Finite Group Databasebeta

← #241323

Subgroups — #241323

Order 512.

850 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37623C211yesmaximal, characteristic
2#21962C212yesmaximal
3#20800C211yesmaximal, characteristic
4#10580C212yesmaximal, complemented
5#10579C212yesmaximal, complemented
6#10577C211yesmaximal, characteristic, complemented
7#10576C211yesmaximal, characteristic, complemented
8#10574C211yesmaximal, characteristic, complemented
9#10573C211yesmaximal, characteristic, complemented
10C2×#1296C211yesmaximal, characteristic
11#7895C212yesmaximal
12C2×#428—22no—
13C22×C4⋊C8C2212yes—
14C2×C4×M16C2211yescharacteristic
15#2843—22no—
16#2660—22no—
17#2647—22no—
18#2627—22no—
19#2571C2211yescharacteristic
20#2552C2212yes—
21#2551C2211yescharacteristic
22#2442—22no—
23#2440—22no—
24#2285C2211yescharacteristic
25C2×#384C2211yescharacteristic
26#2283C2212yes—
27#1949—22no—
28#1949—22no—
29#1948C2212yes—
30#1948C2212yes—
31#1947C2212yes—
32#1947C2212yes—
33#1847—22no—
34#1845C2211yescharacteristic
35#1845C2211yescharacteristic
36#1844C2211yescharacteristic
37#1844C2211yescharacteristic
38#1808—22no—
39#1807—22no—
40#1296C2211yescharacteristic
41#1296C2211yescharacteristic
42#1296C2212yes—
43#1279C2212yes—
44#1279C2212yes—
45#1279C2212yes—
46#1279C2212yes—
47C22×M16—21no—
48C23×C8—22no—
49C2×C4⋊Q8—22no—
50C2×#125—22no—
51C2×C22⋊Q8—22no—
52C2×#117—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4D412yes—
55C22×C42C2311yescharacteristic
56#428—48no—
57C2×#107—22no—
58C2×#107—22no—
59C2×#107—22no—
60C2×#107—22no—
61C2×#106—22no—
62C2×#106—22no—
63C2×#106—22no—
64C2×#106—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—24no—
67C2×C4⋊C8Q818yescomplemented
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2312yes—
70C4×M16—22no—
71C4×M16Q814yescomplemented
72C2×#97—21no—
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2311yescharacteristic
75#400—24no—
76#400—22no—
77#400—22no—
78#398—24no—
79#398—24no—
80#398—24no—
81#398—24no—
82#395—24no—
83#395—22no—
84#395—22no—
85#389—22no—
86#389—22no—
87#389—22no—
88#389—22no—
89#385—24no—
90#384—22no—
91#384—22no—
92#384D412yes—
93#384C2311yescharacteristic
94#384C2311yescharacteristic
95#384C2311yescharacteristic
96#384C2311yescharacteristic
97#383—22no—
98#382—22no—
99#382—21no—
100#382D412yes—
101#382D412yes—
102#381—22no—
103#380—22no—
104#380—22no—
105#380D412yes—
106#380D412yes—
107C4×C4⋊C4—22no—
108C4×C4⋊C4C2312yes—
109C4×C4⋊C4C2312yes—
110C4×C22⋊C4—22no—
111#376—22no—
112#340—22no—
113#340—22no—
114#340—22no—
115#340—22no—
116C22×Q8—22no—
117C23×C4—22no—
118C23×C4C2×D412yes—
119C23×C4C2×D411yescharacteristic
120C2×M16—24no—
121C2×M16—24no—
122C2×M16—24no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—28no—
128C22×C8—82no—
129C22×C8—82no—
130C4⋊Q8—82no—
131C4⋊Q8—82no—
132#125—82no—
133#125—82no—
134C22⋊Q8—84no—
135#117—44no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—82no—
151C2×C4⋊C4—82no—
152C2×C4⋊C4C2×D412yes—
153C2×C4⋊C4C2×D412yes—
154C2×C4⋊C4C2×D412yes—
155C2×C4⋊C4C2×D412yes—
156C2×C4⋊C4C2×D412yes—
157C2×C4⋊C4C2×D412yes—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—41no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—41no—
164C2×C42C2×D412yes—
165C2×C42C2×D411yescharacteristic
166C2×C42C2×Q812yes—
167C2×C42C2×Q814yes—
168C2×C42#4112yes—
169C2×C42C2411yescharacteristic
170#107—44no—
171#107—44no—
172#107—82no—
173#107—82no—
174#106—44no—
175#106—44no—
176#106—82no—
177#106—82no—
178C4⋊C8—28no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8C2×Q814yes—
182C4⋊C8C2×Q814yes—
183#97—24no—
184C4×C8—21no—
185C4×C8—21no—
186C4×C8C2×Q812yes—
187C4×C8C2×Q812yes—
188#95—22no—
189#95—22no—
190#95—24no—
191#95—24no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—42no—
201C24#12011yescharacteristic
202C2×Q8—82no—
203C2×Q8—84no—
204C22×C4—22no—
205C22×C4—22no—
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—44no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—84no—
220C22×C4#12012yes—
221C22×C4#12011yescharacteristic
222C22×C4C22⋊Q814yes—
223C22×C4C22⋊Q812yes—
224C22×C4C4⋊Q812yes—
225C22×C4C4⋊Q814yes—
226C22×C4C22×D411yescharacteristic
227C22×C4C22×D412yes—
228M16—216no—
229C2×C8—22no—
230C2×C8—22no—
231C2×C8—22no—
232C2×C8—22no—
233C2×C8—22no—
234C2×C8—22no—
235C2×C8—24no—
236C2×C8—24no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—88no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—82no—
263C4⋊C4—84no—
264C4⋊C4—82no—
265C4⋊C4—84no—
266C4⋊C4—84no—
267C4⋊C4—84no—
268C4⋊C4—84no—
269C22⋊C4—44no—
270C22⋊C4—44no—
271C22⋊C4—82no—
272C42—42no—
273C42—42no—
274C42—42no—
275C42—42no—
276C42—82no—
277C42C22⋊Q812yes—
278C42C22⋊Q814yes—
279C42C4⋊Q812yes—
280C42C4⋊Q814yes—
281C42C22×Q811yescharacteristic
282C42C22×Q812yes—
283C42C2×#4111yescharacteristic
284C23—22no—
285C23—22no—
286C23#39512yes—
287C23#45312yes—
288C23#45612yes—
289C23C2×#12011yescharacteristic
290Q8—164no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—164no—
324C2×C4#39514yes—
325C2×C4#39512yes—
326C2×C4C2×C22⋊Q811yescharacteristic
327C2×C4C2×C22⋊Q812yes—
328C2×C4C2×C4⋊Q811yescharacteristic
329C2×C4C2×C4⋊Q812yes—
330C2×C4#57612yes—
331C2×C4#57712yes—
332C2×C4#57712yes—
333C2×C4#57812yes—
334C8—24no—
335C8—24no—
336C8—44no—
337C8—44no—
338C22—24no—
339C22—22no—
340C22—22no—
341C22—24no—
342C22#206014yes—
343C22C2×#39511yescharacteristic
344C22#301212yes—
345C22#301312yes—
346C22C2×#45311yescharacteristic
347C22C2×#45611yescharacteristic
348C4—24no—
349C4—21no—
350C4—21no—
351C4—22no—
352C4—82no—
353C4—82no—
354C4—82no—
355C4—82no—
356C4—82no—
357C4—82no—
358C4—161no—
359C4—161no—
360C4#308112yes—
361C4#308212yes—
362C4#316312yes—
363C4#316412yes—
364C2—24no—
365C2#1871211yescharacteristic
366C2#1871311yescharacteristic
367C2#1873312yes—
368C2#2355712yes—
369C2C2×#206011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
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
4. #10580C2semidirect
5. #10579C2semidirect
6. #10577C2semidirect
7. #10576C2semidirect
8. #10574C2semidirect
9. #10573C2semidirect
67. C2×C4⋊C8Q8semidirect
71. C4×M16Q8semidirect

Maximal subgroups

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

subgroupindex
#376232
#219622
#208002
#105802
#105792
#105772
#105762
#105742
#105732
C2×#12962
#78952