Finite Group Databasebeta

← #763438

Subgroups — #763438

Order 512.

2592 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2453C212yesmaximal, complemented
2C2×#2323C212yesmaximal
3#15971C214yesmaximal
4#15725C214yesmaximal
5#15460C214yesmaximal, complemented
6#15454C214yesmaximal, complemented
7#15216C211yesmaximal, characteristic
8#15179C212yesmaximal
9#15178C211yesmaximal, characteristic
10#14132C212yesmaximal, complemented
11#14024C212yesmaximal, complemented
12#14021C212yesmaximal, complemented
13#13844C211yesmaximal, characteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×#125C2211yescharacteristic
16C22×C22⋊Q8C2212yes—
17C22×C4×Q8C2212yes—
18C22×#117C2211yescharacteristic
19#2453—216no—
20C2×#400C2214yes—
21C2×#398C2214yes—
22#2323—216no—
23C2×#391C2212yes—
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#389C2214yes—
28C2×#389C2212yes—
29C2×#389C2214yes—
30C2×#389C2212yes—
31C2×#388C2214yes—
32C2×#388C2214yes—
33C2×#387C2214yes—
34C2×#387C2214yes—
35C2×#380C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C22⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2211yescharacteristic
40#1445C2214yes—
41#1444C2212yes—
42#1444C2214yes—
43#1443C2214yes—
44#1442C2212yes—
45#1442C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1440C2214yes—
53#1439C2212yes—
54#1439C2212yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62C4⋊#95C2211yescharacteristic
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2211yescharacteristic
66#1434C2214yes—
67#1434C2214yes—
68C4×#95C2212yes—
69C23×Q8C2312yes—
70C24×C4C2311yescharacteristic
71C2×C4⋊Q8—28no—
72C2×C4⋊Q8—42no—
73C2×#125—28no—
74C2×#125—42no—
75C2×C22⋊Q8—28no—
76C2×C22⋊Q8—28no—
77C2×C22⋊Q8—28no—
78C2×C4×Q8—28no—
79C2×C4×Q8—28no—
80C2×C4×Q8—44no—
81C2×#117—28no—
82C2×#117—42no—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2311yescharacteristic
97C22×C22⋊C4C2311yescharacteristic
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103C22×C42C2312yes—
104C22×C42C2314yes—
105C22×C42C2312yes—
106C22×C42C2314yes—
107#400—216no—
108#398—216no—
109#391—28no—
110#391—28no—
111#391—28no—
112#391—28no—
113#389—216no—
114#389—28no—
115#389—216no—
116#389D4116yes—
117#388—216no—
118#388—216no—
119#387—216no—
120#387—216no—
121#380—28no—
122C4×C4⋊C4—28no—
123C4×C4⋊C4—28no—
124C4×C22⋊C4—24no—
125C4×C22⋊C4—42no—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2311yescharacteristic
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2311yescharacteristic
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C25C2411yescharacteristic
154C22×Q8—24no—
155C22×Q8—24no—
156C22×Q8—24no—
157C22×Q8—24no—
158C22×Q8—24no—
159C22×Q8—24no—
160C22×Q8#4118yes—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4#4114yes—
166C23×C4C2412yes—
167C23×C4C2414yes—
168C23×C4C2412yes—
169C23×C4C2414yes—
170C23×C4C2414yes—
171C23×C4C2412yes—
172C23×C4C2412yes—
173C23×C4C2414yes—
174C23×C4C2411yescharacteristic
175C23×C4C2412yes—
176C23×C4C2412yes—
177C23×C4C2411yescharacteristic
178C4⋊Q8—416no—
179#125—416no—
180C22⋊Q8—264no—
181C4×Q8—432no—
182#117—416no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
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⋊C4C2×D418yes—
217C2×C4⋊C4C2×D418yes—
218C2×C4⋊C4C2×D4116yes—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
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—28no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—28no—
238C2×C42—28no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—28no—
247C2×C42—44no—
248C2×C42—44no—
249C2×C42—44no—
250C2×C42C2×D418yes—
251#95—24no—
252#95—28no—
253#95—24no—
254#95—28no—
255#95—28no—
256#95—28no—
257#95—28no—
258#95—28no—
259#95—28no—
260#95—28no—
261#95—28no—
262#95—22no—
263#95—22no—
264#95—28no—
265#95—28no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—28no—
270#95—42no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—42no—
277#95C2×D4116yes—
278C24—22no—
279C24—22no—
280C24—21no—
281C24—22no—
282C24—22no—
283C24—21no—
284C24—22no—
285C24—22no—
286C24C2×#4112yes—
287C24C2511yescharacteristic
288C2×Q8—216no—
289C2×Q8—216no—
290C2×Q8—216no—
291C2×Q8—48no—
292C2×Q8—48no—
293C2×Q8—48no—
294C2×Q8—48no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—21no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—216no—
342C22×C4—28no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—44no—
365C22×C4—48no—
366C22×C4—44no—
367C22×C4C22×D418yes—
368C22×C4C22×D414yes—
369C22×C4C22×D418yes—
370C22×C4C22×D414yes—
371C22×C4C22×D414yes—
372C22×C4C2×#4118yes—
373C22×C4C2×#4112yes—
374C22×C4C2×#4112yes—
375C22×C4C2×#4114yes—
376C22×C4H(5,𝔽2)12yes—
377C22×C42−1+414yes—
378C22×C42−1+414yes—
379C22×C42−1+412yes—
380C4⋊C4—232no—
381C4⋊C4—216no—
382C4⋊C4—216no—
383C4⋊C4—416no—
384C4⋊C4—416no—
385C4⋊C4—48no—
386C4⋊C4—48no—
387C4⋊C4—416no—
388C4⋊C4—416no—
389C4⋊C4—416no—
390C4⋊C4—48no—
391C4⋊C4—48no—
392C22⋊C4—232no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—48no—
396C22⋊C4—48no—
397C22⋊C4—88no—
398C42—216no—
399C42—416no—
400C42—48no—
401C42—44no—
402C42—44no—
403C42—88no—
404C23—24no—
405C23—22no—
406C23—24no—
407C23—22no—
408C23—22no—
409C23—41no—
410C23—42no—
411C23—41no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—41no—
424C23—41no—
425C23C23×D412yes—
426C23C22×#4111yescharacteristic
427C23C22×#4112yes—
428C23C2×H(5,𝔽2)11yescharacteristic
429C23C2×2−1+411yescharacteristic
430C23C2×2−1+412yes—
431C23C2×2−1+412yes—
432C23#58512yes—
433C23#58512yes—
434Q8—432no—
435C2×C4—28no—
436C2×C4—28no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—28no—
442C2×C4—28no—
443C2×C4—28no—
444C2×C4—28no—
445C2×C4—28no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—28no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—28no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—416no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4#52518yes—
512C2×C4#53914yes—
513C2×C4D4214yes—
514C2×C4#54818yes—
515C2×C4#55114yes—
516C22—42no—
517C22—42no—
518C22—44no—
519C22—44no—
520C22—42no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22C2×#52512yes—
526C22C2×#53911yescharacteristic
527C22C2×D4211yescharacteristic
528C22C2×#54812yes—
529C22C2×#55111yescharacteristic
530C22#348512yes—
531C22#348912yes—
532C22#349512yes—
533C22#349712yes—
534C22#349912yes—
535C22#350911yescharacteristic
536C22#351211yescharacteristic
537C22#351612yes—
538C22#351612yes—
539C22#352312yes—
540C22#352412yes—
541C22#354112yes—
542C22#354712yes—
543C22#355512yes—
544C22#355611yescharacteristic
545C22#355711yescharacteristic
546C4—44no—
547C4—44no—
548C4—44no—
549C4—48no—
550C4—48no—
551C4—44no—
552C4—42no—
553C4—48no—
554C4—82no—
555C4—84no—
556C4—82no—
557C4#246318yes—
558C2—82no—
559C2C2×#246311yescharacteristic
560C2#3565412yes—
561C2#3567112yes—
562C2#3738211yescharacteristic
563C2#3739411yescharacteristic
564C2#4509112yes—
565C2#4510212yes—
566C2#4532012yes—
567C2#4534312yes—
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
1. C2×#2453C2semidirect
5. #15460C2semidirect
6. #15454C2semidirect
10. #14132C2semidirect
11. #14024C2semidirect
12. #14021C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24532
C2×#23232
#159712
#157252
#154602
#154542
#152162
#151792
#151782
#141322
#140242
#140212
#138442