Finite Group Databasebeta

← #211764

Subgroups — #211764

Order 512.

610 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22234C211yesmaximal, characteristic, complemented
2#20562C211yesmaximal, characteristic, complemented
3#19110C211yesmaximal, characteristic
4#17933C211yesmaximal, characteristic, complemented
5#17928C211yesmaximal, characteristic
6#9754C211yesmaximal, characteristic, complemented
7#9630C211yesmaximal, characteristic, complemented
8#9580C211yesmaximal, characteristic, complemented
9#9577C211yesmaximal, characteristic, complemented
10#9237C211yesmaximal, characteristic
11#9235C211yesmaximal, characteristic, complemented
12#8658C211yesmaximal, characteristic, complemented
13#8270C211yesmaximal, characteristic, complemented
14#8257C211yesmaximal, characteristic, complemented
15#8252C211yesmaximal, characteristic, complemented
16#3131C2211yescharacteristic
17#3101C2211yescharacteristic
18#3100C2211yescharacteristic
19#2960C2211yescharacteristic
20#2931C2211yescharacteristic
21C2×#436C2211yescharacteristic
22C2×C8×D4C2211yescharacteristic
23#2293C2211yescharacteristic
24#1994C2211yescharacteristic
25#1962C2211yescharacteristic
26#1929—21no—
27#1929—21no—
28#1866—21no—
29#1866—21no—
30#1848C2211yescharacteristic
31#1770C2211yescharacteristic
32C8×C4⋊C4C2211yescharacteristic
33C4×C4⋊C8C2211yescharacteristic
34#1753—21no—
35C8×C22⋊C4—21no—
36#1750—21no—
37#1744C2211yescharacteristic
38#1729—21no—
39#1683C412yescomplemented
40#1681C412yescomplemented
41#1679C412yescomplemented
42#1677C412yescomplemented
43#1592—22no—
44#1589—22no—
45#1588—22no—
46#1581—22no—
47#1571C2211yescharacteristic
48#1571C2211yescharacteristic
49#1564C2211yescharacteristic
50#1564C2211yescharacteristic
51#1542C2211yescharacteristic
52#1540C2211yescharacteristic
53#1503C2211yescharacteristic
54#1502C2211yescharacteristic
55#1486C2211yescharacteristic
56#1486C2211yescharacteristic
57#1481C2211yescharacteristic
58#1480C2211yescharacteristic
59#1452C2211yescharacteristic
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
110C2×#108—21no—
111C2×#108—21no—
112#424—21no—
113#423C2311yescharacteristic
114#423C2311yescharacteristic
115C2×C4⋊C8C2311yescharacteristic
116C2×C4⋊C8C2311yescharacteristic
117C2×#104—21no—
118C2×#104—21no—
119#416—21no—
120#416—21no—
121#408—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124#405—21no—
125C2×#97—21no—
126C2×C4×C8—21no—
127C2×C4×C8—21no—
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130#390—21no—
131#389—21no—
132#388—22no—
133C4×C4⋊C4C2311yescharacteristic
134C4×C22⋊C4—21no—
135C43C2311yescharacteristic
136#339—21no—
137#339—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336D411yescharacteristic
142#336D411yescharacteristic
143#336D411yescharacteristic
144#336D411yescharacteristic
145#335—22no—
146#335—22no—
147#335—21no—
148#335C2×C412yes—
149#334—21no—
150#334C2×C412yes—
151#333C2×C412yes—
152#332C2×C412yes—
153#332C2×C412yes—
154#331C2×C412yes—
155#328—22no—
156#327—22no—
157Q8⋊C8—22no—
158D4⋊C8—22no—
159D4⋊C8—22no—
160D4⋊C8—22no—
161#322—21no—
162#322—21no—
163#322—21no—
164C82—21no—
165C2×#41—21no—
166C22×D4#4111yescharacteristic
167C23×C4—21no—
168C2×SD16—22no—
169C2×D8—22no—
170#131—81no—
171#131—81no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—41no—
177C2×M16—81no—
178C2×M16—81no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—41no—
184C22×C8—81no—
185C22×C8—81no—
186C22×C8C2×D411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C22×C8C2×D411yescharacteristic
189C22×C8C2×D411yescharacteristic
190C4⋊Q8C22×C411yescharacteristic
191C4⋊Q8C22×C411yescharacteristic
192C4⋊Q8C22×C411yescharacteristic
193#127C22×C411yescharacteristic
194#125—21no—
195#124—21no—
196C22⋊Q8—21no—
197C22⋊Q8—21no—
198#121—21no—
199#121—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×D4—21no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4C22×C412yes—
207C4×D4C2×D412yes—
208#117—21no—
209#117—81no—
210#117—81no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4#4111yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2411yescharacteristic
223#108—81no—
224#108—81no—
225#107—22no—
226#106—22no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—41no—
236C4⋊C8—41no—
237C4⋊C8C22×C412yes—
238C4⋊C8C22×C412yes—
239C4⋊C8C22×C412yes—
240C4⋊C8C2×D412yes—
241#104—81no—
242#104—81no—
243#103—21no—
244#103—21no—
245#103—21no—
246#103—21no—
247#103—22no—
248#102—22no—
249#102—22no—
250#102—21no—
251#102—21no—
252#102—21no—
253#102—21no—
254#102—22no—
255C22⋊C8—21no—
256C22⋊C8—21no—
257C22⋊C8—22no—
258C22⋊C8—22no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—21no—
262C22⋊C8—21no—
263C22⋊C8—41no—
264C22⋊C8—41no—
265#97—41no—
266#97—81no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—41no—
272C4×C8—81no—
273C4×C8—81no—
274C4×C8C22×C411yescharacteristic
275C4×C8C22×C411yescharacteristic
276C4×C8C2×D411yescharacteristic
277C4×C8C2×D411yescharacteristic
278C4×C8#4111yescharacteristic
279C4×C8#4111yescharacteristic
280C4×C8#4111yescharacteristic
281C4×C8#4111yescharacteristic
282#95—22no—
283#95#4112yes—
284C24—21no—
285#41—41no—
286#41—41no—
287#41—41no—
288#41—41no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—42no—
297C2×D4—41no—
298C2×D4C4×D412yes—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4C22×D411yescharacteristic
311C22×C4C2×#4111yescharacteristic
312C22×C4C2×#4111yescharacteristic
313SD16—82no—
314D8—82no—
315M16—41no—
316M16—41no—
317M16—41no—
318M16—41no—
319M16—82no—
320M16—82no—
321M16—82no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—82no—
343C2×C8—82no—
344C2×C8—82no—
345C2×C8C4×D411yescharacteristic
346C2×C8C4×D411yescharacteristic
347C2×C8C4×D411yescharacteristic
348C2×C8C4×D411yescharacteristic
349C2×C8#12111yescharacteristic
350C2×C8#12111yescharacteristic
351C2×C8#12111yescharacteristic
352C2×C8#12111yescharacteristic
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—42no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—82no—
364C4⋊C4C4×D412yes—
365C22⋊C4—22no—
366C22⋊C4—22no—
367C22⋊C4—44no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C22⋊C4—82no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—42no—
377C42—42no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42C23×C411yescharacteristic
382C42C22×D411yescharacteristic
383C42C2×#4111yescharacteristic
384C42C2×#4111yescharacteristic
385C23—22no—
386C23—42no—
387C23—42no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23#52511yescharacteristic
392Q8—41no—
393Q8—41no—
394Q8—42no—
395Q8—81no—
396D4—22no—
397D4—22no—
398D4—42no—
399D4—41no—
400D4—41no—
401D4—42no—
402D4—81no—
403D4—81no—
404D4—81no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4C2×C4×D411yescharacteristic
429C2×C4C2×C4×D411yescharacteristic
430C2×C4C4×#4111yescharacteristic
431C2×C4C2×#12111yescharacteristic
432C2×C4#52911yescharacteristic
433C2×C4#53211yescharacteristic
434C2×C4#57611yescharacteristic
435C2×C4#57611yescharacteristic
436C2×C4#57711yescharacteristic
437C2×C4#57811yescharacteristic
438C8—21no—
439C8—21no—
440C8—21no—
441C8—21no—
442C8—21no—
443C8—21no—
444C8—21no—
445C8—21no—
446C8—42no—
447C8—42no—
448C8—42no—
449C8—42no—
450C8—81no—
451C8—81no—
452C8—81no—
453C8—81no—
454C22—21no—
455C22—21no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—82no—
460C22—82no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22C4×#12111yescharacteristic
465C22#311811yescharacteristic
466C22#312411yescharacteristic
467C4—21no—
468C4—21no—
469C4—21no—
470C4—42no—
471C4—42no—
472C4—81no—
473C4—82no—
474C4—81no—
475C4—81no—
476C4—81no—
477C4#294611yescharacteristic
478C4#295111yescharacteristic
479C4#295712yes—
480C4#304111yescharacteristic
481C4#304611yescharacteristic
482C2—21no—
483C2—21no—
484C2—82no—
485C2—161no—
486C2#2150911yescharacteristic
487C2#2154711yescharacteristic
488C2#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. #22234C2semidirect
2. #20562C2semidirect
4. #17933C2semidirect
6. #9754C2semidirect
7. #9630C2semidirect
8. #9580C2semidirect
9. #9577C2semidirect
11. #9235C2semidirect
12. #8658C2semidirect
13. #8270C2semidirect
14. #8257C2semidirect
15. #8252C2semidirect
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
#222342
#205622
#191102
#179332
#179282
#97542
#96302
#95802
#95772
#92372
#92352
#86582
#82702
#82572
#82522