Finite Group Databasebeta

← #229357

Subgroups — #229357

Order 512.

739 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23216C211yesmaximal, characteristic, complemented
2#23094C211yesmaximal, characteristic, complemented
3#22995C211yesmaximal, characteristic
4#22713C211yesmaximal, characteristic, complemented
5C2×#1701C211yesmaximal, characteristic
6#9838C211yesmaximal, characteristic, complemented
7#9613C211yesmaximal, characteristic, complemented
8#9563C211yesmaximal, characteristic, complemented
9#9562C211yesmaximal, characteristic, complemented
10#8985C211yesmaximal, characteristic, complemented
11#8980C211yesmaximal, characteristic
12#8924C211yesmaximal, characteristic, complemented
13#8922C211yesmaximal, characteristic, complemented
14#8284C211yesmaximal, characteristic, complemented
15#8210C211yesmaximal, characteristic, complemented
16C2×#486C2211yescharacteristic
17C2×#461C2211yescharacteristic
18C2×#459C2211yescharacteristic
19C2×C4×D8—21no—
20C2×#436C2211yescharacteristic
21#2366C2211yescharacteristic
22#2365C2211yescharacteristic
23#2354C2211yescharacteristic
24D4×C4⋊C4C2211yescharacteristic
25#1990C2211yescharacteristic
26#1988C2211yescharacteristic
27#1982C2211yescharacteristic
28#1968—21no—
29#1957C2211yescharacteristic
30#1957C2211yescharacteristic
31#1956C2211yescharacteristic
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1934—21no—
37#1934—21no—
38#1927—21no—
39#1925—21no—
40#1880—21no—
41#1879—21no—
42#1862—21no—
43#1862—21no—
44#1854—21no—
45#1852—21no—
46#1834C2211yescharacteristic
47#1809—21no—
48C4×#107C2211yescharacteristic
49C8×C4⋊C4C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52C4×#103—21no—
53C4×#102—21no—
54C4×#102C2211yescharacteristic
55#1736—21no—
56#1701C418yescomplemented
57C2×#334C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#331C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×SD16—21no—
69C22×D8—21no—
70C22×M16—21no—
71C2×C4⋊Q8C2311yescharacteristic
72C2×#127C2311yescharacteristic
73C2×C4×Q8—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4C2311yescharacteristic
78C2×#117—21no—
79C22×C4⋊C4—21no—
80#486—22no—
81#486C2×C414yescomplemented
82#461—22no—
83#461C2×C414yescomplemented
84#459—22no—
85#459C2×C414yes—
86C4×D8—44no—
87#436—22no—
88#436C2×C414yescomplemented
89C2×#107—21no—
90C2×#107—21no—
91C2×#107D411yescharacteristic
92C2×#107D411yescharacteristic
93C2×#106—21no—
94C2×#106—21no—
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×#103—21no—
100C2×#103—21no—
101C2×#103—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×#102—41no—
110C2×#102D411yescharacteristic
111C2×#102D411yescharacteristic
112C2×C22⋊C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8—21no—
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119#391—21no—
120#390—21no—
121#390—21no—
122#390—21no—
123#389—21no—
124#389—21no—
125#388—41no—
126#387—41no—
127#386—41no—
128#385—41no—
129#384C2311yescharacteristic
130#384C2311yescharacteristic
131#380—21no—
132#380—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4C2311yescharacteristic
136C4×C4⋊C4C2311yescharacteristic
137C4×C4⋊C4C2311yescharacteristic
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—41no—
147#340D411yescharacteristic
148#340D411yescharacteristic
149#340D411yescharacteristic
150#340D411yescharacteristic
151#336—21no—
152#336—21no—
153#336—21no—
154#334C2×C414yescomplemented
155#332C2×C414yescomplemented
156#331C2×C414yescomplemented
157D4⋊C8—22no—
158#322—22no—
159C22×Q8—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4C2×D411yescharacteristic
164C23×C4—21no—
165C23×C4—41no—
166C23×C4—41no—
167C2×SD16—24no—
168C2×SD16—81no—
169C2×SD16—81no—
170C2×D8—24no—
171C2×D8—42no—
172C2×D8—42no—
173C2×M16—24no—
174C2×M16—81no—
175C2×M16—81no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8C22×C412yes—
189#127—21no—
190#127—21no—
191#127—21no—
192#127C22×C412yes—
193C4×Q8—42no—
194C4×Q8—81no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—42no—
198C4×D4—42no—
199C4×D4—42no—
200C4×D4—81no—
201C4×D4—81no—
202C4×D4—81no—
203C4×D4C22×C412yes—
204C4×D4#4112yes—
205#117—42no—
206#117—81no—
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—42no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—81no—
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—41no—
237C2×C42—41no—
238C2×C42C2411yescharacteristic
239#107—22no—
240#107—22no—
241#107—42no—
242#107—81no—
243#106—81no—
244#106—81no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8C22×C412yes—
251C4⋊C8C22×C412yes—
252C4⋊C8C22×C412yes—
253#103—24no—
254#103—42no—
255#103—81no—
256#102—24no—
257#102—24no—
258#102—24no—
259#102—22no—
260#102—41no—
261#102—42no—
262#102—42no—
263#102—81no—
264#102—81no—
265#102—82no—
266C22⋊C8—24no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—42no—
272C4×C8—81no—
273C4×C8—81no—
274C4×C8C22×C412yes—
275C4×C8#4112yes—
276#95—21no—
277#95—21no—
278#95—21no—
279#95—41no—
280#95—41no—
281#95—41no—
282#95—41no—
283C24—21no—
284C24—41no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—81no—
311C2×D4—81no—
312C2×D4C4×D412yes—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4—161no—
334C22×C4—161no—
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4H(5,𝔽2)11yescharacteristic
338SD16—84no—
339D8—48no—
340M16—84no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8C4×D412yes—
367C2×C8C4×D412yes—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—82no—
391C4⋊C4—82no—
392C4⋊C4—81no—
393C4⋊C4—81no—
394C4⋊C4C4×D412yes—
395C4⋊C4#13512yes—
396C4⋊C4#13512yes—
397C4⋊C4AGL(1,ℤ/8)12yes—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—82no—
401C22⋊C4—82no—
402C22⋊C4—82no—
403C22⋊C4—82no—
404C22⋊C4—161no—
405C22⋊C4—161no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42—82no—
413C42—82no—
414C42C23×C411yescharacteristic
415C42C2×#4111yescharacteristic
416C42C2×#4111yescharacteristic
417C422−1+411yescharacteristic
418C23—22no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—42no—
426C23—81no—
427C23—81no—
428C23—81no—
429C23—81no—
430C23—81no—
431C23—81no—
432C23D4211yescharacteristic
433Q8—42no—
434Q8—42no—
435Q8—81no—
436Q8—81no—
437D4—22no—
438D4—22no—
439D4—41no—
440D4—41no—
441D4—42no—
442D4—42no—
443D4—42no—
444D4—42no—
445D4—82no—
446D4—81no—
447D4—81no—
448D4—81no—
449D4—81no—
450D4—161no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—164no—
488C2×C4—164no—
489C2×C4C2×C4×D411yescharacteristic
490C2×C4C2×C4×D411yescharacteristic
491C2×C4#52011yescharacteristic
492C2×C4#54711yescharacteristic
493C2×C4#54811yescharacteristic
494C2×C4#55311yescharacteristic
495C2×C4C2×#13511yescharacteristic
496C2×C4C2×#13511yescharacteristic
497C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
498C2×C4#57511yescharacteristic
499C2×C4#57611yescharacteristic
500C2×C4#57611yescharacteristic
501C2×C4#57711yescharacteristic
502C2×C4#57711yescharacteristic
503C8—22no—
504C8—22no—
505C8—41no—
506C8—41no—
507C8—42no—
508C8—42no—
509C8—42no—
510C8—81no—
511C8—81no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—81no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22—161no—
523C22—161no—
524C22—161no—
525C22—161no—
526C22#236111yescharacteristic
527C22#326411yescharacteristic
528C22#327111yescharacteristic
529C22#328011yescharacteristic
530C22#328211yescharacteristic
531C22#328811yescharacteristic
532C22#328911yescharacteristic
533C4—21no—
534C4—21no—
535C4—42no—
536C4—42no—
537C4—42no—
538C4—42no—
539C4—81no—
540C4—81no—
541C4—81no—
542C4—82no—
543C4—161no—
544C4—161no—
545C4C4×#13511yescharacteristic
546C4C4×AGL(1,ℤ/8)11yescharacteristic
547C4#294611yescharacteristic
548C4#294611yescharacteristic
549C4#335411yescharacteristic
550C4#335511yescharacteristic
551C4#336011yescharacteristic
552C4#336111yescharacteristic
553C2—82no—
554C2—162no—
555C2#1977111yescharacteristic
556C2#1978611yescharacteristic
557C2#1982011yescharacteristic
558C2#1983311yescharacteristic
559C2#2299911yescharacteristic
560C2#2303711yescharacteristic
561C2#2304911yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#459C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23216C2semidirect
2. #23094C2semidirect
4. #22713C2semidirect
6. #9838C2semidirect
7. #9613C2semidirect
8. #9563C2semidirect
9. #9562C2semidirect
10. #8985C2semidirect
12. #8924C2semidirect
13. #8922C2semidirect
14. #8284C2semidirect
15. #8210C2semidirect
56. #1701C4semidirect
81. #486C2×C4semidirect
83. #461C2×C4semidirect
88. #436C2×C4semidirect
154. #334C2×C4semidirect
155. #332C2×C4semidirect
156. #331C2×C4semidirect

Maximal subgroups

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

subgroupindex
#232162
#230942
#229952
#227132
C2×#17012
#98382
#96132
#95632
#95622
#89852
#89802
#89242
#89222
#82842
#82102