteristic
410C23C23×Q811yescharacteristic
411C23C23×Q811yescharacteristic
412C23C23×Q811yescharacteristic
413C23C22×#4111yescharacteristic
414C23C2×H(5,𝔽2)11yescharacteristic
415C23C2×H(5,𝔽2)11yescharacteristic
416C23C2×H(5,𝔽2)12yes—
417C23C2×2−1+412yes—
418C23#58512yes—
419C23#58512yes—
420Q8—416no—
421Q8—416no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—28no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—28no—
438C2×C4—28no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—28no—
444C2×C4—28no—
445C2×C4—28no—
446C2×C4—28no—
447C2×C4—28no—
448C2×C4—28no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—44no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—44no—
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—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4C2×C22⋊Q818yes—
501C2×C4C2×C22⋊Q814yes—
502C2×C4C2×C4⋊Q818yes—
503C2×C4C2×C4⋊Q814yes—
504C2×C4#53514yes—
505C2×C4#55414yes—
506C2×C4#55414yes—
507C2×C4Q8214yes—
508C2×C4Q8214yes—
509C22C22×C22⋊Q811yescharacteristic
510C22C22×C4⋊Q811yescharacteristic
511C22C2×#53511yescharacteristic
512C22C2×#55411yescharacteristic
513C22C2×#55411yescharacteristic
514C22C2×Q8211yescharacteristic
515C22C2×Q8211yescharacteristic
516C22#348812yes—
517C22#348912yes—
518C22#350112yes—
519C22#350212yes—
520C22#350312yes—
521C22#350312yes—
522C22#350412yes—
523C22#350412yes—
524C22#350612yes—
525C22#350712yes—
526C22#353711yescharacteristic
527C22#353811yescharacteristic
528C22#353911yescharacteristic
529C22#353911yescharacteristic
530C22#354112yes—
531C22#354212yes—
532C4—28no—
533C4—28no—
534C4—48no—
535C4—42no—
536C4—44no—
537C4—48no—
538C4—48no—
539C4—42no—
540C4—44no—
541C4—82no—
542C4—84no—
543C4—82no—
544C4#256718yes—
545C2C2×#256711yescharacteristic
546C2#3551912yes—
547C2#3556612yes—
548C2#3727011yescharacteristic
549C2#3737511yescharacteristic
550C2#4495012yes—
551C2#4496612yes—
552C2#4536211yescharacteristic
553C2#4539511yescharacteristic
554C2#4542711yescharacteristic
555C2#4543511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
73. C2×#125Q8semidirect
122. #384Q8semidirect
123. #382Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23992
C2×#23302
#156822
#156252
#155922
#154382
#152022
#151582
#141602
#141512
#141502
#139102
#137792