Finite Group Databasebeta

← #241291

Subgroups — #241291

Order 512.

944 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34978C211yesmaximal, characteristic
2#21171C212yesmaximal
3#20800C211yesmaximal, characteristic
4#10579C214yesmaximal, complemented
5#10577C212yesmaximal, complemented
6#10574C212yesmaximal, complemented
7#8041C211yesmaximal, characteristic
8#7903C212yesmaximal
9C2×#428—22no—
10C2×#428—22no—
11C2×#423C2212yes—
12C2×C4×M16C2211yescharacteristic
13#2612—22no—
14#2571C2211yescharacteristic
15#2552C2212yes—
16#2551C2211yescharacteristic
17#2441—22no—
18#2439—22no—
19#2396—22no—
20C4×C4⋊Q8—22no—
21C4×#125—22no—
22#2285C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C4×#117C2212yes—
25#1948—22no—
26#1948C2214yes—
27#1947—22no—
28#1947C2214yes—
29#1847—22no—
30#1845C2212yes—
31#1844C2212yes—
32#1736—22no—
33#1296C2212yes—
34#1296C2212yes—
35#1279C2214yes—
36#1279C2214yes—
37C22×M16—21no—
38C22×M16—22no—
39C2×C4⋊Q8—22no—
40C2×#125—22no—
41C2×C4×Q8—22no—
42C2×#117D412yes—
43C2×#117D412yes—
44C2×#117D412yes—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—22no—
47C22×C42C2311yescharacteristic
48#428—48no—
49#428—48no—
50C2×#107—24no—
51C2×#107—24no—
52C2×#106—24no—
53C2×#106—24no—
54#423—24no—
55#423Q818yescomplemented
56C2×C4⋊C8—22no—
57C2×C4⋊C8C2314yes—
58C4×M16—22no—
59C4×M16Q814yescomplemented
60C2×#97—21no—
61C2×C4×C8C2312yes—
62#400—22no—
63#399—22no—
64#398—24no—
65#396—24no—
66#395—22no—
67#393—22no—
68#391—24no—
69#389—24no—
70#389—22no—
71#389—22no—
72#389—22no—
73#389—22no—
74#387—24no—
75#387—24no—
76#384—22no—
77#384—21no—
78#384—22no—
79#384C2312yes—
80#384C2311yescharacteristic
81#384C2311yescharacteristic
82#382—22no—
83#382D412yes—
84#380—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—24no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4D412yes—
89C4×C4⋊C4D412yes—
90C4×C4⋊C4C2312yes—
91C4×C4⋊C4C2312yes—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94#376—22no—
95C43—22no—
96#340—24no—
97#340—24no—
98C22×Q8—22no—
99C23×C4—41no—
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D412yes—
102C2×M16—24no—
103C2×M16—24no—
104C2×M16—24no—
105C2×M16—28no—
106C2×M16—44no—
107C2×M16—44no—
108C22×C8—22no—
109C22×C8—24no—
110C4⋊Q8—44no—
111C4⋊Q8—44no—
112#125—44no—
113#125—44no—
114C4×Q8—84no—
115#117—24no—
116#117—24no—
117#117—42no—
118#117—42no—
119#117—44no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—44no—
130C2×C4⋊C4—44no—
131C2×C4⋊C4—44no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4C2×D412yes—
135C2×C4⋊C4C2×D412yes—
136C2×C4⋊C4C2×D412yes—
137C2×C4⋊C4C2×D412yes—
138C2×C4⋊C4C2×D412yes—
139C2×C4⋊C4C2×D412yes—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42C2×D412yes—
151C2×C42C2×D411yescharacteristic
152C2×C42C2×Q812yes—
153C2×C42C2×Q814yes—
154C2×C42#4112yes—
155C2×C42C2411yescharacteristic
156#107—48no—
157#107—48no—
158#106—48no—
159#106—48no—
160C4⋊C8—28no—
161C4⋊C8—24no—
162C4⋊C8C2×Q818yes—
163#97—24no—
164C4×C8—22no—
165C4×C8C2×Q814yes—
166#95—22no—
167#95—24no—
168#95—22no—
169#95—24no—
170#95—22no—
171#95—22no—
172#95—24no—
173#95—22no—
174#95—22no—
175C24#12011yescharacteristic
176C2×Q8—48no—
177C2×Q8—82no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—42no—
191C22×C4—84no—
192C22×C4—82no—
193C22×C4#12012yes—
194C22×C4#12011yescharacteristic
195C22×C4C22⋊Q814yes—
196C22×C4C22⋊Q812yes—
197C22×C4C4⋊Q812yes—
198C22×C4C4⋊Q814yes—
199C22×C4C22×D411yescharacteristic
200C22×C4C22×D412yes—
201M16—216no—
202M16—416no—
203C2×C8—24no—
204C2×C8—24no—
205C2×C8—24no—
206C2×C8—28no—
207C2×C8—44no—
208C2×C8—44no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—48no—
218C4⋊C4—48no—
219C4⋊C4—48no—
220C4⋊C4—44no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—84no—
228C4⋊C4—84no—
229C4⋊C4—82no—
230C4⋊C4—82no—
231C4⋊C4AGL(1,ℤ/8)14yes—
232C4⋊C4#13714yes—
233C22⋊C4—44no—
234C22⋊C4—44no—
235C22⋊C4—82no—
236C42—44no—
237C42—44no—
238C42—42no—
239C42—42no—
240C42—42no—
241C42—42no—
242C42—44no—
243C42—44no—
244C42—44no—
245C42C22⋊Q812yes—
246C42C22⋊Q814yes—
247C42C4⋊Q812yes—
248C42C4⋊Q814yes—
249C42C22×Q811yescharacteristic
250C42C22×Q812yes—
251C42C2×#4111yescharacteristic
252C23—22no—
253C23—22no—
254C23—21no—
255C23—21no—
256C23#39512yes—
257C23C2×#12011yescharacteristic
258Q8—88no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—44no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—84no—
286C2×C4—82no—
287C2×C4—164no—
288C2×C4#39514yes—
289C2×C4#39512yes—
290C2×C4#45312yes—
291C2×C4#45612yes—
292C2×C4C2×C22⋊Q811yescharacteristic
293C2×C4C2×C22⋊Q812yes—
294C2×C4C2×C4⋊Q811yescharacteristic
295C2×C4C2×C4⋊Q812yes—
296C2×C4C2×AGL(1,ℤ/8)12yes—
297C2×C4C2×#13712yes—
298C2×C4#57512yes—
299C2×C4#57512yes—
300C8—28no—
301C8—48no—
302C22—24no—
303C22—22no—
304C22—41no—
305C22—42no—
306C22—41no—
307C22C2×#39511yescharacteristic
308C22#300612yes—
309C22#300712yes—
310C22C2×#45311yescharacteristic
311C22C2×#45611yescharacteristic
312C4—24no—
313C4—21no—
314C4—21no—
315C4—44no—
316C4—44no—
317C4—82no—
318C4—82no—
319C4—82no—
320C4—82no—
321C4—162no—
322C4#206014yes—
323C4#307712yes—
324C4#307812yes—
325C4#316512yes—
326C4#316612yes—
327C2—42no—
328C2#1852811yescharacteristic
329C2#1854312yes—
330C2#1854811yescharacteristic
331C2#2355412yes—
332C2C2×#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. #10579C2semidirect
5. #10577C2semidirect
6. #10574C2semidirect
55. #423Q8semidirect
59. C4×M16Q8semidirect

Maximal subgroups

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

subgroupindex
#349782
#211712
#208002
#105792
#105772
#105742
#80412
#79032