Finite Group Databasebeta

← #189428

Subgroups — #189428

Order 512.

932 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60211C211yesmaximal, characteristic
2C4×#428C211yesmaximal, characteristic, complemented
3#17797C211yesmaximal, characteristic, complemented
4#17546C211yesmaximal, characteristic
5#17543C211yesmaximal, characteristic
6#8697C212yesmaximal, complemented
7#8696C212yesmaximal, complemented
8#8695C212yesmaximal, complemented
9#8694C212yesmaximal, complemented
10#7949C211yesmaximal, characteristic, complemented
11C2×#1285C211yesmaximal, characteristic, complemented
12#3444C2211yescharacteristic
13#2971C412yescomplemented
14#2970—22no—
15#2970C412yescomplemented
16#2967—22no—
17#2966C2212yes—
18#2966C2212yes—
19#2962C411yescharacteristic, complemented
20#2962C411yescharacteristic, complemented
21#2935C411yescharacteristic, complemented
22#2935C411yescharacteristic, complemented
23#2933—21no—
24#2933—21no—
25#2923—21no—
26#2920C2211yescharacteristic
27C2×#432C2211yescharacteristic
28C2×#432C2211yescharacteristic
29C2×#428—21no—
30#2875—21no—
31C2×C4×M16C2211yescharacteristic
32C4×#117C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1776—22no—
35C4×#107C2212yes—
36C4×#106C2212yes—
37#1736—21no—
38#1569—22no—
39#1568C2212yes—
40#1568C2212yes—
41#1565—21no—
42#1565—21no—
43#1509C2212yes—
44#1509C2212yes—
45#1509C2212yes—
46#1509C2212yes—
47#1506—21no—
48#1505—21no—
49#1504—22no—
50#1503—21no—
51#1502—21no—
52#1501—22no—
53#1285C2212yes—
54#1285C2212yes—
55#1285C2211yescharacteristic
56#1285C2211yescharacteristic
57#1285C2211yescharacteristic
58#1285C2211yescharacteristic
59C2×#131—21no—
60C2×#131—21no—
61C22×M16D411yescharacteristic
62C22×M16D411yescharacteristic
63#533C2×C411yescharacteristic
64#533D411yescharacteristic
65#533Q811yescharacteristic
66#532C2×C411yescharacteristic
67#532D411yescharacteristic
68#532Q811yescharacteristic
69#529—21no—
70#529—21no—
71#529—21no—
72#529—21no—
73#529Q812yes—
74#529C2312yes—
75#525—21no—
76#525—21no—
77#525D412yes—
78C4×#41—21no—
79C4×#41—21no—
80C4×#41C2×C412yes—
81C2×#117—21no—
82C22×C4⋊C4—21no—
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85#446—22no—
86#446—22no—
87#446C2×C412yescomplemented
88#446C2×C412yescomplemented
89C8×Q8C2×C412yescomplemented
90C8×Q8C2×C412yescomplemented
91#436—22no—
92#436—22no—
93#436C2×C412yescomplemented
94#436C2×C412yescomplemented
95#435—22no—
96#435—22no—
97#435—22no—
98#435—22no—
99#435—22no—
100#435—22no—
101C8×D4—22no—
102C8×D4—22no—
103C8×D4C2×C412yescomplemented
104C8×D4C2×C412yescomplemented
105#433—22no—
106#433—22no—
107#432—21no—
108#432—21no—
109#432—21no—
110#432—21no—
111#432—22no—
112#432—22no—
113#432—21no—
114#432—21no—
115#432C2×C411yescharacteristic
116#432C2×C411yescharacteristic
117#432C2312yes—
118#432C2311yescharacteristic
119#432C2311yescharacteristic
120#432C2311yescharacteristic
121#432C2311yescharacteristic
122#431—21no—
123#431—21no—
124#428—28no—
125C2×#107—22no—
126C2×#106—22no—
127#424—21no—
128#424—21no—
129#423—21no—
130C2×C4⋊C8—21no—
131C2×C4⋊C8C2×C411yescharacteristic
132C2×C4⋊C8C2×C411yescharacteristic
133#408—21no—
134#408—21no—
135#407—21no—
136C2×C22⋊C8—21no—
137C2×C22⋊C8—21no—
138#405—21no—
139#405—21no—
140#405—22no—
141#405—22no—
142C4×M16—22no—
143C4×M16—21no—
144C4×M16C2×C411yescharacteristic
145C4×M16C2×C411yescharacteristic
146C4×M16C2×C412yes—
147C4×M16D411yescharacteristic
148C4×M16D411yescharacteristic
149C4×M16D412yes—
150C2×#97—21no—
151C2×#97—22no—
152C2×#97—21no—
153C2×#97D412yes—
154C2×C4×C8C2312yes—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4C2312yes—
160C4×C4⋊C4C2311yescharacteristic
161C4×C4⋊C4C2311yescharacteristic
162C4×C22⋊C4—21no—
163C4×C22⋊C4—21no—
164#376—21no—
165C2×#95—21no—
166C43—21no—
167#340—22no—
168#340—22no—
169#335—22no—
170#335—22no—
171#334—22no—
172#334—22no—
173#328—22no—
174#328—22no—
175#328—22no—
176#328—22no—
177#327—22no—
178#327—22no—
179#327—22no—
180#327—22no—
181C2×#41—21no—
182C2×#41—21no—
183C2×#41C4⋊C412yes—
184C23×C4—41no—
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4#4111yescharacteristic
188#131—44no—
189#131—44no—
190C2×M16—24no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—42no—
194C2×M16—41no—
195C2×M16C22⋊C411yescharacteristic
196C2×M16C22⋊C411yescharacteristic
197C2×M16C22⋊C411yescharacteristic
198C2×M16C22⋊C412yes—
199C2×M16C22⋊C412yes—
200C2×M16C22⋊C411yescharacteristic
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—41no—
206C22×C8C2×D412yes—
207C22×C8C2×D412yes—
208C4⋊Q8C4⋊C411yescharacteristic
209C4⋊Q8C4⋊C411yescharacteristic
210C4⋊Q8C4⋊C411yescharacteristic
211#127C4⋊C411yescharacteristic
212#126—22no—
213#126—22no—
214#125—21no—
215#125—21no—
216#125C2×Q812yes—
217#124—21no—
218#124—21no—
219#124C2×Q812yes—
220#123—22no—
221#123—22no—
222#123—22no—
223C22⋊Q8—21no—
224C22⋊Q8—21no—
225C22⋊Q8—21no—
226C22⋊Q8—21no—
227C22⋊Q8C4⋊C412yes—
228C22⋊Q8C2×D412yes—
229#121—21no—
230#121—21no—
231#121—21no—
232#121—21no—
233#121C4⋊C412yes—
234#121C2×D412yes—
235#120—22no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8C4⋊C412yes—
241C4×Q8C22×C412yes—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4—22no—
248C4×D4—21no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4C4⋊C412yes—
253C4×D4C22×C412yes—
254#117—21no—
255#117—21no—
256#117—22no—
257#117—22no—
258#117—22no—
259#117—24no—
260#117—42no—
261#117C4⋊C412yes—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—41no—
268C2×C22⋊C4—41no—
269C2×C42—22no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—22no—
273C2×C42—41no—
274C2×C42—41no—
275C2×C42—41no—
276C2×C42—82no—
277C2×C42C22×C411yescharacteristic
278C2×C42C22×C411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×Q811yescharacteristic
284C2×C42C2×Q811yescharacteristic
285C2×C42#4111yescharacteristic
286C2×C42#4111yescharacteristic
287C2×C42#4111yescharacteristic
288C2×C42C2411yescharacteristic
289#107—28no—
290#106—28no—
291C4⋊C8—22no—
292C4⋊C8—22no—
293C4⋊C8—22no—
294C4⋊C8—22no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8—22no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300C4⋊C8—41no—
301C4⋊C8—41no—
302C4⋊C8C22×C411yescharacteristic
303C4⋊C8C22×C411yescharacteristic
304C4⋊C8C22×C411yescharacteristic
305C4⋊C8C22×C411yescharacteristic
306C22⋊C8—21no—
307C22⋊C8—21no—
308C22⋊C8—21no—
309C22⋊C8—21no—
310C22⋊C8—22no—
311C22⋊C8—22no—
312C22⋊C8—22no—
313C22⋊C8—22no—
314C22⋊C8—22no—
315C22⋊C8—21no—
316C22⋊C8—21no—
317C22⋊C8—41no—
318C22⋊C8—41no—
319#97—22no—
320#97—22no—
321#97—21no—
322#97—21no—
323#97—21no—
324#97—21no—
325#97—22no—
326#97—21no—
327#97—21no—
328#97—41no—
329#97C22⋊C412yes—
330#97C22⋊C412yes—
331#97C22⋊C412yes—
332#97C22⋊C412yes—
333C4×C8—22no—
334C4×C8—22no—
335C4×C8—41no—
336C4×C8C22×C412yes—
337C4×C8C22×C412yes—
338C4×C8C2×D412yes—
339C4×C8C2×D412yes—
340#95—22no—
341#95—21no—
342#95—21no—
343#95—41no—
344#95—41no—
345C24#12111yescharacteristic
346#41—42no—
347#41—42no—
348#41—81no—
349#41—81no—
350#41—82no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8C2×C4⋊C412yes—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4C2×C4⋊C412yes—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—44no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—81no—
372C22×C4—82no—
373C22×C4—82no—
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C4⋊C411yescharacteristic
377C22×C4C2×C4⋊C411yescharacteristic
378C22×C4#11711yescharacteristic
379C22×C4#11711yescharacteristic
380C22×C4#12111yescharacteristic
381C22×C4#12111yescharacteristic
382C22×C4#12111yescharacteristic
383C22×C4#12111yescharacteristic
384C22×C4#12111yescharacteristic
385C22×C4C22⋊Q811yescharacteristic
386C22×C4C22⋊Q811yescharacteristic
387C22×C4C22⋊Q811yescharacteristic
388C22×C4C22⋊Q811yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C2×#4111yescharacteristic
392M16—22no—
393M16—22no—
394M16—22no—
395M16—24no—
396M16—42no—
397M16—44no—
398M16—44no—
399M16—82no—
400M16#10414yes—
401C2×C8—24no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—42no—
407C2×C8—44no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—44no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—42no—
414C2×C8—82no—
415C2×C8C2×C22⋊C412yes—
416C2×C8C2×C22⋊C412yes—
417C2×C8C2×C22⋊C412yes—
418C2×C8C2×C22⋊C412yes—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—44no—
433C4⋊C4—42no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4C2×C4⋊C412yes—
437C4⋊C4C2×C4⋊C412yes—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—44no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—44no—
452C42—81no—
453C42—81no—
454C42—82no—
455C42—82no—
456C42—82no—
457C42C2×C22⋊C411yescharacteristic
458C42C2×C22⋊C411yescharacteristic
459C42C2×C4⋊C411yescharacteristic
460C42C2×C4⋊C411yescharacteristic
461C42#11711yescharacteristic
462C42#11711yescharacteristic
463C42#12111yescharacteristic
464C42#12111yescharacteristic
465C42C22⋊Q811yescharacteristic
466C42C22⋊Q811yescharacteristic
467C42C22⋊Q811yescharacteristic
468C42C22⋊Q811yescharacteristic
469C42C23×C411yescharacteristic
470C42C22×D411yescharacteristic
471C42C22×Q811yescharacteristic
472C42C2×#4111yescharacteristic
473C23—22no—
474C23—21no—
475C23—21no—
476C23—22no—
477C23—41no—
478C23#38011yescharacteristic
479C23#38011yescharacteristic
480C23C2×#12111yescharacteristic
481Q8—42no—
482Q8—42no—
483Q8—81no—
484D4—42no—
485D4—44no—
486D4—42no—
487D4—42no—
488D4—42no—
489D4—81no—
490D4—82no—
491C2×C4—22no—
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—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—82no—
525C2×C4—88no—
526C2×C4—82no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4#38011yescharacteristic
530C2×C4#38011yescharacteristic
531C2×C4#38011yescharacteristic
532C2×C4#38011yescharacteristic
533C2×C4#38011yescharacteristic
534C2×C4#38011yescharacteristic
535C2×C4C2×#10412yes—
536C2×C4#42112yes—
537C2×C4C22×C22⋊C411yescharacteristic
538C2×C4C22×C4⋊C411yescharacteristic
539C2×C4C2×#11711yescharacteristic
540C2×C4C2×#12111yescharacteristic
541C2×C4C2×C22⋊Q811yescharacteristic
542C2×C4C2×C22⋊Q811yescharacteristic
543C8—22no—
544C8—22no—
545C8—22no—
546C8—41no—
547C8—41no—
548C8—41no—
549C8—41no—
550C8—82no—
551C8—81no—
552C8—81no—
553C8C2×#10414yes—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22#180612yes—
567C22C2×#38011yescharacteristic
568C22#306811yescharacteristic
569C22#306911yescharacteristic
570C4—21no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—21no—
576C4—21no—
577C4—21no—
578C4—21no—
579C4—21no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—82no—
584C4—82no—
585C4—84no—
586C4#180612yes—
587C4C22×#10411yescharacteristic
588C4C2×#42111yescharacteristic
589C2—21no—
590C2—21no—
591C2—22no—
592C2—41no—
593C2—82no—
594C2C2×#180611yescharacteristic
595C2#2067111yescharacteristic
596C2#2067911yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#3444C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#428C2semidirect
3. #17797C2semidirect
6. #8697C2semidirect
7. #8696C2semidirect
8. #8695C2semidirect
9. #8694C2semidirect
10. #7949C2semidirect
11. C2×#1285C2semidirect
13. #2971C4semidirect
15. #2970C4semidirect
19. #2962C4semidirect
20. #2962C4semidirect
21. #2935C4semidirect
22. #2935C4semidirect
87. #446C2×C4semidirect
88. #446C2×C4semidirect
89. C8×Q8C2×C4semidirect
90. C8×Q8C2×C4semidirect
93. #436C2×C4semidirect
94. #436C2×C4semidirect
103. C8×D4C2×C4semidirect
104. C8×D4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#602112
C4×#4282
#177972
#175462
#175432
#86972
#86962
#86952
#86942
#79492
C2×#12852