Finite Group Databasebeta

← #215910

Subgroups — #215910

Order 512.

954 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic
3#14690C211yesmaximal, characteristic
4#9912C214yesmaximal
5#9888C214yesmaximal
6#8717C214yesmaximal
7C2×#533—21no—
8#2868C411yescharacteristic, complemented
9#2868C411yescharacteristic, complemented
10#2862—21no—
11#2859—21no—
12#2859C412yescomplemented
13#2857—21no—
14#2857—21no—
15#2857C2212yes—
16#2857C2212yes—
17C4×C4⋊Q8—21no—
18C4×C4⋊Q8—21no—
19C4×C4⋊Q8C412yescomplemented
20C4×C4⋊Q8C412yescomplemented
21#2285—21no—
22#2285C2212yes—
23C2×C43C2211yescharacteristic
24#1839—21no—
25#1831—21no—
26#1817C2212yes—
27#1797C2212yes—
28#1767—21no—
29#1749C2212yes—
30#1745—21no—
31#1434—21no—
32#1434—21no—
33#1434—21no—
34#1434—21no—
35#1434C2212yes—
36#1434C2212yes—
37#1434C2212yes—
38#1294C22116yes—
39#533—48no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8C2×C412yes—
46C2×#125—22no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×#117—22no—
52C22×C42D411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C2×C4⋊C8—22no—
60C2×C22⋊C8C2314yes—
61C2×#97—22no—
62#391—22no—
63#391—22no—
64#391—22no—
65#391—22no—
66#389—22no—
67#389—22no—
68#389—22no—
69#389—22no—
70#389—22no—
71#389—22no—
72#389C2×C414yes—
73#389C2×C414yes—
74#384—21no—
75#384—21no—
76#384—21no—
77#384—21no—
78#384C2×C412yes—
79#384C2×C412yes—
80#384C2×C412yes—
81#384C2×C412yes—
82#382—22no—
83#382—22no—
84#380—22no—
85#380—22no—
86#380—22no—
87#380C2314yes—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4C2×C414yes—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95C2314yes—
99C43C2×C411yescharacteristic
100C43C2×C411yescharacteristic
101C43C2×C411yescharacteristic
102C43C2×C411yescharacteristic
103C43D411yescharacteristic
104C43D411yescharacteristic
105C43D411yescharacteristic
106C43D411yescharacteristic
107#336—22no—
108#336—22no—
109#336—22no—
110#336—22no—
111C22×Q8—22no—
112C22×Q8—22no—
113C23×C4—24no—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2411yescharacteristic
121C22×C8—24no—
122C4⋊Q8—44no—
123C4⋊Q8—44no—
124C4⋊Q8—44no—
125C4⋊Q8—44no—
126C4⋊Q8—44no—
127C4⋊Q8—44no—
128#125—48no—
129C22⋊Q8—48no—
130C22⋊Q8—48no—
131C4×Q8—48no—
132C4×Q8—48no—
133#117—48no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4C22×C414yes—
147C2×C4⋊C4C22×C414yes—
148C2×C22⋊C4—24no—
149C2×C42—24no—
150C2×C42—24no—
151C2×C42—24no—
152C2×C42—24no—
153C2×C42—42no—
154C2×C42—42no—
155C2×C42—42no—
156C2×C42—42no—
157C2×C42C22⋊C411yescharacteristic
158C2×C42C22⋊C411yescharacteristic
159C2×C42C22⋊C411yescharacteristic
160C2×C42C22⋊C411yescharacteristic
161C2×C42C22⋊C411yescharacteristic
162C2×C42C22⋊C411yescharacteristic
163C2×C42C22⋊C411yescharacteristic
164C2×C42C22⋊C411yescharacteristic
165C2×C42C22⋊C411yescharacteristic
166C2×C42C22⋊C411yescharacteristic
167C2×C42C22⋊C411yescharacteristic
168C2×C42C22⋊C411yescharacteristic
169C2×C42C22⋊C411yescharacteristic
170C2×C42C22⋊C411yescharacteristic
171C2×C42C22⋊C411yescharacteristic
172C2×C42C22⋊C411yescharacteristic
173C2×C42C22×C411yescharacteristic
174C2×C42C22×C411yescharacteristic
175C2×C42C22×C411yescharacteristic
176C2×C42C22×C411yescharacteristic
177C2×C42C22×C411yescharacteristic
178C2×C42C22×C411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C4⋊C8—44no—
186C22⋊C8—28no—
187#97—44no—
188#95—24no—
189#95—24no—
190#95—24no—
191#95—24no—
192#95—28no—
193#95—44no—
194#95—44no—
195#95—44no—
196#95—44no—
197#95—44no—
198#95—44no—
199C24C22×D411yescharacteristic
200C2×Q8—44no—
201C2×Q8—44no—
202C2×Q8—44no—
203C2×Q8—44no—
204C2×Q8—44no—
205C2×Q8—44no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4AGL(1,ℤ/8)14yes—
246C22×C4#13714yes—
247C22×C4C23×C411yescharacteristic
248C22×C4C22×D411yescharacteristic
249C22×C42−1+414yes—
250C2×C8—44no—
251C2×C8—44no—
252C2×C8—44no—
253C4⋊C4—44no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—48no—
258C4⋊C4—44no—
259C4⋊C4—44no—
260C4⋊C4—44no—
261C4⋊C4—44no—
262C4⋊C4—44no—
263C4⋊C4—44no—
264C4⋊C4—48no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C22⋊C4—48no—
268C42—22no—
269C42—22no—
270C42—22no—
271C42—22no—
272C42—22no—
273C42—22no—
274C42—22no—
275C42—22no—
276C42—22no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—41no—
289C42—41no—
290C42—41no—
291C42—41no—
292C42—41no—
293C42—41no—
294C42—41no—
295C42—41no—
296C42—48no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23C22×C22⋊C411yescharacteristic
302C23#53612yes—
303C23C2×AGL(1,ℤ/8)12yes—
304C23C2×#13712yes—
305Q8—88no—
306Q8—88no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—28no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—84no—
353C2×C4—84no—
354C2×C4—84no—
355C2×C4—84no—
356C2×C4—84no—
357C2×C4—84no—
358C2×C4—84no—
359C2×C4#41012yes—
360C2×C4#41012yes—
361C2×C4#41812yes—
362C2×C4#41812yes—
363C2×C4#41912yes—
364C2×C4#41912yes—
365C2×C4#42112yes—
366C2×C4#42112yes—
367C2×C4#51912yes—
368C2×C4#53612yes—
369C8—84no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22#231011yescharacteristic
377C22C2×#41011yescharacteristic
378C22C2×#41011yescharacteristic
379C22C2×#41811yescharacteristic
380C22C2×#41911yescharacteristic
381C22C2×#42111yescharacteristic
382C22C2×#42111yescharacteristic
383C22#323914yes—
384C4—22no—
385C4—22no—
386C4—22no—
387C4—22no—
388C4—22no—
389C4—22no—
390C4—22no—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—84no—
399C4—84no—
400C4—164no—
401C2—24no—
402C2#1756211yescharacteristic
403C2#1758311yescharacteristic
404C2#1759011yescharacteristic
405C2#1759511yescharacteristic
406C2#2182611yescharacteristic
407C2#2184111yescharacteristic
408C2#2184511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
8. #2868C4semidirect
9. #2868C4semidirect
12. #2859C4semidirect
19. C4×C4⋊Q8C4semidirect
20. C4×C4⋊Q8C4semidirect

Maximal subgroups

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

subgroupindex
#366452
#207782
#146902
#99122
#98882
#87172