Finite Group Databasebeta

← #324129

Subgroups — #324129

Order 512.

691 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33209C211yesmaximal, characteristic, complemented
2#24836C212yesmaximal
3#21487C211yesmaximal, characteristic
4#10651C211yesmaximal, characteristic, complemented
5#10615C211yesmaximal, characteristic, complemented
6#10543C211yesmaximal, characteristic, complemented
7#10278C211yesmaximal, characteristic, complemented
8#9575C212yesmaximal, complemented
9#9223C212yesmaximal, complemented
10#9127C212yesmaximal, complemented
11C2×#1277C211yesmaximal, characteristic, complemented
12C2×#500C2211yescharacteristic
13C2×#476C2212yes—
14C2×#436C2211yescharacteristic
15C2×#435—21no—
16C22×C4⋊C8C2212yes—
17C22×C4×C8C2211yescharacteristic
18#2857C2212yes—
19C4×#121C2211yescharacteristic
20#2088—22no—
21#2065—22no—
22#1990C2212yes—
23#1987C2212yes—
24#1954C2211yescharacteristic
25#1948C2212yes—
26#1942—21no—
27#1942—21no—
28#1939C2211yescharacteristic
29#1923—21no—
30#1893—22no—
31#1869—21no—
32#1845C2211yescharacteristic
33#1808—21no—
34#1808—21no—
35#1807—22no—
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×C8—22no—
49C23×C8D411yescharacteristic
50C23×C8D411yescharacteristic
51C2×C4⋊Q8C2312yes—
52C2×C22⋊Q8—22no—
53C2×#121D411yescharacteristic
54C2×C4×D4—21no—
55C2×C4×D4C2311yescharacteristic
56C22×C42C2311yescharacteristic
57#500—42no—
58#476—24no—
59#476—42no—
60#436—42no—
61#435—44no—
62C2×#107—21no—
63C2×#107—21no—
64C2×#107—21no—
65C2×#107—21no—
66C2×#106—22no—
67C2×#106—22no—
68C2×C4⋊C8—24no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—42no—
72C2×C4⋊C8C2311yescharacteristic
73C2×C4⋊C8C2311yescharacteristic
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2312yes—
76C2×#102—22no—
77C2×#102—22no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C2×#97—21no—
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—22no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—41no—
129C22×C8—41no—
130C22×C8—41no—
131C22×C8—44no—
132C22×C8—44no—
133C22×C8—82no—
134C22×C8SD1614yes—
135C22×C8C2×D411yescharacteristic
136C22×C8C2×D411yescharacteristic
137C22×C8C2×D411yescharacteristic
138C22×C8C2×D411yescharacteristic
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C22⋊Q8—84no—
144#121—24no—
145C4×D4—22no—
146C4×D4—22no—
147C4×D4—42no—
148C4×D4—42no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4C2×D411yescharacteristic
157C2×C4⋊C4C2×D411yescharacteristic
158C2×C4⋊C4C2×D411yescharacteristic
159C2×C22⋊C4—21no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—42no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—21no—
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42C2411yescharacteristic
172#107—42no—
173#107—42no—
174#107—42no—
175#107—42no—
176#106—44no—
177#106—82no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—44no—
184C4⋊C8—42no—
185C4⋊C8#4114yes—
186#102—44no—
187#102—82no—
188C22⋊C8—42no—
189C22⋊C8—42no—
190C22⋊C8—42no—
191C22⋊C8—42no—
192#97—42no—
193C4×C8—22no—
194C4×C8—22no—
195C4×C8—41no—
196C4×C8—41no—
197#95—21no—
198#95—21no—
199#95—21no—
200#95—42no—
201#95—42no—
202C24—21no—
203C24#12111yescharacteristic
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—82no—
207C2×Q8—82no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—21no—
211C2×D4—21no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4C2×SD1612yes—
240C22×C4#13712yes—
241C22×C4#13714yes—
242C22×C4C22×D411yescharacteristic
243C22×C4C22×D411yescharacteristic
244C22×C4C2×#4111yescharacteristic
245M16—84no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—24no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—44no—
257C2×C8—44no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—42no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—42no—
267C2×C8—41no—
268C2×C8—42no—
269C2×C8—88no—
270C2×C8C2×SD1612yes—
271C2×C8C2×SD1612yes—
272C2×C8#13512yes—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—82no—
287C4⋊C4—84no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—84no—
294C42—21no—
295C42—21no—
296C42—22no—
297C42—22no—
298C42—41no—
299C42—41no—
300C42C2×#4112yes—
301C42H(5,𝔽2)12yes—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23#46112yes—
314C23#47012yes—
315C23#47212yes—
316C23C2×#12111yescharacteristic
317Q8—162no—
318Q8—162no—
319D4—42no—
320D4—42no—
321D4—84no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—44no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4#46112yes—
358C2×C4#47012yes—
359C2×C4#53211yescharacteristic
360C2×C4#53411yescharacteristic
361C2×C4#54612yes—
362C2×C4#55312yes—
363C2×C4C22×SD1611yescharacteristic
364C2×C4C2×#13511yescharacteristic
365C2×C4C2×#13711yescharacteristic
366C2×C4C2×#13712yes—
367C2×C4#57511yescharacteristic
368C2×C4#57611yescharacteristic
369C2×C4#57811yescharacteristic
370C8—22no—
371C8—22no—
372C8—22no—
373C8—41no—
374C8—44no—
375C8—81no—
376C8—81no—
377C8—82no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22#169514yes—
392C22#254311yescharacteristic
393C22C2×#46111yescharacteristic
394C22#303511yescharacteristic
395C22C2×#47011yescharacteristic
396C22#306111yescharacteristic
397C22C2×#47211yescharacteristic
398C22#306711yescharacteristic
399C4—22no—
400C4—22no—
401C4—22no—
402C4—41no—
403C4—81no—
404C4—81no—
405C4—81no—
406C4—162no—
407C4—162no—
408C4#316711yescharacteristic
409C4#318111yescharacteristic
410C4#318711yescharacteristic
411C4#319011yescharacteristic
412C4#331212yes—
413C4#331412yes—
414C2—24no—
415C2—161no—
416C2C2×#169511yescharacteristic
417C2#1915811yescharacteristic
418C2#1919511yescharacteristic
419C2#1922211yescharacteristic
420C2#2856311yescharacteristic
421C2#2857111yescharacteristic
422C2#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. #10651C2semidirect
5. #10615C2semidirect
6. #10543C2semidirect
7. #10278C2semidirect
8. #9575C2semidirect
9. #9223C2semidirect
10. #9127C2semidirect
11. C2×#1277C2semidirect

Maximal subgroups

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

subgroupindex
#332092
#248362
#214872
#106512
#106152
#105432
#102782
#95752
#92232
#91272
C2×#12772