Finite Group Databasebeta

← #241277

Subgroups — #241277

Order 512.

892 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#34978C211yesmaximal, characteristic
2#21362C212yesmaximal
3#20797C211yesmaximal, characteristic
4#10580C212yesmaximal, complemented
5#10579C212yesmaximal, complemented
6#10577C211yesmaximal, characteristic, complemented
7#10576C211yesmaximal, characteristic, complemented
8#10574C211yesmaximal, characteristic, complemented
9#10573C211yesmaximal, characteristic, complemented
10#8038C211yesmaximal, characteristic
11#7890C212yesmaximal
12C2×#431C2212yes—
13C2×#427—22no—
14C22×C4×C8C2211yescharacteristic
15#2612—22no—
16#2571C2211yescharacteristic
17#2552C2212yes—
18#2551C2211yescharacteristic
19#2441—22no—
20#2439—22no—
21#2396—22no—
22C4×C4⋊Q8—22no—
23C4×#125—22no—
24#2285C2211yescharacteristic
25#2285C2211yescharacteristic
26C4×#117C2212yes—
27#1948C2212yes—
28#1948C2212yes—
29#1947C2212yes—
30#1947C2212yes—
31#1854—22no—
32#1854—22no—
33#1846—22no—
34#1845C2211yescharacteristic
35#1845C2211yescharacteristic
36#1844C2211yescharacteristic
37#1844C2211yescharacteristic
38C4×#107—22no—
39C4×#106—22no—
40#1296C2211yescharacteristic
41#1296C2211yescharacteristic
42#1296C2212yes—
43#1279C2212yes—
44#1279C2212yes—
45#1279C2212yes—
46#1279C2212yes—
47C23×C8—21no—
48C2×C4⋊Q8—22no—
49C2×#125—22no—
50C2×C4×Q8—22no—
51C2×#117—21no—
52C2×#117D412yes—
53C2×#117D412yes—
54C2×#117D412yes—
55C22×C42C2311yescharacteristic
56#431—24no—
57#431Q818yescomplemented
58#427—48no—
59C2×#107—22no—
60C2×#107—22no—
61C2×#107—22no—
62C2×#107—22no—
63C2×#106—22no—
64C2×#106—22no—
65C2×#106—22no—
66C2×#106—22no—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2312yes—
69C2×C22⋊C8—22no—
70C2×C4×C8—21no—
71C2×C4×C8—22no—
72C2×C4×C8—22no—
73C2×C4×C8Q814yescomplemented
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76#400—22no—
77#399—22no—
78#398—24no—
79#396—24no—
80#395—22no—
81#393—22no—
82#391—24no—
83#389—24no—
84#389—22no—
85#389—22no—
86#389—22no—
87#389—22no—
88#387—24no—
89#387—24no—
90#384—22no—
91#384C2311yescharacteristic
92#384C2311yescharacteristic
93#384C2311yescharacteristic
94#384C2311yescharacteristic
95#382—22no—
96#382—21no—
97#382D412yes—
98#380—22no—
99#380—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—24no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4D412yes—
105C4×C4⋊C4D412yes—
106C4×C4⋊C4C2312yes—
107C4×C4⋊C4C2312yes—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110#376—22no—
111C43—22no—
112#340—22no—
113#340—22no—
114#340—22no—
115#340—22no—
116C22×Q8—22no—
117C23×C4C2×D412yes—
118C23×C4C2×D411yescharacteristic
119C22×C8—24no—
120C22×C8—24no—
121C22×C8—24no—
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—22no—
125C22×C8—22no—
126C4⋊Q8—44no—
127C4⋊Q8—44no—
128#125—44no—
129#125—44no—
130C4×Q8—84no—
131#117—24no—
132#117—24no—
133#117—42no—
134#117—42no—
135#117—44no—
136#117—44no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4C2×D412yes—
149C2×C4⋊C4C2×D412yes—
150C2×C4⋊C4C2×D412yes—
151C2×C4⋊C4C2×D412yes—
152C2×C4⋊C4C2×D412yes—
153C2×C4⋊C4C2×D412yes—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—41no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—41no—
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D412yes—
168C2×C42C2×Q812yes—
169C2×C42C2×Q814yes—
170C2×C42#4112yes—
171C2×C42C2411yescharacteristic
172#107—44no—
173#107—44no—
174#107—44no—
175#107—44no—
176#106—44no—
177#106—44no—
178#106—44no—
179#106—44no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8C2×Q814yes—
183C4⋊C8C2×Q814yes—
184C22⋊C8—28no—
185C4×C8—24no—
186C4×C8—28no—
187C4×C8—21no—
188C4×C8—21no—
189C4×C8C2×Q812yes—
190C4×C8C2×Q812yes—
191#95—22no—
192#95—24no—
193#95—22no—
194#95—24no—
195#95—22no—
196#95—22no—
197#95—24no—
198#95—22no—
199#95—22no—
200#95—42no—
201C24#12011yescharacteristic
202C2×Q8—48no—
203C2×Q8—82no—
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—41no—
217C22×C4—41no—
218C22×C4#12012yes—
219C22×C4#12011yescharacteristic
220C22×C4C22⋊Q814yes—
221C22×C4C22⋊Q812yes—
222C22×C4C4⋊Q812yes—
223C22×C4C4⋊Q814yes—
224C22×C4C22×D411yescharacteristic
225C22×C4C22×D412yes—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8—22no—
229C2×C8—22no—
230C2×C8—216no—
231C2×C8—22no—
232C2×C8—22no—
233C2×C8—24no—
234C2×C8—24no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—48no—
248C4⋊C4—48no—
249C4⋊C4—48no—
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—84no—
261C4⋊C4—82no—
262C4⋊C4#13514yes—
263C4⋊C4#13514yes—
264C22⋊C4—44no—
265C22⋊C4—44no—
266C22⋊C4—82no—
267C22⋊C4—82no—
268C42—44no—
269C42—44no—
270C42—42no—
271C42—42no—
272C42—42no—
273C42—42no—
274C42—44no—
275C42—44no—
276C42—44no—
277C42—82no—
278C42C22⋊Q812yes—
279C42C22⋊Q814yes—
280C42C4⋊Q812yes—
281C42C4⋊Q814yes—
282C42C22×Q811yescharacteristic
283C42C22×Q812yes—
284C42C2×#4111yescharacteristic
285C23—22no—
286C23—22no—
287C23—21no—
288C23—21no—
289C23#39512yes—
290C23C2×#12011yescharacteristic
291Q8—88no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4#39514yes—
323C2×C4#39512yes—
324C2×C4#45312yes—
325C2×C4#45612yes—
326C2×C4C2×C22⋊Q811yescharacteristic
327C2×C4C2×C22⋊Q812yes—
328C2×C4C2×C4⋊Q811yescharacteristic
329C2×C4C2×C4⋊Q812yes—
330C2×C4C2×#13512yes—
331C2×C4C2×#13512yes—
332C2×C4#57512yes—
333C2×C4#57512yes—
334C8—24no—
335C8—24no—
336C8—44no—
337C8—44no—
338C22—22no—
339C22—24no—
340C22—41no—
341C22—42no—
342C22—41no—
343C22C2×#39511yescharacteristic
344C22#300212yes—
345C22#300212yes—
346C22C2×#45311yescharacteristic
347C22C2×#45611yescharacteristic
348C4—21no—
349C4—21no—
350C4—24no—
351C4—44no—
352C4—44no—
353C4—82no—
354C4—82no—
355C4—82no—
356C4—82no—
357C4—161no—
358C4—161no—
359C4#206014yes—
360C4#307612yes—
361C4#307612yes—
362C4#316012yes—
363C4#316212yes—
364C2—42no—
365C2#1848011yescharacteristic
366C2#1850312yes—
367C2#1851011yescharacteristic
368C2#2355112yes—
369C2C2×#206011yescharacteristic
subgroupG/H
centerC22×C4#120
commutator, Frattini, cosocleC2×C42C24
socleC23C2×#120
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ 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
57. #431Q8semidirect
73. C2×C4×C8Q8semidirect

Maximal subgroups

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

subgroupindex
#349782
#213622
#207972
#105802
#105792
#105772
#105762
#105742
#105732
#80382
#78902