Finite Group Databasebeta

← #482708

Subgroups — #482708

Order 512.

839 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35749C211yesmaximal, characteristic
2#20639C211yesmaximal, characteristic
3#17371C211yesmaximal, characteristic
4#11905C212yesmaximal, complemented
5#11883C212yesmaximal, complemented
6#11835C212yesmaximal, complemented
7#11831C212yesmaximal, complemented
8#11767C212yesmaximal
9#11763C212yesmaximal
10C2×#557C411yescharacteristic
11C2×#519C2211yescharacteristic
12#2983—21no—
13#2905—21no—
14C22×#101—21no—
15#2634C411yescharacteristic
16#2634C2211yescharacteristic
17#2621C411yescharacteristic
18#2611C411yescharacteristic
19#2611C2211yescharacteristic
20#2599C411yescharacteristic
21#2599C2211yescharacteristic
22#2581C2211yescharacteristic
23#2546C2211yescharacteristic
24#2475—21no—
25#2475—21no—
26#2462—21no—
27#2462—21no—
28#2446—21no—
29#2434—21no—
30#2399C411yescharacteristic
31#2399C2211yescharacteristic
32#2396C411yescharacteristic
33#2365—21no—
34#2362—21no—
35#2356C411yescharacteristic
36#1801—21no—
37#1801—21no—
38#1801C2212yes—
39#1801C2212yes—
40#1742C2212yes—
41#1467—21no—
42#1467C2212yes—
43#1464—21no—
44#1464C2212yes—
45#1461C2212yes—
46#1458—21no—
47#1354C2214yes—
48#1353C2214yes—
49#1346C2214yes—
50#1346C2214yes—
51C23×Q8D411yescharacteristic
52C22×M16—22no—
53#557—44no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#125—21no—
58C2×#125—21no—
59C2×#125C2×C411yescharacteristic
60C2×#125C2×C411yescharacteristic
61C2×C22⋊Q8—21no—
62#519—44no—
63C2×C4×Q8—21no—
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70C2×C4×Q8D411yescharacteristic
71C2×C4×Q8D411yescharacteristic
72C2×C4×Q8D411yescharacteristic
73C2×C4×Q8C2311yescharacteristic
74C2×C4×Q8C2311yescharacteristic
75C2×#117C2×C411yescharacteristic
76C2×#117C2×C411yescharacteristic
77C2×#117D411yescharacteristic
78C2×#117D411yescharacteristic
79C2×#117D411yescharacteristic
80C2×#117C2311yescharacteristic
81#446—44no—
82#432—22no—
83#432—22no—
84#431—22no—
85#424—44no—
86#423—22no—
87#423—22no—
88C2×#101—22no—
89C2×#101—22no—
90C2×#101—44no—
91#400—21no—
92#400—21no—
93#400C2×C411yescharacteristic
94#400C2×C411yescharacteristic
95#400C2311yescharacteristic
96#399C2×C411yescharacteristic
97#398—21no—
98#398—21no—
99#398—21no—
100#398—21no—
101#398C2×C411yescharacteristic
102#398C2×C411yescharacteristic
103#398C2×C411yescharacteristic
104#398C2×C411yescharacteristic
105#398C2311yescharacteristic
106#398C2311yescharacteristic
107#397—21no—
108#397C2×C411yescharacteristic
109#397C2×C411yescharacteristic
110#396—21no—
111#395—21no—
112#395—21no—
113#395—21no—
114#395—21no—
115#395C2×C411yescharacteristic
116#395C2×C411yescharacteristic
117#395C2311yescharacteristic
118#393C2×C411yescharacteristic
119#391—21no—
120#391—21no—
121#391—21no—
122#391—21no—
123#389—21no—
124#389—21no—
125#389—21no—
126#389—21no—
127#389—21no—
128#388—21no—
129#388—21no—
130#388—21no—
131#388—21no—
132#387—21no—
133#387—21no—
134#387—21no—
135#387—21no—
136#387—21no—
137#387—21no—
138#387—21no—
139#387—21no—
140#387—21no—
141#387—21no—
142#387—21no—
143#387—21no—
144#387—21no—
145#387—21no—
146#385—21no—
147#385—21no—
148#385—21no—
149#385—21no—
150#383C2×C411yescharacteristic
151#383C2×C411yescharacteristic
152#382C2×C411yescharacteristic
153C4×C4⋊C4—21no—
154C4×C4⋊C4—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4C2×C411yescharacteristic
160C4×C22⋊C4—21no—
161#376C2×C411yescharacteristic
162#376C2×C411yescharacteristic
163#343—22no—
164#343C2314yes—
165#324—22no—
166#324—22no—
167#324C2314yes—
168#323—22no—
169C22×Q8—22no—
170C22×Q8—22no—
171C22×Q8—21no—
172C22×Q8—42no—
173C22×Q8—42no—
174C22×Q8C22⋊C411yescharacteristic
175C22×Q8C22⋊C411yescharacteristic
176C22×Q8C22×C411yescharacteristic
177C22×Q8C2×D411yescharacteristic
178C22×Q8C2×D411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C2×M16—24no—
183C2×M16—42no—
184C2×M16—44no—
185C2×M16—44no—
186C22×C8—42no—
187C4⋊Q8—42no—
188C4⋊Q8—44no—
189C4⋊Q8—42no—
190C4⋊Q8—42no—
191#125—42no—
192#125—42no—
193#125—44no—
194#125—44no—
195C22⋊Q8—44no—
196C4×Q8—24no—
197C4×Q8—24no—
198C4×Q8—42no—
199C4×Q8—42no—
200C4×Q8—42no—
201C4×Q8—42no—
202C4×Q8—42no—
203C4×Q8—42no—
204C4×Q8—42no—
205C4×Q8—42no—
206C4×Q8—42no—
207C4×Q8—44no—
208#117—24no—
209#117—42no—
210#117—42no—
211#117—42no—
212#117—42no—
213#117—42no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4C22⋊C411yescharacteristic
235C2×C4⋊C4C22⋊C411yescharacteristic
236C2×C4⋊C4C22⋊C411yescharacteristic
237C2×C4⋊C4C22⋊C411yescharacteristic
238C2×C4⋊C4C22⋊C411yescharacteristic
239C2×C4⋊C4C22⋊C411yescharacteristic
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C4⋊C4C22×C411yescharacteristic
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C4⋊C4C2411yescharacteristic
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4C22⋊C411yescharacteristic
254C2×C22⋊C4C22⋊C411yescharacteristic
255C2×C22⋊C4C22×C411yescharacteristic
256C2×C22⋊C4C2×D411yescharacteristic
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22×C411yescharacteristic
268C2×C42C22×C411yescharacteristic
269C2×C42C22×C411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C4⋊C8—44no—
276C4⋊C8—44no—
277C4⋊C8—42no—
278C4⋊C8—42no—
279C4⋊C8—42no—
280#101—44no—
281#101—44no—
282C22⋊C8—24no—
283#97—42no—
284#97—42no—
285C4×C8—42no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95C22×C411yescharacteristic
312#95C22×C411yescharacteristic
313#95C22×C411yescharacteristic
314#95C22×C411yescharacteristic
315C24C2×C22⋊C411yescharacteristic
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—42no—
329C2×Q8—48no—
330C2×Q8—48no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—41no—
334C2×Q8—42no—
335C2×Q8AGL(1,ℤ/8)12yes—
336C2×Q8#13712yes—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C23×C411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369M16—44no—
370M16—44no—
371M16—84no—
372C2×C8—44no—
373C2×C8—44no—
374C2×C8—84no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4#9914yes—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—42no—
419C22⋊C4—42no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—42no—
433C42—42no—
434C42—42no—
435C42—42no—
436C42—42no—
437C42—42no—
438C23—21no—
439C23—21no—
440C23—41no—
441C23—41no—
442C23#42112yes—
443C23C22×C22⋊C411yescharacteristic
444Q8—42no—
445Q8—42no—
446Q8—44no—
447Q8—44no—
448Q8—42no—
449Q8—42no—
450Q8—42no—
451Q8—42no—
452Q8—84no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4C2×#9912yes—
510C2×C4C2×#9912yes—
511C2×C4C2×#9912yes—
512C2×C4#41712yes—
513C2×C4#41712yes—
514C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
515C2×C4C2×#13711yescharacteristic
516C8—84no—
517C8—84no—
518C22—21no—
519C22—21no—
520C22—22no—
521C22—42no—
522C22—41no—
523C22—41no—
524C22—81no—
525C22C22×#9911yescharacteristic
526C22C2×#41711yescharacteristic
527C22C2×#42111yescharacteristic
528C4—22no—
529C4—22no—
530C4—21no—
531C4—21no—
532C4—22no—
533C4—22no—
534C4—22no—
535C4—42no—
536C4—41no—
537C4—41no—
538C4—41no—
539C4—42no—
540C4—41no—
541C4—41no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—81no—
546C4—81no—
547C4—81no—
548C4—81no—
549C4—81no—
550C4—81no—
551C2—21no—
552C2—21no—
553C2—42no—
554C2#1759211yescharacteristic
555C2#3334911yescharacteristic
556C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#519 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #11905C2semidirect
5. #11883C2semidirect
6. #11835C2semidirect
7. #11831C2semidirect

Maximal subgroups

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

subgroupindex
#357492
#206392
#173712
#119052
#118832
#118352
#118312
#117672
#117632