Finite Group Databasebeta

← #460483

Subgroups — #460483

Order 512.

1224 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2453C211yesmaximal, characteristic
2C2×#2083C211yesmaximal, characteristic
3C2×#1803C211yesmaximal, characteristic
4#15151C211yesmaximal, characteristic
5#11385C212yesmaximal
6#11328C212yesmaximal
7#11227C211yesmaximal, characteristic
8#11202C211yesmaximal, characteristic
9#11124C212yesmaximal
10#11067C212yesmaximal
11#10789C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×#107C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18#2453—24no—
19#2453—24no—
20C2×#400—21no—
21C2×#398—21no—
22C2×#398—21no—
23C2×#398—21no—
24C2×#391C2211yescharacteristic
25C2×#391C2211yescharacteristic
26C2×#389—21no—
27C2×#389C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#387—21no—
31C2×#384C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#2083—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×Q8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125—24no—
53C2×#125—42no—
54C2×C4×Q8—22no—
55C2×C4×Q8—24no—
56C2×C4×Q8—42no—
57C2×C4×Q8D414yes—
58C22×C4⋊C4—21no—
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×#107—24no—
78C2×#107—42no—
79C2×#107—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×#103Q818yes—
88#400—28no—
89#398—28no—
90#398—44no—
91#398—44no—
92#391—22no—
93#391—22no—
94#391—22no—
95#391—22no—
96#389—28no—
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—22no—
116C2×#95—22no—
117C2×#95—21no—
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—21no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8C2×D412yes—
138C22×Q8C2×Q814yes—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2411yescharacteristic
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×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8C2×Q814yes—
166#125—48no—
167#125—44no—
168#125—44no—
169C4×Q8—28no—
170C4×Q8—44no—
171C4×Q8—48no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
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—22no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×Q814yes—
224C2×C4⋊C4#4114yes—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—44no—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241#107—48no—
242C4⋊C8—44no—
243#103—48no—
244#103—48no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—22no—
252#95—22no—
253#95—44no—
254#95—42no—
255#95—44no—
256#95—44no—
257#95—42no—
258C24C22×D411yescharacteristic
259C2×Q8—22no—
260C2×Q8—24no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8C4⋊Q814yes—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
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—41no—
327C22×C4—41no—
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#12012yes—
341C22×C4#12012yes—
342C22×C4#13512yes—
343C22×C4AGL(1,ℤ/8)12yes—
344C22×C4AGL(1,ℤ/8)12yes—
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×Q812yes—
348C22×C4C2×#4112yes—
349C22×C4H(5,𝔽2)12yes—
350C22×C4H(5,𝔽2)11yescharacteristic
351C22×C4H(5,𝔽2)11yescharacteristic
352C22×C4H(5,𝔽2)11yescharacteristic
353C22×C42−1+412yes—
354C22×C42−1+411yescharacteristic
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C4⋊C4—22no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—44no—
377C4⋊C4—48no—
378C4⋊C4—48no—
379C4⋊C4—48no—
380C4⋊C4—48no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—44no—
388C4⋊C4—48no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—88no—
394C4⋊C4—84no—
395C4⋊C4—84no—
396C4⋊C4#12314yes—
397C42—22no—
398C42—22no—
399C42—24no—
400C42—48no—
401C42—41no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—42no—
408C23C2×#12011yescharacteristic
409C23#53511yescharacteristic
410C23#53711yescharacteristic
411C23#54612yes—
412C23D4×Q812yes—
413C23C2×#13511yescharacteristic
414C23C2×AGL(1,ℤ/8)11yescharacteristic
415C23C2×AGL(1,ℤ/8)11yescharacteristic
416C23#57511yescharacteristic
417C23#57812yes—
418C23#57812yes—
419Q8—24no—
420Q8—24no—
421Q8—42no—
422Q8—42no—
423Q8—44no—
424Q8—44no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—41no—
444C2×C4—44no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—82no—
476C2×C4—81no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—81no—
481C2×C4—81no—
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#44914yes—
493C2×C4C2×#12311yescharacteristic
494C2×C4C2×C4⋊Q811yescharacteristic
495C2×C4#53411yescharacteristic
496C2×C4#53511yescharacteristic
497C2×C4#53511yescharacteristic
498C2×C4#53611yescharacteristic
499C2×C4#53711yescharacteristic
500C2×C4#53711yescharacteristic
501C2×C4#54112yes—
502C2×C4#54212yes—
503C2×C4#54312yes—
504C2×C4#54412yes—
505C2×C4#54612yes—
506C2×C4D4×Q812yes—
507C2×C4#55412yes—
508C2×C4#55412yes—
509C2×C4#56212yes—
510C2×C4#56212yes—
511C8—44no—
512C8—82no—
513C22#240511yescharacteristic
514C22#243411yescharacteristic
515C22#261511yescharacteristic
516C22#269212yes—
517C22#270412yes—
518C22C2×#44911yescharacteristic
519C22#300611yescharacteristic
520C22#301111yescharacteristic
521C22#301111yescharacteristic
522C22#321211yescharacteristic
523C22#321711yescharacteristic
524C22#322312yes—
525C22#324711yescharacteristic
526C22#325111yescharacteristic
527C22#326112yes—
528C22#330012yes—
529C22#330812yes—
530C22#331512yes—
531C22#333012yes—
532C22#338312yes—
533C22#338612yes—
534C22#338612yes—
535C22#339012yes—
536C4—24no—
537C4—42no—
538C4—44no—
539C4—42no—
540C4—44no—
541C4—84no—
542C4—82no—
543C4—81no—
544C4—82no—
545C4—81no—
546C4—81no—
547C4—162no—
548C4#250211yescharacteristic
549C4#262511yescharacteristic
550C4#265911yescharacteristic
551C4#269212yes—
552C4#270412yes—
553C4#281611yescharacteristic
554C2#1636511yescharacteristic
555C2#2403511yescharacteristic
556C2#2405411yescharacteristic
557C2#2584011yescharacteristic
558C2#2584911yescharacteristic
559C2#2966211yescharacteristic
560C2#2967411yescharacteristic
561C2#3152812yes—
562C2#3154012yes—
563C2#3187612yes—
564C2#3189612yes—
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

Maximal subgroups

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

subgroupindex
C2×#24532
C2×#20832
C2×#18032
#151512
#113852
#113282
#112272
#112022
#111242
#110672
#107892