Finite Group Databasebeta

← #225560

Subgroups — #225560

Order 512.

737 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22766C211yesmaximal, characteristic
2#22758C211yesmaximal, characteristic
3#22707C211yesmaximal, characteristic, complemented
4D4×#106C211yesmaximal, characteristic, complemented
5C2×#1580C211yesmaximal, characteristic
6#9622C211yesmaximal, characteristic, complemented
7#9613C211yesmaximal, characteristic, complemented
8#8893C211yesmaximal, characteristic, complemented
9#8890C211yesmaximal, characteristic, complemented
10#8885C211yesmaximal, characteristic, complemented
11#8882C211yesmaximal, characteristic, complemented
12#8821C211yesmaximal, characteristic, complemented
13#8646C211yesmaximal, characteristic, complemented
14#8642C211yesmaximal, characteristic, complemented
15#8290C211yesmaximal, characteristic, complemented
16C2×C4×D8C2211yescharacteristic
17C2×C8×D4C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C22×#106—21no—
20#2354C2211yescharacteristic
21#2354C2211yescharacteristic
22D4×C4⋊C4C2211yescharacteristic
23D4×C4⋊C4C2211yescharacteristic
24#1948C2211yescharacteristic
25#1948C2211yescharacteristic
26#1947C2211yescharacteristic
27#1947C2211yescharacteristic
28#1936—21no—
29#1934—21no—
30#1921—21no—
31#1918—21no—
32#1879—21no—
33#1879—21no—
34#1854—21no—
35#1854—21no—
36#1853—21no—
37#1853—21no—
38#1845C2211yescharacteristic
39#1844C2211yescharacteristic
40#1834C2211yescharacteristic
41#1834C2211yescharacteristic
42#1834C2211yescharacteristic
43#1834C2211yescharacteristic
44#1808—21no—
45#1807—21no—
46#1802C2211yescharacteristic
47#1802C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50C4×#106C2211yescharacteristic
51#1770C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53#1762—21no—
54C4×#102—21no—
55#1735—21no—
56#1580C418yescomplemented
57C2×#334C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68C22×D8D411yescharacteristic
69C23×C8—21no—
70C23×C8—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4C2311yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C2×#117—21no—
76C2×#117—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C4×D8C2×C414yes—
80C4×D8Q814yes—
81C8×D4—22no—
82C8×D4—22no—
83C8×D4C2×C414yes—
84C8×D4C2×C414yescomplemented
85C2×#107—21no—
86C2×#107—21no—
87C2×#107—41no—
88C2×#107D411yescharacteristic
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—44no—
94C2×#106D411yescharacteristic
95C2×#106D411yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102D411yescharacteristic
105C2×#102D411yescharacteristic
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×#97—21no—
109C2×C4×C8—21no—
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#390—21no—
114#390—21no—
115#389—21no—
116#389—21no—
117#388—41no—
118#388—41no—
119#385—41no—
120#385—41no—
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#380—21no—
126#380—21no—
127#380—21no—
128#380—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133#340—21no—
134#340—21no—
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—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—41no—
148#340D411yescharacteristic
149#340D411yescharacteristic
150#336—21no—
151#336—21no—
152#334—22no—
153#334C2×C414yescomplemented
154D4⋊C8C2×C414yes—
155D4⋊C8C2×C414yescomplemented
156#322C2×C414yescomplemented
157C22×D4C2×D411yescharacteristic
158C22×D4C2×D411yescharacteristic
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—41no—
162C23×C4—41no—
163C2×D8—22no—
164C2×D8—22no—
165C2×D8C4⋊C414yes—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—24no—
170C22×C8—24no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—42no—
174C22×C8—42no—
175C22×C8—81no—
176C22×C8—81no—
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—42no—
186C4×D4—42no—
187C4×D4—81no—
188C4×D4—81no—
189C4×D4C22×C412yes—
190C4×D4C22×C412yes—
191C4×D4C2×Q812yes—
192C4×D4C2×Q812yes—
193#117—42no—
194#117—42no—
195#117—81no—
196#117—81no—
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—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—81no—
214C2×C4⋊C4—81no—
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—41no—
232C2×C42—41no—
233C2×C42C2411yescharacteristic
234#107—81no—
235#107—81no—
236#107—82no—
237#107C4⋊C414yes—
238#106—22no—
239#106—41no—
240#106—42no—
241#106—42no—
242#106—42no—
243#106—81no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8C22×C412yes—
251C4⋊C8C22×C412yes—
252#102—42no—
253#102—42no—
254#102—81no—
255#102—81no—
256#102C4⋊C414yes—
257#102C4⋊C414yes—
258C22⋊C8—24no—
259C22⋊C8—24no—
260#97—81no—
261C4×C8—21no—
262C4×C8—81no—
263C4×C8C22×C412yes—
264C4×C8C22×C412yes—
265C4×C8C22×C412yes—
266C4×C8C2×Q812yes—
267C4×C8#4112yes—
268#95—21no—
269#95—21no—
270#95—41no—
271#95—41no—
272#95—41no—
273#95—41no—
274C24—21no—
275C24—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4C2×C4⋊C412yes—
285C2×D4C2×C4⋊C412yes—
286C2×D4#13512yes—
287C2×D4#13512yes—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4—161no—
315C22×C4—161no—
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4H(5,𝔽2)11yescharacteristic
319D8—28no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—48no—
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×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—84no—
345C2×C8C2×C4⋊C412yes—
346C2×C8C2×C4⋊C412yes—
347C2×C8C4×D412yes—
348C2×C8C4×D412yes—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
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—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—82no—
375C4⋊C4—81no—
376C4⋊C4—81no—
377C4⋊C4—81no—
378C4⋊C4—81no—
379C4⋊C4—82no—
380C4⋊C4C2×C4⋊C412yes—
381C4⋊C4C2×C4⋊C412yes—
382C4⋊C4AGL(1,ℤ/8)12yes—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C22⋊C4—82no—
391C22⋊C4—161no—
392C22⋊C4—161no—
393C42—41no—
394C42—41no—
395C42—41no—
396C42—41no—
397C42—82no—
398C42—82no—
399C42C23×C411yescharacteristic
400C42C22×Q811yescharacteristic
401C42C2×#4111yescharacteristic
402C422−1+411yescharacteristic
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23D4211yescharacteristic
418D4—22no—
419D4—22no—
420D4—22no—
421D4—22no—
422D4—22no—
423D4—41no—
424D4—41no—
425D4#42714yes—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—164no—
467C2×C4—164no—
468C2×C4C22×C4⋊C411yescharacteristic
469C2×C4C2×C4×D411yescharacteristic
470C2×C4#52011yescharacteristic
471C2×C4#54711yescharacteristic
472C2×C4D4×Q811yescharacteristic
473C2×C4#55411yescharacteristic
474C2×C4C2×#13511yescharacteristic
475C2×C4C2×#13511yescharacteristic
476C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
477C2×C4#57511yescharacteristic
478C2×C4#57611yescharacteristic
479C2×C4#57711yescharacteristic
480C2×C4#57711yescharacteristic
481C2×C4#57711yescharacteristic
482C8—22no—
483C8—22no—
484C8—22no—
485C8—22no—
486C8—42no—
487C8—42no—
488C8—42no—
489C8—81no—
490C8—81no—
491C22—24no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22D4×C4⋊C411yescharacteristic
506C22#326411yescharacteristic
507C22#326711yescharacteristic
508C22#328211yescharacteristic
509C22#328211yescharacteristic
510C22#328811yescharacteristic
511C22#328911yescharacteristic
512C4—21no—
513C4—21no—
514C4—42no—
515C4—42no—
516C4—42no—
517C4—81no—
518C4—81no—
519C4—81no—
520C4—161no—
521C4—161no—
522C4—161no—
523C4—161no—
524C4C2×#42711yescharacteristic
525C4#291211yescharacteristic
526C4C4×AGL(1,ℤ/8)11yescharacteristic
527C4#294611yescharacteristic
528C4#333711yescharacteristic
529C4#333711yescharacteristic
530C4#338411yescharacteristic
531C4#338911yescharacteristic
532C2—44no—
533C2—82no—
534C2#1978711yescharacteristic
535C2#1979211yescharacteristic
536C2#1981911yescharacteristic
537C2#1982411yescharacteristic
538C2#2265411yescharacteristic
539C2#2267611yescharacteristic
540C2#2269311yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D8D4
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
3. #22707C2semidirect
4. D4×#106C2semidirect
6. #9622C2semidirect
7. #9613C2semidirect
8. #8893C2semidirect
9. #8890C2semidirect
10. #8885C2semidirect
11. #8882C2semidirect
12. #8821C2semidirect
13. #8646C2semidirect
14. #8642C2semidirect
15. #8290C2semidirect
56. #1580C4semidirect
84. C8×D4C2×C4semidirect
153. #334C2×C4semidirect
155. D4⋊C8C2×C4semidirect
156. #322C2×C4semidirect

Maximal subgroups

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

subgroupindex
#227662
#227582
#227072
D4×#1062
C2×#15802
#96222
#96132
#88932
#88902
#88852
#88822
#88212
#86462
#86422
#82902