Finite Group Databasebeta

← #203353

Subgroups — #203353

Order 512.

762 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21270C211yesmaximal, characteristic, complemented
2C4×#442C211yesmaximal, characteristic
3#20494C211yesmaximal, characteristic, complemented
4#9542C211yesmaximal, characteristic, complemented
5#9541C211yesmaximal, characteristic, complemented
6#9535C211yesmaximal, characteristic, complemented
7#9534C211yesmaximal, characteristic, complemented
8#8951C211yesmaximal, characteristic
9#8951C211yesmaximal, characteristic
10#8849C211yesmaximal, characteristic, complemented
11#8847C211yesmaximal, characteristic, complemented
12#8276C211yesmaximal, characteristic, complemented
13#8172C211yesmaximal, characteristic, complemented
14#8172C211yesmaximal, characteristic, complemented
15#8136C211yesmaximal, characteristic, complemented
16C2×#442C411yescharacteristic
17C2×#442C411yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C2×#436C2211yescharacteristic
21C2×#432—21no—
22C2×C4×M16—21no—
23C42×D4C2211yescharacteristic
24C42×D4C2211yescharacteristic
25#1954C2211yescharacteristic
26#1954C2211yescharacteristic
27#1939C2211yescharacteristic
28#1939C2211yescharacteristic
29#1924—21no—
30#1924—21no—
31#1776C411yescharacteristic
32#1776C411yescharacteristic
33C4×#106C2211yescharacteristic
34C8×C4⋊C4C2211yescharacteristic
35C8×C4⋊C4C2211yescharacteristic
36#1767C2211yescharacteristic
37#1767C2211yescharacteristic
38#1762C411yescharacteristic
39#1762C411yescharacteristic
40#1762C411yescharacteristic
41#1762C411yescharacteristic
42C4×#102C2211yescharacteristic
43C4×#102C2211yescharacteristic
44#1753—21no—
45#1753—21no—
46#1749—21no—
47#1749—21no—
48C4×#97C2211yescharacteristic
49C4×#97C2211yescharacteristic
50C4×#97C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1273C2211yescharacteristic
64#1273C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×D8C2×C411yescharacteristic
68C22×M16—21no—
69C22×M16—21no—
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C22×C42—21no—
77C22×C42—21no—
78#442—24no—
79#442—24no—
80C4×D8—22no—
81C4×D8—22no—
82#436—22no—
83#436—22no—
84#436—22no—
85#436—22no—
86#432—44no—
87C2×#107C2×C411yescharacteristic
88C2×#106C2×C411yescharacteristic
89C2×#106C2×C411yescharacteristic
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×#102C2×C411yescharacteristic
97C2×#102C2×C411yescharacteristic
98C2×#102C2×C411yescharacteristic
99C2×#102C2×C411yescharacteristic
100C2×#102C2×C411yescharacteristic
101C2×#102C2×C411yescharacteristic
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C4×M16—44no—
109C2×#97—21no—
110C2×#97—21no—
111C2×#97—21no—
112C2×#97—21no—
113C2×#97C2×C411yescharacteristic
114C2×#97C2×C411yescharacteristic
115C2×#97C2×C411yescharacteristic
116C2×#97C2×C411yescharacteristic
117C2×#97D411yescharacteristic
118C2×#97D411yescharacteristic
119C2×#97D411yescharacteristic
120C2×#97D411yescharacteristic
121C2×C4×C8—21no—
122C2×C4×C8—21no—
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138C4×C22⋊C4—21no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C43C2311yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#335—22no—
155#334—22no—
156D4⋊C8—22no—
157D4⋊C8—22no—
158D4⋊C8—22no—
159D4⋊C8—22no—
160C22×D4C22×C411yescharacteristic
161C22×D4C22×C411yescharacteristic
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C2×D8—22no—
169C2×D8—22no—
170C2×D8—22no—
171C2×M16—24no—
172C2×M16—24no—
173C2×M16—42no—
174C2×M16—42no—
175C2×M16—42no—
176C2×M16—42no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8C22×C411yescharacteristic
184C22×C8C22×C411yescharacteristic
185C22×C8C22×C411yescharacteristic
186C22×C8C22×C411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C22×C8C2×D411yescharacteristic
189C22×C8C2×D411yescharacteristic
190C22×C8C2×D411yescharacteristic
191C4×D4—22no—
192C4×D4—22no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4—21no—
201C4×D4—41no—
202C4×D4—41no—
203C4×D4—41no—
204C4×D4—41no—
205C4×D4#4112yes—
206C4×D4#4112yes—
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42C22×C411yescharacteristic
230C2×C42C22×C411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42#4111yescharacteristic
234C2×C42#4111yescharacteristic
235C2×C42C2411yescharacteristic
236#107—22no—
237#106—22no—
238#106M1614yescomplemented
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—42no—
246C4⋊C8—42no—
247C4⋊C8#4112yes—
248C4⋊C8#4112yes—
249#102—22no—
250#102—22no—
251#102—22no—
252#102—22no—
253#102—22no—
254#102M1614yescomplemented
255C22⋊C8—24no—
256C22⋊C8—24no—
257C22⋊C8—42no—
258C22⋊C8—42no—
259C22⋊C8—42no—
260C22⋊C8—42no—
261#97—22no—
262#97—22no—
263#97—22no—
264#97—22no—
265#97—41no—
266#97—41no—
267#97—41no—
268#97—41no—
269#97—42no—
270#97—42no—
271#97—42no—
272#97—42no—
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×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—42no—
286C4×C8—42no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293C24—21no—
294C24—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4#11712yes—
306C2×D4C2×M1612yes—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—81no—
334C22×C4—81no—
335C22×C4—81no—
336C22×C4—81no—
337C22×C4C4×D411yescharacteristic
338C22×C4C4×D411yescharacteristic
339C22×C4C4×D411yescharacteristic
340C22×C4C4×D411yescharacteristic
341C22×C4C23×C411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C2×#4111yescharacteristic
344D8—28no—
345M16—48no—
346M16—48no—
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—22no—
356C2×C8—22no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—22no—
362C2×C8—22no—
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—41no—
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×C8C2×M1612yes—
386C2×C8C2×M1612yes—
387C2×C8AGL(1,ℤ/8)12yes—
388C2×C8AGL(1,ℤ/8)12yes—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4#11712yes—
398C4⋊C4C2×M1612yes—
399C4⋊C4C2×M1612yes—
400C4⋊C4C2×M1612yes—
401C22⋊C4—22no—
402C22⋊C4—22no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—41no—
418C42—41no—
419C42—42no—
420C42—42no—
421C42—42no—
422C42—42no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42C2×#4111yescharacteristic
428C42C2×#4111yescharacteristic
429C42C2×#4111yescharacteristic
430C42C2×#4111yescharacteristic
431C23—22no—
432C23—22no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23C2×C4×D411yescharacteristic
446D4—22no—
447D4—22no—
448D4—22no—
449D4—22no—
450D4—41no—
451D4—41no—
452D4—41no—
453D4—41no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—84no—
491C2×C4—84no—
492C2×C4—84no—
493C2×C4—84no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4C2×#11711yescharacteristic
499C2×C4C4×#4111yescharacteristic
500C2×C4#52511yescharacteristic
501C2×C4#52511yescharacteristic
502C2×C4#52911yescharacteristic
503C2×C4#52911yescharacteristic
504C2×C4C22×M1611yescharacteristic
505C2×C4C22×M1611yescharacteristic
506C2×C4#56811yescharacteristic
507C2×C4#56811yescharacteristic
508C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
509C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
510C2×C4#57511yescharacteristic
511C2×C4#57511yescharacteristic
512C8—22no—
513C8—22no—
514C8—22no—
515C8—22no—
516C8—42no—
517C8—42no—
518C8—42no—
519C8—41no—
520C8—42no—
521C8—41no—
522C8—41no—
523C8—41no—
524C8—81no—
525C8—81no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22#229011yescharacteristic
541C22D4×M1611yescharacteristic
542C22D4×M1611yescharacteristic
543C22C4×AGL(1,ℤ/8)11yescharacteristic
544C22C4×AGL(1,ℤ/8)11yescharacteristic
545C22#295611yescharacteristic
546C22#295611yescharacteristic
547C4—21no—
548C4—21no—
549C4—41no—
550C4—42no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—81no—
558C4—81no—
559C4#292011yescharacteristic
560C4#292311yescharacteristic
561C4#297011yescharacteristic
562C4#297111yescharacteristic
563C4#310911yescharacteristic
564C4#311111yescharacteristic
565C4#311411yescharacteristic
566C4#311511yescharacteristic
567C2—82no—
568C2—82no—
569C2#1804611yescharacteristic
570C2#1804811yescharacteristic
571C2#1804811yescharacteristic
572C2#1804811yescharacteristic
573C2#2110811yescharacteristic
574C2#2112211yescharacteristic
575C2#2113911yescharacteristic
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×#97 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21270C2semidirect
3. #20494C2semidirect
4. #9542C2semidirect
5. #9541C2semidirect
6. #9535C2semidirect
7. #9534C2semidirect
10. #8849C2semidirect
11. #8847C2semidirect
12. #8276C2semidirect
13. #8172C2semidirect
14. #8172C2semidirect
15. #8136C2semidirect
238. #106M16semidirect
254. #102M16semidirect

Maximal subgroups

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

subgroupindex
#212702
C4×#4422
#204942
#95422
#95412
#95352
#95342
#89512
#88492
#88472
#82762
#81722
#81362