Finite Group Databasebeta

← #213200

Subgroups — #213200

Order 512.

1038 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22269C211yesmaximal, characteristic, complemented
2#22219C211yesmaximal, characteristic, complemented
3#21690C211yesmaximal, characteristic, complemented
4C4×#493C211yesmaximal, characteristic, complemented
5C4×#492C211yesmaximal, characteristic, complemented
6#21646C211yesmaximal, characteristic, complemented
7C2×#1721C211yesmaximal, characteristic
8#9842C211yesmaximal, characteristic, complemented
9#9838C211yesmaximal, characteristic, complemented
10#9837C211yesmaximal, characteristic, complemented
11#9837C211yesmaximal, characteristic, complemented
12#9832C211yesmaximal, characteristic, complemented
13#9827C211yesmaximal, characteristic, complemented
14#8819C211yesmaximal, characteristic, complemented
15#7842C211yesmaximal, characteristic, complemented
16C2×#493C2211yescharacteristic
17C2×#493C2211yescharacteristic
18C2×#493C2211yescharacteristic
19C2×#492C2211yescharacteristic
20C2×#486C2211yescharacteristic
21C2×#486C2211yescharacteristic
22C2×#442—22no—
23C2×C4×SD16—22no—
24C2×C4×D8—22no—
25C4×C4⋊Q8C2211yescharacteristic
26C4×#127C2211yescharacteristic
27C4×#127C2211yescharacteristic
28C4×#127C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1972—22no—
32#1970—22no—
33#1968—22no—
34#1957C2211yescharacteristic
35#1957C2211yescharacteristic
36#1957C2211yescharacteristic
37#1957C2211yescharacteristic
38#1957C2211yescharacteristic
39#1957C2211yescharacteristic
40#1957C2211yescharacteristic
41#1957C2211yescharacteristic
42#1957C2211yescharacteristic
43#1956C2211yescharacteristic
44#1936—22no—
45#1926—22no—
46#1879—22no—
47#1879—22no—
48#1864—22no—
49#1845C2211yescharacteristic
50#1844C2211yescharacteristic
51#1844C2211yescharacteristic
52#1844C2211yescharacteristic
53#1834C2211yescharacteristic
54#1834C2211yescharacteristic
55#1762—22no—
56#1744C2211yescharacteristic
57C4×#97C2211yescharacteristic
58C42×C8C2211yescharacteristic
59#1721C418yescomplemented
60C2×#322C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63C2×#322C2211yescharacteristic
64C22×SD16—22no—
65C22×D8—22no—
66C22×D8—22no—
67C22×D8—22no—
68C2×C4⋊Q8C2311yescharacteristic
69C2×#127C2311yescharacteristic
70C2×#127C2311yescharacteristic
71C2×#127C2311yescharacteristic
72C2×C4×Q8—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×C4×D4—21no—
80C2×C4×D4—21no—
81C2×C4×D4—21no—
82#493—22no—
83#493—22no—
84#493—22no—
85#493C2×C414yescomplemented
86#493C2×C414yescomplemented
87#493C2×C414yescomplemented
88#492—22no—
89#492C2×C414yescomplemented
90#486—22no—
91#486—22no—
92#486C2×C414yescomplemented
93#486C2×C414yescomplemented
94#442—48no—
95C4×SD16—48no—
96C4×D8—48no—
97C2×#107—22no—
98C2×#106—22no—
99C2×#106—22no—
100C2×#106—22no—
101C2×#103—22no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—22no—
105C2×#102—22no—
106C2×#102—22no—
107C2×#102—22no—
108C2×#102—22no—
109C2×#102—22no—
110C2×#102—22no—
111C2×#102—22no—
112C2×#102—22no—
113C2×#97D412yes—
114C2×#97D412yes—
115C2×C4×C8D412yes—
116C2×C4×C8D412yes—
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123#391—21no—
124#390—21no—
125#390—21no—
126#390—21no—
127#390—21no—
128#390—21no—
129#390—21no—
130#390—21no—
131#390—21no—
132#390—21no—
133#389—21no—
134#389—21no—
135#384C2311yescharacteristic
136#384C2311yescharacteristic
137#384C2311yescharacteristic
138#384C2311yescharacteristic
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C43C2311yescharacteristic
142#340—22no—
143#340—22no—
144#340—22no—
145#340—22no—
146#336D412yes—
147#336D412yes—
148#322—22no—
149#322—22no—
150#322—22no—
151#322C2×C414yescomplemented
152C22×Q8—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C23×C4—41no—
163C23×C4—41no—
164C23×C4—41no—
165C2×SD16—28no—
166C2×SD16—44no—
167C2×SD16—44no—
168C2×D8—28no—
169C2×D8—28no—
170C2×D8—28no—
171C2×D8—44no—
172C2×D8—44no—
173C2×D8—44no—
174C2×D8—44no—
175C2×D8—82no—
176C2×D8—82no—
177C22×C8C2×D412yes—
178C22×C8C2×D412yes—
179C22×C8C2×D412yes—
180C22×C8C2×D412yes—
181C22×C8C2×D412yes—
182C22×C8C2×D412yes—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8C22×C412yes—
187#127—21no—
188#127—21no—
189#127—21no—
190#127—21no—
191#127—21no—
192#127—21no—
193#127—21no—
194#127—21no—
195#127—21no—
196#127C22×C412yes—
197#127C22×C412yes—
198#127C22×C412yes—
199C4×Q8—44no—
200C4×D4—44no—
201C4×D4—44no—
202C4×D4—44no—
203C4×D4—44no—
204C4×D4—44no—
205C4×D4—44no—
206C4×D4—82no—
207C4×D4—82no—
208C4×D4—82no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C42—41no—
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2411yescharacteristic
241#107—44no—
242#106—44no—
243#106—44no—
244#106—82no—
245#103—44no—
246#102—28no—
247#102—28no—
248#102—44no—
249#102—44no—
250#102—44no—
251#102—44no—
252#102—44no—
253#102—44no—
254#102—82no—
255#102—82no—
256#102—82no—
257#97—24no—
258#97—42no—
259C4×C8—24no—
260C4×C8—24no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8C22×C412yes—
268C4×C8C22×C412yes—
269C4×C8C22×C412yes—
270C4×C8#4112yes—
271C4×C8#4112yes—
272C4×C8#4112yes—
273#95—41no—
274#95—41no—
275C24—41no—
276C24—41no—
277C24—41no—
278C2×Q8—22no—
279C2×Q8—41no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4#12711yescharacteristic
333C22×C4#12711yescharacteristic
334C22×C4#12711yescharacteristic
335C22×C4#12711yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339SD16—416no—
340D8—416no—
341D8—416no—
342D8—88no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8C4×D414yes—
357C2×C8C4×D414yes—
358C2×C8C4×D414yes—
359C2×C8AGL(1,ℤ/8)14yes—
360C2×C8AGL(1,ℤ/8)14yes—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—82no—
378C22⋊C4—82no—
379C22⋊C4—82no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C22⋊C4—82no—
383C22⋊C4—82no—
384C22⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—82no—
393C42AGL(1,ℤ/8)14yes—
394C42C23×C411yescharacteristic
395C42C2×#4111yescharacteristic
396C42C2×#4111yescharacteristic
397C42C2×#4111yescharacteristic
398C23—42no—
399C23—42no—
400C23—42no—
401C23—81no—
402C23—81no—
403C23—81no—
404C23—81no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23—81no—
413C23—81no—
414C23—81no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23C2×#12711yescharacteristic
420Q8—44no—
421Q8—82no—
422D4—44no—
423D4—44no—
424D4—44no—
425D4—44no—
426D4—44no—
427D4—44no—
428D4—44no—
429D4—44no—
430D4—44no—
431D4—82no—
432D4—82no—
433D4—82no—
434D4—82no—
435D4—82no—
436D4—82no—
437D4—82no—
438D4—82no—
439D4—82no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—42no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—164no—
466C2×C4—164no—
467C2×C4—164no—
468C2×C4#49612yes—
469C2×C4#49612yes—
470C2×C4C2×C4×D411yescharacteristic
471C2×C4C2×C4×D411yescharacteristic
472C2×C4C2×C4×D411yescharacteristic
473C2×C4#53211yescharacteristic
474C2×C4#53211yescharacteristic
475C2×C4#53211yescharacteristic
476C2×C4C2×AGL(1,ℤ/8)12yes—
477C2×C4C2×AGL(1,ℤ/8)12yes—
478C2×C4C2×AGL(1,ℤ/8)12yes—
479C2×C4#57512yes—
480C2×C4#57612yes—
481C2×C4#57712yes—
482C8—24no—
483C8—24no—
484C8—42no—
485C8—42no—
486C8—42no—
487C8#44218yescomplemented
488C22—82no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22—82no—
496C22—82no—
497C22—161no—
498C22—161no—
499C22—161no—
500C22—161no—
501C22—161no—
502C22—161no—
503C22—161no—
504C22—161no—
505C22—161no—
506C22—161no—
507C22—161no—
508C22—161no—
509C22C4×#12711yescharacteristic
510C22C2×#49611yescharacteristic
511C22#315011yescharacteristic
512C22#315111yescharacteristic
513C22#315111yescharacteristic
514C22#315411yescharacteristic
515C22#315511yescharacteristic
516C4—22no—
517C4—22no—
518C4—41no—
519C4—41no—
520C4—82no—
521C4—161no—
522C4—161no—
523C4—161no—
524C4—161no—
525C4C2×#44212yes—
526C4C4×AGL(1,ℤ/8)12yes—
527C4#294612yes—
528C4#317112yes—
529C4#317912yes—
530C4#318412yes—
531C2—162no—
532C2—162no—
533C2—162no—
534C2#1967111yescharacteristic
535C2#1967611yescharacteristic
536C2#1968911yescharacteristic
537C2#1969411yescharacteristic
538C2#2169011yescharacteristic
539C2C4×#49611yescharacteristic
540C2#2170411yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#1721C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22269C2semidirect
2. #22219C2semidirect
3. #21690C2semidirect
4. C4×#493C2semidirect
5. C4×#492C2semidirect
6. #21646C2semidirect
8. #9842C2semidirect
9. #9838C2semidirect
10. #9837C2semidirect
11. #9837C2semidirect
12. #9832C2semidirect
13. #9827C2semidirect
14. #8819C2semidirect
15. #7842C2semidirect
59. #1721C4semidirect
85. #493C2×C4semidirect
86. #493C2×C4semidirect
87. #493C2×C4semidirect
89. #492C2×C4semidirect
92. #486C2×C4semidirect
93. #486C2×C4semidirect
151. #322C2×C4semidirect
487. C8#442semidirect

Maximal subgroups

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

subgroupindex
#222692
#222192
#216902
C4×#4932
C4×#4922
#216462
C2×#17212
#98422
#98382
#98372
#98322
#98272
#88192
#78422