Finite Group Databasebeta

← #460485

Subgroups — #460485

Order 512.

1278 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2451C211yesmaximal, characteristic
2C2×#2084C211yesmaximal, characteristic
3C2×#1803C211yesmaximal, characteristic
4#15151C211yesmaximal, characteristic
5#11384C212yesmaximal
6#11327C212yesmaximal
7#11228C211yesmaximal, characteristic
8#11202C211yesmaximal, characteristic
9#11122C212yesmaximal
10#11067C212yesmaximal
11#10789C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×#106C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18#2451—24no—
19#2451—24no—
20C2×#398—21no—
21C2×#398—21no—
22C2×#398—21no—
23C2×#395—21no—
24C2×#391—21no—
25C2×#391C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#387—21no—
31C2×#384C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#2084—24no—
34#1803—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1445—21no—
44#1445—21no—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2212yes—
47C23×Q8—21no—
48C23×Q8C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—42no—
55C2×C4×Q8—22no—
56C2×C4×Q8—24no—
57C2×C4×Q8—42no—
58C2×C4×Q8D414yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×#106—24no—
78C2×#106—42no—
79C2×#106—42no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—42no—
82C2×#103—24no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#103Q818yescomplemented
88#398—28no—
89#398—44no—
90#398—44no—
91#395—28no—
92#391—28no—
93#391—22no—
94#391—22no—
95#391—22no—
96#391—22no—
97#389—22no—
98#389—22no—
99#389—22no—
100#389—42no—
101#389—42no—
102#389D414yes—
103#387—28no—
104#384—22no—
105#384—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95D411yescharacteristic
120C2×#95D411yescharacteristic
121#340—24no—
122#340—24no—
123#340—24no—
124#340—42no—
125#340—42no—
126#336—22no—
127#336—42no—
128#336—42no—
129C22×Q8—24no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—44no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8—82no—
142C22×Q8C2×D412yes—
143C22×Q8C2×Q814yes—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2411yescharacteristic
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×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8C2×Q814yes—
171C4⋊Q8—48no—
172C4⋊Q8—44no—
173C4⋊Q8—44no—
174C4×Q8—28no—
175C4×Q8—44no—
176C4×Q8—48no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
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—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4SD1614yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×Q814yes—
225C2×C4⋊C4#4114yes—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—24no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—44no—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242#106—48no—
243C4⋊C8—44no—
244#103—48no—
245#103—48no—
246#95—24no—
247#95—24no—
248#95—24no—
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—44no—
259C24C22×D411yescharacteristic
260C2×Q8—22no—
261C2×Q8—24no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—48no—
271C2×Q8—48no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—88no—
280C2×Q8—84no—
281C2×Q8—88no—
282C2×Q8C4⋊Q814yes—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
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—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4#12012yes—
343C22×C4#12012yes—
344C22×C4C2×SD1612yes—
345C22×C4C2×SD1612yes—
346C22×C4C2×SD1612yes—
347C22×C4AGL(1,ℤ/8)12yes—
348C22×C4#13712yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C22×Q812yes—
352C22×C4C2×#4112yes—
353C22×C4H(5,𝔽2)12yes—
354C22×C4H(5,𝔽2)11yescharacteristic
355C22×C4H(5,𝔽2)11yescharacteristic
356C22×C4H(5,𝔽2)11yescharacteristic
357C22×C42−1+412yes—
358C22×C42−1+411yescharacteristic
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—28no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—44no—
382C4⋊C4—48no—
383C4⋊C4—48no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—84no—
395C4⋊C4—88no—
396C4⋊C4—84no—
397C4⋊C4—84no—
398C4⋊C4—84no—
399C4⋊C4#12314yes—
400C42—22no—
401C42—22no—
402C42—24no—
403C42—48no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—42no—
411C23C2×#12011yescharacteristic
412C23#53511yescharacteristic
413C23#53711yescharacteristic
414C23#54612yes—
415C23D4×Q812yes—
416C23C22×SD1611yescharacteristic
417C23C2×AGL(1,ℤ/8)11yescharacteristic
418C23C2×#13711yescharacteristic
419C23#57511yescharacteristic
420C23#57712yes—
421C23#57812yes—
422Q8—24no—
423Q8—24no—
424Q8—42no—
425Q8—42no—
426Q8—44no—
427Q8—44no—
428Q8—816no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—82no—
480C2×C4—82no—
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—82no—
494C2×C4#44814yes—
495C2×C4C2×#12311yescharacteristic
496C2×C4C2×C4⋊Q811yescharacteristic
497C2×C4#53411yescharacteristic
498C2×C4#53511yescharacteristic
499C2×C4#53511yescharacteristic
500C2×C4#53611yescharacteristic
501C2×C4#53711yescharacteristic
502C2×C4#53711yescharacteristic
503C2×C4#54112yes—
504C2×C4#54212yes—
505C2×C4#54312yes—
506C2×C4#54412yes—
507C2×C4#54612yes—
508C2×C4D4×Q812yes—
509C2×C4#55412yes—
510C2×C4#55412yes—
511C2×C4#56212yes—
512C2×C4#56212yes—
513C8—44no—
514C8—82no—
515C22#240511yescharacteristic
516C22#243411yescharacteristic
517C22#261511yescharacteristic
518C22#269212yes—
519C22#270412yes—
520C22C2×#44811yescharacteristic
521C22#300611yescharacteristic
522C22#300911yescharacteristic
523C22#301111yescharacteristic
524C22#320811yescharacteristic
525C22#321611yescharacteristic
526C22#322512yes—
527C22#324611yescharacteristic
528C22#325411yescharacteristic
529C22#325812yes—
530C22#329812yes—
531C22#330612yes—
532C22#331212yes—
533C22#332312yes—
534C22Q8×SD1612yes—
535C22#338612yes—
536C22#338712yes—
537C22#339012yes—
538C4—24no—
539C4—42no—
540C4—42no—
541C4—44no—
542C4—42no—
543C4—42no—
544C4—44no—
545C4—84no—
546C4—81no—
547C4—82no—
548C4—82no—
549C4—162no—
550C4#250211yescharacteristic
551C4#262511yescharacteristic
552C4#265911yescharacteristic
553C4#269212yes—
554C4#270412yes—
555C4#281611yescharacteristic
556C2#1636511yescharacteristic
557C2#2404111yescharacteristic
558C2#2404811yescharacteristic
559C2#2583711yescharacteristic
560C2#2584911yescharacteristic
561C2#2966111yescharacteristic
562C2#2967211yescharacteristic
563C2#3152012yes—
564C2#3154812yes—
565C2#3186812yes—
566C2#3188812yes—
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
87. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24512
C2×#20842
C2×#18032
#151512
#113842
#113272
#112282
#112022
#111222
#110672
#107892