Finite Group Databasebeta

← #308489

Subgroups — #308489

Order 512.

671 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26462C211yesmaximal, characteristic, complemented
2#26044C211yesmaximal, characteristic
3#22968C211yesmaximal, characteristic
4D4×#106C211yesmaximal, characteristic
5#10685C211yesmaximal, characteristic, complemented
6#10654C211yesmaximal, characteristic, complemented
7#10602C211yesmaximal, characteristic, complemented
8#10574C211yesmaximal, characteristic, complemented
9#9618C211yesmaximal, characteristic, complemented
10#9538C211yesmaximal, characteristic, complemented
11#9219C211yesmaximal, characteristic
12#8981C211yesmaximal, characteristic
13#8925C211yesmaximal, characteristic, complemented
14#8886C211yesmaximal, characteristic
15#8843C211yesmaximal, characteristic
16C2×#475—21no—
17C2×#476—21no—
18C2×#474C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C22×#106—21no—
21#2551C2211yescharacteristic
22#2365C2211yescharacteristic
23D4×C4⋊C4C2211yescharacteristic
24D4×C4⋊C4C2211yescharacteristic
25#2088—21no—
26#2066—21no—
27#2058—21no—
28#2024—21no—
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1981C2211yescharacteristic
32#1956C2211yescharacteristic
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1948C2211yescharacteristic
36#1948C2211yescharacteristic
37#1948C2211yescharacteristic
38#1943—21no—
39#1939C2211yescharacteristic
40#1939C2211yescharacteristic
41#1921—21no—
42#1865—21no—
43#1864—21no—
44#1864—21no—
45#1853—21no—
46#1853—21no—
47#1844C2211yescharacteristic
48#1844C2211yescharacteristic
49#1844C2211yescharacteristic
50#1807—21no—
51#1802C2211yescharacteristic
52C4×#106C2211yescharacteristic
53#1770C2211yescharacteristic
54#1762—21no—
55C4×#102C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1273C2211yescharacteristic
67C23×C8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8C2311yescharacteristic
71C2×C4×Q8—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4C2311yescharacteristic
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77#476—44no—
78#475—44no—
79#474—22no—
80#474Q814yescomplemented
81C8×D4—22no—
82C8×D4—22no—
83C2×#107—21no—
84C2×#106—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—44no—
94C2×#106—41no—
95C2×#106D411yescharacteristic
96C2×#106D411yescharacteristic
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×#103—21no—
104C2×#103—21no—
105C2×#103—21no—
106C2×#103—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×#102D411yescharacteristic
110C2×#102D411yescharacteristic
111C2×C22⋊C8—21no—
112C2×#97—21no—
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116#395—41no—
117#391—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#387—41no—
124#385—41no—
125#385—41no—
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#384C2311yescharacteristic
129#380—21no—
130#380—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4C2311yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—41no—
144#340D411yescharacteristic
145#340D411yescharacteristic
146#340D411yescharacteristic
147#340D411yescharacteristic
148#336—21no—
149#336—21no—
150#336—21no—
151#335—22no—
152#332—22no—
153D4⋊C8—22no—
154C22×Q8—21no—
155C22×Q8—41no—
156C22×D4C2×D411yescharacteristic
157C23×C4—21no—
158C23×C4—41no—
159C23×C4—41no—
160C22×C8—24no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—42no—
167C22×C8—42no—
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—42no—
176C4⋊Q8—42no—
177C4⋊Q8—81no—
178C4⋊Q8—81no—
179C4⋊Q8C2×Q812yes—
180C4×Q8—42no—
181C4×Q8—81no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—42no—
185C4×D4—42no—
186C4×D4—81no—
187C4×D4—81no—
188C4×D4C2×Q812yes—
189C4×D4#4112yes—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—81no—
209C2×C4⋊C4—81no—
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—41no—
224C2×C42C2411yescharacteristic
225#107—24no—
226#106—22no—
227#106—22no—
228#106—42no—
229#106—42no—
230#106—42no—
231#106—42no—
232#106—42no—
233#106—42no—
234#106—81no—
235#106—81no—
236#106—81no—
237#106—82no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—42no—
245C4⋊C8—42no—
246C4⋊C8C2×Q812yes—
247C4⋊C8#4112yes—
248#103—42no—
249#103—42no—
250#103—42no—
251#103—81no—
252#102—24no—
253#102—22no—
254#102—42no—
255#102SD1614yes—
256C22⋊C8—24no—
257#97—81no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—41no—
268#95—41no—
269#95—41no—
270#95—41no—
271#95—41no—
272C24—21no—
273C2×Q8—22no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—82no—
280C2×Q8—82no—
281C2×Q8—161no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4C22⋊Q812yes—
286C2×D4C2×SD1612yes—
287C2×D4#13712yes—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—161no—
310C22×C4—161no—
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4H(5,𝔽2)11yescharacteristic
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—21no—
321C2×C8—22no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—48no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8C2×SD1612yes—
341C2×C8AGL(1,ℤ/8)12yes—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—81no—
372C4⋊C4—81no—
373C4⋊C4—81no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4—82no—
377C4⋊C4C22⋊Q812yes—
378C4⋊C4C2×SD1612yes—
379C4⋊C4AGL(1,ℤ/8)12yes—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—82no—
383C22⋊C4—82no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—82no—
391C42C22×Q811yescharacteristic
392C42C2×#4111yescharacteristic
393C42C2×#4111yescharacteristic
394C422−1+411yescharacteristic
395C23—22no—
396C23—22no—
397C23—22no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23D4211yescharacteristic
403Q8—42no—
404Q8—42no—
405Q8—81no—
406Q8—81no—
407Q8—164no—
408D4—22no—
409D4—22no—
410D4—22no—
411D4#47514yescomplemented
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—164no—
452C2×C4—164no—
453C2×C4C2×C22⋊Q811yescharacteristic
454C2×C4#52511yescharacteristic
455C2×C4#53911yescharacteristic
456C2×C4#54711yescharacteristic
457C2×C4D4×Q811yescharacteristic
458C2×C4#55411yescharacteristic
459C2×C4C22×SD1611yescharacteristic
460C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
461C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
462C2×C4C2×#13711yescharacteristic
463C2×C4#57711yescharacteristic
464C2×C4#57711yescharacteristic
465C2×C4#57711yescharacteristic
466C2×C4#57711yescharacteristic
467C8—22no—
468C8—22no—
469C8—42no—
470C8—42no—
471C8—41no—
472C8—42no—
473C8—41no—
474C8—81no—
475C8—81no—
476C22—24no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22#244911yescharacteristic
484C22#326511yescharacteristic
485C22#326611yescharacteristic
486C22#327211yescharacteristic
487C22D4×SD1611yescharacteristic
488C22#328911yescharacteristic
489C22#329111yescharacteristic
490C4—21no—
491C4—21no—
492C4—42no—
493C4—42no—
494C4—42no—
495C4—82no—
496C4—81no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4—161no—
501C4—161no—
502C4—161no—
503C4C2×#47511yescharacteristic
504C4#308411yescharacteristic
505C4#310911yescharacteristic
506C4#312511yescharacteristic
507C4#333511yescharacteristic
508C4#334411yescharacteristic
509C4#338411yescharacteristic
510C4#338411yescharacteristic
511C2—44no—
512C2#1976411yescharacteristic
513C2#1989011yescharacteristic
514C2#2007311yescharacteristic
515C2#2017911yescharacteristic
516C2#2634011yescharacteristic
517C2#2639211yescharacteristic
518C2#2640111yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×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. #26462C2semidirect
5. #10685C2semidirect
6. #10654C2semidirect
7. #10602C2semidirect
8. #10574C2semidirect
9. #9618C2semidirect
10. #9538C2semidirect
13. #8925C2semidirect
80. #474Q8semidirect
411. D4#475semidirect

Maximal subgroups

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

subgroupindex
#264622
#260442
#229682
D4×#1062
#106852
#106542
#106022
#105742
#96182
#95382
#92192
#89812
#89252
#88862
#88432