Finite Group Databasebeta

← #413250

Subgroups — #413250

Order 512.

1388 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2550C211yesmaximal, characteristic
2C2×#2059C212yesmaximal
3C2×#2057C212yesmaximal
4C2×#1947C211yesmaximal, characteristic
5C2×#1847C211yesmaximal, characteristic
6#13910C211yesmaximal, characteristic
7#11328C212yesmaximal
8#11327C212yesmaximal
9#10850C211yesmaximal, characteristic
10#10835C211yesmaximal, characteristic
11#10703C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×#107C2212yes—
15C22×#107C2212yes—
16C22×#106C2212yes—
17C22×C4⋊C8C2211yescharacteristic
18C22×#97C2211yescharacteristic
19#2550—24no—
20#2550—24no—
21C2×#398—21no—
22C2×#395—21no—
23C2×#391—21no—
24C2×#389—21no—
25C2×#389C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#384—21no—
30C2×#384C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#2059—28no—
36#2057—28no—
37#1947—24no—
38#1847—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46#1444—21no—
47#1444—21no—
48C4⋊#95—21no—
49C23×Q8—21no—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—42no—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125—24no—
59C2×#125—42no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C42—21no—
78C22×C42—21no—
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×#107—24no—
83C2×#107—24no—
84C2×#107—24no—
85C2×#107—24no—
86C2×#107—24no—
87C2×#107—42no—
88C2×#106—24no—
89C2×#106—24no—
90C2×#106—42no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8Q814yescomplemented
94C2×#97—22no—
95C2×#97—22no—
96C2×#97Q814yescomplemented
97#398—28no—
98#395—28no—
99#391—28no—
100#389—28no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—24no—
106#389—24no—
107#389—42no—
108#389—42no—
109#384—22no—
110#384—22no—
111#384—44no—
112#382—22no—
113#382D414yes—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4D414yes—
119C4×C4⋊C4Q814yescomplemented
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95D411yescharacteristic
129C2×#95D411yescharacteristic
130#340—24no—
131#340—24no—
132#340—24no—
133#340D418yes—
134#340Q818yescomplemented
135#336—24no—
136#336—24no—
137C22×Q8—24no—
138C22×Q8—44no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8—82no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2411yescharacteristic
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8C2×D414yes—
166C22×C8C2×Q814yes—
167C22×C8C2×Q814yes—
168C22×C8C2×Q814yes—
169C4⋊Q8—48no—
170C4⋊Q8—44no—
171C4⋊Q8—44no—
172#125—48no—
173#125—44no—
174#125—44no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D414yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D414yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×Q814yes—
229C2×C4⋊C4C2×Q814yes—
230C2×C4⋊C4C2×Q814yes—
231C2×C42—24no—
232C2×C42—22no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—22no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—24no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—82no—
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42C2×Q812yes—
247C2×C42C2×Q812yes—
248C2×C42C2×Q812yes—
249C2×C42#4112yes—
250#107—48no—
251#107—48no—
252#106—48no—
253C4⋊C8—28no—
254#97—28no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—22no—
261#95—22no—
262#95—24no—
263#95—44no—
264#95—44no—
265C24C22×D411yescharacteristic
266C2×Q8—48no—
267C2×Q8—48no—
268C2×Q8—88no—
269C2×Q8—84no—
270C2×Q8—88no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4#12012yes—
330C22×C4#12012yes—
331C22×C4C22⋊Q812yes—
332C22×C4C22⋊Q812yes—
333C22×C4C4⋊Q812yes—
334C22×C4C4⋊Q812yes—
335C22×C4AGL(1,ℤ/8)12yes—
336C22×C4#13712yes—
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D412yes—
340C22×C4C22×Q811yescharacteristic
341C22×C4C22×Q811yescharacteristic
342C22×C4C22×Q811yescharacteristic
343C22×C4C22×Q812yes—
344C22×C4C2×#4111yescharacteristic
345C22×C4H(5,𝔽2)12yes—
346C22×C42−1+412yes—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8C4⋊Q818yes—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—48no—
373C4⋊C4—48no—
374C4⋊C4—48no—
375C4⋊C4—48no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—88no—
385C4⋊C4—84no—
386C4⋊C4—84no—
387C4⋊C4—84no—
388C4⋊C4—88no—
389C4⋊C4#12714yes—
390C4⋊C4C4⋊Q814yes—
391C4⋊C4C4⋊Q818yes—
392C42—48no—
393C42—41no—
394C42—41no—
395C42—41no—
396C42—41no—
397C42—42no—
398C42—42no—
399C42—84no—
400C42C22⋊Q814yes—
401C42C22⋊Q814yes—
402C42C4⋊Q814yes—
403C42C4⋊Q814yes—
404C23C2×#12011yescharacteristic
405C23C2×C22⋊Q811yescharacteristic
406C23C2×C4⋊Q811yescharacteristic
407C23D4212yes—
408C23D4×Q812yes—
409C23C2×AGL(1,ℤ/8)11yescharacteristic
410C23C2×#13711yescharacteristic
411C23#57511yescharacteristic
412C23#57511yescharacteristic
413C23#57612yes—
414C23#57812yes—
415Q8—816no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—28no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4#39514yes—
481C2×C4#39514yes—
482C2×C4C2×C22⋊Q811yescharacteristic
483C2×C4C2×C22⋊Q811yescharacteristic
484C2×C4C2×#12711yescharacteristic
485C2×C4C2×C4⋊Q811yescharacteristic
486C2×C4C2×C4⋊Q811yescharacteristic
487C2×C4C2×C4⋊Q812yes—
488C2×C4C2×C4⋊Q812yes—
489C2×C4C2×C4⋊Q811yescharacteristic
490C2×C4#53511yescharacteristic
491C2×C4#53611yescharacteristic
492C2×C4#54412yes—
493C2×C4#54412yes—
494C2×C4D4212yes—
495C2×C4#54712yes—
496C2×C4D4×Q812yes—
497C2×C4D4×Q812yes—
498C2×C4#55412yes—
499C2×C4Q8212yes—
500C8—28no—
501C8—44no—
502C22C2×#39511yescharacteristic
503C22#243311yescharacteristic
504C22#243411yescharacteristic
505C22#244012yes—
506C22#255112yes—
507C22#300611yescharacteristic
508C22#300711yescharacteristic
509C22#300811yescharacteristic
510C22#301111yescharacteristic
511C22#307711yescharacteristic
512C22#307811yescharacteristic
513C22#308312yes—
514C22#316412yes—
515C22#316511yescharacteristic
516C22#316611yescharacteristic
517C22#326412yes—
518C22#326812yes—
519C22#326912yes—
520C22#327012yes—
521C22#338512yes—
522C22#338612yes—
523C22#338912yes—
524C22#339012yes—
525C4—24no—
526C4—24no—
527C4—42no—
528C4—42no—
529C4—44no—
530C4—48no—
531C4—44no—
532C4—81no—
533C4—82no—
534C4—81no—
535C4—82no—
536C4—162no—
537C4#244012yes—
538C4#254211yescharacteristic
539C4#255112yes—
540C4#256711yescharacteristic
541C4#266011yescharacteristic
542C4#286811yescharacteristic
543C2#1514311yescharacteristic
544C2#2355411yescharacteristic
545C2#2355511yescharacteristic
546C2#2576711yescharacteristic
547C2#2577011yescharacteristic
548C2#2584911yescharacteristic
549C2#2585011yescharacteristic
550C2#2605212yes—
551C2#2605512yes—
552C2#2882012yes—
553C2#2882112yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
93. C2×C4⋊C8Q8semidirect
96. C2×#97Q8semidirect
119. C4×C4⋊C4Q8semidirect
134. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25502
C2×#20592
C2×#20572
C2×#19472
C2×#18472
#139102
#113282
#113272
#108502
#108352
#107032