Finite Group Databasebeta

← #189317

Subgroups — #189317

Order 512.

915 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60236C211yesmaximal, characteristic
2#21985C211yesmaximal, characteristic, complemented
3#17806C211yesmaximal, characteristic, complemented
4#17589C211yesmaximal, characteristic
5#17574C211yesmaximal, characteristic
6#8680C212yesmaximal, complemented
7#8678C212yesmaximal, complemented
8#8676C212yesmaximal, complemented
9#8674C212yesmaximal, complemented
10#7961C211yesmaximal, characteristic, complemented
11#7958C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13#2981—22no—
14#2979C412yescomplemented
15#2979C412yescomplemented
16#2975—21no—
17#2974C414yescomplemented
18#2972—21no—
19#2935—21no—
20#2935—21no—
21#2931C2212yes—
22#2931C2212yes—
23#2920C2211yescharacteristic
24#2920C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27#2880—22no—
28#2285C2211yescharacteristic
29#2283C2211yescharacteristic
30#1948—21no—
31#1947—21no—
32#1854—22no—
33#1846C2212yescomplemented
34#1835C2212yescomplemented
35#1574—21no—
36#1574—21no—
37#1573—22no—
38#1564C2212yescomplemented
39#1564C2212yescomplemented
40#1519—22no—
41#1517C2212yes—
42#1517C2212yes—
43#1517C2212yes—
44#1517C2212yes—
45#1510—22no—
46#1505—22no—
47#1502—22no—
48#1286C2211yescharacteristic, complemented
49#1286C2211yescharacteristic, complemented
50#1286C2211yescharacteristic, complemented
51#1286C2211yescharacteristic, complemented
52#1286C2212yescomplemented
53#1286C2212yescomplemented
54C22×#41—21no—
55C2×#131—22no—
56C2×#131—22no—
57#532—21no—
58#532—21no—
59#532—21no—
60#532—21no—
61#532C2×C412yes—
62#532D412yes—
63#532Q812yes—
64#532C2312yes—
65C2×C4⋊Q8Q811yescharacteristic
66C2×#127Q811yescharacteristic
67C2×#124—21no—
68C2×#121—21no—
69C2×#121D412yes—
70C2×C4×D4—21no—
71C2×C4×D4C2×C412yes—
72C2×#117—21no—
73C22×C42C2311yescharacteristic
74#436—22no—
75#436—22no—
76#436—22no—
77#436—22no—
78#436C2×C414yes—
79#436C2×C414yes—
80#435—24no—
81#435—24no—
82C8×D4—22no—
83C8×D4—22no—
84C8×D4—22no—
85C8×D4—22no—
86C8×D4C2×C414yes—
87C8×D4C2×C414yes—
88#432—21no—
89#432—21no—
90#432—21no—
91#432C2×C412yes—
92#431—21no—
93#431—21no—
94#431—21no—
95#431—21no—
96#431—21no—
97#431C2×C412yes—
98#431C2×C412yes—
99#431C2×C411yescharacteristic
100#431C2×C411yescharacteristic
101#431D412yes—
102#431D411yescharacteristic
103#431D411yescharacteristic
104#431C2311yescharacteristic
105#431C2311yescharacteristic
106#431C2312yes—
107C2×#107—22no—
108C2×#106—22no—
109#423—21no—
110#423C2311yescharacteristic
111#423C2311yescharacteristic
112C2×C4⋊C8—21no—
113C2×C4⋊C8D411yescharacteristic
114C2×C4⋊C8D411yescharacteristic
115#408—22no—
116#408—22no—
117#408—24no—
118#408—24no—
119#407—21no—
120#407—24no—
121#407—21no—
122C2×C22⋊C8—24no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8D412yes—
125C4×M16—21no—
126C4×M16—21no—
127C2×C4×C8—21no—
128C2×C4×C8C2312yes—
129#384—21no—
130#383—21no—
131#382C2311yescharacteristic
132#382C2312yes—
133#382C2311yescharacteristic
134#381—21no—
135#380—21no—
136#380—21no—
137#380—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C22⋊C4—21no—
142#376—21no—
143#340—22no—
144#340—22no—
145#335—22no—
146#335—22no—
147#334—22no—
148#334—22no—
149#327—24no—
150#327—24no—
151#327—24no—
152#327—24no—
153C2×#41—22no—
154C2×#41—22no—
155C2×#41—22no—
156C2×#41—22no—
157C2×#41—81no—
158C2×#41—81no—
159C2×#41—81no—
160C2×#41—81no—
161C22×Q8—21no—
162C22×D4—22no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4C4⋊C412yes—
166C23×C4—22no—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4#4111yescharacteristic
170#131—82no—
171#131—82no—
172#131—82no—
173#131—82no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—24no—
177C2×M16—24no—
178C2×M16—41no—
179C22×C8—24no—
180C22×C8—24no—
181C22×C8—41no—
182C22×C8C2×D412yes—
183C22×C8C2×D412yes—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8C4⋊C412yes—
187C4⋊Q8C2×Q812yes—
188#127—21no—
189#127—21no—
190#127C4⋊C412yes—
191#127C2×Q812yes—
192#124—22no—
193#124—22no—
194#124—22no—
195#124—22no—
196#121—22no—
197#121—22no—
198#121—22no—
199#121—22no—
200#121—22no—
201#121—22no—
202#121C4⋊C414yes—
203#121C2×D414yes—
204C4×D4—22no—
205C4×D4—22no—
206C4×D4—22no—
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4C4⋊C414yes—
211C4×D4C22×C414yes—
212#117—81no—
213#117—81no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4C4⋊C412yes—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C42—41no—
226C2×C42—41no—
227C2×C42C22×C411yescharacteristic
228C2×C42C22×C411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×Q811yescharacteristic
234C2×C42C2×Q811yescharacteristic
235C2×C42#4111yescharacteristic
236C2×C42#4111yescharacteristic
237C2×C42#4111yescharacteristic
238C2×C42C2411yescharacteristic
239#107—82no—
240#106—82no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—22no—
246C4⋊C8—41no—
247C4⋊C8—41no—
248C4⋊C8C22⋊C411yescharacteristic
249C4⋊C8C22⋊C411yescharacteristic
250C4⋊C8C22⋊C412yes—
251C4⋊C8C22⋊C412yes—
252C4⋊C8C22⋊C411yescharacteristic
253C4⋊C8C22⋊C411yescharacteristic
254C4⋊C8C22×C412yes—
255C4⋊C8C22×C412yes—
256C22⋊C8—22no—
257C22⋊C8—22no—
258C22⋊C8—24no—
259C22⋊C8—24no—
260C22⋊C8—24no—
261C22⋊C8—24no—
262C22⋊C8—21no—
263C22⋊C8—21no—
264C22⋊C8—22no—
265C22⋊C8—41no—
266C22⋊C8—41no—
267C22⋊C8C22⋊C412yes—
268C22⋊C8C22⋊C412yes—
269C22⋊C8C22⋊C412yes—
270C22⋊C8C22⋊C412yes—
271#97—22no—
272#97—41no—
273C4×C8—22no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—22no—
277C4×C8—41no—
278C4×C8C22×C412yes—
279C4×C8C22×C412yes—
280C4×C8C2×D412yes—
281C4×C8C2×D412yes—
282#95—21no—
283#95—21no—
284#95—22no—
285#95—21no—
286#95—21no—
287#95—22no—
288#95—41no—
289#95—41no—
290C24—22no—
291C24#12111yescharacteristic
292#41—42no—
293#41—42no—
294#41—42no—
295#41—42no—
296#41—82no—
297#41—82no—
298#41—82no—
299#41—82no—
300#41—82no—
301#41—82no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—81no—
305C2×Q8—81no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—44no—
314C2×D4—81no—
315C2×D4—81no—
316C2×D4—82no—
317C2×D4—82no—
318C2×D4—82no—
319C2×D4C2×C4⋊C414yes—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4—81no—
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C4⋊C411yescharacteristic
337C22×C4C2×C4⋊C411yescharacteristic
338C22×C4#11711yescharacteristic
339C22×C4#11711yescharacteristic
340C22×C4#12111yescharacteristic
341C22×C4#12111yescharacteristic
342C22×C4#12111yescharacteristic
343C22×C4#12111yescharacteristic
344C22×C4#12111yescharacteristic
345C22×C4C22⋊Q811yescharacteristic
346C22×C4C22⋊Q811yescharacteristic
347C22×C4C22⋊Q811yescharacteristic
348C22×C4C22⋊Q811yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C2×#4111yescharacteristic
352M16—41no—
353M16—41no—
354M16—41no—
355M16—41no—
356M16—84no—
357M16—84no—
358M16—82no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—42no—
368C2×C8—84no—
369C2×C8—84no—
370C2×C8—82no—
371C2×C8C2×C22⋊C412yes—
372C2×C8C2×C22⋊C412yes—
373C2×C8C2×C22⋊C412yes—
374C2×C8C2×C22⋊C412yes—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—41no—
379C4⋊C4—42no—
380C4⋊C4—41no—
381C4⋊C4—42no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—82no—
385C4⋊C4—82no—
386C4⋊C4C2×C4⋊C414yes—
387C22⋊C4—24no—
388C22⋊C4—24no—
389C22⋊C4—24no—
390C22⋊C4—24no—
391C22⋊C4—82no—
392C22⋊C4—82no—
393C42—82no—
394C42—82no—
395C42C2×C22⋊C411yescharacteristic
396C42C2×C22⋊C411yescharacteristic
397C42C2×C4⋊C411yescharacteristic
398C42C2×C4⋊C411yescharacteristic
399C42#11711yescharacteristic
400C42#11711yescharacteristic
401C42#12111yescharacteristic
402C42#12111yescharacteristic
403C42C22⋊Q811yescharacteristic
404C42C22⋊Q811yescharacteristic
405C42C22⋊Q811yescharacteristic
406C42C22⋊Q811yescharacteristic
407C42C23×C411yescharacteristic
408C42C22×D411yescharacteristic
409C42C22×Q811yescharacteristic
410C42C2×#4111yescharacteristic
411C23—21no—
412C23—21no—
413C23—24no—
414C23—41no—
415C23—41no—
416C23—44no—
417C23—82no—
418C23—82no—
419C23#38011yescharacteristic
420C23#38011yescharacteristic
421C23C2×#12111yescharacteristic
422Q8—42no—
423Q8—42no—
424Q8—82no—
425D4—44no—
426D4—44no—
427D4—44no—
428D4—44no—
429D4—42no—
430D4—42no—
431D4—44no—
432D4—82no—
433D4—82no—
434D4—84no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—24no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—84no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4#38011yescharacteristic
472C2×C4#38011yescharacteristic
473C2×C4#38011yescharacteristic
474C2×C4#38011yescharacteristic
475C2×C4#38011yescharacteristic
476C2×C4#38011yescharacteristic
477C2×C4C22×C22⋊C411yescharacteristic
478C2×C4C22×C4⋊C411yescharacteristic
479C2×C4C2×#11711yescharacteristic
480C2×C4C2×#12111yescharacteristic
481C2×C4C2×C22⋊Q811yescharacteristic
482C2×C4C2×C22⋊Q811yescharacteristic
483C2×C4#57611yescharacteristic
484C2×C4#57611yescharacteristic
485C2×C4#57711yescharacteristic
486C2×C4#57711yescharacteristic
487C8—22no—
488C8—22no—
489C8—22no—
490C8—22no—
491C8—42no—
492C8—42no—
493C8—81no—
494C8—81no—
495C8—82no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—44no—
505C22—44no—
506C22—44no—
507C22—81no—
508C22—81no—
509C22—164no—
510C22C2×#38011yescharacteristic
511C22#304111yescharacteristic
512C22#304211yescharacteristic
513C4—21no—
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—44no—
523C4—161no—
524C4—162no—
525C4—161no—
526C4#181412yes—
527C4#289711yescharacteristic
528C4#289711yescharacteristic
529C4#308111yescharacteristic
530C4#308111yescharacteristic
531C2—21no—
532C2—21no—
533C2—41no—
534C2—41no—
535C2—84no—
536C2#2065011yescharacteristic
537C2#2067111yescharacteristic
538C2C2×#181411yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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. #21985C2semidirect
3. #17806C2semidirect
6. #8680C2semidirect
7. #8678C2semidirect
8. #8676C2semidirect
9. #8674C2semidirect
10. #7961C2semidirect
11. #7958C2semidirect
14. #2979C4semidirect
15. #2979C4semidirect
17. #2974C4semidirect
33. #1846C22semidirect
34. #1835C22semidirect
38. #1564C22semidirect
39. #1564C22semidirect
48. #1286C22semidirect
49. #1286C22semidirect
50. #1286C22semidirect
51. #1286C22semidirect
52. #1286C22semidirect
53. #1286C22semidirect

Maximal subgroups

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

subgroupindex
#602362
#219852
#178062
#175892
#175742
#86802
#86782
#86762
#86742
#79612
#79582