Finite Group Databasebeta

← #227582

Subgroups — #227582

Order 512.

657 conjugacy classes of subgroups.

The subgroup lattice has 607 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#18485C211yesmaximal, characteristic
2#18021C211yesmaximal, characteristic, complemented
3#18016C211yesmaximal, characteristic, complemented
4#17517C211yesmaximal, characteristic
5#17517C211yesmaximal, characteristic
6#9858C211yesmaximal, characteristic, complemented
7#9588C211yesmaximal, characteristic, complemented
8#9586C211yesmaximal, characteristic, complemented
9#9212C211yesmaximal, characteristic, complemented
10#9212C211yesmaximal, characteristic, complemented
11#9070C211yesmaximal, characteristic, complemented
12#8898C211yesmaximal, characteristic, complemented
13#8898C211yesmaximal, characteristic, complemented
14#8186C211yesmaximal, characteristic, complemented
15#8185C211yesmaximal, characteristic, complemented
16#3481C2211yescharacteristic
17#3131C2211yescharacteristic
18#3107C2211yescharacteristic
19#3107C2211yescharacteristic
20#2990C2211yescharacteristic
21#2990C2211yescharacteristic
22#2986C2211yescharacteristic, complemented
23C4×#131C2211yescharacteristic
24C4×#131C2211yescharacteristic
25#1848C2211yescharacteristic, complemented
26#1644C411yescharacteristic
27#1644C411yescharacteristic
28#1643C411yescharacteristic
29#1643C411yescharacteristic
30#1640C411yescharacteristic
31#1640C411yescharacteristic
32#1639C411yescharacteristic
33#1639C411yescharacteristic
34#1593—21no—
35#1591—21no—
36#1587—21no—
37#1585—21no—
38#1584—21no—
39#1582—21no—
40SD16⋊C8—21no—
41C8×SD16—21no—
42#1574C2211yescharacteristic, complemented
43#1574C2211yescharacteristic, complemented
44#1543C2211yescharacteristic
45#1526C2211yescharacteristic, complemented
46#1494—21no—
47#1494—21no—
48#1493C2211yescharacteristic, complemented
49#1493C2211yescharacteristic, complemented
50#1492C2211yescharacteristic
51#1492C2211yescharacteristic
52#1490—21no—
53#1490—21no—
54#1489C2211yescharacteristic
55#1489C2211yescharacteristic
56#1484C2211yescharacteristic
57#1484C2211yescharacteristic
58#1483—21no—
59#1483—21no—
60#1476C2211yescharacteristic
61#1476C2211yescharacteristic
62#1476C2211yescharacteristic
63#1476C2211yescharacteristic
64C2×D4⋊C8—21no—
65C2×D4⋊C8—21no—
66#1452C2211yescharacteristic, complemented
67#1450C2211yescharacteristic, complemented
68C2×#322C2211yescharacteristic, complemented
69#1299C2211yescharacteristic, complemented
70#1299C2211yescharacteristic, complemented
71#1278C2211yescharacteristic, complemented
72#1278C2211yescharacteristic, complemented
73#1278C2211yescharacteristic, complemented
74#1278C2211yescharacteristic, complemented
75C2×#131—21no—
76C2×#131—21no—
77#559C2×C411yescharacteristic
78#554C2×C411yescharacteristic
79#554C2×C411yescharacteristic
80#553—21no—
81#553—21no—
82#551—41no—
83#546—21no—
84#546—21no—
85D42C2×C411yescharacteristic
86#533C2311yescharacteristic
87#532C2311yescharacteristic
88#529—21no—
89#529—21no—
90#525—41no—
91C4×#41C2311yescharacteristic
92C4×#41C2311yescharacteristic
93C2×C4×D4—21no—
94C2×C4×D4—21no—
95#488—41no—
96#487C2×C411yescharacteristic
97#487C2×C411yescharacteristic
98#486C2×C411yescharacteristic
99#486C2×C411yescharacteristic
100#485—21no—
101#485—21no—
102#475C2×C411yescharacteristic
103#475C2×C411yescharacteristic
104#474C2×C411yescharacteristic
105#474C2×C411yescharacteristic
106#462C2×C411yescharacteristic
107#462C2×C411yescharacteristic
108#461C2×C411yescharacteristic
109#461C2×C411yescharacteristic
110#452—21no—
111#452—21no—
112#446—21no—
113#446—21no—
114#446—21no—
115#446—21no—
116C8×Q8—21no—
117C8×Q8—21no—
118C4×Q16—21no—
119C4×Q16—21no—
120C4×SD16—21no—
121C4×SD16—21no—
122#435—41no—
123#435—41no—
124C8×D4—21no—
125C8×D4—21no—
126#433—41no—
127#432—21no—
128#432—21no—
129#431C2311yescharacteristic
130#431C2311yescharacteristic
131#431C2311yescharacteristic
132#431C2311yescharacteristic
133C2×#108—41no—
134#423C2311yescharacteristic
135#423C2311yescharacteristic
136C2×C4⋊C8—21no—
137C2×C4⋊C8—21no—
138#416—41no—
139#405—21no—
140#405—21no—
141#405—41no—
142#405—41no—
143C4×M16—21no—
144C4×M16—21no—
145C4×M16C2311yescharacteristic
146C4×M16C2311yescharacteristic
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152#335—21no—
153#335—21no—
154#334—21no—
155#334—21no—
156#333C2×C411yescharacteristic
157#333C2×C411yescharacteristic
158#331C2×C411yescharacteristic
159#331C2×C411yescharacteristic
160#330—41no—
161Q8⋊C8—21no—
162Q8⋊C8—21no—
163Q8⋊C8—21no—
164Q8⋊C8—21no—
165Q8⋊C8—21no—
166Q8⋊C8—21no—
167Q8⋊C8C2×C411yescharacteristic
168Q8⋊C8C2×C411yescharacteristic
169Q8⋊C8C2×C411yescharacteristic
170Q8⋊C8C2×C411yescharacteristic
171Q8⋊C8D411yescharacteristic
172Q8⋊C8D411yescharacteristic
173Q8⋊C8D411yescharacteristic
174Q8⋊C8D411yescharacteristic
175D4⋊C8—21no—
176D4⋊C8—21no—
177D4⋊C8—21no—
178D4⋊C8—21no—
179D4⋊C8—21no—
180D4⋊C8—21no—
181D4⋊C8—21no—
182D4⋊C8—21no—
183D4⋊C8—21no—
184D4⋊C8—21no—
185D4⋊C8C2×C411yescharacteristic
186D4⋊C8C2×C411yescharacteristic
187D4⋊C8C2×C411yescharacteristic
188D4⋊C8C2×C411yescharacteristic
189D4⋊C8D411yescharacteristic
190D4⋊C8D411yescharacteristic
191D4⋊C8D411yescharacteristic
192D4⋊C8D411yescharacteristic
193#322—21no—
194#322—21no—
195#322—21no—
196#322—41no—
197C82—21no—
198C2×#41—21no—
199C2×#41#4111yescharacteristic
200C2×#41#4111yescharacteristic
201C22×D4—21no—
202C22×D4—21no—
203C23×C4—41no—
204C2×Q16—21no—
205C2×Q16—21no—
206C2×SD16—21no—
207C2×SD16—21no—
208#131—24no—
209#131—42no—
210#131—42no—
211#131—42no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—41no—
217C2×M16—41no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—41no—
222C22×C8—41no—
223C22×C8—41no—
224C22×C8—41no—
225C4⋊Q8C22×C411yescharacteristic
226C4⋊Q8C22×C411yescharacteristic
227C4⋊Q8C22×C411yescharacteristic
228#127C22×C411yescharacteristic
229#126—41no—
230#126—41no—
231#125—21no—
232#125—21no—
233#125—21no—
234#124—21no—
235#124—21no—
236#124—21no—
237#123—41no—
238#123—41no—
239#123—41no—
240#123—41no—
241C22⋊Q8—21no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—21no—
245C22⋊Q8—41no—
246C22⋊Q8—41no—
247#121—21no—
248#121—21no—
249#121—21no—
250#121—21no—
251#121—41no—
252#121—41no—
253#120—41no—
254#120—41no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8C22×C411yescharacteristic
258C4×Q8C22×C411yescharacteristic
259C4×Q8C2×D411yescharacteristic
260C4×Q8C2×D411yescharacteristic
261C4×D4—21no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4—41no—
272C4×D4—41no—
273C4×D4—41no—
274C4×D4—41no—
275C4×D4C22×C411yescharacteristic
276C4×D4C22×C411yescharacteristic
277C4×D4C2×D411yescharacteristic
278C4×D4C2×D411yescharacteristic
279#117—21no—
280#117—21no—
281#117—21no—
282#117—41no—
283#117#4111yescharacteristic
284#117#4111yescharacteristic
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4—41no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42C2411yescharacteristic
292#108—82no—
293#107—21no—
294#107—21no—
295#106—21no—
296#106—21no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300C4⋊C8—21no—
301C4⋊C8—41no—
302C4⋊C8—41no—
303C4⋊C8—41no—
304C4⋊C8—41no—
305C4⋊C8C22×C411yescharacteristic
306C4⋊C8C22×C411yescharacteristic
307C4⋊C8C22×C411yescharacteristic
308C4⋊C8C22×C411yescharacteristic
309C4⋊C8C2×D411yescharacteristic
310C4⋊C8C2×D411yescharacteristic
311C4⋊C8C2×D411yescharacteristic
312C4⋊C8C2×D411yescharacteristic
313#103—21no—
314#103—21no—
315#103—21no—
316#103—21no—
317#103—21no—
318#103—21no—
319#103—41no—
320#103—41no—
321#102—21no—
322#102—21no—
323#102—41no—
324#102—41no—
325C22⋊C8—21no—
326C22⋊C8—21no—
327C22⋊C8—21no—
328C22⋊C8—21no—
329C22⋊C8—41no—
330C22⋊C8—41no—
331#97—21no—
332#97—21no—
333#97—41no—
334#97—41no—
335#97—41no—
336#97—41no—
337#97—41no—
338C4×C8—21no—
339C4×C8—21no—
340C4×C8—21no—
341C4×C8—21no—
342C4×C8—21no—
343C4×C8—21no—
344C4×C8—21no—
345C4×C8—21no—
346C4×C8—21no—
347C4×C8—41no—
348C4×C8—41no—
349C4×C8—41no—
350C4×C8C22×C411yescharacteristic
351C4×C8C22×C411yescharacteristic
352C4×C8C2×D411yescharacteristic
353C4×C8C2×D411yescharacteristic
354C4×C8C2×D411yescharacteristic
355C4×C8C2×D411yescharacteristic
356C24—41no—
357#41—22no—
358#41—22no—
359#41—41no—
360#41—41no—
361#41—81no—
362#41—81no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8C4×D411yescharacteristic
367C2×Q8C4×D411yescharacteristic
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—41no—
378C2×D4—81no—
379C2×D4—81no—
380C2×D4C4×D411yescharacteristic
381C2×D4C4×D411yescharacteristic
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—161no—
393C22×C4—161no—
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4H(5,𝔽2)11yescharacteristic
397Q16—81no—
398Q16—81no—
399SD16—42no—
400SD16—42no—
401M16—22no—
402M16—22no—
403M16—44no—
404M16—41no—
405M16—41no—
406M16—44no—
407M16—81no—
408M16—81no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—44no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—44no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—161no—
436C2×C8—161no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4C4×D411yescharacteristic
454C4⋊C4C4×D411yescharacteristic
455C4⋊C4C4×D411yescharacteristic
456C4⋊C4C4×D411yescharacteristic
457C22⋊C4—21no—
458C22⋊C4—21no—
459C22⋊C4—21no—
460C22⋊C4—21no—
461C22⋊C4—21no—
462C22⋊C4—21no—
463C22⋊C4—41no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—41no—
479C42C23×C411yescharacteristic
480C42C22×D411yescharacteristic
481C42C22×D411yescharacteristic
482C42H(5,𝔽2)11yescharacteristic
483C23—21no—
484C23—21no—
485C23—41no—
486C23—82no—
487C23—82no—
488C23—161no—
489C23#55111yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—41no—
495Q8—41no—
496Q8—41no—
497Q8—41no—
498Q8—81no—
499D4—21no—
500D4—21no—
501D4—42no—
502D4—42no—
503D4—41no—
504D4—41no—
505D4—41no—
506D4—41no—
507D4—81no—
508D4—82no—
509D4—82no—
510D4—161no—
511D4#44312yes—
512D4#44412yes—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—161no—
541C2×C4—161no—
542C2×C4—161no—
543C2×C4C2×C4×D411yescharacteristic
544C2×C4C2×C4×D411yescharacteristic
545C2×C4#51811yescharacteristic
546C2×C4D4211yescharacteristic
547C2×C4#54611yescharacteristic
548C2×C4#54611yescharacteristic
549C8—22no—
550C8—22no—
551C8—41no—
552C8—41no—
553C8—41no—
554C8—41no—
555C8—42no—
556C8—42no—
557C8—41no—
558C8—41no—
559C8—41no—
560C8—41no—
561C8—41no—
562C8—41no—
563C8—41no—
564C8—41no—
565C8—81no—
566C8—81no—
567C22—21no—
568C22—22no—
569C22—21no—
570C22—22no—
571C22—21no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—164no—
580C22#235811yescharacteristic
581C4—21no—
582C4—21no—
583C4—21no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4—81no—
591C4—81no—
592C4—81no—
593C4C2×#44311yescharacteristic
594C4C2×#44411yescharacteristic
595C4#295611yescharacteristic
596C4#295611yescharacteristic
597C4#328811yescharacteristic
598C4#328911yescharacteristic
599C2—41no—
600C2—42no—
601C2—42no—
602C2—161no—
603C2#2288611yescharacteristic
604C2#2288811yescharacteristic
605C2#2289011yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2358
prime radical#3481C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #18021C2semidirect
3. #18016C2semidirect
6. #9858C2semidirect
7. #9588C2semidirect
8. #9586C2semidirect
9. #9212C2semidirect
10. #9212C2semidirect
11. #9070C2semidirect
12. #8898C2semidirect
13. #8898C2semidirect
14. #8186C2semidirect
15. #8185C2semidirect
22. #2986C22semidirect
25. #1848C22semidirect
42. #1574C22semidirect
43. #1574C22semidirect
45. #1526C22semidirect
48. #1493C22semidirect
49. #1493C22semidirect
66. #1452C22semidirect
67. #1450C22semidirect
68. C2×#322C22semidirect
69. #1299C22semidirect
70. #1299C22semidirect
71. #1278C22semidirect
72. #1278C22semidirect
73. #1278C22semidirect
74. #1278C22semidirect

Maximal subgroups

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

subgroupindex
#184852
#180212
#180162
#175172
#98582
#95882
#95862
#92122
#90702
#88982
#81862
#81852