Finite Group Databasebeta

← #211844

Subgroups — #211844

Order 512.

636 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22234C211yesmaximal, characteristic, complemented
2#20567C211yesmaximal, characteristic, complemented
3#19142C211yesmaximal, characteristic
4#17933C211yesmaximal, characteristic, complemented
5#17928C211yesmaximal, characteristic
6#9751C211yesmaximal, characteristic, complemented
7#9627C211yesmaximal, characteristic, complemented
8#9584C211yesmaximal, characteristic, complemented
9#9581C211yesmaximal, characteristic, complemented
10#9233C211yesmaximal, characteristic
11#9231C211yesmaximal, characteristic, complemented
12#8662C211yesmaximal, 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×#436C2211yescharacteristic
23C42×D4C2211yescharacteristic
24#1994C2211yescharacteristic
25#1962C2211yescharacteristic
26#1929—21no—
27#1929—21no—
28#1866—21no—
29#1866—21no—
30#1848C2211yescharacteristic
31C8×C4⋊C4C2211yescharacteristic
32C8×C4⋊C4C2211yescharacteristic
33#1767C2211yescharacteristic
34#1753—21no—
35#1753—21no—
36C4×C22⋊C8—21no—
37#1744C2211yescharacteristic
38#1728—21no—
39#1703C412yescomplemented
40#1701C412yescomplemented
41#1699C412yescomplemented
42#1697C412yescomplemented
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—
69C22×M16—21no—
70#533C2311yescharacteristic
71#532C2311yescharacteristic
72C2×C4×D4—21no—
73C2×C4×D4C2311yescharacteristic
74C22×C42—21no—
75#488—21no—
76#488—21no—
77#487C2×C411yescharacteristic, complemented
78#487C2×C411yescharacteristic, complemented
79#486C2×C411yescharacteristic, complemented
80#486C2×C411yescharacteristic, complemented
81#483—21no—
82#480—21no—
83#476C2×C412yes—
84#474C2×C412yes—
85#461C2×C412yes—
86#459C2×C412yes—
87#450—21no—
88#447—21no—
89#446—21no—
90#446—21no—
91C8×Q8—21no—
92C8×Q8—21no—
93C4×SD16—22no—
94C4×D8—22no—
95#436—21no—
96#436—21no—
97#436—21no—
98#436—21no—
99#436—21no—
100#436—21no—
101#436C2×C412yes—
102#436C2×C412yes—
103#436D412yes—
104#436D412yes—
105C8×D4—21no—
106C8×D4—21no—
107#431C2311yescharacteristic
108#431C2311yescharacteristic
109#431C2311yescharacteristic
110#431C2311yescharacteristic
111C2×#108—21no—
112C2×#108—21no—
113#424—21no—
114#423C2311yescharacteristic
115#423C2311yescharacteristic
116C2×C4⋊C8C2311yescharacteristic
117C2×C4⋊C8C2311yescharacteristic
118C2×#104—21no—
119C2×#104—21no—
120#416—21no—
121#416—21no—
122#408—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125#405—21no—
126C2×#97—21no—
127C2×C4×C8—21no—
128C2×C4×C8—21no—
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C4×C4⋊C4—21no—
132C4×C4⋊C4C2311yescharacteristic
133C4×C22⋊C4—21no—
134C4×C22⋊C4—22no—
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×C412yescomplemented
149#334—21no—
150#334C2×C412yescomplemented
151#333C2×C412yescomplemented
152#332C2×C412yescomplemented
153#332C2×C412yescomplemented
154#331C2×C412yescomplemented
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—
168C23×C4—22no—
169C2×SD16—22no—
170C2×D8—22no—
171#131—81no—
172#131—81no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—41no—
180C2×M16—81no—
181C2×M16—81no—
182C2×M16—81no—
183C2×M16—81no—
184C22×C8—21no—
185C22×C8—21no—
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×D4—44no—
208C4×D4C22×C412yes—
209C4×D4C2×D412yes—
210#117—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4#4111yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42—81no—
220C2×C42—81no—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2411yescharacteristic
224#108—81no—
225#108—81no—
226#107—22no—
227#106—22no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—41no—
237C4⋊C8—41no—
238C4⋊C8C22×C412yes—
239C4⋊C8C22×C412yes—
240C4⋊C8C22×C412yes—
241C4⋊C8C2×D412yes—
242#104—81no—
243#104—81no—
244#103—21no—
245#103—21no—
246#103—21no—
247#103—21no—
248#103—22no—
249#102—22no—
250#102—22no—
251#102—21no—
252#102—21no—
253#102—21no—
254#102—21no—
255#102—22no—
256C22⋊C8—21no—
257C22⋊C8—21no—
258C22⋊C8—22no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8—21no—
263C22⋊C8—21no—
264C22⋊C8—41no—
265C22⋊C8—41no—
266#97—41no—
267#97—81no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—41no—
273C4×C8—81no—
274C4×C8—81no—
275C4×C8C22×C411yescharacteristic
276C4×C8C22×C411yescharacteristic
277C4×C8C2×D411yescharacteristic
278C4×C8C2×D411yescharacteristic
279C4×C8#4111yescharacteristic
280C4×C8#4111yescharacteristic
281C4×C8#4111yescharacteristic
282C4×C8#4111yescharacteristic
283#95—22no—
284#95#4112yes—
285C24—21no—
286#41—41no—
287#41—41no—
288#41—41no—
289#41—41no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—42no—
298C2×D4—41no—
299C2×D4C4×D412yes—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—82no—
314C22×C4C22×D411yescharacteristic
315C22×C4C2×#4111yescharacteristic
316C22×C4C2×#4111yescharacteristic
317SD16—82no—
318D8—82no—
319M16—41no—
320M16—41no—
321M16—41no—
322M16—41no—
323M16—82no—
324M16—82no—
325M16—82no—
326M16—82no—
327M16—82no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—82no—
349C2×C8C4×D411yescharacteristic
350C2×C8C4×D411yescharacteristic
351C2×C8C4×D411yescharacteristic
352C2×C8C4×D411yescharacteristic
353C2×C8#12111yescharacteristic
354C2×C8#12111yescharacteristic
355C2×C8#12111yescharacteristic
356C2×C8#12111yescharacteristic
357C4⋊C4—24no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4C4×D412yes—
368C22⋊C4—22no—
369C22⋊C4—22no—
370C22⋊C4—44no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—42no—
379C42—42no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—82no—
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—24no—
411C2×C4—22no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—88no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4C2×C4×D411yescharacteristic
433C2×C4C2×C4×D411yescharacteristic
434C2×C4C4×#4111yescharacteristic
435C2×C4C2×#12111yescharacteristic
436C2×C4#52911yescharacteristic
437C2×C4#53211yescharacteristic
438C2×C4#57611yescharacteristic
439C2×C4#57611yescharacteristic
440C2×C4#57711yescharacteristic
441C2×C4#57811yescharacteristic
442C8—21no—
443C8—21no—
444C8—21no—
445C8—21no—
446C8—21no—
447C8—21no—
448C8—21no—
449C8—21no—
450C8—42no—
451C8—42no—
452C8—42no—
453C8—42no—
454C8—81no—
455C8—81no—
456C8—81no—
457C8—81no—
458C22—21no—
459C22—21no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—82no—
464C22—82no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22C4×#12111yescharacteristic
469C22#311811yescharacteristic
470C22#312411yescharacteristic
471C4—21no—
472C4—21no—
473C4—21no—
474C4—44no—
475C4—42no—
476C4—42no—
477C4—81no—
478C4—81no—
479C4—81no—
480C4—81no—
481C4#294611yescharacteristic
482C4#295111yescharacteristic
483C4#295612yes—
484C4#304111yescharacteristic
485C4#304611yescharacteristic
486C2—21no—
487C2—21no—
488C2—82no—
489C2—161no—
490C2#2150911yescharacteristic
491C2#2155111yescharacteristic
492C2#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. #20567C2semidirect
4. #17933C2semidirect
6. #9751C2semidirect
7. #9627C2semidirect
8. #9584C2semidirect
9. #9581C2semidirect
11. #9231C2semidirect
12. #8662C2semidirect
13. #8270C2semidirect
14. #8257C2semidirect
15. #8252C2semidirect
39. #1703C4semidirect
40. #1701C4semidirect
41. #1699C4semidirect
42. #1697C4semidirect
77. #487C2×C4semidirect
78. #487C2×C4semidirect
79. #486C2×C4semidirect
80. #486C2×C4semidirect
148. #335C2×C4semidirect
150. #334C2×C4semidirect
151. #333C2×C4semidirect
152. #332C2×C4semidirect
153. #332C2×C4semidirect
154. #331C2×C4semidirect

Maximal subgroups

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

subgroupindex
#222342
#205672
#191422
#179332
#179282
#97512
#96272
#95842
#95812
#92332
#92312
#86622
#82702
#82572
#82522