Finite Group Databasebeta

← #418699

Subgroups — #418699

Order 512.

1508 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#500C211yesmaximal, characteristic
2C2×#2451C211yesmaximal, characteristic
3C2×#2024C211yesmaximal, characteristic
4C2×#1863C212yesmaximal
5#15128C211yesmaximal, characteristic
6#11384C211yesmaximal, characteristic
7#11328C212yesmaximal
8#11219C212yesmaximal
9#11124C212yesmaximal
10#10926C211yesmaximal, characteristic
11#10874C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C2×#500—28no—
16C2×#500—24no—
17C22×#107C2212yes—
18C22×#107C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×C4×C8C2211yescharacteristic
22#2451—24no—
23#2451—24no—
24C2×#398—21no—
25C2×#395—21no—
26C2×#395—21no—
27C2×#391—21no—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2212yes—
33C2×#387—21no—
34C2×C4×C4⋊C4C2211yescharacteristic
35#2024—24no—
36#1863—28no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C2×#336C2211yescharacteristic
44#1446—21no—
45#1444—21no—
46#1441—21no—
47C4⋊#95C2212yes—
48C4⋊#95C2211yescharacteristic
49C23×Q8—21no—
50C23×Q8—21no—
51C23×Q8C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—42no—
60C2×C4⋊Q8—42no—
61C2×C4⋊Q8D414yes—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—24no—
65C2×C4×Q8—42no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#500—232no—
84C2×#107—24no—
85C2×#107—24no—
86C2×#107—24no—
87C2×#107—24no—
88C2×#107—24no—
89C2×#107D418yes—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—42no—
95C2×#103Q818yescomplemented
96C2×C4×C8—24no—
97C2×C4×C8—22no—
98#398—28no—
99#395—28no—
100#395—44no—
101#391—28no—
102#391—22no—
103#391—22no—
104#391—22no—
105#391D414yes—
106#389—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#389—24no—
111#389—42no—
112#387—28no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95D411yescharacteristic
127C2×#95D411yescharacteristic
128#340—24no—
129#340—24no—
130#340—24no—
131#340—42no—
132#336—24no—
133#336—24no—
134#336—22no—
135C22×Q8—24no—
136C22×Q8—21no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—44no—
143C22×Q8—44no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×Q8—42no—
147C22×Q8—42no—
148C22×Q8—42no—
149C22×Q8—82no—
150C22×Q8—82no—
151C22×Q8C2×D412yes—
152C22×Q8C2×Q814yes—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2411yescharacteristic
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8C2×D414yes—
178C22×C8C2×Q814yes—
179C22×C8#4114yes—
180C4⋊Q8—28no—
181C4⋊Q8—28no—
182C4⋊Q8—44no—
183C4⋊Q8—48no—
184C4⋊Q8—48no—
185C4⋊Q8—44no—
186C4×Q8—48no—
187C4×Q8—44no—
188C4×Q8—44no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
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—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—82no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4Q1614yes—
226C2×C4⋊C4C2×D414yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D414yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×Q814yes—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—24no—
235C2×C42—22no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—44no—
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246#107—216no—
247#107—216no—
248#103—216no—
249#103—48no—
250C4×C8—28no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—22no—
258#95—22no—
259#95—44no—
260#95—44no—
261#95—42no—
262#95—42no—
263#95—82no—
264C24C22×D411yescharacteristic
265C2×Q8—22no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—48no—
274C2×Q8—48no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—48no—
283C2×Q8—84no—
284C2×Q8—88no—
285C2×Q8—88no—
286C2×Q8—88no—
287C2×Q8—88no—
288C2×Q8—84no—
289C2×Q8—84no—
290C2×Q8C4⋊Q814yes—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4#12012yes—
350C22×C4#12012yes—
351C22×C4C2×Q1612yes—
352C22×C4C2×Q1612yes—
353C22×C4C2×Q1612yes—
354C22×C4#13512yes—
355C22×C4AGL(1,ℤ/8)12yes—
356C22×C4#13712yes—
357C22×C4C22×D412yes—
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×Q812yes—
361C22×C4C2×#4112yes—
362C22×C4H(5,𝔽2)12yes—
363C22×C42−1+412yes—
364C22×C42−1+412yes—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8C22⋊Q818yes—
379C4⋊C4—22no—
380C4⋊C4—24no—
381C4⋊C4—28no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—48no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—44no—
399C4⋊C4—84no—
400C4⋊C4—84no—
401C4⋊C4—88no—
402C4⋊C4#12014yes—
403C42—22no—
404C42—22no—
405C42—24no—
406C42—48no—
407C42—41no—
408C42—41no—
409C42—42no—
410C42—42no—
411C42—42no—
412C23C2×#12011yescharacteristic
413C23#53211yescharacteristic
414C23#53611yescharacteristic
415C23D4212yes—
416C23D4×Q812yes—
417C23C22×Q1611yescharacteristic
418C23C2×#13511yescharacteristic
419C23C2×AGL(1,ℤ/8)11yescharacteristic
420C23C2×#13711yescharacteristic
421C23#57612yes—
422C23#57812yes—
423Q8—28no—
424Q8—44no—
425Q8—44no—
426Q8—44no—
427Q8—816no—
428Q8—816no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—44no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4#44914yes—
494C2×C4#45114yes—
495C2×C4C2×#12011yescharacteristic
496C2×C4C2×C22⋊Q812yes—
497C2×C4C2×C4⋊Q811yescharacteristic
498C2×C4#53411yescharacteristic
499C2×C4#53612yes—
500C2×C4#53611yescharacteristic
501C2×C4#54212yes—
502C2×C4#54412yes—
503C2×C4D4212yes—
504C2×C4#54712yes—
505C2×C4#54812yes—
506C2×C4#54812yes—
507C2×C4D4×Q812yes—
508C2×C4D4×Q812yes—
509C2×C4#55412yes—
510C2×C4#56312yes—
511C8—28no—
512C8—82no—
513C22#240311yescharacteristic
514C22#243211yescharacteristic
515C22#243411yescharacteristic
516C22#245212yes—
517C22#267712yes—
518C22C2×#45111yescharacteristic
519C22C2×#44911yescharacteristic
520C22#300811yescharacteristic
521C22#301111yescharacteristic
522C22#317011yescharacteristic
523C22#317512yes—
524C22#318211yescharacteristic
525C22#323011yescharacteristic
526C22#323111yescharacteristic
527C22#324312yes—
528C22#326412yes—
529C22#327012yes—
530C22#328012yes—
531C22D4×Q1612yes—
532C22Q8×Q1612yes—
533C22#338312yes—
534C22#338512yes—
535C22#338612yes—
536C4—24no—
537C4—24no—
538C4—44no—
539C4—42no—
540C4—42no—
541C4—44no—
542C4—81no—
543C4—82no—
544C4—84no—
545C4—81no—
546C4—82no—
547C4—84no—
548C4#240912yes—
549C4#244012yes—
550C4#267412yes—
551C4#272912yes—
552C2#1526511yescharacteristic
553C2#2385111yescharacteristic
554C2#2385611yescharacteristic
555C2#2567411yescharacteristic
556C2#2568111yescharacteristic
557C2#2584011yescharacteristic
558C2#2584111yescharacteristic
559C2#2648512yes—
560C2#2649512yes—
561C2#3094812yes—
562C2#3095712yes—
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
95. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5002
C2×#24512
C2×#20242
C2×#18632
#151282
#113842
#113282
#112192
#111242
#109262
#108742