Finite Group Databasebeta

← #206646

Subgroups — #206646

Order 512.

620 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21341C211yesmaximal, characteristic, complemented
2#21264C211yesmaximal, characteristic
3#21117C211yesmaximal, characteristic
4#9666C211yesmaximal, characteristic
5#9660C211yesmaximal, characteristic
6#9600C211yesmaximal, characteristic, complemented
7#9596C211yesmaximal, characteristic, complemented
8#9560C211yesmaximal, characteristic
9#9559C211yesmaximal, characteristic, complemented
10#9556C211yesmaximal, characteristic
11#9555C211yesmaximal, characteristic, complemented
12#8278C211yesmaximal, characteristic, complemented
13#8182C211yesmaximal, characteristic, complemented
14#8174C211yesmaximal, characteristic
15#7843C211yesmaximal, characteristic, complemented
16C2×C4×SD16C2211yescharacteristic
17C2×#431—21no—
18#2295C2211yescharacteristic
19C42×D4C2211yescharacteristic
20#1984C2211yescharacteristic
21#1984C2211yescharacteristic
22#1954C2211yescharacteristic
23#1954C2211yescharacteristic
24#1939C2211yescharacteristic
25#1939C2211yescharacteristic
26#1939C2211yescharacteristic
27#1939C2211yescharacteristic
28#1927C411yescharacteristic
29#1927C411yescharacteristic
30#1923—21no—
31#1923—21no—
32#1920C411yescharacteristic
33#1920C411yescharacteristic
34#1916—21no—
35#1916—21no—
36#1877C411yescharacteristic
37#1877C411yescharacteristic
38#1864C411yescharacteristic
39#1864C411yescharacteristic
40C4×#106C2211yescharacteristic
41#1768—21no—
42#1767C2211yescharacteristic
43C4×C4⋊C8C2211yescharacteristic
44C4×#103C2211yescharacteristic
45C4×#102C2211yescharacteristic
46C4×C22⋊C8—21no—
47#1746—21no—
48#1744C2211yescharacteristic
49#1744C2211yescharacteristic
50C42×C8C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×#322C2211yescharacteristic
58C2×#322C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×#322C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×SD16C2×C411yescharacteristic
68C2×C4×Q8C2311yescharacteristic
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C22×C42—21no—
73C4×SD16—22no—
74C4×SD16—22no—
75#431—44no—
76C2×#106C2×C411yescharacteristic
77C2×#106C2×C411yescharacteristic
78C2×#106C2×C411yescharacteristic
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×#103C2×C411yescharacteristic
88C2×#103C2×C411yescharacteristic
89C2×#103C2×C411yescharacteristic
90C2×#102C2×C411yescharacteristic
91C2×#102C2×C411yescharacteristic
92C2×#102C2×C411yescharacteristic
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×#97—21no—
98C2×C4×C8—21no—
99C2×C4×C8D411yescharacteristic
100C2×C4×C8D411yescharacteristic
101C2×C4×C8D411yescharacteristic
102C2×C4×C8D411yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109#391C2×C411yescharacteristic
110#391C2×C411yescharacteristic
111#389C2×C411yescharacteristic
112#389C2×C411yescharacteristic
113#387—21no—
114#387—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2311yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
120C4×C22⋊C4—21no—
121C4×C22⋊C4—21no—
122C4×C22⋊C4—21no—
123#376—21no—
124C43C2311yescharacteristic
125#340C2×C411yescharacteristic
126#340C2×C411yescharacteristic
127#340C2×C411yescharacteristic
128#340C2×C411yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#336—21no—
132#336—21no—
133#336—21no—
134#336—21no—
135#336—21no—
136#336—21no—
137#336—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336C2×C411yescharacteristic
144#336C2×C411yescharacteristic
145#336C2×C411yescharacteristic
146#336C2×C411yescharacteristic
147#335—22no—
148#335—22no—
149Q8⋊C8—22no—
150Q8⋊C8—22no—
151D4⋊C8—22no—
152D4⋊C8—22no—
153#322—22no—
154#322—22no—
155#322—22no—
156#322—22no—
157C22×Q8C22×C411yescharacteristic
158C22×D4C22×C411yescharacteristic
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C2×SD16—22no—
163C2×SD16—22no—
164C2×SD16—41no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8C22×C411yescharacteristic
172C22×C8C22×C411yescharacteristic
173C22×C8C22×C411yescharacteristic
174C22×C8C22×C411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—41no—
190C4×D4—41no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42#4111yescharacteristic
215C2×C42#4111yescharacteristic
216C2×C42C2411yescharacteristic
217#106—22no—
218#106—22no—
219#106—41no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—42no—
229C4⋊C8—42no—
230C4⋊C8—42no—
231C4⋊C8—42no—
232#103—22no—
233#103—22no—
234#103—41no—
235#102—22no—
236#102—22no—
237#102—41no—
238C22⋊C8—42no—
239C22⋊C8—42no—
240C22⋊C8—42no—
241C22⋊C8—42no—
242#97—42no—
243C4×C8—22no—
244C4×C8—22no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—41no—
254C4×C8—41no—
255C4×C8—42no—
256C4×C8#4112yes—
257C4×C8#4112yes—
258C4×C8#4112yes—
259C4×C8#4112yes—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270C24—21no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4#13112yes—
283C22×C4—24no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4C4×D411yescharacteristic
303C22×C4C4×D411yescharacteristic
304C22×C4C4×D411yescharacteristic
305C22×C4C4×D411yescharacteristic
306C22×C4C23×C411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C2×#4111yescharacteristic
309SD16—44no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
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—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
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—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8#11712yes—
351C2×C8#11712yes—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
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—42no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4#13112yes—
368C22⋊C4—24no—
369C22⋊C4—42no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42—42no—
387C42—42no—
388C42AGL(1,ℤ/8)12yes—
389C42#13712yes—
390C42C2×#4111yescharacteristic
391C42C2×#4111yescharacteristic
392C42C2×#4111yescharacteristic
393C42C2×#4111yescharacteristic
394C23—22no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23C2×C4×D411yescharacteristic
402Q8—41no—
403Q8—41no—
404Q8—41no—
405Q8—41no—
406D4—22no—
407D4—22no—
408D4—41no—
409D4—41no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—48no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—84no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4#44012yes—
450C2×C4#44012yes—
451C2×C4C2×#11711yescharacteristic
452C2×C4C4×#4111yescharacteristic
453C2×C4#52911yescharacteristic
454C2×C4#52911yescharacteristic
455C2×C4#53211yescharacteristic
456C2×C4#53211yescharacteristic
457C2×C4C2×#13111yescharacteristic
458C2×C4C2×#13111yescharacteristic
459C2×C4#56811yescharacteristic
460C2×C4#56811yescharacteristic
461C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
462C2×C4C2×#13711yescharacteristic
463C2×C4#57511yescharacteristic
464C2×C4#57511yescharacteristic
465C8—22no—
466C8—22no—
467C8—42no—
468C8—42no—
469C8—41no—
470C8—41no—
471C8—41no—
472C8—41no—
473C8—41no—
474C8—41no—
475C8—81no—
476C8—81no—
477C8—81no—
478C8—81no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22#229311yescharacteristic
487C22#292911yescharacteristic
488C22#293011yescharacteristic
489C22C2×#44011yescharacteristic
490C22#294311yescharacteristic
491C22#295511yescharacteristic
492C22#295511yescharacteristic
493C4—22no—
494C4—21no—
495C4—21no—
496C4—22no—
497C4—41no—
498C4—42no—
499C4—41no—
500C4—81no—
501C4—81no—
502C4—81no—
503C4—81no—
504C4—81no—
505C4—81no—
506C4#292411yescharacteristic
507C4#292511yescharacteristic
508C4#296811yescharacteristic
509C4#296911yescharacteristic
510C4#317111yescharacteristic
511C4#317111yescharacteristic
512C4#317211yescharacteristic
513C4#317211yescharacteristic
514C2—82no—
515C2#1790511yescharacteristic
516C2#1790911yescharacteristic
517C2#1793311yescharacteristic
518C2#1793711yescharacteristic
519C2#2127211yescharacteristic
520C2#2127911yescharacteristic
521C2#2129611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21341C2semidirect
6. #9600C2semidirect
7. #9596C2semidirect
9. #9559C2semidirect
11. #9555C2semidirect
12. #8278C2semidirect
13. #8182C2semidirect
15. #7843C2semidirect

Maximal subgroups

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

subgroupindex
#213412
#212642
#211172
#96662
#96602
#96002
#95962
#95602
#95592
#95562
#95552
#82782
#81822
#81742
#78432