Finite Group Databasebeta

← #203607

Subgroups — #203607

Order 512.

700 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21361C211yesmaximal, characteristic
2#21354C211yesmaximal, characteristic
3#21146C211yesmaximal, characteristic
4C2×#1574C211yesmaximal, characteristic
5C2×#1453C211yesmaximal, characteristic
6#9600C211yesmaximal, characteristic, complemented
7#9599C211yesmaximal, characteristic, complemented
8#9598C211yesmaximal, characteristic, complemented
9#9597C211yesmaximal, characteristic, complemented
10#9596C211yesmaximal, characteristic, complemented
11#9595C211yesmaximal, characteristic, complemented
12#9594C211yesmaximal, characteristic, complemented
13#9593C211yesmaximal, characteristic, complemented
14#7899C211yesmaximal, characteristic
15#7892C211yesmaximal, characteristic
16C2×#431C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#423—21no—
20C2×#423C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22#2283C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1949C411yescharacteristic
25#1949C411yescharacteristic
26#1949C411yescharacteristic
27#1949C411yescharacteristic
28#1940—21no—
29#1940—21no—
30#1939C2211yescharacteristic
31#1939C2211yescharacteristic
32#1939C2211yescharacteristic
33#1939C2211yescharacteristic
34#1939C2211yescharacteristic
35#1939C2211yescharacteristic
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1853C411yescharacteristic
39#1853C411yescharacteristic
40#1852C411yescharacteristic
41#1852C411yescharacteristic
42#1851—21no—
43#1851—21no—
44C4×#107C2211yescharacteristic
45C4×#107C2211yescharacteristic
46C4×#106C2211yescharacteristic
47C4×#106C2211yescharacteristic
48#1768—21no—
49#1768—21no—
50#1728—21no—
51#1574—24no—
52C2×#335C2211yescharacteristic
53C2×#335C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×#334C2211yescharacteristic
56#1453—24no—
57C2×#322C2211yescharacteristic
58C2×#322C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68#1279C2211yescharacteristic
69C22×M16—21no—
70C22×M16D411yescharacteristic
71C22×M16D411yescharacteristic
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76#431—22no—
77#431—22no—
78#431—22no—
79#431D414yes—
80#431Q814yes—
81#431Q814yes—
82C2×#107C2×C411yescharacteristic
83C2×#107C2×C411yescharacteristic
84C2×#107C2×C411yescharacteristic
85C2×#107C2×C411yescharacteristic
86C2×#106C2×C411yescharacteristic
87C2×#106C2×C411yescharacteristic
88C2×#106C2×C411yescharacteristic
89C2×#106C2×C411yescharacteristic
90#423—22no—
91#423—22no—
92#423—44no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8C2×C411yescharacteristic
104C2×C4⋊C8C2×C411yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8C2×C411yescharacteristic
112C2×C22⋊C8C2×C411yescharacteristic
113C4×M16—22no—
114C4×M16—22no—
115C2×#97D411yescharacteristic
116C2×#97D411yescharacteristic
117C2×C4×C8—21no—
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124#383—21no—
125#382C2×C411yescharacteristic
126#382C2×C411yescharacteristic
127#381—21no—
128#380C2×C411yescharacteristic
129#380C2×C411yescharacteristic
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135C4×C4⋊C4C2311yescharacteristic
136C4×C4⋊C4C2311yescharacteristic
137C4×C4⋊C4C2311yescharacteristic
138C4×C22⋊C4—21no—
139#376—21no—
140C2×#95—21no—
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#335—22no—
158#335—22no—
159#334—22no—
160#334—22no—
161#322—22no—
162#322—22no—
163#322—22no—
164C82—22no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4C22×C411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4#4111yescharacteristic
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—41no—
175C2×M16—41no—
176C2×M16—42no—
177C2×M16—42no—
178C2×M16—42no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8C22×C411yescharacteristic
186C22×C8C22×C411yescharacteristic
187C22×C8C22×C411yescharacteristic
188C22×C8C22×C411yescharacteristic
189C22×C8C2×D411yescharacteristic
190C22×C8C2×D411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4C22×C411yescharacteristic
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×C4⋊C4C22×C411yescharacteristic
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—42no—
216C2×C42—42no—
217C2×C42C22×C411yescharacteristic
218C2×C42C2×D412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×Q812yes—
221C2×C42C2×Q812yes—
222C2×C42#4112yes—
223C2×C42#4111yescharacteristic
224C2×C42C2411yescharacteristic
225#107—22no—
226#107—22no—
227#107—41no—
228#107—41no—
229#106—22no—
230#106—22no—
231#106—41no—
232#106—41no—
233C4⋊C8—24no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—42no—
243C4⋊C8—42no—
244C4⋊C8—42no—
245C4⋊C8—42no—
246C4⋊C8—42no—
247C4⋊C8—42no—
248C4⋊C8—42no—
249C4⋊C8—42no—
250C4⋊C8—42no—
251C4⋊C8C4⋊C414yes—
252C4⋊C8C4⋊C414yes—
253C22⋊C8—24no—
254C22⋊C8—42no—
255C22⋊C8C4⋊C414yes—
256C22⋊C8C4⋊C414yes—
257#97—22no—
258#97—22no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—42no—
266C4×C8C2×D412yes—
267C4×C8C2×D412yes—
268C4×C8C2×Q812yes—
269C4×C8C2×Q812yes—
270C4×C8C2×Q812yes—
271C4×C8C2×Q812yes—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284C24C4×D411yescharacteristic
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—81no—
308C22×C4—81no—
309C22×C4C2×C4⋊C412yes—
310C22×C4C4×D411yescharacteristic
311C22×C4C4×D411yescharacteristic
312C22×C4C4×D411yescharacteristic
313C22×C4C4×Q812yes—
314C22×C4#12112yes—
315C22×C4C22⋊Q812yes—
316C22×C4#12512yes—
317C22×C4C4⋊Q812yes—
318C22×C4C23×C411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C2×#4111yescharacteristic
321M16—44no—
322M16—44no—
323M16—84no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8C2×C4⋊C412yes—
361C2×C8C2×C4⋊C412yes—
362C2×C8C2×C4⋊C412yes—
363C2×C8C2×C4⋊C412yes—
364C2×C8AGL(1,ℤ/8)12yes—
365C2×C8#13712yes—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4#13112yes—
383C4⋊C4#13112yes—
384C22⋊C4—81no—
385C22⋊C4—81no—
386C42—42no—
387C42—42no—
388C42—41no—
389C42—41no—
390C42—41no—
391C42—41no—
392C42—41no—
393C42—41no—
394C42—41no—
395C42—41no—
396C42C2×C4⋊C412yes—
397C42C4×Q812yes—
398C42#12112yes—
399C42C22⋊Q812yes—
400C42#12512yes—
401C42C4⋊Q812yes—
402C42C22×D411yescharacteristic
403C42C22×Q811yescharacteristic
404C42C22×Q811yescharacteristic
405C42C2×#4111yescharacteristic
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23#38912yes—
413C23C2×C4×D411yescharacteristic
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
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—41no—
455C2×C4—41no—
456C2×C4—84no—
457C2×C4—84no—
458C2×C4#38912yes—
459C2×C4#38912yes—
460C2×C4#38912yes—
461C2×C4C22×C4⋊C411yescharacteristic
462C2×C4C2×C4×Q811yescharacteristic
463C2×C4C2×#12111yescharacteristic
464C2×C4C2×C22⋊Q811yescharacteristic
465C2×C4C2×#12511yescharacteristic
466C2×C4C2×C4⋊Q811yescharacteristic
467C2×C4C2×#13111yescharacteristic
468C2×C4C2×#13111yescharacteristic
469C2×C4#56811yescharacteristic
470C2×C4#56811yescharacteristic
471C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
472C2×C4C2×#13711yescharacteristic
473C2×C4#57511yescharacteristic
474C2×C4#57511yescharacteristic
475C8—22no—
476C8—22no—
477C8—22no—
478C8—22no—
479C8—22no—
480C8—22no—
481C8—42no—
482C8—42no—
483C8—41no—
484C8—41no—
485C8—81no—
486C8—81no—
487C8—81no—
488C8—81no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22C2×#38911yescharacteristic
497C22#293311yescharacteristic
498C22#293311yescharacteristic
499C22C4×AGL(1,ℤ/8)11yescharacteristic
500C22C4×#13711yescharacteristic
501C22#295711yescharacteristic
502C22#295711yescharacteristic
503C4—22no—
504C4—22no—
505C4—21no—
506C4—21no—
507C4—22no—
508C4—42no—
509C4—42no—
510C4—81no—
511C4—81no—
512C4—81no—
513C4—81no—
514C4—81no—
515C4—81no—
516C4#290611yescharacteristic
517C4#290611yescharacteristic
518C4#296011yescharacteristic
519C4#296111yescharacteristic
520C4#305511yescharacteristic
521C4#305611yescharacteristic
522C4#316511yescharacteristic
523C4#316611yescharacteristic
524C2—42no—
525C2#1800611yescharacteristic
526C2#1800811yescharacteristic
527C2#1801011yescharacteristic
528C2#1801211yescharacteristic
529C2#2114811yescharacteristic
530C2#2115911yescharacteristic
531C2#2116311yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9600C2semidirect
7. #9599C2semidirect
8. #9598C2semidirect
9. #9597C2semidirect
10. #9596C2semidirect
11. #9595C2semidirect
12. #9594C2semidirect
13. #9593C2semidirect

Maximal subgroups

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

subgroupindex
#213612
#213542
#211462
C2×#15742
C2×#14532
#96002
#95992
#95982
#95972
#95962
#95952
#95942
#95932
#78992
#78922