Finite Group Databasebeta

← #443541

Subgroups — #443541

Order 512.

1402 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic
2C2×#2552C211yesmaximal, characteristic
3C2×#2059C212yesmaximal
4C2×#2058C211yesmaximal, characteristic
5#15128C211yesmaximal, characteristic
6#11385C211yesmaximal, characteristic
7#11327C212yesmaximal
8#11323C212yesmaximal
9#10907C212yesmaximal
10#10881C211yesmaximal, characteristic
11#10875C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×#125C2211yescharacteristic
15C2×#501—28no—
16C2×#501—24no—
17C22×#107C2212yes—
18C22×#106C2212yes—
19C22×#106C2212yes—
20C22×#97C2211yescharacteristic
21#2552—24no—
22#2552—24no—
23C2×#398—21no—
24C2×#398—21no—
25C2×#395—21no—
26C2×#389—21no—
27C2×#389C2211yescharacteristic
28C2×#389C2212yes—
29C2×#389C2211yescharacteristic
30C2×#384—21no—
31C2×#384C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#2059—28no—
35#2058—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2211yescharacteristic
43C2×#336C2212yes—
44#1446—21no—
45#1444—21no—
46#1441—21no—
47C4⋊#95C2212yes—
48C4⋊#95C2211yescharacteristic
49C23×Q8—21no—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—42no—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125—24no—
59C2×#125—24no—
60C2×#125—22no—
61C2×#125—42no—
62C2×#125—42no—
63C2×#125D414yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86#501—232no—
87C2×#107—24no—
88C2×#107—24no—
89C2×#107D418yes—
90C2×#106—24no—
91C2×#106—24no—
92C2×#106—24no—
93C2×#106—24no—
94C2×#106—24no—
95C2×#106—42no—
96C2×#97—24no—
97C2×#97—22no—
98#398—28no—
99#398—28no—
100#395—44no—
101#389—28no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—24no—
106#389—42no—
107#389D414yes—
108#384—28no—
109#384—22no—
110#384—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95D411yescharacteristic
124C2×#95D411yescharacteristic
125#340—24no—
126#340—24no—
127#340—24no—
128#340—42no—
129#340Q818yescomplemented
130#336—24no—
131#336—22no—
132#336Q818yescomplemented
133C22×Q8—44no—
134C22×Q8—42no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8—82no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2411yescharacteristic
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8C2×D414yes—
162C22×C8C2×Q814yes—
163C22×C8C2×Q814yes—
164C22×C8C2×Q814yes—
165C4⋊Q8—28no—
166C4⋊Q8—48no—
167C4⋊Q8—44no—
168#125—28no—
169#125—44no—
170#125—48no—
171#125—48no—
172#125—44no—
173#125—44no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
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—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—44no—
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×D414yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×Q814yes—
229C2×C4⋊C4C2×Q814yes—
230C2×C42—22no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—22no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—22no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42C2×Q814yes—
245#107—216no—
246#106—216no—
247#106—48no—
248#97—28no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—22no—
253#95—22no—
254#95—44no—
255#95—42no—
256#95—42no—
257#95—44no—
258#95—82no—
259C24C22×D411yescharacteristic
260C2×Q8—48no—
261C2×Q8—84no—
262C2×Q8—84no—
263C2×Q8—88no—
264C2×Q8—88no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4#12012yes—
326C22×C4#12012yes—
327C22×C4C4⋊Q812yes—
328C22×C4C4⋊Q812yes—
329C22×C4AGL(1,ℤ/8)12yes—
330C22×C4#13712yes—
331C22×C4C22×D412yes—
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×Q812yes—
335C22×C4C22×Q812yes—
336C22×C4H(5,𝔽2)12yes—
337C22×C42−1+412yes—
338C22×C42−1+412yes—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8C4⋊Q818yes—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—48no—
367C4⋊C4—48no—
368C4⋊C4—48no—
369C4⋊C4—48no—
370C4⋊C4—48no—
371C4⋊C4—48no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—84no—
380C4⋊C4—84no—
381C4⋊C4—84no—
382C4⋊C4—88no—
383C4⋊C4#12014yes—
384C4⋊C4C4⋊Q814yes—
385C42—22no—
386C42—22no—
387C42—24no—
388C42—24no—
389C42—24no—
390C42—48no—
391C42—41no—
392C42—41no—
393C42—42no—
394C23C2×#12011yescharacteristic
395C23C2×C4⋊Q811yescharacteristic
396C23#53611yescharacteristic
397C23D4212yes—
398C23D4×Q812yes—
399C23C2×AGL(1,ℤ/8)11yescharacteristic
400C23C2×#13711yescharacteristic
401C23#57511yescharacteristic
402C23#57511yescharacteristic
403C23#57712yes—
404C23#57712yes—
405Q8—816no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4#50114yescomplemented
472C2×C4C2×#12011yescharacteristic
473C2×C4C2×C4⋊Q812yes—
474C2×C4C2×C4⋊Q811yescharacteristic
475C2×C4#53411yescharacteristic
476C2×C4#53611yescharacteristic
477C2×C4#53612yes—
478C2×C4#54412yes—
479C2×C4#54412yes—
480C2×C4D4212yes—
481C2×C4D4×Q812yes—
482C2×C4D4×Q812yes—
483C2×C4D4×Q812yes—
484C2×C4D4×Q812yes—
485C2×C4D4×Q812yes—
486C2×C4Q8212yes—
487C2×C4#56312yes—
488C8—28no—
489C8—44no—
490C22#240311yescharacteristic
491C22#243411yescharacteristic
492C22#243411yescharacteristic
493C22#255112yes—
494C22#267712yes—
495C22#300611yescharacteristic
496C22#300711yescharacteristic
497C22#300911yescharacteristic
498C22#301011yescharacteristic
499C22C2×#50111yescharacteristic
500C22#316312yes—
501C22#316611yescharacteristic
502C22#323411yescharacteristic
503C22#323611yescharacteristic
504C22#324412yes—
505C22#326712yes—
506C22#327112yes—
507C22#327212yes—
508C22#327812yes—
509C22#338412yes—
510C22#338712yes—
511C22#338812yes—
512C22#338812yes—
513C4—24no—
514C4—42no—
515C4—44no—
516C4—42no—
517C4—42no—
518C4—44no—
519C4—81no—
520C4—84no—
521C4—81no—
522C4—82no—
523C4—82no—
524C4—84no—
525C4#243412yes—
526C4#255112yes—
527C4#267712yes—
528C4#273412yes—
529C2#1576211yescharacteristic
530C2#2386411yescharacteristic
531C2#2387011yescharacteristic
532C2#2581911yescharacteristic
533C2#2582011yescharacteristic
534C2#2584811yescharacteristic
535C2#2585111yescharacteristic
536C2#2881712yes—
537C2#2882212yes—
538C2#3096812yes—
539C2#3097512yes—
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
129. #340Q8semidirect
132. #336Q8semidirect
471. C2×C4#501semidirect

Maximal subgroups

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

subgroupindex
C22×#5012
C2×#25522
C2×#20592
C2×#20582
#151282
#113852
#113272
#113232
#109072
#108812
#108752