Finite Group Databasebeta

← #418705

Subgroups — #418705

Order 512.

1442 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic
2C2×#2453C211yesmaximal, characteristic
3C2×#2024C211yesmaximal, characteristic
4C2×#1863C212yesmaximal
5#15128C211yesmaximal, characteristic
6#11385C211yesmaximal, characteristic
7#11327C212yesmaximal
8#11219C212yesmaximal
9#11122C212yesmaximal
10#10928C211yesmaximal, characteristic
11#10875C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C2×#501—28no—
16C2×#501—24no—
17C22×#107C2212yes—
18C22×#106C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#97C2211yescharacteristic
22#2453—24no—
23#2453—24no—
24C2×#400—21no—
25C2×#398—21no—
26C2×#395—21no—
27C2×#391C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#389—21no—
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×Q8C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—42no—
57C2×#125—24no—
58C2×#125—22no—
59C2×#125—42no—
60C2×#125D414yes—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—24no—
64C2×C4×Q8—42no—
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—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#501—232no—
84C2×#107—24no—
85C2×#107—24no—
86C2×#107D418yes—
87C2×#106—24no—
88C2×#106—24no—
89C2×#106—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—42no—
95C2×#103Q818yescomplemented
96C2×#97—24no—
97C2×#97—22no—
98#400—28no—
99#398—28no—
100#395—44no—
101#391—22no—
102#391—22no—
103#391—22no—
104#391D414yes—
105#389—28no—
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—21no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—44no—
142C22×Q8—42no—
143C22×Q8—42no—
144C22×Q8—42no—
145C22×Q8—82no—
146C22×Q8C2×D412yes—
147C22×Q8C2×Q814yes—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2411yescharacteristic
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8C2×D414yes—
173C22×C8C2×Q814yes—
174C22×C8#4114yes—
175C4⋊Q8—28no—
176C4⋊Q8—48no—
177C4⋊Q8—44no—
178#125—28no—
179#125—44no—
180#125—48no—
181C4×Q8—48no—
182C4×Q8—44no—
183C4×Q8—44no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
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—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D414yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D414yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×Q814yes—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—24no—
234C2×C42—22no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—44no—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245#107—216no—
246#106—216no—
247#103—216no—
248#103—48no—
249#97—28no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—22no—
257#95—22no—
258#95—44no—
259#95—42no—
260#95—42no—
261#95—44no—
262#95—82no—
263C24C22×D411yescharacteristic
264C2×Q8—22no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—48no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—84no—
281C2×Q8—84no—
282C2×Q8—88no—
283C2×Q8—88no—
284C2×Q8C4⋊Q814yes—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
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×C4AGL(1,ℤ/8)12yes—
352C22×C4#13712yes—
353C22×C4C22×D412yes—
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×Q812yes—
357C22×C4C2×#4112yes—
358C22×C4H(5,𝔽2)12yes—
359C22×C42−1+412yes—
360C22×C42−1+412yes—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8C22⋊Q818yes—
375C4⋊C4—22no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—48no—
387C4⋊C4—48no—
388C4⋊C4—48no—
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—88no—
399C4⋊C4#12014yes—
400C42—22no—
401C42—22no—
402C42—24no—
403C42—48no—
404C42—41no—
405C42—41no—
406C42—42no—
407C42—42no—
408C42—42no—
409C23C2×#12011yescharacteristic
410C23#53211yescharacteristic
411C23#53611yescharacteristic
412C23D4212yes—
413C23D4×Q812yes—
414C23C2×AGL(1,ℤ/8)11yescharacteristic
415C23C2×#13711yescharacteristic
416C23#57511yescharacteristic
417C23#57511yescharacteristic
418C23#57712yes—
419C23#57812yes—
420Q8—28no—
421Q8—44no—
422Q8—44no—
423Q8—44no—
424Q8—816no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—44no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—81no—
480C2×C4—82no—
481C2×C4—81no—
482C2×C4—82no—
483C2×C4—81no—
484C2×C4—81no—
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×C4C2×#12011yescharacteristic
494C2×C4C2×C22⋊Q812yes—
495C2×C4C2×C4⋊Q811yescharacteristic
496C2×C4#53411yescharacteristic
497C2×C4#53612yes—
498C2×C4#53611yescharacteristic
499C2×C4#54212yes—
500C2×C4#54412yes—
501C2×C4D4212yes—
502C2×C4#54712yes—
503C2×C4#54812yes—
504C2×C4#54812yes—
505C2×C4D4×Q812yes—
506C2×C4D4×Q812yes—
507C2×C4#55412yes—
508C2×C4#56312yes—
509C8—28no—
510C8—82no—
511C22#240311yescharacteristic
512C22#243211yescharacteristic
513C22#243411yescharacteristic
514C22#245212yes—
515C22#267712yes—
516C22#300611yescharacteristic
517C22#300711yescharacteristic
518C22#301011yescharacteristic
519C22#301111yescharacteristic
520C22#317211yescharacteristic
521C22#317812yes—
522C22#318011yescharacteristic
523C22#323811yescharacteristic
524C22#324011yescharacteristic
525C22#324212yes—
526C22#326712yes—
527C22#327012yes—
528C22#327212yes—
529C22#327712yes—
530C22#338612yes—
531C22#338712yes—
532C22#338812yes—
533C22#339012yes—
534C4—42no—
535C4—44no—
536C4—42no—
537C4—42no—
538C4—42no—
539C4—44no—
540C4—81no—
541C4—84no—
542C4—82no—
543C4—82no—
544C4—81no—
545C4—84no—
546C4#240912yes—
547C4#244012yes—
548C4#267412yes—
549C4#272912yes—
550C2#1526511yescharacteristic
551C2#2386411yescharacteristic
552C2#2386711yescharacteristic
553C2#2567911yescharacteristic
554C2#2568611yescharacteristic
555C2#2584811yescharacteristic
556C2#2584911yescharacteristic
557C2#2648912yes—
558C2#2649912yes—
559C2#3096412yes—
560C2#3097312yes—
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×#5012
C2×#24532
C2×#20242
C2×#18632
#151282
#113852
#113272
#112192
#111222
#109282
#108752