Finite Group Databasebeta

← #324130

Subgroups — #324130

Order 512.

663 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33209C211yesmaximal, characteristic, complemented
2#24701C212yesmaximal
3#21487C211yesmaximal, characteristic
4#10652C211yesmaximal, characteristic, complemented
5#10616C211yesmaximal, characteristic, complemented
6#10543C211yesmaximal, characteristic, complemented
7#10278C211yesmaximal, characteristic, complemented
8#9575C212yesmaximal, complemented
9#9223C212yesmaximal, complemented
10#9128C212yesmaximal, complemented
11#7872C211yesmaximal, characteristic, complemented
12C2×#500C2211yescharacteristic
13C2×#476C2212yes—
14C2×#436C2211yescharacteristic
15C2×#435—21no—
16C2×#431C2212yes—
17C22×C4×C8C2211yescharacteristic
18#2857C2212yes—
19C4×#121C2211yescharacteristic
20#2077—22no—
21#2034—22no—
22#1990C2212yes—
23#1987C2212yes—
24#1954C2211yescharacteristic
25#1948C2212yes—
26#1942—21no—
27#1942—21no—
28#1941—22no—
29#1939C2211yescharacteristic
30#1934—22no—
31#1923—21no—
32#1869—21no—
33#1845C2211yescharacteristic
34#1808—21no—
35#1808—21no—
36#1802C2212yes—
37#1797C2211yescharacteristic
38#1770C2211yescharacteristic
39#1753—21no—
40C2×#333C2212yes—
41C2×#333C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1277C2212yes—
45#1277C2211yescharacteristic
46#1277C2211yescharacteristic
47C22×M16—21no—
48C23×C8D411yescharacteristic
49C23×C8D411yescharacteristic
50C2×C4⋊Q8C2312yes—
51C2×C22⋊Q8—22no—
52C2×#121D411yescharacteristic
53C2×C4×D4—21no—
54C2×C4×D4C2311yescharacteristic
55C22×C42C2311yescharacteristic
56#500—42no—
57#476—24no—
58#476—42no—
59#436—42no—
60#435—44no—
61#431—24no—
62#431—42no—
63C2×#107—21no—
64C2×#107—21no—
65C2×#107—21no—
66C2×#107—21no—
67C2×#106—22no—
68C2×#106—22no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8C2311yescharacteristic
71C2×C4⋊C8C2311yescharacteristic
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2312yes—
74C2×#102—22no—
75C2×#102—22no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—21no—
81C2×#97—21no—
82C2×C4×C8—22no—
83C2×C4×C8—42no—
84C2×C4×C8D411yescharacteristic
85C2×C4×C8D411yescharacteristic
86C2×C4×C8C2311yescharacteristic
87C2×C4×C8C2311yescharacteristic
88#391—22no—
89#390D411yescharacteristic
90#389—22no—
91#389—22no—
92#389D411yescharacteristic
93#388—21no—
94#386—21no—
95#384C2312yes—
96#380—22no—
97#380—22no—
98#380D411yescharacteristic
99C4×C4⋊C4C2311yescharacteristic
100C4×C22⋊C4—21no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—22no—
106#340—22no—
107#336—21no—
108#336—21no—
109#336—21no—
110#333—24no—
111#333—42no—
112C22×Q8—22no—
113C22×D4—21no—
114C22×D4C2×D411yescharacteristic
115C23×C4—21no—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4#4111yescharacteristic
119C2×M16—42no—
120C2×M16—42no—
121C2×M16—42no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—22no—
128C22×C8—41no—
129C22×C8—41no—
130C22×C8—41no—
131C22×C8SD1614yes—
132C22×C8C2×D411yescharacteristic
133C22×C8C2×D411yescharacteristic
134C22×C8C2×D411yescharacteristic
135C22×C8C2×D411yescharacteristic
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C22⋊Q8—84no—
141#121—42no—
142C4×D4—22no—
143C4×D4—22no—
144C4×D4—42no—
145C4×D4—42no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4C2×D411yescharacteristic
154C2×C4⋊C4C2×D411yescharacteristic
155C2×C4⋊C4C2×D411yescharacteristic
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—42no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—21no—
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42C2411yescharacteristic
169#107—42no—
170#107—42no—
171#107—42no—
172#107—42no—
173#106—44no—
174#106—44no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—42no—
181C4⋊C8#4114yes—
182#102—44no—
183#102—44no—
184C22⋊C8—44no—
185C22⋊C8—42no—
186C22⋊C8—42no—
187C22⋊C8—42no—
188C22⋊C8—42no—
189#97—42no—
190C4×C8—22no—
191C4×C8—22no—
192C4×C8—41no—
193C4×C8—41no—
194C4×C8—44no—
195#95—21no—
196#95—21no—
197#95—21no—
198#95—42no—
199#95—42no—
200C24—21no—
201C24#12111yescharacteristic
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—82no—
205C2×Q8—82no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—21no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—42no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4C2×SD1612yes—
239C22×C4#13712yes—
240C22×C4C22×D411yescharacteristic
241C22×C4C22×D411yescharacteristic
242C22×C4C2×#4111yescharacteristic
243M16—84no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—24no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—44no—
255C2×C8—44no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—42no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—42no—
265C2×C8—41no—
266C2×C8—42no—
267C2×C8C2×SD1612yes—
268C2×C8C2×SD1612yes—
269C2×C8#13512yes—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—82no—
284C4⋊C4—84no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—84no—
291C42—21no—
292C42—21no—
293C42—22no—
294C42—22no—
295C42—41no—
296C42—41no—
297C42C2×#4112yes—
298C42H(5,𝔽2)12yes—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23#46112yes—
313C23C2×#12111yescharacteristic
314Q8—162no—
315Q8—162no—
316D4—42no—
317D4—42no—
318D4—84no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4#46112yes—
355C2×C4#47212yes—
356C2×C4#53211yescharacteristic
357C2×C4#53411yescharacteristic
358C2×C4#54612yes—
359C2×C4#55312yes—
360C2×C4C22×SD1611yescharacteristic
361C2×C4C2×#13511yescharacteristic
362C2×C4C2×#13711yescharacteristic
363C2×C4#57511yescharacteristic
364C2×C4#57512yes—
365C2×C4#57811yescharacteristic
366C2×C4#57811yescharacteristic
367C8—22no—
368C8—22no—
369C8—22no—
370C8—41no—
371C8—44no—
372C8—81no—
373C8—81no—
374C8—82no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22#254311yescharacteristic
390C22C2×#46111yescharacteristic
391C22#303511yescharacteristic
392C22#305411yescharacteristic
393C22#305911yescharacteristic
394C22C2×#47211yescharacteristic
395C22#306711yescharacteristic
396C4—22no—
397C4—22no—
398C4—22no—
399C4—41no—
400C4—81no—
401C4—81no—
402C4—81no—
403C4—162no—
404C4—162no—
405C4#316711yescharacteristic
406C4#318111yescharacteristic
407C4#318711yescharacteristic
408C4#319011yescharacteristic
409C4#331312yes—
410C4#331512yes—
411C2—42no—
412C2—161no—
413C2#1913711yescharacteristic
414C2#1916011yescharacteristic
415C2#1919311yescharacteristic
416C2#1922011yescharacteristic
417C2#2856411yescharacteristic
418C2#2857111yescharacteristic
419C2#2859211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
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
4. #10652C2semidirect
5. #10616C2semidirect
6. #10543C2semidirect
7. #10278C2semidirect
8. #9575C2semidirect
9. #9223C2semidirect
10. #9128C2semidirect
11. #7872C2semidirect

Maximal subgroups

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

subgroupindex
#332092
#247012
#214872
#106522
#106162
#105432
#102782
#95752
#92232
#91282
#78722