Finite Group Databasebeta

← #225733

Subgroups — #225733

Order 512.

554 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#22783C211yesmaximal, characteristic
2#22782C211yesmaximal, characteristic
3#17890C211yesmaximal, characteristic
4#17830C211yesmaximal, characteristic, complemented
5#17830C211yesmaximal, characteristic, complemented
6#9632C211yesmaximal, characteristic, complemented
7#9629C211yesmaximal, characteristic, complemented
8#8905C211yesmaximal, characteristic, complemented
9#8905C211yesmaximal, characteristic, complemented
10#8904C211yesmaximal, characteristic, complemented
11#8904C211yesmaximal, characteristic, complemented
12#8827C211yesmaximal, characteristic, complemented
13#8669C211yesmaximal, characteristic, complemented
14#8669C211yesmaximal, characteristic, complemented
15#8290C211yesmaximal, characteristic, complemented
16#2965C2211yescharacteristic
17#2965C2211yescharacteristic
18C4×#135C2211yescharacteristic
19#2925C2211yescharacteristic
20#2925C2211yescharacteristic
21#2906C2211yescharacteristic
22#2906C2211yescharacteristic
23#1938—21no—
24#1937—21no—
25#1922—21no—
26#1922—21no—
27#1876—21no—
28#1876—21no—
29#1837C2211yescharacteristic
30#1837C2211yescharacteristic
31#1806C2211yescharacteristic
32#1806C2211yescharacteristic
33#1806C2211yescharacteristic
34#1806C2211yescharacteristic
35C4×#106C2211yescharacteristic
36#1770C2211yescharacteristic
37C8×C4⋊C4C2211yescharacteristic
38#1765—21no—
39#1764—21no—
40#1594C411yescharacteristic, complemented
41#1594C411yescharacteristic, complemented
42#1593C411yescharacteristic, complemented
43#1593C411yescharacteristic, complemented
44#1592C411yescharacteristic, complemented
45#1592C411yescharacteristic, complemented
46#1592C411yescharacteristic, complemented
47#1592C411yescharacteristic, complemented
48#1574—21no—
49#1574—21no—
50#1573—21no—
51#1573—21no—
52#1571C2211yescharacteristic
53#1571C2211yescharacteristic
54#1570C2211yescharacteristic
55#1570C2211yescharacteristic
56#1569—21no—
57#1569—21no—
58#1567C2211yescharacteristic
59#1567C2211yescharacteristic
60#1566C2211yescharacteristic
61#1566C2211yescharacteristic
62#1565—21no—
63#1565—21no—
64#1564C2211yescharacteristic
65C2×#334C2211yescharacteristic
66#1476C2211yescharacteristic
67#1476C2211yescharacteristic
68C2×#322C2211yescharacteristic
69#1303C2211yescharacteristic
70#1303C2211yescharacteristic
71#1303C2211yescharacteristic
72#1303C2211yescharacteristic
73#1279C2211yescharacteristic
74#1279C2211yescharacteristic
75C2×#135D411yescharacteristic
76C2×#131—21no—
77C2×#131—21no—
78C4×#41C2311yescharacteristic
79C4×#41C2311yescharacteristic
80#446C2×C411yescharacteristic
81#446C2×C411yescharacteristic
82#446C2×C411yescharacteristic
83#446C2×C411yescharacteristic
84C4×Q16C2×C411yescharacteristic
85C4×Q16Q811yescharacteristic
86C4×SD16C2×C411yescharacteristic
87C4×SD16C2×C411yescharacteristic
88C4×SD16Q811yescharacteristic
89C4×SD16Q811yescharacteristic
90C4×D8C2×C411yescharacteristic
91C4×D8Q811yescharacteristic
92#436C2×C411yescharacteristic
93#436C2×C411yescharacteristic
94#436C2×C411yescharacteristic
95#436C2×C411yescharacteristic
96#435—21no—
97#435—21no—
98#435—21no—
99#435—21no—
100#433—41no—
101#433—41no—
102#432—21no—
103#432—21no—
104#431—21no—
105#431—21no—
106#431—21no—
107#431—21no—
108#431C2311yescharacteristic
109#431C2311yescharacteristic
110#427D411yescharacteristic
111C2×#106—21no—
112#424—21no—
113#424—21no—
114#424—41no—
115#424—41no—
116#423—21no—
117#423—21no—
118#423C2311yescharacteristic
119#423C2311yescharacteristic
120#423C2311yescharacteristic
121#423C2311yescharacteristic
122C2×C4⋊C8—21no—
123C2×C4⋊C8—21no—
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×#104—21no—
127C2×#104—21no—
128C2×#104—21no—
129C2×#104—21no—
130#416D411yescharacteristic
131#416D411yescharacteristic
132#408—21no—
133#408—21no—
134#405—21no—
135#405—21no—
136C4×M16—21no—
137C4×M16—21no—
138C2×#97—21no—
139C2×C4×C8—21no—
140C2×C4×C8C2311yescharacteristic
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2311yescharacteristic
143C4×C4⋊C4C2311yescharacteristic
144C4×C4⋊C4C2311yescharacteristic
145#341—21no—
146#341—21no—
147#341—21no—
148#341—21no—
149#340—21no—
150#339—21no—
151#339—21no—
152#339—21no—
153#339—21no—
154#336—21no—
155#336—21no—
156#335—21no—
157#335—21no—
158#335—21no—
159#335—21no—
160#335—21no—
161#335—21no—
162#335—21no—
163#335—21no—
164#335C2×C411yescharacteristic
165#335C2×C411yescharacteristic
166#334—21no—
167#334—21no—
168#334—21no—
169#334—21no—
170#334—41no—
171#334—41no—
172#334C2×C411yescharacteristic
173#334C2×C411yescharacteristic
174#334D411yescharacteristic
175#334D411yescharacteristic
176#334D411yescharacteristic
177#334D411yescharacteristic
178Q8⋊C8C2×C411yescharacteristic
179Q8⋊C8C2×C411yescharacteristic
180Q8⋊C8C2×C411yescharacteristic
181Q8⋊C8C2×C411yescharacteristic
182D4⋊C8C2×C411yescharacteristic
183D4⋊C8C2×C411yescharacteristic
184D4⋊C8C2×C411yescharacteristic
185D4⋊C8C2×C411yescharacteristic
186#322C2×C411yescharacteristic
187#322C2×C411yescharacteristic
188#322C2×C411yescharacteristic
189#322C2×C411yescharacteristic
190C2×#41C2×D411yescharacteristic
191C2×#41C2×D411yescharacteristic
192#135—41no—
193#135—41no—
194C2×Q16C4⋊C411yescharacteristic
195C2×SD16C4⋊C411yescharacteristic
196C2×SD16C4⋊C411yescharacteristic
197C2×D8C4⋊C411yescharacteristic
198#131—42no—
199#131—42no—
200#131—81no—
201#131—81no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—41no—
211C2×M16—41no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—41no—
216C22×C8—41no—
217C22×C8C2×D411yescharacteristic
218C22×C8C2×D411yescharacteristic
219C22×C8#4111yescharacteristic
220C22×C8#4111yescharacteristic
221C4×Q8C22×C411yescharacteristic
222C4×Q8C22×C411yescharacteristic
223C4×Q8C2×Q811yescharacteristic
224C4×Q8C2×Q811yescharacteristic
225C4×D4—21no—
226C4×D4—21no—
227C4×D4—21no—
228C4×D4—21no—
229C4×D4C22×C411yescharacteristic
230C4×D4C22×C411yescharacteristic
231C4×D4C2×Q811yescharacteristic
232C4×D4C2×Q811yescharacteristic
233#117—21no—
234#117—21no—
235#117C2×D411yescharacteristic
236#117C2×D411yescharacteristic
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42C2411yescharacteristic
244#107C4⋊C411yescharacteristic
245#107C4⋊C411yescharacteristic
246#106—42no—
247#106C4⋊C411yescharacteristic
248#106C4⋊C411yescharacteristic
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—41no—
264C4⋊C8—41no—
265C4⋊C8—41no—
266C4⋊C8—41no—
267C4⋊C8C22×C411yescharacteristic
268C4⋊C8C22×C411yescharacteristic
269C4⋊C8C22×C411yescharacteristic
270C4⋊C8C22×C411yescharacteristic
271C4⋊C8C2×D411yescharacteristic
272C4⋊C8C2×D411yescharacteristic
273C4⋊C8C2×D411yescharacteristic
274C4⋊C8C2×D411yescharacteristic
275#104—42no—
276#104—42no—
277#104—81no—
278#104—81no—
279#103C4⋊C411yescharacteristic
280#103C4⋊C411yescharacteristic
281#103C4⋊C411yescharacteristic
282#103C4⋊C411yescharacteristic
283#102C4⋊C411yescharacteristic
284#102C4⋊C411yescharacteristic
285#102C4⋊C411yescharacteristic
286#102C4⋊C411yescharacteristic
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—21no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293C22⋊C8—41no—
294C22⋊C8—41no—
295C22⋊C8—41no—
296C22⋊C8—41no—
297#97—21no—
298#97—21no—
299#97—21no—
300#97—21no—
301#97—41no—
302#97—41no—
303#97—81no—
304C4×C8—21no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—41no—
308C4×C8—41no—
309C4×C8—81no—
310C4×C8C22×C411yescharacteristic
311C4×C8C22×C411yescharacteristic
312C4×C8C22×C411yescharacteristic
313C4×C8C22×C411yescharacteristic
314C4×C8C22×C411yescharacteristic
315C4×C8C22×C411yescharacteristic
316C4×C8C2×D411yescharacteristic
317C4×C8C2×D411yescharacteristic
318C4×C8C2×Q811yescharacteristic
319C4×C8C2×Q811yescharacteristic
320#95—21no—
321#95—21no—
322#41—21no—
323#41—21no—
324#41—21no—
325#41—21no—
326#41—21no—
327#41—21no—
328#41—21no—
329#41—21no—
330C2×Q8C2×C4⋊C411yescharacteristic
331C2×Q8C2×C4⋊C411yescharacteristic
332C2×D4—21no—
333C2×D4—21no—
334C2×D4C2×C4⋊C411yescharacteristic
335C2×D4C2×C4⋊C411yescharacteristic
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4C22×D411yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C42−1+411yescharacteristic
347Q16—41no—
348SD16—41no—
349SD16—41no—
350D8—41no—
351M16—42no—
352M16—42no—
353M16—41no—
354M16—41no—
355M16—41no—
356M16—41no—
357M16—41no—
358M16—41no—
359M16—41no—
360M16—41no—
361M16—81no—
362M16—81no—
363M16—82no—
364M16—82no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—81no—
390C2×C8—82no—
391C2×C8—82no—
392C2×C8C2×C4⋊C411yescharacteristic
393C2×C8C2×C4⋊C411yescharacteristic
394C2×C8C2×C4⋊C411yescharacteristic
395C2×C8C2×C4⋊C411yescharacteristic
396C2×C8C4×D411yescharacteristic
397C2×C8C4×D411yescharacteristic
398C2×C8C4×D411yescharacteristic
399C2×C8C4×D411yescharacteristic
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—22no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4C2×C4⋊C411yescharacteristic
409C4⋊C4C2×C4⋊C411yescharacteristic
410C4⋊C4C2×C4⋊C411yescharacteristic
411C4⋊C4C2×C4⋊C411yescharacteristic
412C22⋊C4—21no—
413C22⋊C4—21no—
414C22⋊C4—21no—
415C22⋊C4—21no—
416C22⋊C4—21no—
417C22⋊C4—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42—41no—
428C42C23×C411yescharacteristic
429C42C22×D411yescharacteristic
430C42C22×Q811yescharacteristic
431C42H(5,𝔽2)11yescharacteristic
432C23—21no—
433C23—21no—
434C23#54711yescharacteristic
435Q8—21no—
436Q8—21no—
437Q8—21no—
438Q8—21no—
439D4—21no—
440D4—21no—
441D4—21no—
442D4—21no—
443D4—41no—
444D4—41no—
445D4—41no—
446D4—41no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4C22×C4⋊C411yescharacteristic
473C2×C4C2×C4×D411yescharacteristic
474C2×C4#52011yescharacteristic
475C2×C4D4211yescharacteristic
476C2×C4D4×Q811yescharacteristic
477C2×C4#55411yescharacteristic
478C2×C4#57511yescharacteristic
479C2×C4#57511yescharacteristic
480C2×C4#57711yescharacteristic
481C2×C4#57711yescharacteristic
482C8—21no—
483C8—21no—
484C8—21no—
485C8—21no—
486C8—21no—
487C8—21no—
488C8—21no—
489C8—21no—
490C8—41no—
491C8—41no—
492C8—41no—
493C8—41no—
494C8—41no—
495C8—41no—
496C8—41no—
497C8—41no—
498C8—41no—
499C8—41no—
500C8—81no—
501C8—81no—
502C8—81no—
503C8—81no—
504C22—21no—
505C22—21no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22D4×C4⋊C411yescharacteristic
513C22#335111yescharacteristic
514C22#335111yescharacteristic
515C4—21no—
516C4—21no—
517C4—21no—
518C4—41no—
519C4—41no—
520C4—41no—
521C4—41no—
522C4—41no—
523C4—41no—
524C4—82no—
525C4—161no—
526C4#291111yescharacteristic
527C4#291211yescharacteristic
528C4#295511yescharacteristic
529C4#295611yescharacteristic
530C4#327511yescharacteristic
531C4#327511yescharacteristic
532C2—21no—
533C2—21no—
534C2—81no—
535C2—81no—
536C2#2266611yescharacteristic
537C2#2268711yescharacteristic
538C2#2269511yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22D4×C4⋊C4
prime radicalC2×#135D4
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
4. #17830C2semidirect
5. #17830C2semidirect
6. #9632C2semidirect
7. #9629C2semidirect
8. #8905C2semidirect
9. #8905C2semidirect
10. #8904C2semidirect
11. #8904C2semidirect
12. #8827C2semidirect
13. #8669C2semidirect
14. #8669C2semidirect
15. #8290C2semidirect
40. #1594C4semidirect
41. #1594C4semidirect
42. #1593C4semidirect
43. #1593C4semidirect
44. #1592C4semidirect
45. #1592C4semidirect
46. #1592C4semidirect
47. #1592C4semidirect

Maximal subgroups

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

subgroupindex
#227832
#227822
#178902
#178302
#96322
#96292
#89052
#89042
#88272
#86692
#82902