Finite Group Databasebeta

← #482424

Subgroups — #482424

Order 512.

1155 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2#20722C211yesmaximal, characteristic
3#17364C211yesmaximal, characteristic
4#11899C214yesmaximal, complemented
5#11809C214yesmaximal, complemented
6#11786C214yesmaximal, complemented
7C22×C4×D4C2211yescharacteristic
8C23×C4⋊C4C411yescharacteristic, complemented
9#2920—21no—
10#2487—21no—
11#2487—21no—
12#2487—21no—
13#2487—21no—
14#2487C412yescomplemented
15#2487C412yescomplemented
16#2487C2212yes—
17#2487C2212yes—
18C2×#400C412yescomplemented
19C2×#399C2212yes—
20C2×#389—21no—
21C2×#386—21no—
22C2×#385—21no—
23#2280—21no—
24C2×#380—21no—
25C2×#380C411yescharacteristic, complemented
26#1819C2212yes—
27#1799—21no—
28#1799C2212yes—
29#1781—21no—
30C4×#99—21no—
31C4×#99C2212yes—
32#1466—21no—
33#1466C2212yes—
34#1462—21no—
35#1462C2212yes—
36#1460—21no—
37#1460C2212yes—
38#1352C22116yescomplemented
39C23×D4D411yescharacteristic
40C24×C4—21no—
41C24×C4C2×C411yescharacteristic
42C24×C4D411yescharacteristic
43C2×C4×D4—41no—
44C2×C4×D4—41no—
45C2×C4×D4—41no—
46C2×C4×D4—41no—
47C2×C4×D4C2×C411yescharacteristic
48C2×C4×D4C2×C411yescharacteristic
49C2×C4×D4D411yescharacteristic
50C2×C4×D4D411yescharacteristic
51C2×C4×D4D411yescharacteristic
52C2×C4×D4D411yescharacteristic
53C2×C4×D4C2311yescharacteristic
54C2×C4×D4C2311yescharacteristic
55C2×#117—41no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—41no—
62C22×C4⋊C4—41no—
63C22×C4⋊C4—41no—
64C22×C4⋊C4—41no—
65C22×C4⋊C4C2×C412yes—
66C22×C4⋊C4C2×C412yes—
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4D411yescharacteristic
72C22×C22⋊C4C2311yescharacteristic
73C22×C42D411yescharacteristic
74#432—22no—
75#432—41no—
76#431—22no—
77#431—41no—
78#423—41no—
79C2×#99—22no—
80C2×#99—22no—
81#407—22no—
82C4×M16—41no—
83#400—22no—
84#400—22no—
85#400C2×C414yes—
86#400C2×C414yes—
87#399—22no—
88#399—22no—
89#399C2×C414yes—
90#399C2314yes—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#386—22no—
96#386—22no—
97#386—22no—
98#386—22no—
99#385—22no—
100#385—22no—
101#385—22no—
102#385—22no—
103#381—41no—
104#380—21no—
105#380—21no—
106#380—22no—
107#380—22no—
108#380—22no—
109#380—22no—
110#380—22no—
111#380—41no—
112#380C2×C412yes—
113#380C2×C412yes—
114#379—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—41no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95C2×C412yes—
120#342—22no—
121#342C2314yes—
122#323—22no—
123#323—22no—
124#323—22no—
125#323C2314yes—
126C25C22⋊C411yescharacteristic
127C25C2×D411yescharacteristic
128C22×D4—24no—
129C22×D4—41no—
130C22×D4—41no—
131C22×D4—42no—
132C22×D4C22⋊C411yescharacteristic
133C22×D4C22⋊C411yescharacteristic
134C22×D4C2×D411yescharacteristic
135C22×D4C2×D411yescharacteristic
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—44no—
149C23×C4—44no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—84no—
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C2×M16—42no—
167C4×D4—41no—
168C4×D4—41no—
169C4×D4—41no—
170C4×D4—41no—
171C4×D4—41no—
172C4×D4—41no—
173C4×D4—41no—
174C4×D4—41no—
175C4×D4—41no—
176C4×D4—41no—
177C4×D4—41no—
178C4×D4—41no—
179C4×D4—41no—
180C4×D4—41no—
181C4×D4—41no—
182C4×D4—41no—
183#117—84no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—84no—
210C2×C4⋊C4C22⋊C411yescharacteristic
211C2×C4⋊C4C22⋊C411yescharacteristic
212C2×C4⋊C4C22×C414yes—
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—84no—
228C2×C22⋊C4C22⋊C411yescharacteristic
229C2×C22⋊C4C22⋊C411yescharacteristic
230C2×C22⋊C4C22×C411yescharacteristic
231C2×C22⋊C4C2×D411yescharacteristic
232C2×C22⋊C4C2×D411yescharacteristic
233C2×C22⋊C4C2×D411yescharacteristic
234C2×C22⋊C4C2×D411yescharacteristic
235C2×C22⋊C4C2411yescharacteristic
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—42no—
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22⋊C411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C4⋊C8—42no—
246C4⋊C8—42no—
247#99—44no—
248#99—44no—
249C22⋊C8—24no—
250C22⋊C8—42no—
251#97—42no—
252C4×C8—42no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—22no—
258#95—42no—
259#95—42no—
260#95C22×C414yes—
261C24—21no—
262C24—21no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—21no—
267C24—21no—
268C24—41no—
269C24—42no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24C2×C22⋊C411yescharacteristic
274C24C2×C22⋊C411yescharacteristic
275C24C2×C22⋊C411yescharacteristic
276C24C2×C22⋊C411yescharacteristic
277C24C22×D411yescharacteristic
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—48no—
290C2×D4—48no—
291C2×D4#13714yes—
292C22×C4—24no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—41no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—44no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4—84no—
351C22×C4—84no—
352C22×C4—88no—
353C22×C4—88no—
354C22×C4#9911yescharacteristic
355C22×C4#9911yescharacteristic
356C22×C4#9911yescharacteristic
357C22×C4#9911yescharacteristic
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
368M16—84no—
369C2×C8—44no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—84no—
383C4⋊C4—84no—
384C4⋊C4—84no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4—84no—
398C22⋊C4—84no—
399C22⋊C4—84no—
400C22⋊C4—84no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—84no—
410C23—22no—
411C23—21no—
412C23—22no—
413C23—21no—
414C23—24no—
415C23—24no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—42no—
426C23—42no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—42no—
432C23—42no—
433C23—48no—
434C23—48no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—48no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—81no—
445C23—81no—
446C23—84no—
447C23C2×#9911yescharacteristic
448C23C2×#9911yescharacteristic
449C23#41912yes—
450C23#42112yes—
451C23C22×C22⋊C411yescharacteristic
452C23C2×#13712yes—
453D4—44no—
454D4—44no—
455D4—44no—
456D4—44no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—21no—
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—42no—
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—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—88no—
510C2×C4—84no—
511C2×C4—88no—
512C2×C4—88no—
513C2×C4—84no—
514C2×C4—84no—
515C2×C4—84no—
516C2×C4—84no—
517C2×C4#35214yes—
518C2×C4C2×#9911yescharacteristic
519C2×C4C2×#9911yescharacteristic
520C2×C4C2×#9911yescharacteristic
521C2×C4C2×#9911yescharacteristic
522C2×C4#41912yes—
523C8—84no—
524C22—24no—
525C22—24no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—48no—
541C22—48no—
542C22—42no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—81no—
553C22—84no—
554C22—84no—
555C22—81no—
556C22C2×#35212yes—
557C22#212812yes—
558C22C22×#9911yescharacteristic
559C22C2×#41911yescharacteristic
560C22C2×#42111yescharacteristic
561C4—21no—
562C4—21no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—84no—
572C4—84no—
573C4—164no—
574C4C2×#35212yes—
575C4C2×#35212yes—
576C2—21no—
577C2—21no—
578C2—22no—
579C2—22no—
580C2—42no—
581C2—42no—
582C2—42no—
583C2—81no—
584C2#1757811yescharacteristic
585C2C22×#35211yescharacteristic
586C2C2×#212811yescharacteristic
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. #11899C2semidirect
5. #11809C2semidirect
6. #11786C2semidirect
8. C23×C4⋊C4C4semidirect
14. #2487C4semidirect
15. #2487C4semidirect
18. C2×#400C4semidirect
25. C2×#380C4semidirect
38. #1352C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
#207222
#173642
#118992
#118092
#117862