Finite Group Databasebeta

← #225653

Subgroups — #225653

Order 512.

689 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22805C211yesmaximal, characteristic
2#22743C211yesmaximal, characteristic, complemented
3#22671C211yesmaximal, characteristic
4#22663C211yesmaximal, characteristic
5C2×#1592C211yesmaximal, characteristic
6#9618C211yesmaximal, characteristic, complemented
7#9614C211yesmaximal, characteristic, complemented
8#8897C211yesmaximal, characteristic
9#8896C211yesmaximal, characteristic, complemented
10#8887C211yesmaximal, characteristic, complemented
11#8886C211yesmaximal, characteristic
12#8819C211yesmaximal, characteristic, complemented
13#8654C211yesmaximal, characteristic
14#8651C211yesmaximal, characteristic, complemented
15#8291C211yesmaximal, characteristic, complemented
16C2×#446C2211yescharacteristic
17C2×#440—21no—
18C2×C4×SD16C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C2×#428—21no—
21#2355C2211yescharacteristic
22#2354C2211yescharacteristic
23Q8×C4⋊C4C2211yescharacteristic
24D4×C4⋊C4C2211yescharacteristic
25#1968—21no—
26#1949—21no—
27#1948C2211yescharacteristic
28#1948C2211yescharacteristic
29#1947—21no—
30#1947C2211yescharacteristic
31#1947C2211yescharacteristic
32#1944—21no—
33#1927—21no—
34#1926—21no—
35#1863—21no—
36#1862—21no—
37#1854—21no—
38#1853—21no—
39#1845C2211yescharacteristic
40#1844C2211yescharacteristic
41#1844C2211yescharacteristic
42#1844C2211yescharacteristic
43#1836—21no—
44#1834C2211yescharacteristic
45#1834C2211yescharacteristic
46#1809—21no—
47#1803C2211yescharacteristic
48#1803C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51#1776—21no—
52C4×#107C2211yescharacteristic
53C4×#106—21no—
54#1770C2211yescharacteristic
55C8×C4⋊C4C2211yescharacteristic
56#1592C418yescomplemented
57C2×#335C2211yescharacteristic
58C2×#335C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68C22×SD16D411yescharacteristic
69C22×M16—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8C2311yescharacteristic
72C2×C4×D4—21no—
73C2×C4×D4C2311yescharacteristic
74C2×#117—21no—
75C22×C4⋊C4—21no—
76#446—22no—
77#446C2×C414yescomplemented
78#440—44no—
79C4×SD16C2×C414yes—
80C4×SD16Q814yes—
81#436—22no—
82#436C2×C414yes—
83#428—44no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107D411yescharacteristic
89C2×#107D411yescharacteristic
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—21no—
94C2×#106—21no—
95C2×#106—21no—
96C2×#106—41no—
97C2×#106D411yescharacteristic
98C2×C4⋊C8—21no—
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×#103—21no—
104C2×#103—21no—
105C2×#103D411yescharacteristic
106C2×#102—21no—
107C2×#102—21no—
108C2×#102D411yescharacteristic
109C2×C22⋊C8—21no—
110C2×#97—21no—
111C2×#97—21no—
112C2×C4×C8—21no—
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116#391—21no—
117#390—21no—
118#389—21no—
119#389—21no—
120#389—41no—
121#388—41no—
122#387—41no—
123#385—41no—
124#384—21no—
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#384C2311yescharacteristic
129#382—21no—
130#380—21no—
131#380—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4C2311yescharacteristic
137C4×C4⋊C4C2311yescharacteristic
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—41no—
149#340D411yescharacteristic
150#340D411yescharacteristic
151#336—21no—
152#336—21no—
153#335—22no—
154#335C2×C414yescomplemented
155Q8⋊C8C2×C414yescomplemented
156D4⋊C8C2×C414yes—
157#322C2×C414yescomplemented
158C22×Q8C2×D411yescharacteristic
159C22×D4C2×D411yescharacteristic
160C23×C4—21no—
161C23×C4—41no—
162C2×SD16—22no—
163C2×SD16—22no—
164C2×SD16C4⋊C414yes—
165C2×M16—24no—
166C2×M16—42no—
167C2×M16—42no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C4×Q8—21no—
178C4×Q8—21no—
179C4×Q8—42no—
180C4×Q8—81no—
181C4×Q8C22×C412yes—
182C4×Q8C2×Q812yes—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—42no—
186C4×D4—81no—
187C4×D4C22×C412yes—
188C4×D4C2×Q812yes—
189#117—42no—
190#117—81no—
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—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—81no—
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—41no—
223C2×C42—41no—
224C2×C42—41no—
225C2×C42C2411yescharacteristic
226#107—22no—
227#107—41no—
228#107—42no—
229#107—42no—
230#107—42no—
231#107—81no—
232#106—42no—
233#106—42no—
234#106—42no—
235#106—42no—
236#106—81no—
237#106—81no—
238#106—82no—
239#106C4⋊C414yes—
240C4⋊C8—24no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8C22×C412yes—
248C4⋊C8C22×C412yes—
249#103—42no—
250#103—81no—
251#103C4⋊C414yes—
252#102—42no—
253#102—81no—
254#102C4⋊C414yes—
255C22⋊C8—24no—
256#97—24no—
257#97—42no—
258C4×C8—21no—
259C4×C8—42no—
260C4×C8C22×C412yes—
261C4×C8C22×C412yes—
262C4×C8C22×C412yes—
263C4×C8C2×Q812yes—
264C4×C8#4112yes—
265#95—21no—
266#95—21no—
267#95—41no—
268#95—41no—
269#95—41no—
270#95—41no—
271#95—41no—
272#95—41no—
273C24—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8C2×C4⋊C412yes—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4C2×C4⋊C412yes—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4—161no—
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4H(5,𝔽2)11yescharacteristic
309SD16—28no—
310M16—48no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8C2×C4⋊C412yes—
333C2×C8C2×C4⋊C412yes—
334C2×C8C4×D412yes—
335C2×C8C4×D412yes—
336C2×C8AGL(1,ℤ/8)12yes—
337C2×C8#13712yes—
338C4⋊C4—22no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—22no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
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—81no—
366C4⋊C4—81no—
367C4⋊C4—81no—
368C4⋊C4—81no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—161no—
372C4⋊C4C2×C4⋊C412yes—
373C4⋊C4C2×C4⋊C412yes—
374C4⋊C4AGL(1,ℤ/8)12yes—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—82no—
378C22⋊C4—82no—
379C22⋊C4—161no—
380C42—22no—
381C42—22no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42—82no—
387C42—82no—
388C42—161no—
389C42C23×C411yescharacteristic
390C42C22×Q811yescharacteristic
391C42C2×#4111yescharacteristic
392C422−1+411yescharacteristic
393C23—22no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23D4211yescharacteristic
401Q8—22no—
402Q8—22no—
403Q8—41no—
404Q8—41no—
405D4—22no—
406D4—22no—
407D4—41no—
408D4—41no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—41no—
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—41no—
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—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
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—164no—
449C2×C4C22×C4⋊C411yescharacteristic
450C2×C4C2×C4×D411yescharacteristic
451C2×C4#52011yescharacteristic
452C2×C4#54711yescharacteristic
453C2×C4D4×Q811yescharacteristic
454C2×C4#55411yescharacteristic
455C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
456C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
457C2×C4C2×#13711yescharacteristic
458C2×C4#57511yescharacteristic
459C2×C4#57711yescharacteristic
460C2×C4#57711yescharacteristic
461C2×C4#57711yescharacteristic
462C2×C4#57811yescharacteristic
463C8—22no—
464C8—22no—
465C8—22no—
466C8—42no—
467C8—42no—
468C8—42no—
469C8—81no—
470C8—81no—
471C8#42814yescomplemented
472C22—42no—
473C22—42no—
474C22—42no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22—81no—
479C22D4×C4⋊C411yescharacteristic
480C22#326511yescharacteristic
481C22#326911yescharacteristic
482C22#327211yescharacteristic
483C22#327811yescharacteristic
484C22#328811yescharacteristic
485C22#328911yescharacteristic
486C4—21no—
487C4—21no—
488C4—42no—
489C4—42no—
490C4—42no—
491C4—42no—
492C4—81no—
493C4—81no—
494C4—81no—
495C4—161no—
496C4—161no—
497C4—161no—
498C4—161no—
499C4C2×#42811yescharacteristic
500C4#291211yescharacteristic
501C4C4×AGL(1,ℤ/8)11yescharacteristic
502C4#294711yescharacteristic
503C4#335411yescharacteristic
504C4#335611yescharacteristic
505C4#338611yescharacteristic
506C4#338811yescharacteristic
507C2—82no—
508C2#2019511yescharacteristic
509C2#2020411yescharacteristic
510C2#2022711yescharacteristic
511C2#2023611yescharacteristic
512C2#2266311yescharacteristic
513C2#2267711yescharacteristic
514C2#2269311yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ 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
2. #22743C2semidirect
6. #9618C2semidirect
7. #9614C2semidirect
9. #8896C2semidirect
10. #8887C2semidirect
12. #8819C2semidirect
14. #8651C2semidirect
15. #8291C2semidirect
56. #1592C4semidirect
77. #446C2×C4semidirect
154. #335C2×C4semidirect
155. Q8⋊C8C2×C4semidirect
157. #322C2×C4semidirect
471. C8#428semidirect

Maximal subgroups

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

subgroupindex
#228052
#227432
#226712
#226632
C2×#15922
#96182
#96142
#88972
#88962
#88872
#88862
#88192
#86542
#86512
#82912