Finite Group Databasebeta

← #225910

Subgroups — #225910

Order 512.

732 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22701C211yesmaximal, characteristic, complemented
2#22696C211yesmaximal, characteristic, complemented
3#21113C211yesmaximal, characteristic
4C2×#1586C211yesmaximal, characteristic
5C2×#1580C211yesmaximal, characteristic
6#9599C211yesmaximal, characteristic, complemented
7#9595C211yesmaximal, characteristic, complemented
8#8847C211yesmaximal, characteristic, complemented
9#8843C211yesmaximal, characteristic, complemented
10#8843C211yesmaximal, characteristic, complemented
11#8843C211yesmaximal, characteristic, complemented
12#8811C211yesmaximal, characteristic, complemented
13#8569C211yesmaximal, characteristic, complemented
14#8565C211yesmaximal, characteristic, complemented
15#8265C211yesmaximal, characteristic, complemented
16C2×C4×D8C2211yescharacteristic
17C2×#436C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21#2293C2211yescharacteristic
22C42×D4C2211yescharacteristic
23#1939C2211yescharacteristic
24#1939C2211yescharacteristic
25#1939C2211yescharacteristic
26#1939C2211yescharacteristic
27#1925C411yescharacteristic
28#1925C411yescharacteristic
29#1918C411yescharacteristic
30#1918C411yescharacteristic
31#1879C411yescharacteristic
32#1879C411yescharacteristic
33#1862C411yescharacteristic
34#1862C411yescharacteristic
35#1851—21no—
36#1850—21no—
37#1850—21no—
38#1850—21no—
39#1831C2211yescharacteristic
40#1831C2211yescharacteristic
41#1798—21no—
42#1797—21no—
43C4×#107C2211yescharacteristic
44C4×C4⋊C8C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1744C2211yescharacteristic
49#1729—21no—
50C2×#336—21no—
51#1586—24no—
52#1580—24no—
53C2×#335C2211yescharacteristic
54C2×#335C2211yescharacteristic
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×D8C2×C411yescharacteristic
70C22×M16—21no—
71C23×C8—21no—
72C23×C8—21no—
73C23×C8—21no—
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C2×C4×D4C2311yescharacteristic
78C2×#117—21no—
79C22×C42—21no—
80C4×D8D414yes—
81C4×D8Q814yes—
82#436—22no—
83#436—41no—
84C8×D4—22no—
85C8×D4—22no—
86C8×D4—22no—
87C8×D4—41no—
88C8×D4—41no—
89C8×D4—41no—
90C2×#107C2×C411yescharacteristic
91C2×#107C2×C411yescharacteristic
92C2×#107C2×C411yescharacteristic
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8D411yescharacteristic
99C2×C4⋊C8D411yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108C2×#102C2×C411yescharacteristic
109C2×#102C2×C411yescharacteristic
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120#390C2×C411yescharacteristic
121#390C2×C411yescharacteristic
122#389C2×C411yescharacteristic
123#389C2×C411yescharacteristic
124#388—21no—
125#388—21no—
126C4×C4⋊C4C2×C411yescharacteristic
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130C4×C22⋊C4—21no—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134C43C2311yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
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#336C2×C411yescharacteristic
153#336C2×C411yescharacteristic
154#336D411yescharacteristic
155#336D411yescharacteristic
156#335—41no—
157#335—41no—
158#334—22no—
159#334—22no—
160D4⋊C8—22no—
161D4⋊C8—22no—
162D4⋊C8—22no—
163D4⋊C8—22no—
164#322—22no—
165#322—22no—
166C22×D4C22×C411yescharacteristic
167C22×D4C22×C411yescharacteristic
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C2×D8—22no—
173C2×D8—22no—
174C2×D8C4⋊C414yes—
175C2×M16—42no—
176C2×M16—42no—
177C2×M16—81no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—42no—
185C22×C8—42no—
186C22×C8—42no—
187C22×C8—42no—
188C22×C8—42no—
189C22×C8—42no—
190C22×C8—81no—
191C22×C8—81no—
192C22×C8—81no—
193C22×C8C22×C411yescharacteristic
194C22×C8C22×C411yescharacteristic
195C22×C8C22×C411yescharacteristic
196C22×C8C22×C411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C22×C8C2×D411yescharacteristic
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—41no—
209C4×D4C2×D412yes—
210C4×D4C2×D412yes—
211C4×D4C2×Q812yes—
212C4×D4C2×Q812yes—
213#117—81no—
214#117—81no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—81no—
236C2×C42—81no—
237C2×C42C22×C411yescharacteristic
238C2×C42C22×C411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42#4111yescharacteristic
243C2×C42C2411yescharacteristic
244#107—22no—
245#107—41no—
246#107C4⋊C414yes—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—41no—
256C4⋊C8—41no—
257C4⋊C8—42no—
258C4⋊C8—81no—
259C4⋊C8—81no—
260C4⋊C8—81no—
261C4⋊C8—81no—
262#102—22no—
263#102—22no—
264#102—22no—
265#102—41no—
266#102C4⋊C414yes—
267#102C4⋊C414yes—
268C22⋊C8—42no—
269C22⋊C8—42no—
270C22⋊C8—42no—
271C22⋊C8—42no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8C2×D412yes—
283C4×C8C2×Q812yes—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290C24—21no—
291C24—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4C2×C4⋊C412yes—
301C2×D4C2×C4⋊C412yes—
302C2×D4#13112yes—
303C2×D4#13112yes—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—81no—
326C22×C4—81no—
327C22×C4—81no—
328C22×C4C4×D411yescharacteristic
329C22×C4C4×D411yescharacteristic
330C22×C4C4×D411yescharacteristic
331C22×C4C4×D411yescharacteristic
332C22×C4C23×C411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C2×#4111yescharacteristic
335D8—28no—
336M16—84no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—22no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
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—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—84no—
378C2×C8—84no—
379C2×C8—84no—
380C2×C8C2×C4⋊C412yes—
381C2×C8C2×C4⋊C412yes—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—81no—
392C4⋊C4—81no—
393C4⋊C4C2×C4⋊C412yes—
394C4⋊C4C2×C4⋊C412yes—
395C4⋊C4#13112yes—
396C22⋊C4—24no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—22no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—81no—
404C22⋊C4—81no—
405C22⋊C4—81no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—81no—
421C42—81no—
422C42C22×D411yescharacteristic
423C42C22×Q811yescharacteristic
424C42H(5,𝔽2)11yescharacteristic
425C422−1+411yescharacteristic
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23C2×C4×D411yescharacteristic
441D4—22no—
442D4—22no—
443D4—22no—
444D4—22no—
445D4—22no—
446D4—41no—
447D4—41no—
448D4#42414yes—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
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—84no—
490C2×C4—84no—
491C2×C4C22×C4⋊C411yescharacteristic
492C2×C4#52011yescharacteristic
493C2×C4D4211yescharacteristic
494C2×C4#54711yescharacteristic
495C2×C4D4×Q811yescharacteristic
496C2×C4#55411yescharacteristic
497C2×C4C2×#13111yescharacteristic
498C2×C4C2×#13111yescharacteristic
499C2×C4C2×#13111yescharacteristic
500C2×C4#56811yescharacteristic
501C2×C4#57511yescharacteristic
502C2×C4#57511yescharacteristic
503C2×C4#57611yescharacteristic
504C2×C4#57611yescharacteristic
505C8—22no—
506C8—22no—
507C8—42no—
508C8—41no—
509C8—41no—
510C8—41no—
511C8—41no—
512C8—41no—
513C8—41no—
514C8—81no—
515C8—81no—
516C8—81no—
517C8—81no—
518C8—81no—
519C22—24no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22D4×C4⋊C411yescharacteristic
534C22#292911yescharacteristic
535C22#293311yescharacteristic
536C22#294311yescharacteristic
537C22#294811yescharacteristic
538C22#295511yescharacteristic
539C22#295611yescharacteristic
540C4—21no—
541C4—21no—
542C4—41no—
543C4—42no—
544C4—41no—
545C4—41no—
546C4—42no—
547C4—42no—
548C4—41no—
549C4—81no—
550C4—81no—
551C4—81no—
552C4C2×#42411yescharacteristic
553C4#290611yescharacteristic
554C4#299211yescharacteristic
555C4#299411yescharacteristic
556C4#327311yescharacteristic
557C4#327311yescharacteristic
558C4#338911yescharacteristic
559C4#338911yescharacteristic
560C2—44no—
561C2—82no—
562C2#1790911yescharacteristic
563C2#1792211yescharacteristic
564C2#1800211yescharacteristic
565C2#1802611yescharacteristic
566C2#2264711yescharacteristic
567C2#2267211yescharacteristic
568C2#2268711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D8C2×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
1. #22701C2semidirect
2. #22696C2semidirect
6. #9599C2semidirect
7. #9595C2semidirect
8. #8847C2semidirect
9. #8843C2semidirect
10. #8843C2semidirect
11. #8843C2semidirect
12. #8811C2semidirect
13. #8569C2semidirect
14. #8565C2semidirect
15. #8265C2semidirect

Maximal subgroups

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

subgroupindex
#227012
#226962
#211132
C2×#15862
C2×#15802
#95992
#95952
#88472
#88432
#88112
#85692
#85652
#82652