Finite Group Databasebeta

← #352019

Subgroups — #352019

Order 512.

921 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33210C211yesmaximal, characteristic, complemented
2#33207C211yesmaximal, characteristic, complemented
3#33174C211yesmaximal, characteristic
4#33170C211yesmaximal, characteristic
5#29877C211yesmaximal, characteristic, complemented
6#29807C211yesmaximal, characteristic
7C2×#1705C211yesmaximal, characteristic, complemented
8C2×C8×M16C211yesmaximal, characteristic, complemented
9#9842C211yesmaximal, characteristic, complemented
10#9841C211yesmaximal, characteristic, complemented
11#9829C211yesmaximal, characteristic, complemented
12#9828C211yesmaximal, characteristic, complemented
13#9820C211yesmaximal, characteristic, complemented
14#9819C211yesmaximal, characteristic, complemented
15#8814C211yesmaximal, characteristic, complemented
16C2×#498C2211yescharacteristic
17C2×#498C2211yescharacteristic
18C2×#498C2211yescharacteristic
19C2×#492C2211yescharacteristic
20C2×#492C2211yescharacteristic
21C2×#492C2211yescharacteristic
22C2×#468—22no—
23C2×#465—22no—
24C2×#431C2211yescharacteristic
25C2×C4×M16C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27#2857C2211yescharacteristic
28#2857C2211yescharacteristic
29#2857C2211yescharacteristic
30#2856C2211yescharacteristic
31#2088—22no—
32#2077—22no—
33#2055—22no—
34#2000—22no—
35#1990C2211yescharacteristic
36#1990C2211yescharacteristic
37#1990C2211yescharacteristic
38#1990C2211yescharacteristic
39#1990C2211yescharacteristic
40#1990C2211yescharacteristic
41#1973—22no—
42#1968—22no—
43#1957C2211yescharacteristic
44#1957C2211yescharacteristic
45#1957C2211yescharacteristic
46#1956C2211yescharacteristic
47#1956C2211yescharacteristic
48#1956C2211yescharacteristic
49#1943—22no—
50#1941—22no—
51#1845C2211yescharacteristic
52#1845C2211yescharacteristic
53#1844C2211yescharacteristic
54#1844C2211yescharacteristic
55#1834C2211yescharacteristic
56#1834C2211yescharacteristic
57#1809—22no—
58#1807—22no—
59#1705—24no—
60C8×M16—24no—
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63C2×C82C2211yescharacteristic
64C2×C82C2211yescharacteristic
65C22×SD16—22no—
66C22×SD16—22no—
67C22×SD16—22no—
68C22×M16D412yes—
69C23×C8D412yes—
70C2×C4⋊Q8C2311yescharacteristic
71C2×C4⋊Q8C2311yescharacteristic
72C2×C4⋊Q8C2311yescharacteristic
73C2×#127C2311yescharacteristic
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8—21no—
77C2×#121—21no—
78C2×#121—21no—
79C2×#121—21no—
80C22×C42C2311yescharacteristic
81#498—22no—
82#498—22no—
83#498—22no—
84#498—22no—
85#498—22no—
86#498—22no—
87#492—22no—
88#492—22no—
89#492—22no—
90#492—22no—
91#492—22no—
92#492—22no—
93#468—48no—
94#465—48no—
95#431—22no—
96#431—22no—
97C2×#107—22no—
98C2×#107—22no—
99C2×#106—22no—
100C2×#106—22no—
101C2×#106—22no—
102C2×#106—22no—
103C2×#106—22no—
104C2×C4⋊C8D412yes—
105C2×#103—22no—
106C2×#103—22no—
107C2×#103—22no—
108C2×#102—22no—
109C2×#102—22no—
110C2×#102—22no—
111C2×C22⋊C8D412yes—
112C4×M16—22no—
113C4×M16—22no—
114C2×#97D412yes—
115C2×C4×C8—22no—
116C2×C4×C8—22no—
117C2×C4×C8D412yes—
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124#391—21no—
125#391—21no—
126#391—21no—
127#390—21no—
128#390—21no—
129#390—21no—
130#389—21no—
131#389—21no—
132#389—21no—
133#389—21no—
134#389—21no—
135#389—21no—
136#384C2311yescharacteristic
137#384C2311yescharacteristic
138#384C2311yescharacteristic
139#384C2311yescharacteristic
140#380—21no—
141#380—21no—
142#380—21no—
143#380—21no—
144#380—21no—
145#380—21no—
146#340—22no—
147#340—22no—
148#340—22no—
149#340—22no—
150#340—22no—
151#340—22no—
152#340—22no—
153#340—22no—
154#322—22no—
155#322—22no—
156C82—22no—
157C82D414yescomplemented
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—41no—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C2×SD16—44no—
169C2×SD16—44no—
170C2×SD16—44no—
171C2×SD16—44no—
172C2×SD16—44no—
173C2×SD16—44no—
174C2×SD16—44no—
175C2×SD16—82no—
176C2×SD16—82no—
177C2×M16—24no—
178C2×M16—24no—
179C2×M16SD1618yescomplemented
180C22×C8—24no—
181C22×C8—24no—
182C22×C8—24no—
183C22×C8C2×D412yes—
184C22×C8C2×D412yes—
185C22×C8C2×D412yes—
186C22×C8C2×D412yes—
187C22×C8C2×D412yes—
188C22×C8C2×D412yes—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201#127—21no—
202#127—21no—
203#127—21no—
204#127—21no—
205C22⋊Q8—44no—
206C22⋊Q8—44no—
207C22⋊Q8—82no—
208#121—44no—
209#121—44no—
210#121—82no—
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—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—41no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2411yescharacteristic
246#107—44no—
247#107—44no—
248#106—44no—
249#106—44no—
250#106—44no—
251#106—44no—
252#106—44no—
253C4⋊C8—24no—
254#103—44no—
255#103—44no—
256#103—82no—
257#102—44no—
258#102—44no—
259#102—82no—
260C22⋊C8—24no—
261#97—24no—
262C4×C8—24no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8C2×D412yes—
270C4×C8C2×D412yes—
271C4×C8C2×D412yes—
272C4×C8#4112yes—
273C4×C8#4112yes—
274C4×C8#4112yes—
275#95—41no—
276#95—41no—
277#95—41no—
278#95—41no—
279#95—41no—
280#95—41no—
281C24—41no—
282C24#12711yescharacteristic
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—82no—
308C2×D4—82no—
309C2×D4—82no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4#12711yescharacteristic
329C22×C4#12711yescharacteristic
330C22×C4#12711yescharacteristic
331C22×C4C2×SD1614yes—
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335SD16—88no—
336SD16—88no—
337SD16—88no—
338M16—216no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—48no—
350C2×C8#12114yes—
351C2×C8#12114yes—
352C2×C8#12114yes—
353C2×C8C2×SD1614yes—
354C2×C8C2×SD1614yes—
355C2×C8#13514yes—
356C2×C8AGL(1,ℤ/8)14yes—
357C2×C8#13714yes—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—82no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C22⋊C4—82no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42C22×D411yescharacteristic
398C42C2×#4111yescharacteristic
399C42C2×#4111yescharacteristic
400C42C2×#4111yescharacteristic
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23—81no—
413C23—81no—
414C23#49212yes—
415C23C2×#12711yescharacteristic
416Q8—82no—
417Q8—82no—
418Q8—82no—
419Q8—82no—
420Q8—82no—
421Q8—82no—
422D4—82no—
423D4—82no—
424D4—82no—
425D4—82no—
426D4—82no—
427D4—82no—
428D4—164no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
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#49212yes—
465C2×C4C2×#12111yescharacteristic
466C2×C4C2×#12111yescharacteristic
467C2×C4C2×#12111yescharacteristic
468C2×C4#53211yescharacteristic
469C2×C4#53211yescharacteristic
470C2×C4#53211yescharacteristic
471C2×C4C22×SD1612yes—
472C2×C4C2×#13512yes—
473C2×C4C2×AGL(1,ℤ/8)12yes—
474C2×C4C2×#13712yes—
475C2×C4#57712yes—
476C2×C4#57712yes—
477C8—24no—
478C8—24no—
479C8—42no—
480C8—42no—
481C8#46518yescomplemented
482C8#46818yescomplemented
483C22—22no—
484C22—22no—
485C22—22no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22—161no—
491C22—161no—
492C22—161no—
493C22—161no—
494C22—161no—
495C22—161no—
496C22—161no—
497C22#285611yescharacteristic
498C22C2×#49211yescharacteristic
499C22#314711yescharacteristic
500C22#315111yescharacteristic
501C22#315211yescharacteristic
502C22#315411yescharacteristic
503C22#315411yescharacteristic
504C4—22no—
505C4—41no—
506C4—41no—
507C4—161no—
508C4—161no—
509C4—161no—
510C4—161no—
511C4—161no—
512C4—161no—
513C4—161no—
514C4C2×#46512yes—
515C4C2×#46812yes—
516C4#305712yes—
517C4#316712yes—
518C4#317412yes—
519C4#318212yes—
520C2—42no—
521C2—321no—
522C2#1962111yescharacteristic
523C2#1962211yescharacteristic
524C2#1964111yescharacteristic
525C2#1964211yescharacteristic
526C2#3317011yescharacteristic
527C2#3317411yescharacteristic
528C2#3317811yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33210C2semidirect
2. #33207C2semidirect
5. #29877C2semidirect
7. C2×#1705C2semidirect
8. C2×C8×M16C2semidirect
9. #9842C2semidirect
10. #9841C2semidirect
11. #9829C2semidirect
12. #9828C2semidirect
13. #9820C2semidirect
14. #9819C2semidirect
15. #8814C2semidirect
157. C82D4semidirect
179. C2×M16SD16semidirect
481. C8#465semidirect
482. C8#468semidirect

Maximal subgroups

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

subgroupindex
#332102
#332072
#331742
#331702
#298772
#298072
C2×#17052
C2×C8×M162
#98422
#98412
#98292
#98282
#98202
#98192
#88142