Finite Group Databasebeta

← #189439

Subgroups — #189439

Order 512.

851 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60214C211yesmaximal, characteristic
2#20819C211yesmaximal, characteristic, complemented
3#17797C211yesmaximal, characteristic, complemented
4#17581C211yesmaximal, characteristic
5#17580C211yesmaximal, characteristic
6#8697C212yesmaximal, complemented
7#8696C212yesmaximal, complemented
8#8695C212yesmaximal, complemented
9#8694C212yesmaximal, complemented
10#7953C211yesmaximal, characteristic, complemented
11#7951C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13#2969C412yescomplemented
14#2969C412yescomplemented
15#2968—21no—
16#2968—21no—
17#2967—22no—
18#2966C2212yes—
19#2966C2212yes—
20Q8×M16C412yescomplemented
21D4×M16—22no—
22D4×M16C412yescomplemented
23#2921—21no—
24#2920C2211yescharacteristic
25#2920C2211yescharacteristic
26C2×#432C2211yescharacteristic
27#2873—21no—
28C2×C4×M16C2211yescharacteristic
29#2285C2211yescharacteristic
30#2283C2211yescharacteristic
31#1847—21no—
32#1836—21no—
33#1809—22no—
34C4×#107C2212yescomplemented
35C4×#106C2212yescomplemented
36#1569—21no—
37#1569—21no—
38#1568C2212yescomplemented
39#1568C2212yescomplemented
40#1565—22no—
41#1521—22no—
42#1520—21no—
43#1519—21no—
44#1512—22no—
45#1511—21no—
46#1510—21no—
47#1509C2212yes—
48#1509C2212yes—
49#1509C2212yes—
50#1509C2212yes—
51#1285C2211yescharacteristic, complemented
52#1285C2211yescharacteristic, complemented
53#1285C2211yescharacteristic, complemented
54#1285C2211yescharacteristic, complemented
55#1285C2212yescomplemented
56#1285C2212yescomplemented
57C22×M16—22no—
58C22×M16D412yes—
59#529—21no—
60#529—21no—
61#529—21no—
62#529—21no—
63#529C2×C412yes—
64#529D412yes—
65#529Q812yes—
66#529C2312yes—
67C2×#126—21no—
68C2×#125Q811yescharacteristic
69C2×#124Q811yescharacteristic
70C2×#123—21no—
71C2×C22⋊Q8D411yescharacteristic
72C2×#121D411yescharacteristic
73C2×C4×Q8C2×C411yescharacteristic
74C2×C4×D4—21no—
75C2×C4×D4C2×C411yescharacteristic
76C2×#117—21no—
77C2×#117—21no—
78C22×C42C2311yescharacteristic
79#446—22no—
80#446—22no—
81#446C2×C412yes—
82#446C2×C412yes—
83C8×Q8C2×C412yes—
84C8×Q8C2×C412yes—
85#436—22no—
86#436—22no—
87#436C2×C412yes—
88#436C2×C412yes—
89#435—22no—
90#435—22no—
91#435—22no—
92#435—22no—
93#435—22no—
94#435—22no—
95C8×D4—22no—
96C8×D4—22no—
97C8×D4C2×C412yes—
98C8×D4C2×C412yes—
99#433—22no—
100#433—22no—
101#433—22no—
102#433—22no—
103#432—21no—
104#432—21no—
105#432—21no—
106#432—21no—
107#432—21no—
108#432C2311yescharacteristic
109#432C2311yescharacteristic
110#432C2312yes—
111#432C2311yescharacteristic
112#432C2311yescharacteristic
113#431—21no—
114#431—21no—
115#431—21no—
116#431C2×C412yes—
117C2×#107—22no—
118C2×#106—22no—
119#423—22no—
120#423—21no—
121#423C2×C412yes—
122C2×C4⋊C8—21no—
123#407—21no—
124#407—22no—
125#407—21no—
126C2×C22⋊C8—21no—
127#405—22no—
128#405—22no—
129#405—22no—
130#405—22no—
131C4×M16—22no—
132C4×M16—21no—
133C4×M16—21no—
134C4×M16C2×C412yes—
135C4×M16C2×C411yescharacteristic
136C4×M16C2×C411yescharacteristic
137C4×M16D412yes—
138C4×M16D411yescharacteristic
139C4×M16D411yescharacteristic
140C2×#97—21no—
141C2×#97D411yescharacteristic
142C2×#97D411yescharacteristic
143C2×C4×C8C2312yes—
144#384—21no—
145#383—21no—
146#382—21no—
147#382—21no—
148#381—21no—
149#380—21no—
150#380—21no—
151#380—21no—
152C4×C4⋊C4—21no—
153C4×C4⋊C4C2311yescharacteristic
154C4×C4⋊C4C2311yescharacteristic
155C4×C4⋊C4C2312yes—
156C4×C22⋊C4—21no—
157#376—21no—
158#340—22no—
159#340—22no—
160#335—22no—
161#335—22no—
162#334—22no—
163#334—22no—
164#328—22no—
165#328—22no—
166#328—22no—
167#328—22no—
168#327—22no—
169#327—22no—
170#327—22no—
171#327—22no—
172C22×Q8C4⋊C411yescharacteristic
173C22×D4—21no—
174C22×D4C4⋊C411yescharacteristic
175C23×C4—21no—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4#4111yescharacteristic
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—44no—
183C2×M16—48no—
184C2×M16—41no—
185C2×M16C22⋊C412yes—
186C2×M16C22⋊C412yes—
187C2×M16C22⋊C412yes—
188C2×M16C22⋊C412yes—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—41no—
192C22×C8C2×D412yes—
193C22×C8C2×D412yes—
194#126—22no—
195#126—22no—
196#126—22no—
197#126—22no—
198#125—21no—
199#125—21no—
200#125C4⋊C412yes—
201#125C2×Q812yes—
202#124—21no—
203#124—21no—
204#124C4⋊C412yes—
205#124C2×Q812yes—
206#123—22no—
207#123—22no—
208#123—22no—
209#123—22no—
210C22⋊Q8—21no—
211C22⋊Q8—21no—
212C22⋊Q8C4⋊C412yes—
213C22⋊Q8C2×D412yes—
214#121—21no—
215#121—21no—
216#121C4⋊C412yes—
217#121C2×D412yes—
218C4×Q8—21no—
219C4×Q8—21no—
220C4×Q8C4⋊C412yes—
221C4×Q8C22×C412yes—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4—21no—
227C4×D4—21no—
228C4×D4C4⋊C412yes—
229C4×D4C22×C412yes—
230#117—22no—
231#117—22no—
232#117—22no—
233#117—22no—
234#117—82no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4C4⋊C411yescharacteristic
245C2×C4⋊C4C4⋊C411yescharacteristic
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42C22×C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×Q811yescharacteristic
264C2×C42C2×Q811yescharacteristic
265C2×C42#4111yescharacteristic
266C2×C42#4111yescharacteristic
267C2×C42#4111yescharacteristic
268C2×C42C2411yescharacteristic
269#107—44no—
270#106—44no—
271C4⋊C8—22no—
272C4⋊C8—22no—
273C4⋊C8—22no—
274C4⋊C8—22no—
275C4⋊C8—22no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—22no—
279C4⋊C8—41no—
280C4⋊C8—41no—
281C4⋊C8C22×C412yes—
282C4⋊C8C22×C412yes—
283C22⋊C8—22no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286C22⋊C8—22no—
287C22⋊C8—22no—
288C22⋊C8—22no—
289C22⋊C8—22no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292C22⋊C8—22no—
293C22⋊C8—41no—
294C22⋊C8—41no—
295#97—22no—
296#97—22no—
297#97—21no—
298#97—21no—
299#97—21no—
300#97—21no—
301#97—22no—
302#97—41no—
303#97C22⋊C411yescharacteristic
304#97C22⋊C411yescharacteristic
305#97C22⋊C411yescharacteristic
306#97C22⋊C411yescharacteristic
307#97C22⋊C412yes—
308#97C22⋊C412yes—
309C4×C8—22no—
310C4×C8—22no—
311C4×C8—22no—
312C4×C8—41no—
313C4×C8C22×C412yes—
314C4×C8C22×C412yes—
315C4×C8C2×D412yes—
316C4×C8C2×D412yes—
317#95—22no—
318#95—21no—
319#95—21no—
320#95—41no—
321#95—41no—
322#95—41no—
323#95—41no—
324C24—21no—
325C24#12111yescharacteristic
326C2×Q8—21no—
327C2×Q8—81no—
328C2×Q8C2×C4⋊C412yes—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—21no—
332C2×D4—81no—
333C2×D4—81no—
334C2×D4—81no—
335C2×D4C2×C4⋊C412yes—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—44no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—81no—
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C4⋊C411yescharacteristic
352C22×C4C2×C4⋊C411yescharacteristic
353C22×C4#11711yescharacteristic
354C22×C4#11711yescharacteristic
355C22×C4#12111yescharacteristic
356C22×C4#12111yescharacteristic
357C22×C4#12111yescharacteristic
358C22×C4#12111yescharacteristic
359C22×C4#12111yescharacteristic
360C22×C4C22⋊Q811yescharacteristic
361C22×C4C22⋊Q811yescharacteristic
362C22×C4C22⋊Q811yescharacteristic
363C22×C4C22⋊Q811yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C2×#4111yescharacteristic
367M16—24no—
368M16—24no—
369M16—42no—
370M16—42no—
371M16—44no—
372M16—44no—
373M16—44no—
374M16—82no—
375C2×C8—24no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—42no—
379C2×C8—44no—
380C2×C8—44no—
381C2×C8—42no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—82no—
385C2×C8C2×C22⋊C412yes—
386C2×C8C2×C22⋊C412yes—
387C2×C8C2×C22⋊C412yes—
388C2×C8C2×C22⋊C412yes—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—82no—
404C4⋊C4—82no—
405C4⋊C4C2×C4⋊C412yes—
406C4⋊C4C2×C4⋊C412yes—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—82no—
418C22⋊C4—82no—
419C42—22no—
420C42—22no—
421C42—81no—
422C42—81no—
423C42—82no—
424C42C2×C22⋊C411yescharacteristic
425C42C2×C22⋊C411yescharacteristic
426C42C2×C4⋊C411yescharacteristic
427C42C2×C4⋊C411yescharacteristic
428C42#11711yescharacteristic
429C42#11711yescharacteristic
430C42#12111yescharacteristic
431C42#12111yescharacteristic
432C42C22⋊Q811yescharacteristic
433C42C22⋊Q811yescharacteristic
434C42C22⋊Q811yescharacteristic
435C42C22⋊Q811yescharacteristic
436C42C23×C411yescharacteristic
437C42C22×D411yescharacteristic
438C42C22×Q811yescharacteristic
439C42C2×#4111yescharacteristic
440C23—21no—
441C23—21no—
442C23—22no—
443C23—41no—
444C23—41no—
445C23—42no—
446C23—81no—
447C23—81no—
448C23#38011yescharacteristic
449C23#38011yescharacteristic
450C23C2×#12111yescharacteristic
451Q8—42no—
452Q8—81no—
453D4—42no—
454D4—42no—
455D4—42no—
456D4—81no—
457D4—82no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4#38011yescharacteristic
497C2×C4#38011yescharacteristic
498C2×C4#38011yescharacteristic
499C2×C4#38011yescharacteristic
500C2×C4#38011yescharacteristic
501C2×C4#38011yescharacteristic
502C2×C4C22×C22⋊C411yescharacteristic
503C2×C4C22×C4⋊C411yescharacteristic
504C2×C4C2×#11711yescharacteristic
505C2×C4C2×#12111yescharacteristic
506C2×C4C2×C22⋊Q811yescharacteristic
507C2×C4C2×C22⋊Q811yescharacteristic
508C8—22no—
509C8—22no—
510C8—22no—
511C8—42no—
512C8—42no—
513C8—81no—
514C8—81no—
515C8—82no—
516C8#42114yes—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—81no—
529C22—81no—
530C22—162no—
531C22C2×#38011yescharacteristic
532C22#306811yescharacteristic
533C22#306911yescharacteristic
534C4—21no—
535C4—21no—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—42no—
546C4—42no—
547C4—42no—
548C4—161no—
549C4—162no—
550C4—161no—
551C4C2×#42112yes—
552C2—21no—
553C2—21no—
554C2—41no—
555C2—41no—
556C2—82no—
557C2#2065411yescharacteristic
558C2#2067111yescharacteristic
559C2#2067911yescharacteristic
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. #20819C2semidirect
3. #17797C2semidirect
6. #8697C2semidirect
7. #8696C2semidirect
8. #8695C2semidirect
9. #8694C2semidirect
10. #7953C2semidirect
11. #7951C2semidirect
13. #2969C4semidirect
14. #2969C4semidirect
20. Q8×M16C4semidirect
22. D4×M16C4semidirect
34. C4×#107C22semidirect
35. C4×#106C22semidirect
38. #1568C22semidirect
39. #1568C22semidirect
51. #1285C22semidirect
52. #1285C22semidirect
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
#602142
#208192
#177972
#175812
#175802
#86972
#86962
#86952
#86942
#79532
#79512