Finite Group Databasebeta

← #244041

Subgroups — #244041

Order 512.

882 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21984C211yesmaximal, characteristic, complemented
2#21982C211yesmaximal, characteristic, complemented
3#21859C211yesmaximal, characteristic
4#21805C211yesmaximal, characteristic
5#21410C211yesmaximal, characteristic
6#20800C211yesmaximal, characteristic, complemented
7#10684C211yesmaximal, characteristic, complemented
8#10681C211yesmaximal, characteristic, complemented
9#10680C211yesmaximal, characteristic, complemented
10#10677C211yesmaximal, characteristic, complemented
11#10674C211yesmaximal, characteristic, complemented
12#10673C211yesmaximal, characteristic, complemented
13#10672C211yesmaximal, characteristic, complemented
14#10672C211yesmaximal, characteristic, complemented
15#7850C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#428—22no—
20C2×C4×M16C2211yescharacteristic
21#2285C2211yescharacteristic
22#2285C2211yescharacteristic
23C2×#384C2211yescharacteristic
24#1990C2211yescharacteristic
25#1990C2211yescharacteristic
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1983—22no—
29#1957C2211yescharacteristic
30#1957C2211yescharacteristic
31#1957C2211yescharacteristic
32#1957C2211yescharacteristic
33#1957C2211yescharacteristic
34#1957C2211yescharacteristic
35#1956C2211yescharacteristic
36#1956C2211yescharacteristic
37#1949—22no—
38#1947—22no—
39#1895—22no—
40#1874—22no—
41#1854—22no—
42#1852—22no—
43#1847—22no—
44#1845C2211yescharacteristic
45#1845C2211yescharacteristic
46#1845C2211yescharacteristic
47#1845C2211yescharacteristic
48#1845C2211yescharacteristic
49#1844C2211yescharacteristic
50#1844C2211yescharacteristic
51#1844C2211yescharacteristic
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1834C2211yescharacteristic
55#1834C2211yescharacteristic
56#1802—22no—
57#1788—22no—
58#1762—22no—
59#1273C2211yescharacteristic
60#1273C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C22×#41—21no—
64C22×M16D412yes—
65#532—22no—
66#532—22no—
67#532—22no—
68#532Q814yes—
69C2×C4⋊Q8C2311yescharacteristic
70C2×#127C2311yescharacteristic
71C2×#124—21no—
72C2×#121—21no—
73C2×#121—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×#117—21no—
77C2×#117—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C42C2311yescharacteristic
82#431—22no—
83#431—22no—
84#431—22no—
85#431—22no—
86#428—48no—
87C2×#107—22no—
88C2×#107—22no—
89C2×#107—22no—
90C2×#107—22no—
91C2×#107—22no—
92C2×#106—22no—
93C2×#106—22no—
94C2×#106—22no—
95C2×C4⋊C8D412yes—
96C2×C4⋊C8D412yes—
97C2×#103—22no—
98C2×#103—22no—
99C2×#102—22no—
100C2×#102—22no—
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—22no—
105C2×C22⋊C8D412yes—
106C2×C22⋊C8D412yes—
107C4×M16—22no—
108C4×M16—22no—
109C2×#97D412yes—
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116#384—21no—
117#384—21no—
118#384—21no—
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#382—21no—
126#382—21no—
127#380—21no—
128#380—21no—
129#380—21no—
130#380—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135#340—22no—
136#340—22no—
137#340—22no—
138#340—22no—
139#340—22no—
140#340—22no—
141#340—22no—
142#340—22no—
143C2×#41—24no—
144C2×#41—42no—
145C2×#41—82no—
146C2×#41—82no—
147C22×Q8—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—41no—
152C23×C4—41no—
153C23×C4—41no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C2×M16—24no—
158C2×M16—24no—
159C2×M16—24no—
160C22×C8C2×D412yes—
161C22×C8C2×D412yes—
162C22×C8C2×D412yes—
163C22×C8C2×D412yes—
164C22×C8C2×D412yes—
165C22×C8C2×D412yes—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8C2×Q812yes—
170#127—21no—
171#127—21no—
172#127—21no—
173#127C2×Q812yes—
174#124—24no—
175#124—42no—
176#121—24no—
177#121—24no—
178#121—42no—
179#121—42no—
180C4×D4—24no—
181C4×D4—24no—
182C4×D4—42no—
183C4×D4—42no—
184#117—44no—
185#117—82no—
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⋊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—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4—82no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C42—41no—
217C2×C42—41no—
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×Q812yes—
222C2×C42#4112yes—
223C2×C42#4112yes—
224C2×C42#4112yes—
225C2×C42C2411yescharacteristic
226#107—44no—
227#107—44no—
228#107—82no—
229#107—82no—
230#107—82no—
231#106—44no—
232#106—44no—
233#106—82no—
234C4⋊C8—24no—
235C4⋊C8—24no—
236#103—82no—
237#103—82no—
238#102—44no—
239#102—44no—
240#102—82no—
241#102—82no—
242#102—82no—
243#102—82no—
244C22⋊C8—24no—
245C22⋊C8—24no—
246#97—24no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259#95—41no—
260#95—41no—
261#95—41no—
262#95—41no—
263C24—41no—
264C24#12711yescharacteristic
265#41—88no—
266#41—88no—
267C2×Q8—22no—
268C2×Q8—41no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—44no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—82no—
285C2×D4—82no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—41no—
293C22×C4—44no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4C22⋊Q812yes—
306C22×C4C22⋊Q812yes—
307C22×C4C22⋊Q812yes—
308C22×C4#12312yes—
309C22×C4#12312yes—
310C22×C4#12312yes—
311C22×C4#12711yescharacteristic
312C22×C4#12711yescharacteristic
313C22×C4#12711yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317M16—216no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8AGL(1,ℤ/8)14yes—
335C2×C8#13714yes—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C22⋊C4—44no—
362C22⋊C4—44no—
363C22⋊C4—82no—
364C22⋊C4—82no—
365C42—82no—
366C42—82no—
367C42C22⋊Q812yes—
368C42C22⋊Q812yes—
369C42C22⋊Q812yes—
370C42#12312yes—
371C42#12312yes—
372C42#12312yes—
373C42C22×Q811yescharacteristic
374C42C2×#4111yescharacteristic
375C42C2×#4111yescharacteristic
376C42C2×#4111yescharacteristic
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—42no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—81no—
389C23—81no—
390C23#39912yes—
391C23C2×#12711yescharacteristic
392Q8—44no—
393Q8—82no—
394D4—44no—
395D4—44no—
396D4—44no—
397D4—88no—
398D4—82no—
399D4—82no—
400D4—82no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—42no—
414C2×C4—88no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
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—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—164no—
436C2×C4#39912yes—
437C2×C4#39912yes—
438C2×C4#39912yes—
439C2×C4C2×C22⋊Q811yescharacteristic
440C2×C4C2×C22⋊Q811yescharacteristic
441C2×C4C2×C22⋊Q811yescharacteristic
442C2×C4C2×#12311yescharacteristic
443C2×C4C2×#12311yescharacteristic
444C2×C4C2×#12311yescharacteristic
445C2×C4C2×AGL(1,ℤ/8)12yes—
446C2×C4C2×#13712yes—
447C2×C4#57612yes—
448C2×C4#57612yes—
449C2×C4#57712yes—
450C2×C4#57812yes—
451C8—24no—
452C8—24no—
453C8—42no—
454C8—42no—
455C8—42no—
456C8—42no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—161no—
468C22—161no—
469C22—161no—
470C22—161no—
471C22C2×#39911yescharacteristic
472C22#315111yescharacteristic
473C22#315211yescharacteristic
474C22#315411yescharacteristic
475C22#315511yescharacteristic
476C22#315511yescharacteristic
477C22#315611yescharacteristic
478C4—22no—
479C4—22no—
480C4—22no—
481C4—82no—
482C4—161no—
483C4—161no—
484C4—161no—
485C4—161no—
486C4—161no—
487C4—161no—
488C4#307912yes—
489C4#308112yes—
490C4#308312yes—
491C4#309212yes—
492C4#309612yes—
493C4#309812yes—
494C2—42no—
495C2—162no—
496C2#1972511yescharacteristic
497C2#1972611yescharacteristic
498C2#1972711yescharacteristic
499C2#1972811yescharacteristic
500C2#2373411yescharacteristic
501C2#2373511yescharacteristic
502C2#2373711yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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
1. #21984C2semidirect
2. #21982C2semidirect
6. #20800C2semidirect
7. #10684C2semidirect
8. #10681C2semidirect
9. #10680C2semidirect
10. #10677C2semidirect
11. #10674C2semidirect
12. #10673C2semidirect
13. #10672C2semidirect
14. #10672C2semidirect
15. #7850C2semidirect

Maximal subgroups

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

subgroupindex
#219842
#219822
#218592
#218052
#214102
#208002
#106842
#106812
#106802
#106772
#106742
#106732
#106722
#78502