Finite Group Databasebeta

← #229423

Subgroups — #229423

Order 512.

748 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23215C211yesmaximal, characteristic, complemented
2#23001C211yesmaximal, characteristic
3#22969C211yesmaximal, characteristic, complemented
4#22703C211yesmaximal, characteristic, complemented
5C2×#1677C211yesmaximal, characteristic
6#9822C211yesmaximal, characteristic, complemented
7#9613C211yesmaximal, characteristic, complemented
8#9575C211yesmaximal, characteristic, complemented
9#9572C211yesmaximal, characteristic, complemented
10#8977C211yesmaximal, characteristic
11#8973C211yesmaximal, characteristic, complemented
12#8928C211yesmaximal, characteristic, complemented
13#8921C211yesmaximal, characteristic, complemented
14#8283C211yesmaximal, characteristic, complemented
15C8×#102C211yesmaximal, characteristic, complemented
16C2×#486C2211yescharacteristic
17C2×#476C2211yescharacteristic
18C2×#459C2211yescharacteristic
19C2×#442—21no—
20C2×C8×D4C2211yescharacteristic
21#2369C2211yescharacteristic
22#2366C2211yescharacteristic
23D4×C4⋊C4C2211yescharacteristic
24D4×C4⋊C4C2211yescharacteristic
25#1990C2211yescharacteristic
26#1987C2211yescharacteristic
27#1982C2211yescharacteristic
28#1957C2211yescharacteristic
29#1957C2211yescharacteristic
30#1956C2211yescharacteristic
31#1948C2211yescharacteristic
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1947C2211yescharacteristic
35#1944—21no—
36#1936—21no—
37#1934—21no—
38#1926—21no—
39#1920—21no—
40#1879—21no—
41#1864—21no—
42#1864—21no—
43#1862—21no—
44#1853—21no—
45#1853—21no—
46#1834C2211yescharacteristic
47#1807—21no—
48C4×#106C2211yescharacteristic
49#1770C2211yescharacteristic
50#1770C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52#1762—21no—
53C4×#102—21no—
54C4×#102C2211yescharacteristic
55C2×#340—21no—
56#1677C418yescomplemented
57C2×#334C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#331C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×C82C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×D8—21no—
69C23×C8—21no—
70C2×C4⋊Q8C2311yescharacteristic
71C2×#127C2311yescharacteristic
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4C2311yescharacteristic
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79#486—22no—
80#486C2×C414yescomplemented
81#476—22no—
82#476C2×C414yescomplemented
83#459—22no—
84#459C2×C414yes—
85#442—44no—
86C8×D4—22no—
87C8×D4C2×C414yescomplemented
88C2×#107—21no—
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—21no—
93C2×#106D411yescharacteristic
94C2×#106D411yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×#103—41no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#102—41no—
109C2×#102D411yescharacteristic
110C2×#102D411yescharacteristic
111C2×C22⋊C8—21no—
112C2×#97—21no—
113C2×#97—21no—
114C2×C4×C8—21no—
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118#391—41no—
119#390—21no—
120#390—21no—
121#389—21no—
122#389—21no—
123#389—21no—
124#389—21no—
125#386—41no—
126#385—41no—
127#385—41no—
128#384C2311yescharacteristic
129#384C2311yescharacteristic
130#380—21no—
131#380—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4C2311yescharacteristic
135C4×C4⋊C4C2311yescharacteristic
136C4×C4⋊C4C2311yescharacteristic
137#340—21no—
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—21no—
147#340D411yescharacteristic
148#340D411yescharacteristic
149#340D411yescharacteristic
150#340D411yescharacteristic
151#336—21no—
152#336—21no—
153#336—21no—
154#334C2×C414yescomplemented
155#333C2×C414yescomplemented
156#331C2×C414yescomplemented
157D4⋊C8—22no—
158C82—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—
167C23×C4—41no—
168C2×D8—24no—
169C2×D8—42no—
170C2×D8—42no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—24no—
176C22×C8—21no—
177C22×C8—81no—
178C22×C8—81no—
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8C22×C412yes—
187#127—21no—
188#127—21no—
189#127—21no—
190#127C22×C412yes—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—42no—
194C4×D4—42no—
195C4×D4—42no—
196C4×D4—42no—
197C4×D4—81no—
198C4×D4—81no—
199C4×D4—81no—
200C4×D4—81no—
201C4×D4C22×C412yes—
202C4×D4#4112yes—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—81no—
217C2×C4⋊C4—81no—
218C2×C4⋊C4C2×D411yescharacteristic
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×C22⋊C4—21no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—41no—
235C2×C42C2411yescharacteristic
236#107—81no—
237#106—24no—
238#106—41no—
239#106—41no—
240#106—42no—
241#106—81no—
242#106—81no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8C22×C412yes—
249C4⋊C8C22×C412yes—
250C4⋊C8C22×C412yes—
251#103—82no—
252#102—24no—
253#102—22no—
254#102—24no—
255#102—24no—
256#102—24no—
257#102—42no—
258#102—42no—
259#102—81no—
260#102—81no—
261#102—82no—
262#102SD1614yes—
263C22⋊C8—24no—
264#97—42no—
265#97—81no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—81no—
271C4×C8C22×C412yes—
272C4×C8#4112yes—
273#95—21no—
274#95—21no—
275#95—21no—
276#95—41no—
277#95—41no—
278#95—41no—
279#95—41no—
280C24—21no—
281C24—41no—
282C2×Q8—22no—
283C2×Q8—41no—
284C2×Q8—81no—
285C2×Q8—81no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—81no—
304C2×D4—81no—
305C2×D4C4×D412yes—
306C2×D4C2×SD1612yes—
307C2×D4#13512yes—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—161no—
331C22×C4—161no—
332C22×C4—161no—
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4H(5,𝔽2)11yescharacteristic
336D8—48no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—84no—
362C2×C8C4×D412yes—
363C2×C8C4×D412yes—
364C2×C8C2×SD1612yes—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—41no—
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—81no—
390C4⋊C4—81no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C4⋊C4—161no—
394C4⋊C4C4×D412yes—
395C4⋊C4C2×SD1612yes—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C22⋊C4—82no—
399C22⋊C4—82no—
400C22⋊C4—82no—
401C22⋊C4—82no—
402C22⋊C4—161no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—161no—
410C42C23×C411yescharacteristic
411C42C2×#4111yescharacteristic
412C42C2×#4111yescharacteristic
413C422−1+411yescharacteristic
414C23—22no—
415C23—22no—
416C23—22no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—42no—
422C23—81no—
423C23—81no—
424C23—81no—
425C23—81no—
426C23—81no—
427C23—81no—
428C23D4211yescharacteristic
429Q8—82no—
430Q8—161no—
431D4—22no—
432D4—22no—
433D4—22no—
434D4—42no—
435D4—42no—
436D4—42no—
437D4—42no—
438D4—82no—
439D4—81no—
440D4—81no—
441D4—81no—
442D4—81no—
443D4—161no—
444D4C4×SD1614yes—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—164no—
482C2×C4—164no—
483C2×C4—164no—
484C2×C4C2×C4×D411yescharacteristic
485C2×C4C2×C4×D411yescharacteristic
486C2×C4#52011yescharacteristic
487C2×C4#54711yescharacteristic
488C2×C4#54811yescharacteristic
489C2×C4#55311yescharacteristic
490C2×C4C22×SD1611yescharacteristic
491C2×C4C2×#13511yescharacteristic
492C2×C4#57511yescharacteristic
493C2×C4#57511yescharacteristic
494C2×C4#57611yescharacteristic
495C2×C4#57711yescharacteristic
496C2×C4#57711yescharacteristic
497C2×C4#57711yescharacteristic
498C8—22no—
499C8—22no—
500C8—22no—
501C8—42no—
502C8—41no—
503C8—42no—
504C8—42no—
505C8—42no—
506C8—81no—
507C22—24no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—161no—
518C22—161no—
519C22—161no—
520C22—161no—
521C22#236111yescharacteristic
522C22#327311yescharacteristic
523C22#327511yescharacteristic
524C22D4×SD1611yescharacteristic
525C22#328211yescharacteristic
526C22#329011yescharacteristic
527C22#329211yescharacteristic
528C4—21no—
529C4—21no—
530C4—42no—
531C4—42no—
532C4—82no—
533C4—81no—
534C4—81no—
535C4—81no—
536C4—81no—
537C4—81no—
538C4—161no—
539C4—161no—
540C4C2×C4×SD1611yescharacteristic
541C4#294311yescharacteristic
542C4#294611yescharacteristic
543C4#295211yescharacteristic
544C4#333511yescharacteristic
545C4#333711yescharacteristic
546C4#337011yescharacteristic
547C4#337611yescharacteristic
548C2—44no—
549C2—162no—
550C2#1989111yescharacteristic
551C2#1989311yescharacteristic
552C2#1991111yescharacteristic
553C2#1991611yescharacteristic
554C2#2300511yescharacteristic
555C2#2302211yescharacteristic
556C2#2304611yescharacteristic
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. #23215C2semidirect
3. #22969C2semidirect
4. #22703C2semidirect
6. #9822C2semidirect
7. #9613C2semidirect
8. #9575C2semidirect
9. #9572C2semidirect
11. #8973C2semidirect
12. #8928C2semidirect
13. #8921C2semidirect
14. #8283C2semidirect
15. C8×#102C2semidirect
56. #1677C4semidirect
80. #486C2×C4semidirect
82. #476C2×C4semidirect
87. C8×D4C2×C4semidirect
154. #334C2×C4semidirect
155. #333C2×C4semidirect
156. #331C2×C4semidirect

Maximal subgroups

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

subgroupindex
#232152
#230012
#229692
#227032
C2×#16772
#98222
#96132
#95752
#95722
#89772
#89732
#89282
#89212
#82832
C8×#1022