Finite Group Databasebeta

← #243210

Subgroups — #243210

Order 512.

640 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21362C211yesmaximal, characteristic
2#21356C211yesmaximal, characteristic
3#21169C211yesmaximal, characteristic
4#20824C211yesmaximal, characteristic
5#10657C211yesmaximal, characteristic, complemented
6#10657C211yesmaximal, characteristic, complemented
7#10656C211yesmaximal, characteristic, complemented
8#10655C211yesmaximal, characteristic, complemented
9#10654C211yesmaximal, characteristic, complemented
10#10653C211yesmaximal, characteristic, complemented
11#10652C211yesmaximal, characteristic, complemented
12#10649C211yesmaximal, characteristic, complemented
13#8046C211yesmaximal, characteristic
14#8044C211yesmaximal, characteristic
15#7873C211yesmaximal, characteristic
16C2×#432C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#427—21no—
19C22×C4⋊C8C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21#2285C2211yescharacteristic
22#2285C2211yescharacteristic
23C4×#117C2211yescharacteristic
24#1949—21no—
25#1949—21no—
26#1948C2211yescharacteristic
27#1948C2211yescharacteristic
28#1948C2211yescharacteristic
29#1948C2211yescharacteristic
30#1947C2211yescharacteristic
31#1947C2211yescharacteristic
32#1947C2211yescharacteristic
33#1947C2211yescharacteristic
34#1939C2211yescharacteristic
35#1939C2211yescharacteristic
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1854—21no—
39#1854—21no—
40#1851—21no—
41#1851—21no—
42#1845C2211yescharacteristic
43#1844C2211yescharacteristic
44#1834C2211yescharacteristic
45#1834C2211yescharacteristic
46#1809—21no—
47#1809—21no—
48#1776—21no—
49#1776—21no—
50C4×#107—21no—
51C4×#106—21no—
52#1767—21no—
53#1767—21no—
54#1735—21no—
55#1296C2211yescharacteristic
56#1296C2211yescharacteristic
57#1296C2211yescharacteristic
58#1296C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16D411yescharacteristic
68C22×M16D411yescharacteristic
69C23×C8—21no—
70C2×#117—21no—
71C2×#117—21no—
72C2×#117D411yescharacteristic
73C2×#117D411yescharacteristic
74C22×C42C2311yescharacteristic
75#432—22no—
76#432—22no—
77#431—22no—
78#431Q814yes—
79#427—44no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8Q814yes—
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C4×M16—22no—
108C4×M16—22no—
109C2×#97—21no—
110C2×#97D411yescharacteristic
111C2×#97D411yescharacteristic
112C2×C4×C8—21no—
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#384C2311yescharacteristic
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#382—21no—
120#382—21no—
121#380—21no—
122#380—21no—
123#380—21no—
124#380—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4D411yescharacteristic
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135#376—21no—
136C43—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—21no—
148#340—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4#4111yescharacteristic
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—41no—
161C2×M16—41no—
162C22×C8—24no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—42no—
170C22×C8—42no—
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175#117—22no—
176#117—22no—
177#117—41no—
178#117—41no—
179#117—42no—
180#117—42no—
181#117—82no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—41no—
209C2×C42—41no—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×Q812yes—
213C2×C42C2×Q812yes—
214C2×C42#4112yes—
215C2×C42#4112yes—
216C2×C42#4111yescharacteristic
217C2×C42C2411yescharacteristic
218#107—42no—
219#107—42no—
220#107—42no—
221#107—42no—
222#107—42no—
223#107—42no—
224#106—42no—
225#106—42no—
226#106—42no—
227#106—42no—
228#106—42no—
229#106—42no—
230C4⋊C8—24no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—42no—
240C4⋊C8—42no—
241C4⋊C8—42no—
242C4⋊C8—42no—
243C4⋊C8C2×Q812yes—
244C4⋊C8C2×Q812yes—
245C4⋊C8C2×Q812yes—
246C4⋊C8C2×Q812yes—
247C22⋊C8—24no—
248C22⋊C8—42no—
249#97—22no—
250#97—22no—
251#97—42no—
252C4×C8—24no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—41no—
262#95—41no—
263#95—41no—
264#95—41no—
265C24#12111yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4#12111yescharacteristic
285C22×C4#12111yescharacteristic
286C22×C4#12111yescharacteristic
287C22×C4C22⋊Q812yes—
288C22×C4C22⋊Q812yes—
289C22×C4#12312yes—
290C22×C4#12512yes—
291C22×C4#12512yes—
292C22×C4C4⋊Q812yes—
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C2×#4111yescharacteristic
296M16—44no—
297M16—44no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—48no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8#13512yes—
328C2×C8#13512yes—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—81no—
352C4⋊C4—81no—
353C4⋊C4—81no—
354C4⋊C4—81no—
355C4⋊C4AGL(1,ℤ/8)12yes—
356C4⋊C4#13712yes—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—82no—
360C22⋊C4—82no—
361C42—42no—
362C42—42no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—81no—
368C42—81no—
369C42—82no—
370C42—82no—
371C42C22⋊Q812yes—
372C42C22⋊Q812yes—
373C42#12312yes—
374C42#12512yes—
375C42#12512yes—
376C42C4⋊Q812yes—
377C42C22×Q811yescharacteristic
378C42C22×Q811yescharacteristic
379C42C2×#4111yescharacteristic
380C42C2×#4111yescharacteristic
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23#39812yes—
388C23C2×#12111yescharacteristic
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4#39812yes—
432C2×C4#39812yes—
433C2×C4#39812yes—
434C2×C4C2×C22⋊Q811yescharacteristic
435C2×C4C2×C22⋊Q811yescharacteristic
436C2×C4C2×#12311yescharacteristic
437C2×C4C2×#12511yescharacteristic
438C2×C4C2×#12511yescharacteristic
439C2×C4C2×C4⋊Q811yescharacteristic
440C2×C4C2×#13511yescharacteristic
441C2×C4C2×#13511yescharacteristic
442C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
443C2×C4C2×#13711yescharacteristic
444C2×C4#57511yescharacteristic
445C2×C4#57511yescharacteristic
446C2×C4#57511yescharacteristic
447C2×C4#57511yescharacteristic
448C8—22no—
449C8—22no—
450C8—42no—
451C8—42no—
452C8—42no—
453C8—42no—
454C8—41no—
455C8—41no—
456C8—81no—
457C8—81no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22C2×#39811yescharacteristic
466C22#303511yescharacteristic
467C22#303511yescharacteristic
468C22#305511yescharacteristic
469C22#305611yescharacteristic
470C22#306811yescharacteristic
471C22#306911yescharacteristic
472C4—22no—
473C4—22no—
474C4—21no—
475C4—21no—
476C4—22no—
477C4—42no—
478C4—42no—
479C4—81no—
480C4—81no—
481C4—161no—
482C4—161no—
483C4—161no—
484C4—161no—
485C4#307711yescharacteristic
486C4#307811yescharacteristic
487C4#307911yescharacteristic
488C4#308011yescharacteristic
489C4#309111yescharacteristic
490C4#309111yescharacteristic
491C4#316011yescharacteristic
492C4#316211yescharacteristic
493C2—42no—
494C2#1917911yescharacteristic
495C2#1918011yescharacteristic
496C2#1918211yescharacteristic
497C2#1918511yescharacteristic
498C2#2365611yescharacteristic
499C2#2367011yescharacteristic
500C2#2367511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10657C2semidirect
6. #10657C2semidirect
7. #10656C2semidirect
8. #10655C2semidirect
9. #10654C2semidirect
10. #10653C2semidirect
11. #10652C2semidirect
12. #10649C2semidirect

Maximal subgroups

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

subgroupindex
#213622
#213562
#211692
#208242
#106572
#106562
#106552
#106542
#106532
#106522
#106492
#80462
#80442
#78732