Finite Group Databasebeta

← #213330

Subgroups — #213330

Order 512.

733 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21706C211yesmaximal, characteristic, complemented
2#21660C211yesmaximal, characteristic, complemented
3#19619C211yesmaximal, characteristic
4#17976C211yesmaximal, characteristic, complemented
5#17975C211yesmaximal, characteristic, complemented
6#17957C211yesmaximal, characteristic, complemented
7#17956C211yesmaximal, characteristic, complemented
8#9860C211yesmaximal, characteristic, complemented
9#9858C211yesmaximal, characteristic, complemented
10#9851C211yesmaximal, characteristic, complemented
11#9851C211yesmaximal, characteristic, complemented
12#9849C211yesmaximal, characteristic, complemented
13#9849C211yesmaximal, characteristic, complemented
14#8828C211yesmaximal, characteristic, complemented
15#7846C211yesmaximal, characteristic, complemented
16#3169C2211yescharacteristic
17#3169C2211yescharacteristic
18#3167C2211yescharacteristic
19#3167C2211yescharacteristic
20#3147C2211yescharacteristic
21#3131C2211yescharacteristic
22#2960C2211yescharacteristic
23C2×#444—22no—
24#2931C2211yescharacteristic
25#2931C2211yescharacteristic
26#2931C2211yescharacteristic
27#1994C2211yescharacteristic
28#1975—22no—
29#1962C2211yescharacteristic
30#1962C2211yescharacteristic
31#1962C2211yescharacteristic
32#1938—22no—
33#1848C2211yescharacteristic
34#1837C2211yescharacteristic
35#1765—22no—
36C4×#97C2211yescharacteristic
37C4×#97C2211yescharacteristic
38C4×#97C2211yescharacteristic
39#1723C411yescharacteristic, complemented
40#1721C411yescharacteristic, complemented
41#1720C411yescharacteristic, complemented
42#1719C411yescharacteristic, complemented
43#1717C411yescharacteristic, complemented
44#1717C411yescharacteristic, complemented
45#1715C411yescharacteristic, complemented
46#1714C411yescharacteristic, complemented
47#1590—22no—
48#1589—22no—
49#1583—22no—
50#1582—22no—
51#1581—22no—
52#1581—22no—
53C8×SD16—22no—
54C8×D8—22no—
55#1567C2211yescharacteristic
56#1567C2211yescharacteristic
57#1566C2211yescharacteristic
58#1566C2211yescharacteristic
59#1493C2211yescharacteristic
60#1493C2211yescharacteristic
61#1492C2211yescharacteristic
62#1492C2211yescharacteristic
63#1492C2211yescharacteristic
64#1492C2211yescharacteristic
65#1492C2211yescharacteristic
66#1492C2211yescharacteristic
67C2×#322C2211yescharacteristic
68C2×#322C2211yescharacteristic
69C2×#322C2211yescharacteristic
70C2×C82C2211yescharacteristic
71C2×#135—22no—
72C2×#131—21no—
73C2×#131—21no—
74C2×#131—21no—
75#533C2311yescharacteristic
76#532C2311yescharacteristic
77#532C2311yescharacteristic
78#532C2311yescharacteristic
79#500C2×C411yescharacteristic
80#500C2×C411yescharacteristic
81#498C2×C411yescharacteristic
82#498C2×C411yescharacteristic
83#495—22no—
84#494C2×C411yescharacteristic
85#493C2×C411yescharacteristic
86#493C2×C411yescharacteristic
87#493C2×C411yescharacteristic
88#492C2×C411yescharacteristic
89#492C2×C411yescharacteristic
90#492C2×C411yescharacteristic
91#492C2×C411yescharacteristic
92#488—22no—
93#487C2×C411yescharacteristic
94#487C2×C411yescharacteristic
95#487C2×C411yescharacteristic
96#487C2×C411yescharacteristic
97#486C2×C411yescharacteristic
98#486C2×C411yescharacteristic
99#486C2×C411yescharacteristic
100#486C2×C411yescharacteristic
101#486C2×C411yescharacteristic
102#486C2×C411yescharacteristic
103#486C2×C411yescharacteristic
104#486C2×C411yescharacteristic
105#466—22no—
106#466—22no—
107#465—22no—
108#465—22no—
109#446—21no—
110#446—21no—
111C8×Q8—21no—
112C8×Q8—21no—
113#444—48no—
114C4×SD16—22no—
115C4×SD16—22no—
116C4×D8—22no—
117C4×D8—22no—
118#436—21no—
119#436—21no—
120#436—21no—
121#436—21no—
122#436—21no—
123#436—21no—
124C8×D4—21no—
125C8×D4—21no—
126C8×D4—21no—
127C8×D4—21no—
128C8×D4—21no—
129C8×D4—21no—
130C2×#108—22no—
131#424—21no—
132#423C2311yescharacteristic
133#423C2311yescharacteristic
134#423C2311yescharacteristic
135#423C2311yescharacteristic
136C2×#104—22no—
137C2×#104—22no—
138C2×#104—22no—
139#416—22no—
140#408—21no—
141#408—21no—
142#408—21no—
143#405—21no—
144C2×#97—22no—
145C2×#97—22no—
146C2×#97D412yes—
147C2×#97D412yes—
148C2×C4×C8C2311yescharacteristic
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C43C2311yescharacteristic
155#341—22no—
156#339—22no—
157#335—22no—
158#335—22no—
159#334—22no—
160#334—22no—
161Q8⋊C8—22no—
162Q8⋊C8—22no—
163D4⋊C8—22no—
164D4⋊C8—22no—
165D4⋊C8—22no—
166D4⋊C8—22no—
167D4⋊C8—22no—
168D4⋊C8—22no—
169#322—22no—
170#322C2×C411yescharacteristic
171#322C2×C411yescharacteristic
172#322C2×C411yescharacteristic
173#322C2×C411yescharacteristic
174#322D412yes—
175#322D412yes—
176C82D412yes—
177C82D412yes—
178C2×#41—21no—
179C2×#41—21no—
180C2×#41—21no—
181#135—44no—
182#135—44no—
183C2×Q16—22no—
184C2×SD16—22no—
185C2×SD16—22no—
186C2×SD16—22no—
187C2×SD16—22no—
188C2×D8—22no—
189C2×D8—22no—
190C2×D8—22no—
191#131—44no—
192#131—44no—
193#131—44no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—41no—
199C2×M16—41no—
200C2×M16—41no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—41no—
204C22×C8—41no—
205C22×C8—41no—
206C22×C8C2×D412yes—
207C22×C8C2×D411yescharacteristic
208C22×C8C2×D411yescharacteristic
209C22×C8#4112yes—
210C22×C8#4112yes—
211C4⋊Q8C22×C411yescharacteristic
212C4⋊Q8C22×C411yescharacteristic
213C4⋊Q8C22×C411yescharacteristic
214C4⋊Q8C22×C411yescharacteristic
215C4⋊Q8C22×C411yescharacteristic
216#127C22×C411yescharacteristic
217#127C22×C411yescharacteristic
218#127C22×C411yescharacteristic
219#125—21no—
220#124—21no—
221#124—21no—
222#124—21no—
223C22⋊Q8—21no—
224C22⋊Q8—21no—
225#121—21no—
226#121—21no—
227#121—21no—
228#121—21no—
229#121—21no—
230#121—21no—
231C4×Q8—21no—
232C4×Q8—21no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—21no—
239#117—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42C2×D412yes—
243C2×C42C2411yescharacteristic
244#108—44no—
245#107—22no—
246#107—22no—
247#106—22no—
248#106—22no—
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—41no—
258C4⋊C8—41no—
259#104—44no—
260#104—44no—
261#104—44no—
262#103—22no—
263#103—22no—
264#103—22no—
265#103—22no—
266#102—22no—
267#102—22no—
268#102—22no—
269#102—22no—
270#102—22no—
271#102—22no—
272#102—22no—
273#102—22no—
274C22⋊C8—41no—
275C22⋊C8—41no—
276C22⋊C8—41no—
277C22⋊C8—41no—
278C22⋊C8—41no—
279C22⋊C8—41no—
280#97—24no—
281#97—42no—
282#97—44no—
283#97—44no—
284#97—41no—
285C4×C8—21no—
286C4×C8—21no—
287C4×C8—41no—
288C4×C8C22×C411yescharacteristic
289C4×C8C22×C411yescharacteristic
290C4×C8C22×C411yescharacteristic
291C4×C8C22×C411yescharacteristic
292C4×C8C22×C411yescharacteristic
293C4×C8C22×C411yescharacteristic
294C4×C8C2×D412yes—
295C4×C8C2×D412yes—
296C4×C8C2×D412yes—
297C4×C8C2×D412yes—
298C4×C8C2×D411yescharacteristic
299C4×C8C2×D411yescharacteristic
300C4×C8C2×D411yescharacteristic
301C4×C8C2×D411yescharacteristic
302C4×C8#4112yes—
303#41—42no—
304#41—42no—
305#41—42no—
306#41—42no—
307#41—42no—
308#41—42no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C22×C4—22no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4C22×D411yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4111yescharacteristic
333Q16—44no—
334SD16—44no—
335SD16—44no—
336SD16—44no—
337SD16—82no—
338D8—44no—
339D8—44no—
340D8—82no—
341M16—42no—
342M16—42no—
343M16—42no—
344M16—42no—
345M16—82no—
346M16—82no—
347M16—82no—
348C2×C8—24no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—24no—
354C2×C8—22no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—82no—
362C2×C8—82no—
363C2×C8—82no—
364C2×C8C4×D412yes—
365C2×C8C4×D412yes—
366C2×C8C4×D412yes—
367C2×C8C4×D412yes—
368C2×C8C4×D412yes—
369C2×C8C4×D412yes—
370C2×C8#12711yescharacteristic
371C2×C8#12711yescharacteristic
372C2×C8#12711yescharacteristic
373C2×C8#12711yescharacteristic
374C2×C8AGL(1,ℤ/8)14yes—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C42—22no—
395C42—22no—
396C42—42no—
397C42—42no—
398C42—41no—
399C42C23×C411yescharacteristic
400C42C22×D411yescharacteristic
401C42C22×D411yescharacteristic
402C42C2×#4111yescharacteristic
403C23—41no—
404C23—41no—
405C23—41no—
406C23#53211yescharacteristic
407Q8—42no—
408Q8—42no—
409Q8—42no—
410Q8—42no—
411Q8—42no—
412D4—42no—
413D4—42no—
414D4—42no—
415D4—42no—
416D4—42no—
417D4—42no—
418D4—42no—
419D4—42no—
420D4—42no—
421D4—82no—
422D4—82no—
423D4—82no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4C2×C4×D411yescharacteristic
441C2×C4C2×C4×D411yescharacteristic
442C2×C4C2×C4×D411yescharacteristic
443C2×C4C2×#12711yescharacteristic
444C2×C4#53211yescharacteristic
445C2×C4#53211yescharacteristic
446C2×C4C2×AGL(1,ℤ/8)12yes—
447C2×C4#57512yes—
448C8—24no—
449C8—21no—
450C8—21no—
451C8—22no—
452C8—22no—
453C8—22no—
454C8—22no—
455C8—22no—
456C8—42no—
457C8—81no—
458C8—81no—
459C8—81no—
460C8—81no—
461C8#44414yescomplemented
462C8#49612yes—
463C8#49612yes—
464C22—21no—
465C22—21no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22C4×#12711yescharacteristic
476C22#317112yes—
477C4—21no—
478C4—21no—
479C4—82no—
480C4—81no—
481C4—81no—
482C4—81no—
483C4—81no—
484C4—81no—
485C4C4×AGL(1,ℤ/8)12yes—
486C4C2×#44412yes—
487C4#295612yes—
488C4C2×#49611yescharacteristic
489C4#315011yescharacteristic
490C2—22no—
491C2—161no—
492C2—161no—
493C2—161no—
494C2C4×#49611yescharacteristic
495C2#2170611yescharacteristic
496C2#2171011yescharacteristic
subgroupG/H
centerC2×C4C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#127
prime radical#19619C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21706C2semidirect
2. #21660C2semidirect
4. #17976C2semidirect
5. #17975C2semidirect
6. #17957C2semidirect
7. #17956C2semidirect
8. #9860C2semidirect
9. #9858C2semidirect
10. #9851C2semidirect
11. #9851C2semidirect
12. #9849C2semidirect
13. #9849C2semidirect
14. #8828C2semidirect
15. #7846C2semidirect
39. #1723C4semidirect
40. #1721C4semidirect
41. #1720C4semidirect
42. #1719C4semidirect
43. #1717C4semidirect
44. #1717C4semidirect
45. #1715C4semidirect
46. #1714C4semidirect
461. C8#444semidirect

Maximal subgroups

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

subgroupindex
#217062
#216602
#196192
#179762
#179752
#179572
#179562
#98602
#98582
#98512
#98492
#88282
#78462