Finite Group Databasebeta

← #244072

Subgroups — #244072

Order 512.

950 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21410C211yesmaximal, characteristic
2#21410C211yesmaximal, characteristic
3#21200C211yesmaximal, characteristic
4#20824C211yesmaximal, characteristic, complemented
5#20800C211yesmaximal, characteristic, complemented
6#20764C211yesmaximal, characteristic, complemented
7#10684C211yesmaximal, characteristic, complemented
8#10681C211yesmaximal, characteristic, complemented
9#10679C211yesmaximal, characteristic, complemented
10#10677C211yesmaximal, characteristic, complemented
11#10675C211yesmaximal, characteristic, complemented
12#10672C211yesmaximal, characteristic, complemented
13#10672C211yesmaximal, characteristic, complemented
14#10671C211yesmaximal, characteristic, complemented
15#7852C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#428—22no—
18C2×#417—22no—
19C2×C4×M16C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22#2285C2211yescharacteristic
23#2285C2211yescharacteristic
24C2×#384C2211yescharacteristic
25#1990C2211yescharacteristic
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1959—22no—
30#1957C2211yescharacteristic
31#1957C2211yescharacteristic
32#1957C2211yescharacteristic
33#1957C2211yescharacteristic
34#1957C2211yescharacteristic
35#1957C2211yescharacteristic
36#1956C2211yescharacteristic
37#1956C2211yescharacteristic
38#1895—22no—
39#1895—22no—
40#1847—22no—
41#1845C2211yescharacteristic
42#1845C2211yescharacteristic
43#1845C2211yescharacteristic
44#1844C2211yescharacteristic
45#1844C2211yescharacteristic
46#1844C2211yescharacteristic
47#1836—22no—
48#1834C2211yescharacteristic
49#1834C2211yescharacteristic
50#1834C2211yescharacteristic
51#1834C2211yescharacteristic
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1809—22no—
55#1776—22no—
56#1762—22no—
57#1762—22no—
58#1736—22no—
59#1273C2211yescharacteristic
60#1273C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C22×#41—21no—
64C22×M16D412yes—
65C22×M16D412yes—
66C22×M16D412yes—
67#532—22no—
68#532—22no—
69#532—22no—
70#532Q814yes—
71C2×C4⋊Q8C2311yescharacteristic
72C2×#127C2311yescharacteristic
73C2×#124—21no—
74C2×#121—21no—
75C2×#121—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×#117—21no—
79C2×#117—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C42C2311yescharacteristic
84#428—48no—
85C2×#107—22no—
86C2×#107—22no—
87C2×#107—22no—
88C2×#106—22no—
89C2×#106—22no—
90C2×#106—22no—
91#417—48no—
92C2×#103—22no—
93C2×#103—22no—
94C2×#102—22no—
95C2×#102—22no—
96C2×#102—22no—
97C2×#102—22no—
98C2×#102—22no—
99C2×#102—22no—
100C4×M16—22no—
101C4×M16—22no—
102C4×M16—22no—
103C4×M16—22no—
104C4×M16—22no—
105C4×M16—22no—
106C2×#97D412yes—
107C2×#97D412yes—
108C2×#97D412yes—
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#384—21no—
116#384—21no—
117#384—21no—
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#382—21no—
125#382—21no—
126#380—21no—
127#380—21no—
128#380—21no—
129#380—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134#340—22no—
135#340—22no—
136#340—22no—
137#340—22no—
138#340—22no—
139#340—22no—
140#340—22no—
141#340—22no—
142#340—22no—
143#340—22no—
144C2×#41—24no—
145C2×#41—44no—
146C2×#41—44no—
147C2×#41—42no—
148C22×Q8—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—41no—
153C23×C4—41no—
154C23×C4—41no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C2×M16—24no—
159C2×M16—24no—
160C2×M16—24no—
161C2×M16—24no—
162C2×M16—24no—
163C2×M16—24no—
164C2×M16—42no—
165C2×M16—42no—
166C2×M16—42no—
167C22×C8C2×D412yes—
168C22×C8C2×D412yes—
169C22×C8C2×D412yes—
170C22×C8C2×D412yes—
171C22×C8C2×D412yes—
172C22×C8C2×D412yes—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8C2×Q812yes—
177#127—21no—
178#127—21no—
179#127—21no—
180#127C2×Q812yes—
181#124—24no—
182#124—42no—
183#121—24no—
184#121—24no—
185#121—42no—
186#121—42no—
187C4×D4—24no—
188C4×D4—24no—
189C4×D4—42no—
190C4×D4—42no—
191#117—44no—
192#117—44no—
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—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C42—41no—
224C2×C42—41no—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×Q812yes—
229C2×C42#4112yes—
230C2×C42#4112yes—
231C2×C42#4112yes—
232C2×C42C2411yescharacteristic
233#107—44no—
234#107—44no—
235#107—44no—
236#106—44no—
237#106—44no—
238#106—44no—
239#103—44no—
240#103—44no—
241#102—44no—
242#102—44no—
243#102—44no—
244#102—44no—
245#102—44no—
246#102—44no—
247#97—24no—
248#97—24no—
249#97—24no—
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—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262#95—41no—
263#95—41no—
264#95—41no—
265#95—41no—
266C24—41no—
267C24#12711yescharacteristic
268#41—416no—
269#41—88no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—44no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—82no—
288C2×D4—82no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—41no—
296C22×C4—44no—
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—82no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4C22⋊Q812yes—
309C22×C4C22⋊Q812yes—
310C22×C4C22⋊Q812yes—
311C22×C4#12312yes—
312C22×C4#12312yes—
313C22×C4#12312yes—
314C22×C4#12711yescharacteristic
315C22×C4#12711yescharacteristic
316C22×C4#12711yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320M16—216no—
321M16—48no—
322M16—48no—
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×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8AGL(1,ℤ/8)14yes—
339C2×C8AGL(1,ℤ/8)14yes—
340C2×C8#13714yes—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—41no—
344C4⋊C4—41no—
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—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—82no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C22⋊C4—44no—
367C22⋊C4—44no—
368C22⋊C4—82no—
369C22⋊C4—82no—
370C42—82no—
371C42—82no—
372C42C22⋊Q812yes—
373C42C22⋊Q812yes—
374C42C22⋊Q812yes—
375C42#12312yes—
376C42#12312yes—
377C42#12312yes—
378C42C22×Q811yescharacteristic
379C42C2×#4111yescharacteristic
380C42C2×#4111yescharacteristic
381C42C2×#4111yescharacteristic
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—42no—
389C23—81no—
390C23—81no—
391C23—81no—
392C23—81no—
393C23—81no—
394C23—81no—
395C23#39912yes—
396C23C2×#12711yescharacteristic
397Q8—44no—
398Q8—82no—
399D4—44no—
400D4—44no—
401D4—44no—
402D4—88no—
403D4—82no—
404D4—82no—
405D4—82no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—42no—
419C2×C4—88no—
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—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—164no—
441C2×C4#39912yes—
442C2×C4#39912yes—
443C2×C4#39912yes—
444C2×C4C2×C22⋊Q811yescharacteristic
445C2×C4C2×C22⋊Q811yescharacteristic
446C2×C4C2×C22⋊Q811yescharacteristic
447C2×C4C2×#12311yescharacteristic
448C2×C4C2×#12311yescharacteristic
449C2×C4C2×#12311yescharacteristic
450C2×C4C2×AGL(1,ℤ/8)12yes—
451C2×C4C2×AGL(1,ℤ/8)12yes—
452C2×C4C2×#13712yes—
453C2×C4#57512yes—
454C2×C4#57512yes—
455C2×C4#57512yes—
456C8—24no—
457C8—24no—
458C8—24no—
459C8—42no—
460C8—42no—
461C8—42no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—82no—
470C22—82no—
471C22—82no—
472C22—161no—
473C22—161no—
474C22—161no—
475C22—161no—
476C22C2×#39911yescharacteristic
477C22#315111yescharacteristic
478C22#315111yescharacteristic
479C22#315211yescharacteristic
480C22#315311yescharacteristic
481C22#315311yescharacteristic
482C22#315311yescharacteristic
483C4—22no—
484C4—22no—
485C4—22no—
486C4—82no—
487C4—161no—
488C4—161no—
489C4—161no—
490C4—161no—
491C4—161no—
492C4—161no—
493C4#307712yes—
494C4#307912yes—
495C4#308012yes—
496C4#309212yes—
497C4#309312yes—
498C4#309412yes—
499C2—42no—
500C2—162no—
501C2#1974711yescharacteristic
502C2#1974811yescharacteristic
503C2#1975011yescharacteristic
504C2#1975211yescharacteristic
505C2#2373111yescharacteristic
506C2#2373311yescharacteristic
507C2#2373411yescharacteristic
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
4. #20824C2semidirect
5. #20800C2semidirect
6. #20764C2semidirect
7. #10684C2semidirect
8. #10681C2semidirect
9. #10679C2semidirect
10. #10677C2semidirect
11. #10675C2semidirect
12. #10672C2semidirect
13. #10672C2semidirect
14. #10671C2semidirect
15. #7852C2semidirect

Maximal subgroups

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

subgroupindex
#214102
#212002
#208242
#208002
#207642
#106842
#106812
#106792
#106772
#106752
#106722
#106712
#78522