Finite Group Databasebeta

← #229196

Subgroups — #229196

Order 512.

610 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#471C211yesmaximal, characteristic
2#17896C211yesmaximal, characteristic, complemented
3#17876C211yesmaximal, characteristic, complemented
4#17875C211yesmaximal, characteristic
5#17788C211yesmaximal, characteristic, complemented
6#9655C211yesmaximal, characteristic, complemented
7#9607C211yesmaximal, characteristic, complemented
8#9552C211yesmaximal, characteristic, complemented
9#9551C211yesmaximal, characteristic, complemented
10#8954C211yesmaximal, characteristic
11#8953C211yesmaximal, characteristic, complemented
12#8838C211yesmaximal, characteristic, complemented
13#8837C211yesmaximal, characteristic, complemented
14#8293C211yesmaximal, characteristic, complemented
15#8184C211yesmaximal, characteristic, complemented
16C2×#471C411yescharacteristic
17#2988C2211yescharacteristic
18#2988C2211yescharacteristic
19#2983C2211yescharacteristic
20#2975C2211yescharacteristic
21C4×#137C2211yescharacteristic
22C4×AGL(1,ℤ/8)C2211yescharacteristic
23C2×C4×SD16C2211yescharacteristic
24C4×#131C2211yescharacteristic
25#1984C2211yescharacteristic
26#1977C411yescharacteristic
27#1954C2211yescharacteristic
28#1951C411yescharacteristic
29#1931C411yescharacteristic
30#1930C411yescharacteristic
31#1908C411yescharacteristic
32#1907C411yescharacteristic
33#1815C411yescharacteristic
34C4×#108C2211yescharacteristic
35C4×#101C2211yescharacteristic
36C4×#100C2211yescharacteristic
37#1592—21no—
38#1592—21no—
39#1590—21no—
40#1590—21no—
41#1588—21no—
42#1587—21no—
43#1586—21no—
44#1585—21no—
45#1584—21no—
46#1583—21no—
47#1582—21no—
48#1581—21no—
49#1565—21no—
50#1565—21no—
51#1565C2211yescharacteristic
52#1565C2211yescharacteristic
53#1565C2211yescharacteristic
54#1565C2211yescharacteristic
55#1564—21no—
56#1564C2211yescharacteristic
57C2×#334—21no—
58C2×#334C2211yescharacteristic
59#1479C2211yescharacteristic
60#1478C2211yescharacteristic
61#1476C2211yescharacteristic
62#1476C2211yescharacteristic
63C2×Q8⋊C8C2211yescharacteristic
64C2×D4⋊C8C2211yescharacteristic
65#1451C2211yescharacteristic
66#1451C2211yescharacteristic
67#1450C2211yescharacteristic
68C2×C82C2211yescharacteristic
69#1283C2211yescharacteristic
70#1283C2211yescharacteristic
71#1278C2211yescharacteristic
72#1278C2211yescharacteristic
73#1278C2211yescharacteristic
74#1278C2211yescharacteristic
75C2×#137C2×C411yescharacteristic
76C2×AGL(1,ℤ/8)C2×C411yescharacteristic
77C22×SD16C2×C411yescharacteristic
78C2×#131C2×C411yescharacteristic
79C2×#131C2×C411yescharacteristic
80C4×#41C2311yescharacteristic
81C2×C4×Q8C2311yescharacteristic
82C2×C4×D4C2311yescharacteristic
83#471—24no—
84#446—21no—
85#446—21no—
86#446—21no—
87#446—21no—
88C8×Q8—21no—
89C8×Q8—21no—
90#442—21no—
91#441—21no—
92#440—21no—
93#440—21no—
94C4×Q16—21no—
95C4×SD16—21no—
96C4×SD16—21no—
97C4×SD16—21no—
98C4×SD16—21no—
99C4×SD16—21no—
100C4×SD16—21no—
101C4×D8—21no—
102#436—21no—
103#436—21no—
104#436—21no—
105#436—21no—
106C8×D4—21no—
107C8×D4—21no—
108C8×D4—41no—
109C8×D4—41no—
110#432—41no—
111#431—21no—
112#431—21no—
113#431C2311yescharacteristic
114#431C2311yescharacteristic
115#431C2311yescharacteristic
116#431C2311yescharacteristic
117C2×#108C2×C411yescharacteristic
118C2×#108C2×C411yescharacteristic
119#428C2×C411yescharacteristic
120#428C2×C411yescharacteristic
121C2×#106C2×C411yescharacteristic
122C2×C4⋊C8—21no—
123C2×C4⋊C8—21no—
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126#419C2×C411yescharacteristic
127#418C2×C411yescharacteristic
128#417C2×C411yescharacteristic
129#417C2×C411yescharacteristic
130C2×#103C2×C411yescharacteristic
131C2×#102C2×C411yescharacteristic
132C2×#101C2×C411yescharacteristic
133C2×#101C2×C411yescharacteristic
134C2×#100C2×C411yescharacteristic
135C2×#100C2×C411yescharacteristic
136#407—41no—
137C2×C22⋊C8—41no—
138#405—21no—
139#405—21no—
140#405C2×C411yescharacteristic
141#405C2×C411yescharacteristic
142C4×M16—21no—
143C4×M16C2311yescharacteristic
144C4×M16C2311yescharacteristic
145C4×M16C2311yescharacteristic
146C2×C4×C8—21no—
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149C2×C4×C8C2311yescharacteristic
150#343C2×C411yescharacteristic
151#343C2×C411yescharacteristic
152#343C2×C411yescharacteristic
153#343C2×C411yescharacteristic
154#341C2×C411yescharacteristic
155#341C2×C411yescharacteristic
156#336—41no—
157#335—21no—
158#335—21no—
159#335—21no—
160#335—21no—
161#335D411yescharacteristic
162#335D411yescharacteristic
163#334—21no—
164#334—21no—
165#334—21no—
166#334—21no—
167#334—21no—
168#334—21no—
169#334D411yescharacteristic
170#334D411yescharacteristic
171Q8⋊C8—21no—
172Q8⋊C8—21no—
173Q8⋊C8—21no—
174Q8⋊C8—21no—
175Q8⋊C8—21no—
176Q8⋊C8D411yescharacteristic
177Q8⋊C8D411yescharacteristic
178D4⋊C8—21no—
179D4⋊C8—21no—
180D4⋊C8—21no—
181D4⋊C8—21no—
182D4⋊C8—21no—
183D4⋊C8D411yescharacteristic
184D4⋊C8D411yescharacteristic
185#322—21no—
186#322—21no—
187#322—21no—
188#322—21no—
189#322—21no—
190C82—21no—
191C82—21no—
192C82—21no—
193C2×#41C22×C411yescharacteristic
194C22×Q8C22×C411yescharacteristic
195C22×D4C22×C411yescharacteristic
196C23×C4—21no—
197#137—22no—
198#137—22no—
199AGL(1,ℤ/8)—22no—
200AGL(1,ℤ/8)—22no—
201C2×Q16—21no—
202C2×SD16—21no—
203C2×SD16—21no—
204C2×SD16—21no—
205C2×SD16—21no—
206C2×SD16—22no—
207C2×SD16—22no—
208C2×D8—21no—
209#131—22no—
210#131—22no—
211#131—41no—
212#131M1614yescomplemented
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—41no—
216C2×M16—41no—
217C2×M16C22×C411yescharacteristic
218C2×M16C22×C411yescharacteristic
219C2×M16C22×C411yescharacteristic
220C2×M16C22×C411yescharacteristic
221C2×M16C22×C411yescharacteristic
222C2×M16C22×C411yescharacteristic
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—41no—
230C22×C8—41no—
231C22×C8C22×C411yescharacteristic
232C22×C8C22×C411yescharacteristic
233C4×Q8—21no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×Q8—21no—
237C4×Q8C2×D411yescharacteristic
238C4×Q8#4111yescharacteristic
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4C2×D411yescharacteristic
246C4×D4#4111yescharacteristic
247#117—21no—
248#117C22×C411yescharacteristic
249C2×C4⋊C4—21no—
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C22⋊C4—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42C2411yescharacteristic
256#108—22no—
257#108—22no—
258#107—21no—
259#107—21no—
260#106—21no—
261#106—21no—
262#106—21no—
263#106—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—41no—
270C4⋊C8—41no—
271C4⋊C8—41no—
272C4⋊C8—41no—
273C4⋊C8C2×D411yescharacteristic
274C4⋊C8C2×D411yescharacteristic
275C4⋊C8C2×D411yescharacteristic
276C4⋊C8C2×D411yescharacteristic
277C4⋊C8C2×D411yescharacteristic
278C4⋊C8C2×D411yescharacteristic
279#103—21no—
280#103—21no—
281#103—21no—
282#103—21no—
283#103—21no—
284#103—21no—
285#102—21no—
286#102—21no—
287#102—21no—
288#102—21no—
289#102—21no—
290#102—21no—
291#101—22no—
292#101—41no—
293#100—22no—
294#100—41no—
295C22⋊C8—21no—
296C22⋊C8—21no—
297C22⋊C8—21no—
298C22⋊C8—21no—
299C22⋊C8—41no—
300C22⋊C8—41no—
301C22⋊C8—41no—
302C22⋊C8—41no—
303C22⋊C8—82no—
304#97—21no—
305#97—21no—
306#97—21no—
307#97—21no—
308#97—21no—
309#97—41no—
310#97—41no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—41no—
324C4×C8—41no—
325C4×C8C2×D411yescharacteristic
326C4×C8C2×D411yescharacteristic
327C4×C8C2×D411yescharacteristic
328C4×C8C2×D411yescharacteristic
329C4×C8#4111yescharacteristic
330C4×C8#4111yescharacteristic
331C24—21no—
332#41—21no—
333#41—21no—
334#41C2×M1612yes—
335#41#13112yes—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8C4×D411yescharacteristic
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4C4×D411yescharacteristic
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—81no—
360C22×C4—81no—
361C22×C4C23×C411yescharacteristic
362C22×C4H(5,𝔽2)11yescharacteristic
363C22×C42−1+411yescharacteristic
364Q16—42no—
365SD16—42no—
366SD16—42no—
367SD16—42no—
368SD16—42no—
369D8—42no—
370M16—21no—
371M16—24no—
372M16—21no—
373M16—21no—
374M16—21no—
375M16—21no—
376M16—21no—
377M16—21no—
378M16—21no—
379M16—21no—
380M16—21no—
381M16—21no—
382M16—42no—
383M16—161no—
384M16—161no—
385M16C2×M1612yes—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—24no—
391C2×C8—22no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—81no—
412C2×C8—81no—
413C2×C8—161no—
414C2×C8—161no—
415C2×C8C4×D411yescharacteristic
416C2×C8C4×D411yescharacteristic
417C2×C8C4×D411yescharacteristic
418C2×C8C4×D411yescharacteristic
419C2×C8C2×M1612yes—
420C2×C8#13112yes—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4C4×D411yescharacteristic
430C4⋊C4C4×D411yescharacteristic
431C22⋊C4—21no—
432C22⋊C4—21no—
433C22⋊C4—21no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—41no—
440C42—41no—
441C42C22×D411yescharacteristic
442C42C22×D411yescharacteristic
443C42C2×#4111yescharacteristic
444C42C2×#4111yescharacteristic
445C23—21no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—81no—
450C23—81no—
451C23#52011yescharacteristic
452Q8—21no—
453Q8—41no—
454Q8—41no—
455Q8—41no—
456Q8—41no—
457Q8#43612yes—
458D4—22no—
459D4—22no—
460D4—21no—
461D4—41no—
462D4—41no—
463D4—41no—
464D4—41no—
465D4#43612yes—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—161no—
487C2×C4C2×C4×D411yescharacteristic
488C2×C4C2×C4×D411yescharacteristic
489C2×C4D4211yescharacteristic
490C2×C4#54711yescharacteristic
491C2×C4#54811yescharacteristic
492C2×C4#55311yescharacteristic
493C2×C4C22×M1611yescharacteristic
494C2×C4C2×#13111yescharacteristic
495C2×C4C2×#13111yescharacteristic
496C2×C4#56811yescharacteristic
497C8—22no—
498C8—22no—
499C8—21no—
500C8—21no—
501C8—41no—
502C8—42no—
503C8—41no—
504C8—41no—
505C8—41no—
506C8—41no—
507C8—41no—
508C8—41no—
509C8—41no—
510C8—41no—
511C8—41no—
512C8—41no—
513C8—41no—
514C8—81no—
515C8—81no—
516C8—81no—
517C8#43512yes—
518C8#43512yes—
519C22—21no—
520C22—21no—
521C22—22no—
522C22—41no—
523C22—41no—
524C22—82no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22—161no—
529C22#236111yescharacteristic
530C22#298911yescharacteristic
531C22#299411yescharacteristic
532C4—21no—
533C4—21no—
534C4—21no—
535C4—22no—
536C4—41no—
537C4—41no—
538C4—81no—
539C4—81no—
540C4—81no—
541C4C2×#43511yescharacteristic
542C4C2×#43611yescharacteristic
543C4#293311yescharacteristic
544C4#293511yescharacteristic
545C4#329011yescharacteristic
546C4#329111yescharacteristic
547C2—21no—
548C2—21no—
549C2—42no—
550C2—161no—
551C2#2298911yescharacteristic
552C2#2304211yescharacteristic
553C2#2304411yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22#2361
prime radicalC2×AGL(1,ℤ/8)C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ 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. #17896C2semidirect
3. #17876C2semidirect
5. #17788C2semidirect
6. #9655C2semidirect
7. #9607C2semidirect
8. #9552C2semidirect
9. #9551C2semidirect
11. #8953C2semidirect
12. #8838C2semidirect
13. #8837C2semidirect
14. #8293C2semidirect
15. #8184C2semidirect
212. #131M16semidirect

Maximal subgroups

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

subgroupindex
C4×#4712
#178962
#178762
#178752
#177882
#96552
#96072
#95522
#95512
#89542
#89532
#88382
#88372
#82932
#81842