Finite Group Databasebeta

← #319006

Subgroups — #319006

Order 512.

1138 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33209C211yesmaximal, characteristic, complemented
2#27960C211yesmaximal, characteristic, complemented
3#27905C211yesmaximal, characteristic, complemented
4#25855C211yesmaximal, characteristic
5#25717C211yesmaximal, characteristic
6C2×#1957C211yesmaximal, characteristic
7#10672C212yesmaximal, complemented
8#10308C212yesmaximal, complemented
9#9822C212yesmaximal, complemented
10#9820C212yesmaximal, complemented
11C8×C22⋊C8C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C2×#500C2211yescharacteristic, complemented
14C2×#492C2212yes—
15C2×#486C2212yes—
16C2×#483—22no—
17C2×#450—22no—
18C22×#102—22no—
19C22×C4×C8C2211yescharacteristic
20#2857C2211yescharacteristic
21#2857C2211yescharacteristic
22C2×#384C2211yescharacteristic
23#2099—22no—
24#2096—22no—
25#2041—22no—
26#2014—22no—
27#1990C2212yescomplemented
28#1990C2211yescharacteristic, complemented
29#1990C2211yescharacteristic, complemented
30#1957—22no—
31#1957C2212yes—
32#1957C2212yes—
33#1957C2212yes—
34#1957C2212yes—
35#1956C2212yescomplemented
36#1942—22no—
37#1845C2211yescharacteristic, complemented
38#1844C2212yescomplemented
39#1834C2212yescomplemented
40#1823—22no—
41#1808—22no—
42#1786—22no—
43C4×C22⋊C8C2211yescharacteristic
44C4×C22⋊C8C2211yescharacteristic
45C2×C82C2212yescomplemented
46#1273C2212yescomplemented
47C23×D4—21no—
48C23×D4—21no—
49C23×D4—21no—
50C22×SD16—24no—
51C23×C8D412yes—
52C2×C4⋊Q8C2311yescharacteristic
53C2×C4⋊Q8C2311yescharacteristic
54C2×#127—24no—
55C2×#127—21no—
56C2×#127—21no—
57C2×#127—21no—
58C2×#127—42no—
59C2×#127C2312yes—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C42C2311yescharacteristic
66#500—42no—
67#492—24no—
68#492—42no—
69#486—24no—
70#486—42no—
71#483—48no—
72#450—48no—
73C2×#107—22no—
74C2×#107—22no—
75C2×#106—24no—
76C2×#103—24no—
77C2×#102—24no—
78C2×#102—24no—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—48no—
83C2×C22⋊C8D412yes—
84C2×C22⋊C8D412yes—
85C2×C4×C8—42no—
86C2×C4×C8D412yes—
87C2×C4×C8C2312yes—
88C2×C4×C8C2312yes—
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91#391—21no—
92#391—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#384—21no—
98#384—21no—
99#384—21no—
100#384C2311yescharacteristic
101#384C2311yescharacteristic
102#384C2312yes—
103#380—21no—
104#380—21no—
105#380—21no—
106#380—21no—
107#340—22no—
108#340—22no—
109#340—24no—
110#340—24no—
111#336D412yes—
112#336D412yes—
113C82—42no—
114C25—41no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×D4—24no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—44no—
122C22×D4—44no—
123C22×D4—44no—
124C22×D4—42no—
125C22×D4—41no—
126C22×D4—41no—
127C22×D4—41no—
128C22×D4—44no—
129C22×D4—44no—
130C22×D4—44no—
131C22×D4—82no—
132C22×D4—82no—
133C23×C4—41no—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C2×SD16—48no—
138C2×SD16—48no—
139C2×SD16—84no—
140C22×C8—42no—
141C22×C8—42no—
142C22×C8SD1618yescomplemented
143C22×C8C2×D414yes—
144C22×C8C2×D414yes—
145C22×C8C2×D412yes—
146C22×C8C2×D412yes—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151#127—24no—
152#127—22no—
153#127—22no—
154#127—22no—
155#127—44no—
156#127—44no—
157#127—42no—
158#127#4114yes—
159C22⋊Q8—44no—
160C22⋊Q8—44no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
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—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2411yescharacteristic
190#107—44no—
191#107—44no—
192#106—48no—
193#103—48no—
194#102—48no—
195#102—48no—
196#102—48no—
197#102—48no—
198#102—84no—
199C22⋊C8—24no—
200C22⋊C8D818yes—
201C4×C8—22no—
202C4×C8—22no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8—22no—
207C4×C8—42no—
208#95—41no—
209#95—41no—
210#95—41no—
211#95—41no—
212C24—44no—
213C24—42no—
214C24—82no—
215C24—82no—
216C24—82no—
217C24—82no—
218C24—82no—
219C24—82no—
220C24#12711yescharacteristic
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×D4—24no—
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—416no—
231C2×D4—44no—
232C2×D4—44no—
233C2×D4—44no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—44no—
238C2×D4—44no—
239C2×D4—44no—
240C2×D4—44no—
241C2×D4—44no—
242C2×D4—44no—
243C2×D4—816no—
244C2×D4—816no—
245C2×D4—82no—
246C2×D4—88no—
247C2×D4—82no—
248C2×D4—82no—
249C2×D4—82no—
250C2×D4—82no—
251C2×D4—82no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4#12711yescharacteristic
268C22×C4#12711yescharacteristic
269C22×C4#12711yescharacteristic
270C22×C4C2×D814yes—
271C22×C4C2×SD1614yes—
272C22×C4#13514yes—
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276SD16—816no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—24no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—24no—
286C2×C8—48no—
287C2×C8C2×D818yes—
288C2×C8C2×SD1614yes—
289C2×C8C2×SD1614yes—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—44no—
298C4⋊C4—42no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C42—21no—
313C42—21no—
314C42—22no—
315C42—22no—
316C42C2×#4112yes—
317C42H(5,𝔽2)12yes—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—44no—
323C23—88no—
324C23—88no—
325C23—88no—
326C23—82no—
327C23—82no—
328C23—82no—
329C23—82no—
330C23—82no—
331C23—82no—
332C23—164no—
333C23—164no—
334C23—164no—
335C23—164no—
336C23#49212yes—
337C23#49312yes—
338C23#49512yes—
339C23C2×#12711yescharacteristic
340Q8—84no—
341Q8—84no—
342D4—48no—
343D4—48no—
344D4—48no—
345D4—88no—
346D4—88no—
347D4—88no—
348D4—84no—
349D4—84no—
350D4—84no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—28no—
360C2×C4—28no—
361C2×C4—82no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—164no—
377C2×C4#49212yes—
378C2×C4#49312yes—
379C2×C4#49512yes—
380C2×C4#52511yescharacteristic
381C2×C4#53411yescharacteristic
382C2×C4#54612yes—
383C2×C4#54612yes—
384C2×C4C22×D812yes—
385C2×C4C22×SD1612yes—
386C2×C4C2×#13512yes—
387C2×C4#57512yes—
388C2×C4#57712yes—
389C2×C4#57712yes—
390C8—24no—
391C8—24no—
392C8—28no—
393C8—44no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—84no—
401C22—84no—
402C22—84no—
403C22—1616no—
404C22—162no—
405C22—162no—
406C22—162no—
407C22—162no—
408C22#170614yes—
409C22#253911yescharacteristic
410C22C2×#49211yescharacteristic
411C22C2×#49311yescharacteristic
412C22C2×#49511yescharacteristic
413C22#315311yescharacteristic
414C22#315411yescharacteristic
415C22#315411yescharacteristic
416C4—22no—
417C4—22no—
418C4—22no—
419C4—162no—
420C4—161no—
421C4—161no—
422C4—161no—
423C4—161no—
424C4#310112yes—
425C4#318712yes—
426C4#329414yes—
427C4#329614yes—
428C2—24no—
429C2—164no—
430C2C2×#170611yescharacteristic
431C2#1963111yescharacteristic
432C2#1963711yescharacteristic
433C2#1965111yescharacteristic
434C2#2830211yescharacteristic
435C2#2832911yescharacteristic
436C2#2833011yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
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. #33209C2semidirect
2. #27960C2semidirect
3. #27905C2semidirect
7. #10672C2semidirect
8. #10308C2semidirect
9. #9822C2semidirect
10. #9820C2semidirect
11. C8×C22⋊C8C2semidirect
13. C2×#500C22semidirect
27. #1990C22semidirect
28. #1990C22semidirect
29. #1990C22semidirect
35. #1956C22semidirect
37. #1845C22semidirect
38. #1844C22semidirect
39. #1834C22semidirect
45. C2×C82C22semidirect
46. #1273C22semidirect
142. C22×C8SD16semidirect

Maximal subgroups

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

subgroupindex
#332092
#279602
#279052
#258552
#257172
C2×#19572
#106722
#103082
#98222
#98202
C8×C22⋊C82