Finite Group Databasebeta

← #482571

Subgroups — #482571

Order 512.

910 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35325C211yesmaximal, characteristic
2C2×#1820C211yesmaximal, characteristic
3#17357C211yesmaximal, characteristic
4#11842C214yesmaximal, complemented
5#11812C212yesmaximal, complemented
6#11801C212yesmaximal, complemented
7#11759C214yesmaximal
8C2×#533C411yescharacteristic, complemented
9C22×#117C2211yescharacteristic
10#2920—21no—
11#2855—21no—
12#2848—21no—
13#2628C411yescharacteristic, complemented
14#2628C2211yescharacteristic
15#2621C411yescharacteristic, complemented
16#2621C2211yescharacteristic
17#2611—21no—
18#2611—21no—
19#2599C412yescomplemented
20#2599C2212yes—
21#2583C412yescomplemented
22#2561C2212yes—
23#2484—21no—
24#2482—21no—
25#2323—21no—
26#2323—21no—
27#2283C411yescharacteristic, complemented
28C2×#381—21no—
29#1820—21no—
30#1820—21no—
31#1820C2212yes—
32#1820C2212yes—
33C2×#342—21no—
34C2×#342C2212yes—
35#1464—21no—
36#1464C2212yes—
37#1463—21no—
38#1462C2212yes—
39#1458—21no—
40#1458C2212yes—
41#1351C2218yes—
42#1344C2218yes—
43C24×C4D411yescharacteristic
44#533—41no—
45#533—41no—
46#533—41no—
47#533—41no—
48C2×C4⋊Q8C2×C411yescharacteristic, complemented
49C2×C4⋊Q8C2×C411yescharacteristic, complemented
50C2×#125—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8C2×C412yes—
55C2×C4×Q8—21no—
56C2×C4×Q8C2×C412yescomplemented
57C2×#117—21no—
58C2×#117—41no—
59C2×#117—41no—
60C2×#117—41no—
61C2×#117—41no—
62C2×#117C2×C411yescharacteristic
63C2×#117C2×C411yescharacteristic
64C2×#117D411yescharacteristic
65C2×#117D411yescharacteristic
66C2×#117D411yescharacteristic
67C2×#117D411yescharacteristic
68C2×#117C2311yescharacteristic
69C2×#117C2311yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C22⋊C4—22no—
73C22×C22⋊C4D411yescharacteristic
74C22×C22⋊C4C2311yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42D411yescharacteristic
77#432—22no—
78#432—41no—
79#431—22no—
80#431—41no—
81#423—41no—
82#407—22no—
83C4×M16—41no—
84#400—21no—
85#400—21no—
86#399—21no—
87#399—21no—
88#398—21no—
89#398—21no—
90#398C2×C412yescomplemented
91#398C2×C412yescomplemented
92#397—21no—
93#397—21no—
94#397C2×C412yes—
95#397C2312yes—
96#395C2×C412yescomplemented
97#395C2×C412yescomplemented
98#393C2×C412yes—
99#393C2312yes—
100#391—21no—
101#391—21no—
102#389—22no—
103#389—22no—
104#389—21no—
105#389—21no—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#387—22no—
111#387—22no—
112#387—22no—
113#387—22no—
114#385—22no—
115#385—22no—
116#385—22no—
117#385—22no—
118#383C2×C412yescomplemented
119#382C2×C411yescharacteristic, complemented
120#382C2×C411yescharacteristic, complemented
121#381—21no—
122#381—22no—
123#381—22no—
124#381—22no—
125#381—41no—
126#381—41no—
127#380—21no—
128#380—21no—
129#380—21no—
130#379—22no—
131C4×C4⋊C4—21no—
132C4×C22⋊C4C2×C412yes—
133#376—21no—
134C2×#95—41no—
135C2×#95—41no—
136#342—22no—
137#342—22no—
138#342—22no—
139#342C2314yes—
140#324—22no—
141#324—22no—
142#323—22no—
143#323C2314yes—
144C25C2×D411yescharacteristic
145C22×Q8—21no—
146C22×Q8C22×C412yes—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C2×M16—42no—
165C4⋊Q8—41no—
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C4⋊Q8—41no—
169#125—42no—
170#125—42no—
171C22⋊Q8—44no—
172C22⋊Q8—44no—
173C22⋊Q8—41no—
174C22⋊Q8—41no—
175C22⋊Q8—41no—
176C22⋊Q8—41no—
177C22⋊Q8—42no—
178C22⋊Q8—42no—
179C4×Q8—42no—
180C4×Q8—42no—
181C4×Q8—42no—
182C4×Q8—42no—
183#117—41no—
184#117—41no—
185#117—41no—
186#117—41no—
187#117—41no—
188#117—41no—
189#117—41no—
190#117—41no—
191#117—41no—
192#117—41no—
193#117—41no—
194#117—41no—
195#117—41no—
196#117—41no—
197#117—41no—
198#117—41no—
199#117—42no—
200#117—42no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4C22⋊C411yescharacteristic
217C2×C4⋊C4C22⋊C411yescharacteristic
218C2×C4⋊C4C22⋊C411yescharacteristic
219C2×C4⋊C4C22⋊C411yescharacteristic
220C2×C4⋊C4C22×C412yes—
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4C22⋊C411yescharacteristic
240C2×C22⋊C4C22⋊C411yescharacteristic
241C2×C22⋊C4C22×C411yescharacteristic
242C2×C22⋊C4C2×D411yescharacteristic
243C2×C22⋊C4C2×D411yescharacteristic
244C2×C22⋊C4C2×D411yescharacteristic
245C2×C22⋊C4C2×D411yescharacteristic
246C2×C22⋊C4C2411yescharacteristic
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22×C411yescharacteristic
262C2×C42C22×C411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C4⋊C8—42no—
266C4⋊C8—42no—
267C22⋊C8—24no—
268C22⋊C8—42no—
269#97—42no—
270C4×C8—42no—
271#95—22no—
272#95—21no—
273#95—21no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—42no—
284#95—42no—
285#95—42no—
286#95—42no—
287#95C22×C412yes—
288#95C22×C412yes—
289C24—21no—
290C24—22no—
291C24—21no—
292C24—22no—
293C24—42no—
294C24—81no—
295C24C2×C22⋊C411yescharacteristic
296C24C2×C22⋊C411yescharacteristic
297C24C22×D411yescharacteristic
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—82no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—44no—
345C22×C4—81no—
346C22×C4—81no—
347C22×C4—84no—
348C22×C4—84no—
349C22×C4—84no—
350C22×C4#9911yescharacteristic
351C22×C4#9911yescharacteristic
352C22×C4#9911yescharacteristic
353C22×C4#9911yescharacteristic
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×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C23×C411yescharacteristic
365C22×C4C22×D411yescharacteristic
366M16—84no—
367C2×C8—44no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—44no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—82no—
389C4⋊C4—82no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—82no—
406C22⋊C4—82no—
407C22⋊C4—84no—
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—44no—
425C42—82no—
426C23—24no—
427C23—24no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—22no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—41no—
436C23—42no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—42no—
442C23—81no—
443C23—81no—
444C23—84no—
445C23—84no—
446C23C2×#9911yescharacteristic
447C23C2×#9911yescharacteristic
448C23#42112yes—
449C23C22×C22⋊C411yescharacteristic
450C23#57512yes—
451Q8—82no—
452Q8—82no—
453Q8—84no—
454C2×C4—21no—
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—21no—
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—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
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—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—84no—
515C2×C4—84no—
516C2×C4—84no—
517C2×C4—168no—
518C2×C4C2×#9911yescharacteristic
519C2×C4C2×#9911yescharacteristic
520C2×C4C2×#9911yescharacteristic
521C2×C4C2×#9911yescharacteristic
522C8—84no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—22no—
530C22—21no—
531C22—22no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—48no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—81no—
542C22—81no—
543C22—84no—
544C22#212912yes—
545C22C22×#9911yescharacteristic
546C22#289611yescharacteristic
547C22C2×#42111yescharacteristic
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4—41no—
557C4—41no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—82no—
563C4—82no—
564C4—82no—
565C4—82no—
566C4—164no—
567C2—21no—
568C2—21no—
569C2—22no—
570C2—42no—
571C2—41no—
572C2—42no—
573C2#1758711yescharacteristic
574C2#3334711yescharacteristic
575C2C2×#212911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11842C2semidirect
5. #11812C2semidirect
6. #11801C2semidirect
8. C2×#533C4semidirect
13. #2628C4semidirect
15. #2621C4semidirect
19. #2599C4semidirect
21. #2583C4semidirect
27. #2283C4semidirect
48. C2×C4⋊Q8C2×C4semidirect
49. C2×C4⋊Q8C2×C4semidirect
56. C2×C4×Q8C2×C4semidirect
90. #398C2×C4semidirect
91. #398C2×C4semidirect
96. #395C2×C4semidirect
97. #395C2×C4semidirect
118. #383C2×C4semidirect
119. #382C2×C4semidirect
120. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
#353252
C2×#18202
#173572
#118422
#118122
#118012
#117592