Finite Group Databasebeta

← #482661

Subgroups — #482661

Order 512.

863 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35248C211yesmaximal, characteristic
2C2×#1801C211yesmaximal, characteristic
3#17365C211yesmaximal, characteristic
4#11907C212yesmaximal, complemented
5#11883C212yesmaximal, complemented
6#11835C214yesmaximal, complemented
7#11764C214yesmaximal
8C2×#533C411yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2920—21no—
11C2×#423—21no—
12C22×#101—21no—
13#2855—21no—
14#2855—21no—
15#2634C412yes—
16#2634C2212yes—
17#2581—21no—
18#2581—21no—
19#2568C411yescharacteristic
20#2568C2211yescharacteristic
21#2546C411yescharacteristic
22#2546C2211yescharacteristic
23#2503—21no—
24#2484C412yes—
25#2484C2212yes—
26#2434—21no—
27#2327—21no—
28#2327—21no—
29#2283C411yescharacteristic
30#1801—21no—
31#1801—21no—
32#1801C2212yes—
33#1801C2212yes—
34C2×#343C2212yes—
35#1467—21no—
36#1467C2212yes—
37#1463—21no—
38#1463C2212yes—
39#1461—21no—
40#1461C2212yes—
41#1353C2218yes—
42#1346C2218yes—
43C23×Q8D411yescharacteristic
44C22×M16—22no—
45#533—41no—
46#533—41no—
47#533—41no—
48#533—41no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#125C2×C412yes—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—41no—
59C2×C4×Q8—41no—
60C2×C4×Q8—41no—
61C2×C4×Q8—41no—
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×Q8D411yescharacteristic
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8C2311yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C2×#117C2×C412yes—
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77#432—22no—
78#432—41no—
79#431—22no—
80#431—41no—
81#423—22no—
82#423—22no—
83#423—41no—
84#423—41no—
85#423—41no—
86C2×C4⋊C8—41no—
87C2×C4⋊C8—41no—
88C2×#101—22no—
89C2×#101—22no—
90C2×#101—44no—
91#407—22no—
92C4×M16—41no—
93#400—22no—
94#400—22no—
95#400C2×C412yes—
96#400C2×C412yes—
97#400C2×C412yes—
98#400C2312yes—
99#398C2×C412yes—
100#398C2×C412yes—
101#398C2×C412yes—
102#398C2312yes—
103#395—22no—
104#395—22no—
105#391—21no—
106#391—21no—
107#391—21no—
108#391—21no—
109#391—21no—
110#391—21no—
111#391—21no—
112#391—21no—
113#389—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#387—22no—
118#387—22no—
119#387—22no—
120#387—22no—
121#387—22no—
122#387—22no—
123#387—22no—
124#387—22no—
125#385—22no—
126#385—22no—
127#383C2×C412yes—
128#382—21no—
129#382—21no—
130#382—21no—
131#381—22no—
132#381—22no—
133#381C2×C412yes—
134#380—21no—
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C4⋊C4C2×C411yescharacteristic
137C4×C22⋊C4—21no—
138#376—21no—
139#343—22no—
140#343C2314yes—
141#324—22no—
142#324—22no—
143#324—22no—
144#324C2314yes—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—41no—
150C22×Q8—41no—
151C22×Q8—42no—
152C22×Q8C22⋊C411yescharacteristic
153C22×Q8C22⋊C411yescharacteristic
154C22×Q8C2×D411yescharacteristic
155C22×Q8C2×D411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C2×M16—24no—
164C2×M16—42no—
165C2×M16—42no—
166C2×M16—44no—
167C2×M16—44no—
168C22×C8—42no—
169C4⋊Q8—41no—
170C4⋊Q8—41no—
171C4⋊Q8—41no—
172C4⋊Q8—41no—
173C4⋊Q8—44no—
174C4⋊Q8—42no—
175C4⋊Q8—42no—
176#125—42no—
177#125—42no—
178C22⋊Q8—42no—
179C22⋊Q8—42no—
180C22⋊Q8—41no—
181C22⋊Q8—41no—
182C22⋊Q8—41no—
183C22⋊Q8—41no—
184C4×Q8—41no—
185C4×Q8—41no—
186C4×Q8—41no—
187C4×Q8—41no—
188C4×Q8—41no—
189C4×Q8—41no—
190C4×Q8—41no—
191C4×Q8—41no—
192C4×Q8—41no—
193C4×Q8—41no—
194C4×Q8—41no—
195C4×Q8—41no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×Q8—41no—
199C4×Q8—41no—
200C4×Q8—42no—
201C4×Q8—42no—
202C4×Q8—42no—
203C4×Q8—42no—
204#117—42no—
205#117—42no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4C22⋊C411yescharacteristic
223C2×C4⋊C4C22⋊C411yescharacteristic
224C2×C4⋊C4C22⋊C411yescharacteristic
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C22×C412yes—
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2411yescharacteristic
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22×C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C4⋊C8—42no—
266C4⋊C8—42no—
267C4⋊C8—42no—
268C4⋊C8—42no—
269C4⋊C8—42no—
270C4⋊C8—42no—
271#101—44no—
272#101—44no—
273C22⋊C8—24no—
274C22⋊C8—42no—
275#97—42no—
276C4×C8—42no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95C22×C412yes—
293#95C22×C412yes—
294C24C2×C22⋊C411yescharacteristic
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—44no—
308C2×Q8—44no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—44no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—42no—
317C2×Q8AGL(1,ℤ/8)12yes—
318C2×Q8#13712yes—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4#9911yescharacteristic
347C22×C4#9911yescharacteristic
348C22×C4#9911yescharacteristic
349C22×C4#9911yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
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×C4C23×C411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364M16—44no—
365M16—44no—
366M16—84no—
367M16—84no—
368C2×C8—44no—
369C2×C8—44no—
370C2×C8—84no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—44no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—44no—
426C42—42no—
427C42—42no—
428C23—22no—
429C23—21no—
430C23—21no—
431C23—41no—
432C23C2×#9911yescharacteristic
433C23C2×#9911yescharacteristic
434C23C22×C22⋊C411yescharacteristic
435Q8—42no—
436Q8—42no—
437Q8—44no—
438Q8—44no—
439Q8—42no—
440Q8—42no—
441Q8—82no—
442Q8—84no—
443Q8—82no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4C2×#9911yescharacteristic
502C2×C4C2×#9911yescharacteristic
503C2×C4C2×#9911yescharacteristic
504C2×C4C2×#9911yescharacteristic
505C2×C4#41712yes—
506C2×C4#41712yes—
507C2×C4#42112yes—
508C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
509C2×C4C2×#13711yescharacteristic
510C8—84no—
511C8—84no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—42no—
520C22—42no—
521C22#212312yes—
522C22C22×#9911yescharacteristic
523C22C2×#41711yescharacteristic
524C22C2×#42111yescharacteristic
525C4—21no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—22no—
531C4—22no—
532C4—22no—
533C4—41no—
534C4—41no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—82no—
542C4—82no—
543C4—82no—
544C4—82no—
545C4#213312yes—
546C2—21no—
547C2—21no—
548C2—22no—
549C2—41no—
550C2#1759211yescharacteristic
551C2C2×#212311yescharacteristic
552C2C2×#213311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11907C2semidirect
5. #11883C2semidirect
6. #11835C2semidirect

Maximal subgroups

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

subgroupindex
#352482
C2×#18012
#173652
#119072
#118832
#118352
#117642