Finite Group Databasebeta

← #226836

Subgroups — #226836

Order 512.

689 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22839C211yesmaximal, characteristic, complemented
2#22799C211yesmaximal, characteristic, complemented
3C4×#442C211yesmaximal, characteristic
4#9659C211yesmaximal, characteristic
5#9657C211yesmaximal, characteristic
6#9546C211yesmaximal, characteristic, complemented
7#9546C211yesmaximal, characteristic, complemented
8#9545C211yesmaximal, characteristic, complemented
9#9545C211yesmaximal, characteristic, complemented
10#8808C211yesmaximal, characteristic, complemented
11#8565C211yesmaximal, characteristic, complemented
12#8563C211yesmaximal, characteristic, complemented
13#8286C211yesmaximal, characteristic, complemented
14#8283C211yesmaximal, characteristic, complemented
15#8262C211yesmaximal, characteristic, complemented
16C2×#442C411yescharacteristic
17C2×#442C411yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×#432—21no—
20C42×D4C2211yescharacteristic
21C42×D4C2211yescharacteristic
22#1954C2211yescharacteristic
23#1954C2211yescharacteristic
24#1954C2211yescharacteristic
25#1954C2211yescharacteristic
26#1924—21no—
27#1924—21no—
28#1924—21no—
29#1924—21no—
30#1842C2211yescharacteristic
31#1831C2211yescharacteristic
32#1817—21no—
33#1797—21no—
34#1776C411yescharacteristic
35#1776C411yescharacteristic
36C4×#106C2211yescharacteristic
37#1770C2211yescharacteristic
38#1770C2211yescharacteristic
39#1770C2211yescharacteristic
40#1770C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42C4×C4⋊C8C2211yescharacteristic
43#1762C411yescharacteristic
44#1762C411yescharacteristic
45#1762C411yescharacteristic
46#1762C411yescharacteristic
47C4×#102C2211yescharacteristic
48C4×#102C2211yescharacteristic
49#1749—21no—
50C4×#97C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×#334C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×D8C2×C411yescharacteristic
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C2×C4×D4C2311yescharacteristic
74C22×C42—21no—
75C22×C42—21no—
76#442—24no—
77#442—24no—
78C4×D8—22no—
79C4×D8—22no—
80#432—44no—
81C2×#107C2×C411yescharacteristic
82C2×#106C2×C411yescharacteristic
83C2×#106C2×C411yescharacteristic
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8D411yescharacteristic
88C2×C4⋊C8D411yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×#102C2×C411yescharacteristic
94C2×#102C2×C411yescharacteristic
95C2×#102C2×C411yescharacteristic
96C2×#102C2×C411yescharacteristic
97C2×#102C2×C411yescharacteristic
98C2×#102C2×C411yescharacteristic
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×#97—21no—
108C2×#97—21no—
109C2×#97—21no—
110C2×#97—21no—
111C2×#97C2×C411yescharacteristic
112C2×#97C2×C411yescharacteristic
113C2×#97C2×C411yescharacteristic
114C2×#97C2×C411yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—21no—
133C43C2311yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336D411yescharacteristic
148#336D411yescharacteristic
149#335—22no—
150#335—41no—
151#334—22no—
152#334—41no—
153D4⋊C8—41no—
154D4⋊C8—41no—
155D4⋊C8—41no—
156D4⋊C8—41no—
157#322—22no—
158C82—22no—
159C22×D4C22×C411yescharacteristic
160C22×D4C22×C411yescharacteristic
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C2×D8—22no—
168C2×D8—22no—
169C2×D8M1614yes—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8C22×C411yescharacteristic
177C22×C8C22×C411yescharacteristic
178C22×C8C22×C411yescharacteristic
179C22×C8C22×C411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—41no—
199C4×D4—41no—
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42—81no—
221C2×C42—81no—
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42#4111yescharacteristic
227C2×C42#4111yescharacteristic
228C2×C42C2411yescharacteristic
229#107—22no—
230#106—22no—
231#106—22no—
232C4⋊C8—22no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—41no—
242C4⋊C8—42no—
243C4⋊C8—42no—
244C4⋊C8—81no—
245#102—22no—
246#102—22no—
247#102—22no—
248#102—22no—
249#102—22no—
250#102—22no—
251C22⋊C8—42no—
252C22⋊C8—42no—
253C22⋊C8—42no—
254C22⋊C8—42no—
255C22⋊C8—81no—
256C22⋊C8—81no—
257C22⋊C8—81no—
258C22⋊C8—81no—
259#97—22no—
260#97—22no—
261#97—41no—
262#97—42no—
263#97—42no—
264#97—81no—
265#97—81no—
266#97M1614yes—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8#4112yes—
278C4×C8#4112yes—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285C24—21no—
286C24—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4C2×M1612yes—
298C2×D4C2×M1612yes—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—81no—
326C22×C4—81no—
327C22×C4—81no—
328C22×C4—81no—
329C22×C4C4×D411yescharacteristic
330C22×C4C4×D411yescharacteristic
331C22×C4C4×D411yescharacteristic
332C22×C4C4×D411yescharacteristic
333C22×C4C23×C411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C2×#4111yescharacteristic
336D8—28no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—22no—
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—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×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8#11712yes—
374C2×C8#11712yes—
375C2×C8C2×M1612yes—
376C2×C8C2×M1612yes—
377C2×C8C2×M1612yes—
378C2×C8AGL(1,ℤ/8)12yes—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
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—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C42—21no—
400C42—21no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—41no—
406C42—41no—
407C42—42no—
408C42—42no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42—41no—
413C42—81no—
414C42—81no—
415C42C2×M1612yes—
416C42#13112yes—
417C42C2×#4111yescharacteristic
418C42C2×#4111yescharacteristic
419C42H(5,𝔽2)11yescharacteristic
420C422−1+411yescharacteristic
421C23—22no—
422C23—22no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23C2×C4×D411yescharacteristic
436D4—22no—
437D4—22no—
438D4—22no—
439D4—22no—
440D4—41no—
441D4—41no—
442D4—41no—
443D4—41no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—41no—
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—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—84no—
475C2×C4—84no—
476C2×C4—84no—
477C2×C4—84no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4#43612yes—
487C2×C4#43612yes—
488C2×C4C2×#11711yescharacteristic
489C2×C4#52011yescharacteristic
490C2×C4#54711yescharacteristic
491C2×C4#55011yescharacteristic
492C2×C4#55511yescharacteristic
493C2×C4#55611yescharacteristic
494C2×C4C22×M1611yescharacteristic
495C2×C4C22×M1611yescharacteristic
496C2×C4C2×#13111yescharacteristic
497C2×C4#56811yescharacteristic
498C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
499C2×C4#57511yescharacteristic
500C2×C4#57611yescharacteristic
501C2×C4#57711yescharacteristic
502C8—22no—
503C8—22no—
504C8—22no—
505C8—42no—
506C8—41no—
507C8—41no—
508C8—41no—
509C8—41no—
510C8—81no—
511C8—81no—
512C8—81no—
513C8—81no—
514C8—81no—
515C8#43214yescomplemented
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22#235411yescharacteristic
531C22C2×#43611yescharacteristic
532C22D4×M1611yescharacteristic
533C22C4×AGL(1,ℤ/8)11yescharacteristic
534C22#294611yescharacteristic
535C22#295511yescharacteristic
536C22#295611yescharacteristic
537C4—21no—
538C4—21no—
539C4—22no—
540C4—22no—
541C4—41no—
542C4—41no—
543C4—81no—
544C4—81no—
545C4—81no—
546C4—81no—
547C4—81no—
548C4—81no—
549C4C2×#43211yescharacteristic
550C4#292511yescharacteristic
551C4#298911yescharacteristic
552C4#299011yescharacteristic
553C4#334511yescharacteristic
554C4#334711yescharacteristic
555C4#339911yescharacteristic
556C4#340111yescharacteristic
557C2—82no—
558C2—82no—
559C2#1789111yescharacteristic
560C2#1789311yescharacteristic
561C2#1804611yescharacteristic
562C2#1805011yescharacteristic
563C2#2279911yescharacteristic
564C2#2280411yescharacteristic
565C2#2281711yescharacteristic
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. #22839C2semidirect
2. #22799C2semidirect
6. #9546C2semidirect
7. #9546C2semidirect
8. #9545C2semidirect
9. #9545C2semidirect
10. #8808C2semidirect
11. #8565C2semidirect
12. #8563C2semidirect
13. #8286C2semidirect
14. #8283C2semidirect
15. #8262C2semidirect
515. C8#432semidirect

Maximal subgroups

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

subgroupindex
#228392
#227992
C4×#4422
#96592
#96572
#95462
#95452
#88082
#85652
#85632
#82862
#82832
#82622