Finite Group Databasebeta

← #226169

Subgroups — #226169

Order 512.

557 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#22802C211yesmaximal, characteristic
2#22783C211yesmaximal, characteristic
3#17863C211yesmaximal, characteristic
4#17832C211yesmaximal, characteristic, complemented
5#17829C211yesmaximal, characteristic, complemented
6#9629C211yesmaximal, 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#8285C211yesmaximal, characteristic, complemented
16#2965C2211yescharacteristic
17C8×#41C2211yescharacteristic
18C4×#135C2211yescharacteristic
19#2925C2211yescharacteristic
20#2925C2211yescharacteristic
21#2925C2211yescharacteristic
22#2906C2211yescharacteristic
23#1974—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#1772—21no—
37C8×C4⋊C4C2211yescharacteristic
38C8×C4⋊C4C2211yescharacteristic
39#1764—21no—
40#1584C411yescharacteristic, complemented
41#1584C411yescharacteristic, complemented
42#1583C411yescharacteristic, complemented
43#1583C411yescharacteristic, complemented
44#1582C411yescharacteristic, complemented
45#1582C411yescharacteristic, complemented
46#1581C411yescharacteristic, complemented
47#1581C411yescharacteristic, 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×C4×C8—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—24no—
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—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—41no—
306C4×C8—41no—
307C4×C8—41no—
308C4×C8—42no—
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—22no—
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—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—42no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—81no—
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—22no—
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—42no—
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—21no—
481C8—21no—
482C8—21no—
483C8—21no—
484C8—21no—
485C8—21no—
486C8—21no—
487C8—41no—
488C8—41no—
489C8—41no—
490C8—41no—
491C8—41no—
492C8—41no—
493C8—41no—
494C8—41no—
495C8—41no—
496C8—41no—
497C8—81no—
498C8—81no—
499C8—81no—
500C8—81no—
501C8#44312yes—
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—82no—
524C4#291111yescharacteristic
525C4#291211yescharacteristic
526C4C2×#44311yescharacteristic
527C4#295711yescharacteristic
528C4#326711yescharacteristic
529C4#327111yescharacteristic
530C2—21no—
531C2—21no—
532C2—81no—
533C2—81no—
534C2#2266211yescharacteristic
535C2#2268811yescharacteristic
536C2#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. #17832C2semidirect
5. #17829C2semidirect
6. #9629C2semidirect
7. #9628C2semidirect
8. #8905C2semidirect
9. #8904C2semidirect
10. #8903C2semidirect
11. #8903C2semidirect
12. #8828C2semidirect
13. #8669C2semidirect
14. #8666C2semidirect
15. #8285C2semidirect
40. #1584C4semidirect
41. #1584C4semidirect
42. #1583C4semidirect
43. #1583C4semidirect
44. #1582C4semidirect
45. #1582C4semidirect
46. #1581C4semidirect
47. #1581C4semidirect

Maximal subgroups

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

subgroupindex
#228022
#227832
#178632
#178322
#178292
#96292
#96282
#89052
#89042
#89032
#88282
#86692
#86662
#82852