Finite Group Databasebeta

← #482416

Subgroups — #482416

Order 512.

1336 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2#20722C211yesmaximal, characteristic
3#17362C211yesmaximal, characteristic
4#11901C214yesmaximal, complemented
5#11809C214yesmaximal, complemented
6#11788C214yesmaximal, complemented
7C22×C4⋊Q8C411yescharacteristic, complemented
8C22×C22⋊Q8—21no—
9C22×C22⋊Q8—21no—
10C22×C4×D4C2211yescharacteristic
11C2×#407—21no—
12#2542—21no—
13#2542—21no—
14#2542—21no—
15#2542—21no—
16#2542C412yescomplemented
17#2542C412yescomplemented
18#2542C2212yes—
19#2542C2212yes—
20C2×#399C412yescomplemented
21C2×#399C2212yes—
22C2×#390—21no—
23C2×#384—21no—
24#2280—21no—
25C2×C4×C4⋊C4C411yescharacteristic, complemented
26#1819C2212yes—
27#1799—21no—
28#1799C2212yes—
29#1781—21no—
30C4×#99—21no—
31C4×#99C2212yes—
32#1465—21no—
33#1465C2212yes—
34#1463—21no—
35#1462C2212yes—
36#1459—21no—
37#1459C2212yes—
38#1352C22116yescomplemented
39C23×Q8—21no—
40C23×D4D411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4D411yescharacteristic
43C22×M16—41no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8—41no—
48C2×C4⋊Q8—41no—
49C2×C4⋊Q8—41no—
50C2×C4⋊Q8C2×C412yes—
51C2×C4⋊Q8C2×C412yes—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—48no—
61C2×C22⋊Q8—48no—
62C2×C4×D4—41no—
63C2×C4×D4—41no—
64C2×C4×D4—41no—
65C2×C4×D4—41no—
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C2×C4×D4C2311yescharacteristic
74C2×#117—41no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4C2×C412yes—
80C22×C4⋊C4D411yescharacteristic
81C22×C22⋊C4—22no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4C2×C411yescharacteristic
84C22×C22⋊C4C2311yescharacteristic
85C22×C42—21no—
86C22×C42C2×C411yescharacteristic
87C2×#99—22no—
88C2×#99—22no—
89#407—22no—
90#407—22no—
91#407—41no—
92#407—41no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—41no—
95#399—22no—
96#399—22no—
97#399—22no—
98#399—22no—
99#399C2×C414yes—
100#399C2×C414yes—
101#399C2×C414yes—
102#399C2314yes—
103#390—22no—
104#390—22no—
105#390—22no—
106#390—22no—
107#384—22no—
108#384—22no—
109#384—22no—
110#384—22no—
111#381—41no—
112#380—22no—
113#380—41no—
114#379—22no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C412yes—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—41no—
121C2×#95C2×C412yes—
122#342—22no—
123#342C2314yes—
124#324—22no—
125#324—22no—
126#323—22no—
127#323C2314yes—
128C25C22⋊C411yescharacteristic
129C25C2×D411yescharacteristic
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—44no—
133C22×Q8—44no—
134C22×Q8—41no—
135C22×Q8—41no—
136C22×Q8—41no—
137C22×Q8—41no—
138C22×D4—24no—
139C22×D4—42no—
140C22×D4—41no—
141C22×D4—41no—
142C22×D4C22⋊C411yescharacteristic
143C22×D4C22⋊C411yescharacteristic
144C22×D4C2×D411yescharacteristic
145C22×D4C2×D411yescharacteristic
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C2×M16—42no—
172C2×M16—42no—
173C2×M16—84no—
174C22×C8—42no—
175C4⋊Q8—42no—
176C4⋊Q8—42no—
177C4⋊Q8—42no—
178C4⋊Q8—42no—
179C4⋊Q8—42no—
180C4⋊Q8—42no—
181C4⋊Q8—42no—
182C4⋊Q8—42no—
183C22⋊Q8—48no—
184C22⋊Q8—48no—
185C22⋊Q8—48no—
186C22⋊Q8—48no—
187C22⋊Q8—48no—
188C22⋊Q8—48no—
189C22⋊Q8—48no—
190C22⋊Q8—48no—
191C4×D4—41no—
192C4×D4—41no—
193C4×D4—41no—
194C4×D4—41no—
195C4×D4—41no—
196C4×D4—41no—
197C4×D4—41no—
198C4×D4—41no—
199C4×D4—41no—
200C4×D4—41no—
201C4×D4—41no—
202C4×D4—41no—
203C4×D4—41no—
204C4×D4—41no—
205C4×D4—41no—
206C4×D4—41no—
207#117—84no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—48no—
223C2×C4⋊C4—48no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4C22⋊C411yescharacteristic
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22×C414yes—
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—41no—
241C2×C22⋊C4—48no—
242C2×C22⋊C4—48no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4C22⋊C411yescharacteristic
246C2×C22⋊C4C22⋊C411yescharacteristic
247C2×C22⋊C4C22×C411yescharacteristic
248C2×C22⋊C4C22×C411yescharacteristic
249C2×C22⋊C4C22×C411yescharacteristic
250C2×C22⋊C4C2×D411yescharacteristic
251C2×C22⋊C4C2×D411yescharacteristic
252C2×C22⋊C4C2411yescharacteristic
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—44no—
260C2×C42—42no—
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265#99—44no—
266#99—44no—
267C22⋊C8—24no—
268C22⋊C8—42no—
269C22⋊C8—42no—
270C22⋊C8—42no—
271#95—22no—
272#95—42no—
273#95—42no—
274#95C22×C414yes—
275C24—22no—
276C24—22no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—41no—
282C24—41no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24—42no—
287C24C2×C22⋊C411yescharacteristic
288C24C2×C22⋊C411yescharacteristic
289C24C2×C22⋊C411yescharacteristic
290C24C2×C22⋊C411yescharacteristic
291C24C22×D411yescharacteristic
292C2×Q8—44no—
293C2×Q8—44no—
294C2×Q8—44no—
295C2×Q8—44no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—84no—
301C2×Q8—84no—
302C2×Q8—84no—
303C2×Q8—84no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—48no—
316C2×D4—48no—
317C2×D4#13714yes—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—28no—
324C22×C4—24no—
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—21no—
335C22×C4—21no—
336C22×C4—21no—
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—24no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4—82no—
369C22×C4—82no—
370C22×C4—82no—
371C22×C4—82no—
372C22×C4—82no—
373C22×C4—82no—
374C22×C4#9911yescharacteristic
375C22×C4#9911yescharacteristic
376C22×C4#9911yescharacteristic
377C22×C4#9911yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
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×C4C23×C411yescharacteristic
387C22×C4C22×D411yescharacteristic
388M16—84no—
389M16—84no—
390C2×C8—44no—
391C2×C8—84no—
392C4⋊C4—28no—
393C4⋊C4—28no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—44no—
399C4⋊C4—48no—
400C4⋊C4—48no—
401C4⋊C4—48no—
402C4⋊C4—48no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—84no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—48no—
416C22⋊C4—48no—
417C22⋊C4—48no—
418C22⋊C4—48no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—84no—
422C22⋊C4—84no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—44no—
432C42—44no—
433C42—84no—
434C23—22no—
435C23—22no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—48no—
461C23—48no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—81no—
467C23—81no—
468C23—81no—
469C23—81no—
470C23—84no—
471C23—84no—
472C23C2×#9911yescharacteristic
473C23C2×#9911yescharacteristic
474C23#41912yes—
475C23C22×C22⋊C411yescharacteristic
476C23C2×#13712yes—
477Q8—84no—
478Q8—84no—
479Q8—84no—
480Q8—84no—
481D4—44no—
482D4—44no—
483D4—44no—
484D4—44no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—28no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—416no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—84no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4—84no—
541C2×C4—84no—
542C2×C4#35314yes—
543C2×C4C2×#9911yescharacteristic
544C2×C4C2×#9911yescharacteristic
545C2×C4C2×#9911yescharacteristic
546C2×C4C2×#9911yescharacteristic
547C2×C4#41912yes—
548C2×C4#42112yes—
549C8—84no—
550C22—24no—
551C22—24no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—22no—
559C22—22no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—81no—
575C22—81no—
576C22—84no—
577C22—84no—
578C22—84no—
579C22—84no—
580C22—81no—
581C22—81no—
582C22#212312yes—
583C22C2×#35312yes—
584C22C22×#9911yescharacteristic
585C22C2×#41911yescharacteristic
586C22C2×#42111yescharacteristic
587C4—21no—
588C4—21no—
589C4—22no—
590C4—22no—
591C4—48no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—84no—
599C4—164no—
600C4C2×#35312yes—
601C4C2×#35312yes—
602C2—21no—
603C2—21no—
604C2—22no—
605C2—22no—
606C2—42no—
607C2—42no—
608C2—81no—
609C2—81no—
610C2#1757811yescharacteristic
611C2C2×#212311yescharacteristic
612C2C22×#35311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11901C2semidirect
5. #11809C2semidirect
6. #11788C2semidirect
7. C22×C4⋊Q8C4semidirect
16. #2542C4semidirect
17. #2542C4semidirect
20. C2×#399C4semidirect
25. C2×C4×C4⋊C4C4semidirect
38. #1352C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
#207222
#173622
#119012
#118092
#117882