Finite Group Databasebeta

← #189443

Subgroups — #189443

Order 512.

825 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60288C211yesmaximal, characteristic
2#21360C211yesmaximal, characteristic, complemented
3#17819C211yesmaximal, characteristic, complemented
4#17580C211yesmaximal, characteristic
5#17537C211yesmaximal, characteristic
6#8697C212yesmaximal, complemented
7#8696C212yesmaximal, complemented
8#8695C212yesmaximal, complemented
9#8694C212yesmaximal, complemented
10#7954C211yesmaximal, characteristic, complemented
11#7949C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13#2994C412yescomplemented
14#2994C412yescomplemented
15#2992—21no—
16#2992—21no—
17#2984C412yescomplemented
18#2974C412yescomplemented
19#2973—21no—
20#2972—21no—
21#2968—22no—
22#2966C2212yes—
23#2966C2212yes—
24#2924—22no—
25#2920C2211yescharacteristic
26#2920C2211yescharacteristic
27C2×#432C2211yescharacteristic
28C2×#431C2211yescharacteristic
29#2283C2211yescharacteristic
30C4×#117C2211yescharacteristic
31#1949—21no—
32#1949—21no—
33#1854—21no—
34#1854—21no—
35C4×#107C2211yescharacteristic, complemented
36C4×#107C2211yescharacteristic, complemented
37C4×#106C2211yescharacteristic, complemented
38C4×#106C2211yescharacteristic, complemented
39#1574—22no—
40#1574—22no—
41#1568C2212yescomplemented
42#1568C2212yescomplemented
43#1521—22no—
44#1511—21no—
45#1510—21no—
46#1509C2212yes—
47#1509C2212yes—
48#1509C2212yes—
49#1509C2212yes—
50#1508—21no—
51#1507—21no—
52#1500—22no—
53#1285C2212yescomplemented
54#1285C2212yescomplemented
55#1285C2212yescomplemented
56#1285C2212yescomplemented
57#529—21no—
58#529—21no—
59#529—21no—
60#529—21no—
61#529C2×C412yes—
62#529D412yes—
63#529Q812yes—
64#529C2312yes—
65C2×#126—21no—
66C2×#125Q811yescharacteristic
67C2×#124Q811yescharacteristic
68C2×#123—21no—
69C2×C22⋊Q8D411yescharacteristic
70C2×#121D411yescharacteristic
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×D4—21no—
73C2×C4×D4C2×C411yescharacteristic
74C2×#117—21no—
75C2×#117—21no—
76C22×C42C2311yescharacteristic
77#446—22no—
78#446—22no—
79#446C2×C412yes—
80#446C2×C412yes—
81C8×Q8C2×C412yes—
82C8×Q8C2×C412yes—
83#436C2×C412yes—
84#436C2×C412yes—
85#435—22no—
86#435—22no—
87#435—22no—
88#435—22no—
89#435—22no—
90#435—22no—
91C8×D4—22no—
92C8×D4—22no—
93C8×D4—22no—
94C8×D4—22no—
95C8×D4C2×C412yes—
96C8×D4C2×C412yes—
97#433—22no—
98#433—22no—
99#433—22no—
100#433—22no—
101#432—21no—
102#432—21no—
103#432—22no—
104#432—21no—
105#432—21no—
106#432C2×C412yes—
107#432C2312yes—
108#432C2312yes—
109#432C2312yes—
110#431—22no—
111#431—21no—
112#431—21no—
113#431—21no—
114#431—21no—
115#431C2×C412yes—
116#431C2×C412yes—
117#431D412yes—
118#431D412yes—
119C2×#107—21no—
120C2×#107—21no—
121C2×#106—21no—
122C2×#106—21no—
123#424—22no—
124#424—22no—
125#423—21no—
126#423—21no—
127C2×C4⋊C8C2×C411yescharacteristic
128C2×C4⋊C8C2×C411yescharacteristic
129C2×C4⋊C8D411yescharacteristic
130C2×C4⋊C8D411yescharacteristic
131#407—22no—
132#407—21no—
133#407—21no—
134C2×C22⋊C8—21no—
135C2×C22⋊C8—22no—
136C2×C22⋊C8D411yescharacteristic
137C2×C22⋊C8D411yescharacteristic
138#405—22no—
139#405—22no—
140C4×M16—21no—
141C4×M16—21no—
142C2×#97—21no—
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145#383—21no—
146#382—21no—
147#381—21no—
148#380—21no—
149C4×C4⋊C4—21no—
150C4×C4⋊C4C2311yescharacteristic
151C4×C4⋊C4C2311yescharacteristic
152C4×C4⋊C4C2311yescharacteristic
153C4×C4⋊C4C2311yescharacteristic
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157#376—21no—
158#376—21no—
159C43—21no—
160#340—21no—
161#340—21no—
162#340—21no—
163#340—21no—
164#335—22no—
165#335—22no—
166#334—22no—
167#334—22no—
168#328—22no—
169#328—22no—
170#328—22no—
171#328—22no—
172#327—22no—
173#327—22no—
174#327—22no—
175#327—22no—
176C22×Q8C4⋊C411yescharacteristic
177C22×D4—21no—
178C22×D4C4⋊C411yescharacteristic
179C23×C4—21no—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4#4111yescharacteristic
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—41no—
186C2×M16—41no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195#126—22no—
196#126—22no—
197#126—22no—
198#126—22no—
199#125—21no—
200#125—21no—
201#125C4⋊C412yes—
202#125C2×Q812yes—
203#124—21no—
204#124—21no—
205#124C4⋊C412yes—
206#124C2×Q812yes—
207#123—22no—
208#123—22no—
209#123—22no—
210#123—22no—
211C22⋊Q8—21no—
212C22⋊Q8—21no—
213C22⋊Q8C4⋊C412yes—
214C22⋊Q8C2×D412yes—
215#121—21no—
216#121—21no—
217#121C4⋊C412yes—
218#121C2×D412yes—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8C4⋊C412yes—
222C4×Q8C22×C412yes—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—21no—
228C4×D4—21no—
229C4×D4C4⋊C412yes—
230C4×D4C22×C412yes—
231#117—22no—
232#117—22no—
233#117—22no—
234#117—22no—
235#117—42no—
236#117—42no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4C4⋊C411yescharacteristic
246C2×C4⋊C4C4⋊C411yescharacteristic
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
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—41no—
260C2×C42—41no—
261C2×C42C22×C411yescharacteristic
262C2×C42C22×C411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×Q811yescharacteristic
268C2×C42C2×Q811yescharacteristic
269C2×C42#4111yescharacteristic
270C2×C42#4111yescharacteristic
271C2×C42#4111yescharacteristic
272C2×C42C2411yescharacteristic
273#107—42no—
274#107—42no—
275#106—42no—
276#106—42no—
277C4⋊C8—22no—
278C4⋊C8—22no—
279C4⋊C8—22no—
280C4⋊C8—22no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—22no—
284C4⋊C8—22no—
285C4⋊C8—41no—
286C4⋊C8—41no—
287C4⋊C8C22⋊C412yes—
288C4⋊C8C22⋊C412yes—
289C4⋊C8C22⋊C412yes—
290C4⋊C8C22⋊C412yes—
291C4⋊C8C22×C412yes—
292C4⋊C8C22×C412yes—
293C22⋊C8—22no—
294C22⋊C8—22no—
295C22⋊C8—22no—
296C22⋊C8—22no—
297C22⋊C8—22no—
298C22⋊C8—22no—
299C22⋊C8—22no—
300C22⋊C8—22no—
301C22⋊C8—41no—
302C22⋊C8—41no—
303C22⋊C8C22⋊C412yes—
304C22⋊C8C22⋊C412yes—
305C22⋊C8C22⋊C412yes—
306C22⋊C8C22⋊C412yes—
307#97—22no—
308#97—22no—
309#97—22no—
310#97—22no—
311#97—22no—
312#97—22no—
313C4×C8—22no—
314C4×C8—22no—
315C4×C8—41no—
316C4×C8—41no—
317C4×C8C22×C412yes—
318C4×C8C22×C412yes—
319C4×C8C2×D412yes—
320C4×C8C2×D412yes—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—41no—
326#95—41no—
327#95—41no—
328#95—41no—
329C24—21no—
330C24#12111yescharacteristic
331C2×Q8—21no—
332C2×Q8—81no—
333C2×Q8C2×C4⋊C412yes—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—21no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—81no—
340C2×D4C2×C4⋊C412yes—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—81no—
355C22×C4—81no—
356C22×C4—81no—
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C4⋊C411yescharacteristic
360C22×C4C2×C4⋊C411yescharacteristic
361C22×C4#11711yescharacteristic
362C22×C4#11711yescharacteristic
363C22×C4#12111yescharacteristic
364C22×C4#12111yescharacteristic
365C22×C4#12111yescharacteristic
366C22×C4#12111yescharacteristic
367C22×C4#12111yescharacteristic
368C22×C4C22⋊Q811yescharacteristic
369C22×C4C22⋊Q811yescharacteristic
370C22×C4C22⋊Q811yescharacteristic
371C22×C4C22⋊Q811yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C2×#4111yescharacteristic
375M16—82no—
376M16—82no—
377M16—82no—
378M16—82no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—82no—
388C2×C8—82no—
389C2×C8—81no—
390C2×C8—81no—
391C2×C8C2×C22⋊C412yes—
392C2×C8C2×C22⋊C412yes—
393C2×C8C2×C22⋊C412yes—
394C2×C8C2×C22⋊C412yes—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4C2×C4⋊C412yes—
414C4⋊C4C2×C4⋊C412yes—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—44no—
426C22⋊C4—82no—
427C42—22no—
428C42—22no—
429C42—44no—
430C42—81no—
431C42—81no—
432C42—81no—
433C42—81no—
434C42—82no—
435C42C2×C22⋊C411yescharacteristic
436C42C2×C22⋊C411yescharacteristic
437C42C2×C4⋊C411yescharacteristic
438C42C2×C4⋊C411yescharacteristic
439C42#11711yescharacteristic
440C42#11711yescharacteristic
441C42#12111yescharacteristic
442C42#12111yescharacteristic
443C42C22⋊Q811yescharacteristic
444C42C22⋊Q811yescharacteristic
445C42C22⋊Q811yescharacteristic
446C42C22⋊Q811yescharacteristic
447C42C23×C411yescharacteristic
448C42C22×D411yescharacteristic
449C42C22×Q811yescharacteristic
450C42C2×#4111yescharacteristic
451C23—21no—
452C23—21no—
453C23—22no—
454C23—41no—
455C23—41no—
456C23—42no—
457C23—81no—
458C23—81no—
459C23#38011yescharacteristic
460C23#38011yescharacteristic
461C23C2×#12111yescharacteristic
462Q8—42no—
463Q8—81no—
464D4—42no—
465D4—44no—
466D4—42no—
467D4—42no—
468D4—81no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—82no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4#38011yescharacteristic
514C2×C4#38011yescharacteristic
515C2×C4#38011yescharacteristic
516C2×C4#38011yescharacteristic
517C2×C4#38011yescharacteristic
518C2×C4#38011yescharacteristic
519C2×C4C22×C22⋊C411yescharacteristic
520C2×C4C22×C4⋊C411yescharacteristic
521C2×C4C2×#11711yescharacteristic
522C2×C4C2×#12111yescharacteristic
523C2×C4C2×C22⋊Q811yescharacteristic
524C2×C4C2×C22⋊Q811yescharacteristic
525C8—22no—
526C8—22no—
527C8—22no—
528C8—22no—
529C8—42no—
530C8—42no—
531C8—82no—
532C8—82no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—81no—
545C22—81no—
546C22—162no—
547C22C2×#38011yescharacteristic
548C22#306711yescharacteristic
549C22#306711yescharacteristic
550C4—21no—
551C4—21no—
552C4—21no—
553C4—21no—
554C4—21no—
555C4—21no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—82no—
565C4—82no—
566C4—161no—
567C4—161no—
568C4#290111yescharacteristic
569C4#290111yescharacteristic
570C2—21no—
571C2—21no—
572C2—41no—
573C2—41no—
574C2—82no—
575C2#2065511yescharacteristic
576C2#2067111yescharacteristic
577C2#2067811yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
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. #21360C2semidirect
3. #17819C2semidirect
6. #8697C2semidirect
7. #8696C2semidirect
8. #8695C2semidirect
9. #8694C2semidirect
10. #7954C2semidirect
11. #7949C2semidirect
13. #2994C4semidirect
14. #2994C4semidirect
17. #2984C4semidirect
18. #2974C4semidirect
35. C4×#107C22semidirect
36. C4×#107C22semidirect
37. C4×#106C22semidirect
38. C4×#106C22semidirect
41. #1568C22semidirect
42. #1568C22semidirect
53. #1285C22semidirect
54. #1285C22semidirect
55. #1285C22semidirect
56. #1285C22semidirect

Maximal subgroups

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

subgroupindex
#602882
#213602
#178192
#175802
#175372
#86972
#86962
#86952
#86942
#79542
#79492