Finite Group Databasebeta

← #315749

Subgroups — #315749

Order 512.

744 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33212C211yesmaximal, characteristic
2#27948C211yesmaximal, characteristic
3#27905C211yesmaximal, characteristic
4#27889C211yesmaximal, characteristic
5#27875C211yesmaximal, characteristic
6#20800C211yesmaximal, characteristic
7#10685C211yesmaximal, characteristic, complemented
8#10685C211yesmaximal, characteristic, complemented
9#10680C211yesmaximal, characteristic, complemented
10#10675C211yesmaximal, characteristic, complemented
11#9836C211yesmaximal, characteristic, complemented
12#9823C211yesmaximal, characteristic, complemented
13#8819C211yesmaximal, characteristic, complemented
14#8813C211yesmaximal, characteristic, complemented
15#8146C211yesmaximal, characteristic
16C2×#500C2211yescharacteristic
17C2×#483—22no—
18C2×#428—22no—
19C2×C4×M16C2211yescharacteristic
20#2857C2211yescharacteristic
21#2857C2211yescharacteristic
22#2285C2211yescharacteristic
23C2×#384C2211yescharacteristic
24#2096—22no—
25#2084—22no—
26#2077—22no—
27#2035—22no—
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1990C2211yescharacteristic
34#1956C2211yescharacteristic
35#1956C2211yescharacteristic
36#1956C2211yescharacteristic
37#1956C2211yescharacteristic
38#1944—22no—
39#1927—22no—
40#1920—22no—
41#1853—22no—
42#1847—22no—
43#1845C2211yescharacteristic
44#1845C2211yescharacteristic
45#1844C2211yescharacteristic
46#1844C2211yescharacteristic
47#1844C2211yescharacteristic
48#1844C2211yescharacteristic
49#1844C2211yescharacteristic
50#1844C2211yescharacteristic
51#1844C2211yescharacteristic
52#1844C2211yescharacteristic
53#1844C2211yescharacteristic
54#1834C2211yescharacteristic
55#1834C2211yescharacteristic
56#1809—22no—
57C4×C22⋊C8C2211yescharacteristic
58C4×C22⋊C8C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C22×M16D412yes—
64C2×C4⋊Q8C2311yescharacteristic
65C2×C4⋊Q8C2311yescharacteristic
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×#117—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C42C2311yescharacteristic
73#500—22no—
74#500—22no—
75#483—48no—
76#428—48no—
77C2×#107—22no—
78C2×#107—22no—
79C2×#107—22no—
80C2×#106—22no—
81C2×#106—22no—
82C2×#106—22no—
83C2×#106—22no—
84C2×#106—22no—
85C2×#106—22no—
86C2×#106—22no—
87C2×#106—22no—
88C2×#106—22no—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—22no—
92C2×#103—22no—
93C2×C22⋊C8D412yes—
94C2×C22⋊C8D412yes—
95C4×M16—22no—
96C4×M16—22no—
97C2×#97D412yes—
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#391—21no—
105#391—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#384—21no—
111#384—21no—
112#384—21no—
113#384C2311yescharacteristic
114#384C2311yescharacteristic
115#384C2311yescharacteristic
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#382—21no—
120#380—21no—
121#380—21no—
122#380—21no—
123#380—21no—
124#380—21no—
125#380—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#336D412yes—
137#336D412yes—
138#322—22no—
139C82—22no—
140C22×Q8—21no—
141C22×Q8—21no—
142C23×C4—41no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C2×M16—24no—
147C2×M16—24no—
148C2×M16SD1618yes—
149C22×C8C2×D412yes—
150C22×C8C2×D412yes—
151C22×C8C2×D412yes—
152C22×C8C2×D412yes—
153C22×C8C2×D412yes—
154C22×C8C2×D412yes—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C22⋊Q8—44no—
164C22⋊Q8—82no—
165#117—44no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
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—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—82no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—41no—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2411yescharacteristic
208#107—44no—
209#107—44no—
210#107—44no—
211#106—44no—
212#106—44no—
213#106—44no—
214#106—44no—
215#106—44no—
216#106—44no—
217#106—82no—
218#106—82no—
219#106—82no—
220#103—44no—
221#103—44no—
222#103—44no—
223#103—82no—
224C22⋊C8—24no—
225C22⋊C8—24no—
226#97—24no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8#4112yes—
238C4×C8#4112yes—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95—41no—
245C24#12711yescharacteristic
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4#12711yescharacteristic
277C22×C4#12711yescharacteristic
278C22×C4#12711yescharacteristic
279C22×C4C2×SD1614yes—
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283M16—216no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8#12314yes—
297C2×C8C2×SD1614yes—
298C2×C8C2×SD1614yes—
299C2×C8AGL(1,ℤ/8)14yes—
300C2×C8#13714yes—
301C2×C8#13714yes—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42—21no—
342C42—82no—
343C42C2×#4111yescharacteristic
344C42C2×#4111yescharacteristic
345C422−1+411yescharacteristic
346C422−1+411yescharacteristic
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23#49212yes—
354C23C2×#12711yescharacteristic
355Q8—82no—
356Q8—82no—
357Q8—82no—
358Q8—82no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—164no—
398C2×C4#49212yes—
399C2×C4C2×#12311yescharacteristic
400C2×C4#53611yescharacteristic
401C2×C4#54711yescharacteristic
402C2×C4#54711yescharacteristic
403C2×C4#54711yescharacteristic
404C2×C4#54711yescharacteristic
405C2×C4C22×SD1612yes—
406C2×C4C2×AGL(1,ℤ/8)12yes—
407C2×C4C2×#13712yes—
408C2×C4C2×#13712yes—
409C2×C4#57712yes—
410C2×C4#57712yes—
411C8—24no—
412C8—24no—
413C8—24no—
414C8—42no—
415C8—42no—
416C8#48318yescomplemented
417C22—22no—
418C22—22no—
419C22—22no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22#250111yescharacteristic
425C22C2×#49211yescharacteristic
426C22#315111yescharacteristic
427C22#315211yescharacteristic
428C22#315211yescharacteristic
429C22#315411yescharacteristic
430C22#315411yescharacteristic
431C4—22no—
432C4—41no—
433C4—41no—
434C4—161no—
435C4—161no—
436C4—161no—
437C4—161no—
438C4—161no—
439C4—161no—
440C4—161no—
441C4—161no—
442C4C2×#48312yes—
443C4#322912yes—
444C4#334412yes—
445C4#334512yes—
446C4#335412yes—
447C4#335612yes—
448C2—42no—
449C2#1962111yescharacteristic
450C2#1962211yescharacteristic
451C2#1971411yescharacteristic
452C2#1974611yescharacteristic
453C2#2790511yescharacteristic
454C2#2794011yescharacteristic
455C2#2794111yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
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
7. #10685C2semidirect
8. #10685C2semidirect
9. #10680C2semidirect
10. #10675C2semidirect
11. #9836C2semidirect
12. #9823C2semidirect
13. #8819C2semidirect
14. #8813C2semidirect
416. C8#483semidirect

Maximal subgroups

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

subgroupindex
#332122
#279482
#279052
#278892
#278752
#208002
#106852
#106802
#106752
#98362
#98232
#88192
#88132
#81462