Finite Group Databasebeta

← #226165

Subgroups — #226165

Order 512.

552 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22803C211yesmaximal, characteristic
2#22783C211yesmaximal, characteristic
3#17868C211yesmaximal, characteristic
4#17832C211yesmaximal, characteristic, complemented
5#17829C211yesmaximal, characteristic, complemented
6#9632C211yesmaximal, characteristic, complemented
7#9628C211yesmaximal, characteristic, complemented
8#8905C211yesmaximal, characteristic, complemented
9#8904C211yesmaximal, characteristic, complemented
10#8903C211yesmaximal, characteristic, complemented
11#8903C211yesmaximal, characteristic, complemented
12#8828C211yesmaximal, characteristic, complemented
13#8669C211yesmaximal, characteristic, complemented
14#8666C211yesmaximal, characteristic, complemented
15#8289C211yesmaximal, characteristic, complemented
16#2965C2211yescharacteristic
17C8×#41C2211yescharacteristic
18C4×#135C2211yescharacteristic
19#2925C2211yescharacteristic
20#2925C2211yescharacteristic
21#2925C2211yescharacteristic
22#2906C2211yescharacteristic
23#1975—21no—
24#1937—21no—
25#1929—21no—
26#1929—21no—
27#1922—21no—
28#1876—21no—
29#1848C2211yescharacteristic
30#1837C2211yescharacteristic
31#1806C2211yescharacteristic
32#1806C2211yescharacteristic
33#1806C2211yescharacteristic
34#1806C2211yescharacteristic
35C4×#106C2211yescharacteristic
36#1773—21no—
37#1770C2211yescharacteristic
38C8×C4⋊C4C2211yescharacteristic
39#1764—21no—
40#1588C411yescharacteristic, complemented
41#1588C411yescharacteristic, complemented
42#1587C411yescharacteristic, complemented
43#1587C411yescharacteristic, complemented
44#1586C411yescharacteristic, complemented
45#1586C411yescharacteristic, complemented
46#1585C411yescharacteristic, complemented
47#1585C411yescharacteristic, complemented
48#1574—21no—
49#1573—21no—
50#1572—21no—
51#1572—21no—
52#1571C2211yescharacteristic
53#1570C2211yescharacteristic
54#1570C2211yescharacteristic
55#1570C2211yescharacteristic
56#1569—21no—
57#1568—21no—
58#1567C2211yescharacteristic
59#1567C2211yescharacteristic
60#1566C2211yescharacteristic
61#1566C2211yescharacteristic
62#1565—21no—
63#1564—21no—
64#1564C2211yescharacteristic
65C2×#335C2211yescharacteristic
66#1476C2211yescharacteristic
67#1476C2211yescharacteristic
68C2×#322C2211yescharacteristic
69#1303C2211yescharacteristic
70#1303C2211yescharacteristic
71#1303C2211yescharacteristic
72#1303C2211yescharacteristic
73#1279C2211yescharacteristic
74#1279C2211yescharacteristic
75C2×#135D411yescharacteristic
76C2×#131—21no—
77C4×#41C2311yescharacteristic
78C4×#41C2311yescharacteristic
79#446C2×C411yescharacteristic
80#446C2×C411yescharacteristic
81C8×Q8C2×C411yescharacteristic
82C8×Q8C2×C411yescharacteristic
83C4×Q16C2×C411yescharacteristic
84C4×Q16Q811yescharacteristic
85C4×SD16C2×C411yescharacteristic
86C4×SD16C2×C411yescharacteristic
87C4×SD16Q811yescharacteristic
88C4×SD16Q811yescharacteristic
89C4×D8C2×C411yescharacteristic
90C4×D8Q811yescharacteristic
91#436C2×C411yescharacteristic
92#436C2×C411yescharacteristic
93#435—21no—
94#435—21no—
95C8×D4—21no—
96C8×D4—21no—
97C8×D4C2×C411yescharacteristic
98C8×D4C2×C411yescharacteristic
99#433—41no—
100#433—41no—
101#433—41no—
102#432—21no—
103#432—21no—
104#432—21no—
105#431—21no—
106#431—21no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431C2311yescharacteristic
111#431C2311yescharacteristic
112C2×#108—21no—
113C2×#108—21no—
114#427D411yescharacteristic
115C2×#106—21no—
116#424—21no—
117#424—41no—
118#423—21no—
119#423C2311yescharacteristic
120#423C2311yescharacteristic
121#423C2311yescharacteristic
122#423C2311yescharacteristic
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#408—21no—
135#405—21no—
136#405—21no—
137#405—21no—
138C4×M16—21no—
139C2×#97—21no—
140C2×C4×C8—21no—
141C2×C4×C8—21no—
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145C4×C4⋊C4C2311yescharacteristic
146C4×C4⋊C4C2311yescharacteristic
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—41no—
161#335C2×C411yescharacteristic
162#335C2×C411yescharacteristic
163#335D411yescharacteristic
164#335D411yescharacteristic
165#335D411yescharacteristic
166#335D411yescharacteristic
167#334—21no—
168#334—21no—
169#334—21no—
170#334—21no—
171#334—21no—
172#334—21no—
173#334—21no—
174#334—21no—
175#334—41no—
176#334C2×C411yescharacteristic
177#334C2×C411yescharacteristic
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—81no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—41no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—41no—
213C22×C8C2×D411yescharacteristic
214C22×C8C2×D411yescharacteristic
215C22×C8#4111yescharacteristic
216C22×C8#4111yescharacteristic
217C4×Q8C22×C411yescharacteristic
218C4×Q8C22×C411yescharacteristic
219C4×Q8C2×Q811yescharacteristic
220C4×Q8C2×Q811yescharacteristic
221C4×D4—21no—
222C4×D4—21no—
223C4×D4—21no—
224C4×D4—21no—
225C4×D4C22×C411yescharacteristic
226C4×D4C22×C411yescharacteristic
227C4×D4C2×Q811yescharacteristic
228C4×D4C2×Q811yescharacteristic
229#117—21no—
230#117—21no—
231#117C2×D411yescharacteristic
232#117C2×D411yescharacteristic
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42C2411yescharacteristic
240#108—42no—
241#108—81no—
242#107C4⋊C411yescharacteristic
243#107C4⋊C411yescharacteristic
244#106—42no—
245#106C4⋊C411yescharacteristic
246#106C4⋊C411yescharacteristic
247C4⋊C8—21no—
248C4⋊C8—21no—
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—41no—
262C4⋊C8—41no—
263C4⋊C8C22×C411yescharacteristic
264C4⋊C8C22×C411yescharacteristic
265C4⋊C8C22×C411yescharacteristic
266C4⋊C8C22×C411yescharacteristic
267C4⋊C8C2×D411yescharacteristic
268C4⋊C8C2×D411yescharacteristic
269C4⋊C8C2×D411yescharacteristic
270C4⋊C8C2×D411yescharacteristic
271#104—42no—
272#104—81no—
273#104—81no—
274#104—81no—
275#103C4⋊C411yescharacteristic
276#103C4⋊C411yescharacteristic
277#103C4⋊C411yescharacteristic
278#103C4⋊C411yescharacteristic
279#102C4⋊C411yescharacteristic
280#102C4⋊C411yescharacteristic
281#102C4⋊C411yescharacteristic
282#102C4⋊C411yescharacteristic
283C22⋊C8—21no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—41no—
290C22⋊C8—41no—
291C22⋊C8—41no—
292C22⋊C8—41no—
293C22⋊C8—41no—
294C22⋊C8—41no—
295#97—21no—
296#97—21no—
297#97—41no—
298#97—41no—
299#97—41no—
300#97—42no—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—21no—
306C4×C8—41no—
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—41no—
353M16—41no—
354M16—41no—
355M16—41no—
356M16—41no—
357M16—41no—
358M16—41no—
359M16—41no—
360M16—81no—
361M16—82no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
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—41no—
377C2×C8—41no—
378C2×C8—42no—
379C2×C8—42no—
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—81no—
389C2×C8—82no—
390C2×C8C2×C4⋊C411yescharacteristic
391C2×C8C2×C4⋊C411yescharacteristic
392C2×C8C2×C4⋊C411yescharacteristic
393C2×C8C2×C4⋊C411yescharacteristic
394C2×C8C4×D411yescharacteristic
395C2×C8C4×D411yescharacteristic
396C2×C8C4×D411yescharacteristic
397C2×C8C4×D411yescharacteristic
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4C2×C4⋊C411yescharacteristic
407C4⋊C4C2×C4⋊C411yescharacteristic
408C4⋊C4C2×C4⋊C411yescharacteristic
409C4⋊C4C2×C4⋊C411yescharacteristic
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—21no—
414C22⋊C4—21no—
415C22⋊C4—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42C23×C411yescharacteristic
427C42C22×D411yescharacteristic
428C42C22×Q811yescharacteristic
429C42H(5,𝔽2)11yescharacteristic
430C23—21no—
431C23—21no—
432C23#54711yescharacteristic
433Q8—21no—
434Q8—21no—
435Q8—21no—
436Q8—21no—
437D4—21no—
438D4—21no—
439D4—21no—
440D4—21no—
441D4—41no—
442D4—41no—
443D4—41no—
444D4—41no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4C22×C4⋊C411yescharacteristic
471C2×C4C2×C4×D411yescharacteristic
472C2×C4#52011yescharacteristic
473C2×C4D4211yescharacteristic
474C2×C4D4×Q811yescharacteristic
475C2×C4#55411yescharacteristic
476C2×C4#57511yescharacteristic
477C2×C4#57511yescharacteristic
478C2×C4#57711yescharacteristic
479C2×C4#57711yescharacteristic
480C8—22no—
481C8—21no—
482C8—21no—
483C8—21no—
484C8—21no—
485C8—21no—
486C8—21no—
487C8—21no—
488C8—21no—
489C8—41no—
490C8—41no—
491C8—41no—
492C8—41no—
493C8—41no—
494C8—41no—
495C8—41no—
496C8—41no—
497C8—41no—
498C8—81no—
499C8—81no—
500C8—81no—
501C8—81no—
502C22—21no—
503C22—21no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22D4×C4⋊C411yescharacteristic
511C22#334811yescharacteristic
512C22#335011yescharacteristic
513C4—21no—
514C4—21no—
515C4—21no—
516C4—41no—
517C4—41no—
518C4—41no—
519C4—41no—
520C4—41no—
521C4—41no—
522C4—82no—
523C4—161no—
524C4#291111yescharacteristic
525C4#291211yescharacteristic
526C4#295511yescharacteristic
527C4#295611yescharacteristic
528C4#326711yescharacteristic
529C4#327111yescharacteristic
530C2—21no—
531C2—21no—
532C2—81no—
533C2—81no—
534C2#2266211yescharacteristic
535C2#2268711yescharacteristic
536C2#2269411yescharacteristic
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. #17832C2semidirect
5. #17829C2semidirect
6. #9632C2semidirect
7. #9628C2semidirect
8. #8905C2semidirect
9. #8904C2semidirect
10. #8903C2semidirect
11. #8903C2semidirect
12. #8828C2semidirect
13. #8669C2semidirect
14. #8666C2semidirect
15. #8289C2semidirect
40. #1588C4semidirect
41. #1588C4semidirect
42. #1587C4semidirect
43. #1587C4semidirect
44. #1586C4semidirect
45. #1586C4semidirect
46. #1585C4semidirect
47. #1585C4semidirect

Maximal subgroups

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

subgroupindex
#228032
#227832
#178682
#178322
#178292
#96322
#96282
#89052
#89042
#89032
#88282
#86692
#86662
#82892