Finite Group Databasebeta

← #213148

Subgroups — #213148

Order 512.

958 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22269C211yesmaximal, characteristic, complemented
2#22269C211yesmaximal, characteristic, complemented
3#22268C211yesmaximal, characteristic, complemented
4#22219C211yesmaximal, characteristic, complemented
5C4×#493C211yesmaximal, characteristic, complemented
6C4×#486C211yesmaximal, characteristic, complemented
7C2×#1721C211yesmaximal, characteristic
8#9838C211yesmaximal, characteristic, complemented
9#9837C211yesmaximal, characteristic, complemented
10#9832C211yesmaximal, characteristic, complemented
11#9832C211yesmaximal, characteristic, complemented
12#9830C211yesmaximal, characteristic, complemented
13#9829C211yesmaximal, characteristic, complemented
14#8821C211yesmaximal, characteristic, complemented
15#7843C211yesmaximal, characteristic, complemented
16C2×#493C2211yescharacteristic
17C2×#493C2211yescharacteristic
18C2×#493C2211yescharacteristic
19C2×#492C2211yescharacteristic
20C2×#486C2211yescharacteristic
21C2×#486C2211yescharacteristic
22C2×C4×D8—22no—
23C4×C4⋊Q8C2211yescharacteristic
24C4×#127C2211yescharacteristic
25C4×#127C2211yescharacteristic
26C4×#127C2211yescharacteristic
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1970—22no—
30#1957C2211yescharacteristic
31#1957C2211yescharacteristic
32#1957C2211yescharacteristic
33#1957C2211yescharacteristic
34#1957C2211yescharacteristic
35#1957C2211yescharacteristic
36#1957C2211yescharacteristic
37#1957C2211yescharacteristic
38#1957C2211yescharacteristic
39#1956C2211yescharacteristic
40#1934—22no—
41#1926—22no—
42#1880—22no—
43#1879—22no—
44#1879—22no—
45#1879—22no—
46#1879—22no—
47#1877—22no—
48#1864—22no—
49#1845C2211yescharacteristic
50#1844C2211yescharacteristic
51#1834C2211yescharacteristic
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1834C2211yescharacteristic
55C4×#102—22no—
56#1744C2211yescharacteristic
57#1744C2211yescharacteristic
58C42×C8C2211yescharacteristic
59#1721C418yescomplemented
60C2×#322C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63C2×#322C2211yescharacteristic
64C22×SD16—22no—
65C22×D8—22no—
66C22×D8—22no—
67C22×D8—22no—
68C2×C4⋊Q8C2311yescharacteristic
69C2×#127C2311yescharacteristic
70C2×#127C2311yescharacteristic
71C2×#127C2311yescharacteristic
72C2×C4×Q8—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×C4×D4—21no—
80C2×C4×D4—21no—
81C2×C4×D4—21no—
82#493—22no—
83#493—22no—
84#493—22no—
85#493C2×C414yescomplemented
86#493C2×C414yescomplemented
87#493C2×C414yescomplemented
88#492—22no—
89#492C2×C414yescomplemented
90#486—22no—
91#486—22no—
92#486C2×C414yescomplemented
93#486C2×C414yescomplemented
94C4×D8—48no—
95C2×#107—22no—
96C2×#106—22no—
97C2×#103—22no—
98C2×#102—22no—
99C2×#102—22no—
100C2×#102—22no—
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—22no—
105C2×#102—22no—
106C2×#102—22no—
107C2×#102—22no—
108C2×#102—22no—
109C2×C4×C8D412yes—
110C2×C4×C8D412yes—
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117#391—21no—
118#390—21no—
119#390—21no—
120#390—21no—
121#390—21no—
122#390—21no—
123#390—21no—
124#390—21no—
125#390—21no—
126#390—21no—
127#389—21no—
128#389—21no—
129#384C2311yescharacteristic
130#384C2311yescharacteristic
131#384C2311yescharacteristic
132#384C2311yescharacteristic
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C43C2311yescharacteristic
136#340—22no—
137#340—22no—
138#340—22no—
139#340—22no—
140#340—22no—
141#340—22no—
142#336D412yes—
143#336D412yes—
144#336D412yes—
145#336D412yes—
146#322—22no—
147#322—22no—
148#322—22no—
149#322C2×C414yescomplemented
150C22×Q8—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—21no—
160C23×C4—41no—
161C23×C4—41no—
162C23×C4—41no—
163C2×SD16—28no—
164C2×SD16—82no—
165C2×SD16—82no—
166C2×D8—28no—
167C2×D8—28no—
168C2×D8—28no—
169C2×D8—44no—
170C2×D8—44no—
171C2×D8—82no—
172C2×D8—82no—
173C2×D8—82no—
174C2×D8—82no—
175C22×C8C2×D412yes—
176C22×C8C2×D412yes—
177C22×C8C2×D412yes—
178C22×C8C2×D412yes—
179C22×C8C2×D412yes—
180C22×C8C2×D412yes—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8C22×C412yes—
185#127—21no—
186#127—21no—
187#127—21no—
188#127—21no—
189#127—21no—
190#127—21no—
191#127—21no—
192#127—21no—
193#127—21no—
194#127C22×C412yes—
195#127C22×C412yes—
196#127C22×C412yes—
197C4×Q8—82no—
198C4×D4—44no—
199C4×D4—44no—
200C4×D4—44no—
201C4×D4—82no—
202C4×D4—82no—
203C4×D4—82no—
204C4×D4—82no—
205C4×D4—82no—
206C4×D4—82no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C42—41no—
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2411yescharacteristic
239#107—44no—
240#106—82no—
241#103—82no—
242#102—28no—
243#102—28no—
244#102—44no—
245#102—44no—
246#102—44no—
247#102—82no—
248#102—82no—
249#102—82no—
250#102—82no—
251#102—82no—
252#102—82no—
253C4×C8—24no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—42no—
261C4×C8C22×C412yes—
262C4×C8C22×C412yes—
263C4×C8C22×C412yes—
264C4×C8#4112yes—
265C4×C8#4112yes—
266C4×C8#4112yes—
267#95—41no—
268#95—41no—
269C24—41no—
270C24—41no—
271C24—41no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4#12711yescharacteristic
327C22×C4#12711yescharacteristic
328C22×C4#12711yescharacteristic
329C22×C4#12711yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333SD16—88no—
334D8—416no—
335D8—88no—
336D8—88no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8C4×D414yes—
353C2×C8C4×D414yes—
354C2×C8C4×D414yes—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—82no—
372C22⋊C4—82no—
373C22⋊C4—82no—
374C22⋊C4—82no—
375C22⋊C4—82no—
376C22⋊C4—82no—
377C22⋊C4—82no—
378C22⋊C4—82no—
379C22⋊C4—82no—
380C22⋊C4—82no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—82no—
387C42AGL(1,ℤ/8)14yes—
388C42C23×C411yescharacteristic
389C42C2×#4111yescharacteristic
390C42C2×#4111yescharacteristic
391C42C2×#4111yescharacteristic
392C23—42no—
393C23—42no—
394C23—42no—
395C23—81no—
396C23—81no—
397C23—81no—
398C23—81no—
399C23—81no—
400C23—81no—
401C23—81no—
402C23—81no—
403C23—81no—
404C23—81no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23—81no—
413C23C2×#12711yescharacteristic
414Q8—44no—
415Q8—82no—
416D4—44no—
417D4—44no—
418D4—44no—
419D4—44no—
420D4—44no—
421D4—44no—
422D4—44no—
423D4—44no—
424D4—44no—
425D4—82no—
426D4—82no—
427D4—82no—
428D4—82no—
429D4—82no—
430D4—82no—
431D4—82no—
432D4—82no—
433D4—82no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—42no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—164no—
460C2×C4—164no—
461C2×C4—164no—
462C2×C4#49612yes—
463C2×C4#49612yes—
464C2×C4C2×C4×D411yescharacteristic
465C2×C4C2×C4×D411yescharacteristic
466C2×C4C2×C4×D411yescharacteristic
467C2×C4#53211yescharacteristic
468C2×C4#53211yescharacteristic
469C2×C4#53211yescharacteristic
470C2×C4C2×AGL(1,ℤ/8)12yes—
471C2×C4#57512yes—
472C2×C4#57612yes—
473C2×C4#57612yes—
474C2×C4#57612yes—
475C2×C4#57712yes—
476C8—24no—
477C8—24no—
478C8—24no—
479C8—42no—
480C8—42no—
481C8—42no—
482C22—82no—
483C22—82no—
484C22—82no—
485C22—82no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22—161no—
492C22—161no—
493C22—161no—
494C22—161no—
495C22—161no—
496C22—161no—
497C22—161no—
498C22—161no—
499C22—161no—
500C22—161no—
501C22—161no—
502C22—161no—
503C22C4×#12711yescharacteristic
504C22C2×#49611yescharacteristic
505C22#315011yescharacteristic
506C22#315411yescharacteristic
507C22#315511yescharacteristic
508C22#315511yescharacteristic
509C22#315511yescharacteristic
510C4—22no—
511C4—22no—
512C4—41no—
513C4—41no—
514C4—82no—
515C4—161no—
516C4—161no—
517C4—161no—
518C4—161no—
519C4C4×AGL(1,ℤ/8)12yes—
520C4#294612yes—
521C4#294812yes—
522C4#317112yes—
523C4#317612yes—
524C4#317912yes—
525C2—162no—
526C2—162no—
527C2—162no—
528C2#1969811yescharacteristic
529C2#1970111yescharacteristic
530C2#1970811yescharacteristic
531C2#1971011yescharacteristic
532C2#2169711yescharacteristic
533C2C4×#49611yescharacteristic
534C2#2170411yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#1721C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22269C2semidirect
2. #22269C2semidirect
3. #22268C2semidirect
4. #22219C2semidirect
5. C4×#493C2semidirect
6. C4×#486C2semidirect
8. #9838C2semidirect
9. #9837C2semidirect
10. #9832C2semidirect
11. #9832C2semidirect
12. #9830C2semidirect
13. #9829C2semidirect
14. #8821C2semidirect
15. #7843C2semidirect
59. #1721C4semidirect
85. #493C2×C4semidirect
86. #493C2×C4semidirect
87. #493C2×C4semidirect
89. #492C2×C4semidirect
92. #486C2×C4semidirect
93. #486C2×C4semidirect
149. #322C2×C4semidirect

Maximal subgroups

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

subgroupindex
#222692
#222682
#222192
C4×#4932
C4×#4862
C2×#17212
#98382
#98372
#98322
#98302
#98292
#88212
#78432