Finite Group Databasebeta

← #211702

Subgroups — #211702

Order 512.

651 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21658C211yesmaximal, characteristic, complemented
2#20485C211yesmaximal, characteristic, complemented
3#19110C211yesmaximal, characteristic
4#17930C211yesmaximal, characteristic, complemented
5#17928C211yesmaximal, characteristic
6#9751C211yesmaximal, characteristic, complemented
7#9629C211yesmaximal, characteristic, complemented
8#9584C211yesmaximal, characteristic, complemented
9#9582C211yesmaximal, characteristic, complemented
10#9237C211yesmaximal, characteristic
11#9235C211yesmaximal, characteristic, complemented
12#8660C211yesmaximal, characteristic, complemented
13#8266C211yesmaximal, characteristic, complemented
14#8256C211yesmaximal, characteristic, complemented
15#8255C211yesmaximal, characteristic, complemented
16#3131C2211yescharacteristic
17#3101C2211yescharacteristic
18#3100C2211yescharacteristic
19#2960C2211yescharacteristic
20#2931C2211yescharacteristic
21C2×#436C2211yescharacteristic
22C2×C8×D4C2211yescharacteristic
23C2×#405—21no—
24#2293C2211yescharacteristic
25#1994C2211yescharacteristic
26#1962C2211yescharacteristic
27#1937—21no—
28#1937—21no—
29#1837C2211yescharacteristic
30#1770C2211yescharacteristic
31C8×C4⋊C4C2211yescharacteristic
32#1767C2211yescharacteristic
33#1764—21no—
34#1764—21no—
35#1753—21no—
36C8×C22⋊C4—21no—
37#1750—21no—
38C42×C8C2211yescharacteristic
39#1683C412yescomplemented
40#1681C412yescomplemented
41#1679C412yescomplemented
42#1677C412yescomplemented
43#1590—22no—
44#1589—22no—
45#1583—22no—
46#1581—22no—
47#1571C2211yescharacteristic
48#1570C2211yescharacteristic
49#1564C2211yescharacteristic
50#1564C2211yescharacteristic
51#1542C2211yescharacteristic
52#1540C2211yescharacteristic
53#1503C2211yescharacteristic
54#1502C2211yescharacteristic
55#1486C2211yescharacteristic
56#1486C2211yescharacteristic
57#1481C2211yescharacteristic
58#1480C2211yescharacteristic
59#1450C2211yescharacteristic
60#1450C2211yescharacteristic
61#1303C2211yescharacteristic
62#1303C2211yescharacteristic
63#1301C2211yescharacteristic
64#1301C2211yescharacteristic
65#1276C2211yescharacteristic
66#1276C2211yescharacteristic
67C2×#131—21no—
68C22×M16—21no—
69C23×C8—21no—
70#533C2311yescharacteristic
71#532C2311yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C2×#117—21no—
74#488—21no—
75#488—21no—
76#487C2×C411yescharacteristic
77#487C2×C411yescharacteristic
78#486C2×C411yescharacteristic
79#486C2×C411yescharacteristic
80#483—21no—
81#480—21no—
82#476C2×C412yes—
83#474C2×C412yes—
84#461C2×C412yes—
85#459C2×C412yes—
86#450—21no—
87#447—21no—
88#446—21no—
89#446—21no—
90C8×Q8—21no—
91C8×Q8—21no—
92C4×SD16—22no—
93C4×D8—22no—
94#436—21no—
95#436—21no—
96#436—21no—
97#436—21no—
98#436D412yes—
99#436D412yes—
100C8×D4—21no—
101C8×D4—21no—
102C8×D4—21no—
103C8×D4—21no—
104C8×D4C2×C412yes—
105C8×D4C2×C412yes—
106#431C2311yescharacteristic
107#431C2311yescharacteristic
108#431C2311yescharacteristic
109#431C2311yescharacteristic
110#424—21no—
111#423C2311yescharacteristic
112#423C2311yescharacteristic
113C2×C4⋊C8C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115C2×#104—21no—
116C2×#104—21no—
117#416—21no—
118#416—21no—
119#408—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122#405—21no—
123#405—44no—
124C2×#97—21no—
125C2×#97—21no—
126C2×C4×C8—21no—
127C2×C4×C8D411yescharacteristic
128C2×C4×C8D411yescharacteristic
129C2×C4×C8D411yescharacteristic
130C2×C4×C8D411yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133#390—21no—
134#389—21no—
135#388—22no—
136C4×C4⋊C4C2311yescharacteristic
137C4×C22⋊C4—21no—
138C43C2311yescharacteristic
139#341—21no—
140#341—21no—
141#339—21no—
142#339—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#335—22no—
147#335—21no—
148#335C2×C412yes—
149#334—22no—
150#334—21no—
151#334C2×C412yes—
152#333C2×C412yes—
153#332C2×C412yes—
154#332C2×C412yes—
155#331C2×C412yes—
156#328—22no—
157#327—22no—
158Q8⋊C8—22no—
159D4⋊C8—22no—
160D4⋊C8—22no—
161D4⋊C8—22no—
162#322—21no—
163#322—21no—
164C82—21no—
165C82—21no—
166C2×#41—21no—
167C22×D4#4111yescharacteristic
168C23×C4—21no—
169C2×SD16—22no—
170C2×D8—22no—
171#131—42no—
172#131—42no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—41no—
178C2×M16—42no—
179C2×M16—42no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—42no—
185C22×C8—42no—
186C22×C8—41no—
187C22×C8C2×D411yescharacteristic
188C22×C8C2×D411yescharacteristic
189C22×C8C2×D411yescharacteristic
190C22×C8C2×D411yescharacteristic
191C4⋊Q8C22×C411yescharacteristic
192C4⋊Q8C22×C411yescharacteristic
193C4⋊Q8C22×C411yescharacteristic
194#127C22×C411yescharacteristic
195#125—21no—
196#124—21no—
197C22⋊Q8—21no—
198C22⋊Q8—21no—
199#121—21no—
200#121—21no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4C22×C412yes—
208C4×D4C2×D412yes—
209#117—21no—
210#117—42no—
211#117—42no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4#4111yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2411yescharacteristic
224#107—22no—
225#106—22no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—41no—
235C4⋊C8—41no—
236C4⋊C8C22×C412yes—
237C4⋊C8C22×C412yes—
238C4⋊C8C22×C412yes—
239C4⋊C8C2×D412yes—
240#104—42no—
241#104—42no—
242#103—21no—
243#103—21no—
244#103—21no—
245#103—21no—
246#103—22no—
247#102—22no—
248#102—22no—
249#102—21no—
250#102—21no—
251#102—21no—
252#102—21no—
253#102—22no—
254C22⋊C8—21no—
255C22⋊C8—21no—
256C22⋊C8—22no—
257C22⋊C8—22no—
258C22⋊C8—22no—
259C22⋊C8—22no—
260C22⋊C8—21no—
261C22⋊C8—21no—
262C22⋊C8—41no—
263C22⋊C8—41no—
264#97—42no—
265#97—41no—
266#97—42no—
267C4×C8—22no—
268C4×C8—22no—
269C4×C8—22no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—41no—
275C4×C8—42no—
276C4×C8—41no—
277C4×C8C22×C411yescharacteristic
278C4×C8C22×C411yescharacteristic
279C4×C8C2×D411yescharacteristic
280C4×C8C2×D411yescharacteristic
281C4×C8#4111yescharacteristic
282C4×C8#4111yescharacteristic
283C4×C8#4111yescharacteristic
284C4×C8#4111yescharacteristic
285#95—22no—
286#95#4112yes—
287C24—21no—
288#41—41no—
289#41—41no—
290#41—41no—
291#41—41no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—42no—
300C2×D4—41no—
301C2×D4C4×D412yes—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4C22×D411yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C22×C4C2×#4111yescharacteristic
316SD16—82no—
317D8—82no—
318M16—44no—
319M16—44no—
320M16—41no—
321M16—41no—
322M16—41no—
323M16—41no—
324M16—82no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—44no—
336C2×C8—44no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—82no—
346C2×C8C4×D411yescharacteristic
347C2×C8C4×D411yescharacteristic
348C2×C8C4×D411yescharacteristic
349C2×C8C4×D411yescharacteristic
350C2×C8#12111yescharacteristic
351C2×C8#12111yescharacteristic
352C2×C8#12111yescharacteristic
353C2×C8#12111yescharacteristic
354C2×C8#13512yes—
355C2×C8AGL(1,ℤ/8)12yes—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—42no—
364C4⋊C4—44no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4C4×D412yes—
368C22⋊C4—22no—
369C22⋊C4—22no—
370C22⋊C4—44no—
371C22⋊C4—44no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—42no—
379C42—42no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42#13512yes—
384C42C23×C411yescharacteristic
385C42C22×D411yescharacteristic
386C42C2×#4111yescharacteristic
387C42C2×#4111yescharacteristic
388C23—22no—
389C23—42no—
390C23—42no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23#52511yescharacteristic
395Q8—41no—
396Q8—41no—
397Q8—42no—
398Q8—81no—
399D4—22no—
400D4—22no—
401D4—42no—
402D4—41no—
403D4—41no—
404D4—42no—
405D4—81no—
406D4—81no—
407D4—81no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4C2×C4×D411yescharacteristic
432C2×C4C2×C4×D411yescharacteristic
433C2×C4C4×#4111yescharacteristic
434C2×C4C2×#12111yescharacteristic
435C2×C4#52911yescharacteristic
436C2×C4#53211yescharacteristic
437C2×C4C2×#13511yescharacteristic
438C2×C4C2×#13511yescharacteristic
439C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
440C2×C4#57511yescharacteristic
441C8—21no—
442C8—21no—
443C8—21no—
444C8—21no—
445C8—21no—
446C8—21no—
447C8—42no—
448C8—42no—
449C8—42no—
450C8—42no—
451C8—81no—
452C8—81no—
453C8—81no—
454C8—81no—
455C8#46312yes—
456C8#46312yes—
457C22—21no—
458C22—21no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—82no—
463C22—82no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22C4×#12111yescharacteristic
468C22#310211yescharacteristic
469C22#310511yescharacteristic
470C4—21no—
471C4—21no—
472C4—21no—
473C4—42no—
474C4—42no—
475C4—42no—
476C4—82no—
477C4—81no—
478C4—81no—
479C4—81no—
480C4C4×#13511yescharacteristic
481C4C4×AGL(1,ℤ/8)11yescharacteristic
482C4#295712yes—
483C4C2×#46311yescharacteristic
484C4#303511yescharacteristic
485C2—21no—
486C2—21no—
487C2—82no—
488C2—161no—
489C2C4×#46311yescharacteristic
490C2#2154511yescharacteristic
491C2#2155811yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radical#3100C22
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. #21658C2semidirect
2. #20485C2semidirect
4. #17930C2semidirect
6. #9751C2semidirect
7. #9629C2semidirect
8. #9584C2semidirect
9. #9582C2semidirect
11. #9235C2semidirect
12. #8660C2semidirect
13. #8266C2semidirect
14. #8256C2semidirect
15. #8255C2semidirect
39. #1683C4semidirect
40. #1681C4semidirect
41. #1679C4semidirect
42. #1677C4semidirect

Maximal subgroups

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

subgroupindex
#216582
#204852
#191102
#179302
#179282
#97512
#96292
#95842
#95822
#92372
#92352
#86602
#82662
#82562
#82552