Finite Group Databasebeta

← #230767

Subgroups — #230767

Order 512.

699 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23173C211yesmaximal, characteristic
2#22832C211yesmaximal, characteristic, complemented
3C4×#460C211yesmaximal, characteristic
4#9666C211yesmaximal, characteristic, complemented
5#9661C211yesmaximal, characteristic, complemented
6#9642C211yesmaximal, characteristic
7#9603C211yesmaximal, characteristic, complemented
8#9546C211yesmaximal, characteristic, complemented
9#9545C211yesmaximal, characteristic, complemented
10#8945C211yesmaximal, characteristic, complemented
11#8945C211yesmaximal, characteristic, complemented
12#8265C211yesmaximal, characteristic, complemented
13#8204C211yesmaximal, characteristic, complemented
14#8202C211yesmaximal, characteristic, complemented
15#8171C211yesmaximal, characteristic, complemented
16C2×#460C411yescharacteristic
17C2×C4×SD16C2211yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×C8×D4—21no—
20C2×C8×D4C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22C4×#127C2211yescharacteristic
23C42×Q8C2211yescharacteristic
24#1984C2211yescharacteristic
25#1984C2211yescharacteristic
26#1982C411yescharacteristic
27#1954—21no—
28#1954C2211yescharacteristic
29#1954C2211yescharacteristic
30#1948C411yescharacteristic
31#1926C411yescharacteristic
32#1926C411yescharacteristic
33#1917—21no—
34#1917—21no—
35#1880C411yescharacteristic
36#1880C411yescharacteristic
37#1869—21no—
38#1869—21no—
39#1839—21no—
40#1831C2211yescharacteristic
41#1803C411yescharacteristic
42#1770C2211yescharacteristic
43#1770C2211yescharacteristic
44C4×C4⋊C8C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×#102C2211yescharacteristic
49#1745—21no—
50#1744C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×#322C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×SD16C2×C411yescharacteristic
68C22×SD16C2×C411yescharacteristic
69C23×C8—21no—
70C23×C8—21no—
71C2×#127C2×C411yescharacteristic
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×Q8C2×C411yescharacteristic
74C2×C4×Q8C2311yescharacteristic
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4C2311yescharacteristic
78C2×C4×D4C2311yescharacteristic
79#460—24no—
80C4×SD16—41no—
81C4×SD16—41no—
82C4×SD16—41no—
83C4×SD16—41no—
84C8×D4—22no—
85C8×D4—22no—
86C8×D4—41no—
87C8×D4—41no—
88C8×D4—44no—
89C2×#106C2×C411yescharacteristic
90C2×#106C2×C411yescharacteristic
91C2×C4⋊C8C2×C411yescharacteristic
92C2×C4⋊C8C2×C411yescharacteristic
93C2×C4⋊C8D411yescharacteristic
94C2×C4⋊C8D411yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×#103C2×C411yescharacteristic
100C2×#103C2×C411yescharacteristic
101C2×#102C2×C411yescharacteristic
102C2×#102C2×C411yescharacteristic
103C2×#102C2×C411yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×#97—21no—
114C2×#97—21no—
115C2×C4×C8—21no—
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122#390—21no—
123#390—21no—
124#390C2×C411yescharacteristic
125#390C2×C411yescharacteristic
126#384C2×C411yescharacteristic
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C43—21no—
134C43C2311yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336C2×C411yescharacteristic
151#336C2×C411yescharacteristic
152#336D411yescharacteristic
153#336D411yescharacteristic
154#335—41no—
155#334—41no—
156Q8⋊C8—22no—
157Q8⋊C8—22no—
158D4⋊C8—22no—
159D4⋊C8—22no—
160D4⋊C8—41no—
161D4⋊C8—41no—
162#322—41no—
163#322—41no—
164C22×Q8C22×C411yescharacteristic
165C22×D4—21no—
166C22×D4—21no—
167C22×D4C22×C411yescharacteristic
168C22×D4C22×C411yescharacteristic
169C23×C4—21no—
170C23×C4—21no—
171C2×SD16—22no—
172C2×SD16—22no—
173C2×SD16—22no—
174C2×SD16—22no—
175C2×SD16—41no—
176C2×SD16—41no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—42no—
184C22×C8—42no—
185C22×C8—42no—
186C22×C8—42no—
187C22×C8—81no—
188C22×C8—81no—
189C22×C8C22×C411yescharacteristic
190C22×C8C22×C411yescharacteristic
191C22×C8C22×C411yescharacteristic
192C22×C8C22×C411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22×C8C2×D411yescharacteristic
196C22×C8C2×D411yescharacteristic
197#127—22no—
198#127M1614yes—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—21no—
203C4×Q8—21no—
204C4×Q8—21no—
205C4×Q8—21no—
206C4×Q8—41no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—21no—
215C4×D4—42no—
216C4×D4—42no—
217C4×D4—81no—
218C4×D4—81no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42#4111yescharacteristic
246C2×C42#4111yescharacteristic
247C2×C42C2411yescharacteristic
248#106—41no—
249#106—41no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—22no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—41no—
261C4⋊C8#4112yes—
262#103—41no—
263#103—41no—
264#102—22no—
265#102—22no—
266#102—41no—
267#102—41no—
268#102—41no—
269#102—41no—
270C22⋊C8—42no—
271C22⋊C8—42no—
272C22⋊C8—42no—
273C22⋊C8—42no—
274C22⋊C8—81no—
275C22⋊C8—81no—
276#97—42no—
277#97—42no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—42no—
290C4×C8#4112yes—
291#95—21no—
292#95—21no—
293#95—21no—
294C24—21no—
295C24—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8#13112yes—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—81no—
317C2×D4—81no—
318C2×D4C2×M1612yes—
319C2×D4C2×M1612yes—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4C4×D411yescharacteristic
339C22×C4C4×D411yescharacteristic
340C22×C4C4×D411yescharacteristic
341C22×C4C4×D411yescharacteristic
342C22×C4C23×C411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C2×#4111yescharacteristic
345SD16—44no—
346SD16—44no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—84no—
389C2×C8—84no—
390C2×C8AGL(1,ℤ/8)12yes—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—42no—
406C4⋊C4—81no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—81no—
414C22⋊C4—81no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—41no—
422C42—41no—
423C42—42no—
424C42—42no—
425C42—41no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—42no—
430C42—81no—
431C42C2×M1612yes—
432C42#13112yes—
433C42C2×#4111yescharacteristic
434C42C2×#4111yescharacteristic
435C42H(5,𝔽2)11yescharacteristic
436C42H(5,𝔽2)11yescharacteristic
437C23—22no—
438C23—22no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23C2×C4×D411yescharacteristic
452Q8—22no—
453Q8—22no—
454Q8—41no—
455Q8—41no—
456D4—22no—
457D4—22no—
458D4—22no—
459D4—22no—
460D4—48no—
461D4—41no—
462D4—41no—
463D4—41no—
464D4—41no—
465D4—84no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4#43512yes—
505C2×C4#43512yes—
506C2×C4C4×#4111yescharacteristic
507C2×C4#51811yescharacteristic
508C2×C4#54611yescharacteristic
509C2×C4#55011yescharacteristic
510C2×C4#55311yescharacteristic
511C2×C4#55511yescharacteristic
512C2×C4C22×M1611yescharacteristic
513C2×C4C2×#13111yescharacteristic
514C2×C4C2×#13111yescharacteristic
515C2×C4#56811yescharacteristic
516C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
517C2×C4#57511yescharacteristic
518C2×C4#57711yescharacteristic
519C2×C4#57711yescharacteristic
520C8—22no—
521C8—42no—
522C8—41no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—41no—
527C8—42no—
528C8—41no—
529C8—41no—
530C8—81no—
531C8—81no—
532C8—81no—
533C8—81no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22#236311yescharacteristic
549C22C2×#43511yescharacteristic
550C22#293311yescharacteristic
551C22C4×AGL(1,ℤ/8)11yescharacteristic
552C22#294911yescharacteristic
553C22#295511yescharacteristic
554C22#295711yescharacteristic
555C4—21no—
556C4—21no—
557C4—22no—
558C4—22no—
559C4—42no—
560C4—41no—
561C4—41no—
562C4—81no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4#296611yescharacteristic
568C4#297011yescharacteristic
569C4#297411yescharacteristic
570C4#298111yescharacteristic
571C4#331011yescharacteristic
572C4#331711yescharacteristic
573C4#339711yescharacteristic
574C4#340111yescharacteristic
575C2—82no—
576C2—82no—
577C2#1786911yescharacteristic
578C2#1787911yescharacteristic
579C2#1801111yescharacteristic
580C2#1803011yescharacteristic
581C2#2308411yescharacteristic
582C2#2310611yescharacteristic
583C2#2311511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#127C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22832C2semidirect
4. #9666C2semidirect
5. #9661C2semidirect
7. #9603C2semidirect
8. #9546C2semidirect
9. #9545C2semidirect
10. #8945C2semidirect
11. #8945C2semidirect
12. #8265C2semidirect
13. #8204C2semidirect
14. #8202C2semidirect
15. #8171C2semidirect

Maximal subgroups

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

subgroupindex
#231732
#228322
C4×#4602
#96662
#96612
#96422
#96032
#95462
#95452
#89452
#82652
#82042
#82022
#81712