Finite Group Databasebeta

← #482380

Subgroups — #482380

Order 512.

1150 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2536C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic, complemented
3#17365C211yesmaximal, characteristic, complemented
4#11879C214yesmaximal, complemented
5#11803C212yesmaximal, complemented
6#11802C212yesmaximal
7#11778C214yesmaximal, complemented
8C22×C22⋊Q8—21no—
9C22×C4×Q8C2211yescharacteristic
10C22×C4×D4C411yescharacteristic
11#2920—21no—
12#2920—21no—
13#2536—21no—
14#2536—21no—
15#2536—21no—
16#2536—21no—
17#2536C412yes—
18#2536C412yes—
19#2536C2212yes—
20#2536C2212yes—
21C2×#398C411yescharacteristic
22C2×#398C2211yescharacteristic
23C2×#394C411yescharacteristic
24C2×#394C2211yescharacteristic
25C2×#388—21no—
26C2×#388—21no—
27C2×#382C411yescharacteristic
28C2×#380—21no—
29#1821C2212yes—
30#1801—21no—
31#1801C2212yescomplemented
32#1784—21no—
33C4×#101—21no—
34C4×#101C2212yescomplemented
35#1467—21no—
36#1467C2212yescomplemented
37#1463—21no—
38#1463C2212yes—
39#1461—21no—
40#1461C2212yescomplemented
41#1346C2218yescomplemented
42#1345C2218yes—
43C23×Q8D411yescharacteristic
44C23×D4C2×C411yescharacteristic
45C24×C4—21no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—48no—
51C2×C4×Q8—41no—
52C2×C4×Q8—41no—
53C2×C4×Q8—41no—
54C2×C4×Q8—41no—
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8D411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8C2311yescharacteristic
62C2×C4×Q8C2311yescharacteristic
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4—41no—
66C2×C4×D4—41no—
67C2×C4×D4—41no—
68C2×C4×D4—41no—
69C2×C4×D4C2×C412yes—
70C2×C4×D4C2×C412yes—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C42C2×C411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86#432—22no—
87#432—41no—
88#432—41no—
89#432—41no—
90#431—22no—
91#431—41no—
92#431—41no—
93#431—41no—
94#423—22no—
95#423—41no—
96C2×#101—22no—
97C2×#101—22no—
98#407—22no—
99#407—22no—
100C4×M16—22no—
101C4×M16—41no—
102#398—21no—
103#398—21no—
104#398—21no—
105#398—21no—
106#398C2×C412yes—
107#398C2×C412yes—
108#398C2×C412yes—
109#398C2312yes—
110#394—21no—
111#394—21no—
112#394—21no—
113#394—21no—
114#394C2×C412yes—
115#394C2×C412yes—
116#394C2×C412yes—
117#394C2312yes—
118#388—22no—
119#388—22no—
120#388—22no—
121#388—22no—
122#388—22no—
123#388—22no—
124#388—22no—
125#388—22no—
126#382—21no—
127#382—21no—
128#382C2×C412yes—
129#382C2×C412yes—
130#380—22no—
131#380—22no—
132#380—22no—
133#380—22no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95C2×C411yescharacteristic
138C2×#95C2×C411yescharacteristic
139#343—22no—
140#343C2314yes—
141#324—22no—
142#324—22no—
143#324—22no—
144#324C2314yes—
145C25—21no—
146C22×Q8—24no—
147C22×Q8—41no—
148C22×Q8—41no—
149C22×Q8—42no—
150C22×Q8C22⋊C411yescharacteristic
151C22×Q8C22⋊C411yescharacteristic
152C22×Q8C2×D411yescharacteristic
153C22×Q8C2×D411yescharacteristic
154C22×D4—21no—
155C22×D4—41no—
156C22×D4—41no—
157C22×D4—44no—
158C22×D4C22×C412yes—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C2×M16—24no—
176C2×M16—42no—
177C22⋊Q8—48no—
178C22⋊Q8—48no—
179C22⋊Q8—48no—
180C22⋊Q8—48no—
181C4×Q8—41no—
182C4×Q8—41no—
183C4×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×D4—42no—
198C4×D4—42no—
199C4×D4—42no—
200C4×D4—42no—
201C4×D4—42no—
202C4×D4—42no—
203C4×D4—42no—
204C4×D4—42no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4C22⋊C411yescharacteristic
224C2×C4⋊C4C22⋊C411yescharacteristic
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C22⋊C411yescharacteristic
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×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22×C411yescharacteristic
273C2×C42C22×C411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C4⋊C8—42no—
279C4⋊C8—42no—
280C4⋊C8—42no—
281C4⋊C8—42no—
282#101—44no—
283#101—44no—
284C22⋊C8—24no—
285C22⋊C8—42no—
286C22⋊C8—42no—
287C22⋊C8—42no—
288#97—42no—
289#97—42no—
290C4×C8—42no—
291C4×C8—42no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—21no—
298#95—21no—
299#95—22no—
300#95C22×C412yes—
301#95C22×C412yes—
302C24—24no—
303C24—22no—
304C24—42no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24—41no—
310C24—41no—
311C24C2×C22⋊C411yescharacteristic
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—48no—
324C2×Q8—48no—
325C2×Q8AGL(1,ℤ/8)14yes—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—48no—
331C2×D4—44no—
332C2×D4—44no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—88no—
375C22×C4—88no—
376C22×C4#9911yescharacteristic
377C22×C4#9911yescharacteristic
378C22×C4#9911yescharacteristic
379C22×C4#9911yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C23×C411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394M16—44no—
395M16—44no—
396M16—84no—
397C2×C8—44no—
398C2×C8—44no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—44no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—44no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C23—22no—
456C23—21no—
457C23—21no—
458C23—22no—
459C23—24no—
460C23—42no—
461C23—44no—
462C23—48no—
463C23—48no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—44no—
468C23—42no—
469C23—42no—
470C23—82no—
471C23—82no—
472C23—84no—
473C23—84no—
474C23—82no—
475C23—82no—
476C23C2×#9911yescharacteristic
477C23C2×#9911yescharacteristic
478C23C22×C22⋊C411yescharacteristic
479Q8—44no—
480Q8—44no—
481Q8—44no—
482Q8—44no—
483D4—44no—
484D4—44no—
485D4—48no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—88no—
538C2×C4—88no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—84no—
542C2×C4—84no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4C2×#9911yescharacteristic
546C2×C4C2×#9911yescharacteristic
547C2×C4C2×#9911yescharacteristic
548C2×C4C2×#9911yescharacteristic
549C2×C4#41812yes—
550C2×C4#41812yes—
551C2×C4#42112yes—
552C2×C4C2×AGL(1,ℤ/8)12yes—
553C8—84no—
554C8—84no—
555C22—24no—
556C22—24no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—82no—
567C22—82no—
568C22—88no—
569C22—88no—
570C22—82no—
571C22—82no—
572C22#212012yes—
573C22#213312yes—
574C22C22×#9911yescharacteristic
575C22C2×#41811yescharacteristic
576C22C2×#42111yescharacteristic
577C4—21no—
578C4—21no—
579C4—21no—
580C4—21no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—22no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—82no—
594C4—82no—
595C4—82no—
596C2—21no—
597C2—21no—
598C2—22no—
599C2—22no—
600C2—84no—
601C2#1757711yescharacteristic
602C2C2×#212011yescharacteristic
603C2C2×#213311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC23×D4C2×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
2. #20727C2semidirect
3. #17365C2semidirect
4. #11879C2semidirect
5. #11803C2semidirect
7. #11778C2semidirect
31. #1801C22semidirect
34. C4×#101C22semidirect
36. #1467C22semidirect
40. #1461C22semidirect
41. #1346C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25362
#207272
#173652
#118792
#118032
#118022
#117782