Finite Group Databasebeta

← #490702

Subgroups — #490702

Order 512.

453 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 453 conjugacy classes of subgroups — too many to draw. Only its 140 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33628C211yesmaximal, characteristic
2C2×#2209C211yesmaximal, characteristic
3#19653C211yesmaximal, characteristic
4#17897C211yesmaximal, characteristic
5#12490C214yesmaximal, complemented
6#12488C214yesmaximal, complemented
7#12005C211yesmaximal, characteristic
8#11939C211yesmaximal, characteristic
9#11932C211yesmaximal, characteristic
10C22×Q32—21no—
11#3229—21no—
12#3229—21no—
13#3160C2211yescharacteristic
14C2×#494C2211yescharacteristic
15#2983—21no—
16C2×C4×Q16C2211yescharacteristic
17#2229—22no—
18#2227C2212yes—
19#2225C2212yes—
20#2209—22no—
21#2209C2214yes—
22#2159—21no—
23#2150C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25#2141—21no—
26C2×#358—21no—
27#1716—22no—
28#1707C2212yes—
29#1707C2212yes—
30#1593—22no—
31#1591C2214yes—
32#1564C2211yescharacteristic
33#1479—21no—
34#1450C2211yescharacteristic
35#1370C2214yes—
36#1365C2212yes—
37#1364C2212yes—
38#1362C2214yes—
39C22×Q16—21no—
40C22×Q16—21no—
41C22×Q16D411yescharacteristic
42#533—21no—
43C2×C4⋊Q8—21no—
44C2×C4×Q8—21no—
45C2×Q32—24no—
46C2×Q32—44no—
47C2×M32—21no—
48C22×C16—21no—
49#500C2311yescharacteristic
50#500C2311yescharacteristic
51#499—21no—
52#499—21no—
53#498—21no—
54#498—22no—
55#498—22no—
56#494—21no—
57#494—21no—
58#494—21no—
59#494C2312yes—
60#487—22no—
61#487—22no—
62#483—42no—
63#483—42no—
64#475—42no—
65#475—42no—
66#446—24no—
67#446—42no—
68C4×Q16—21no—
69C4×Q16—21no—
70C4×Q16D412yes—
71C4×Q16C2312yes—
72#432—42no—
73#431—21no—
74#431D411yescharacteristic
75#431D411yescharacteristic
76#427—21no—
77#427—21no—
78C2×#107D411yescharacteristic
79C2×#103—21no—
80C2×C4×C8C2311yescharacteristic
81#367—22no—
82#366—22no—
83C4⋊C16—21no—
84C4⋊C16—21no—
85C4⋊C16C2312yes—
86C4⋊C16C2312yes—
87#358—22no—
88#358—22no—
89#358—24no—
90#335C2312yes—
91#334—21no—
92Q8⋊C8—24no—
93#322D412yes—
94C82C2312yes—
95C22×Q8—21no—
96C22×Q8—41no—
97C2×Q16—21no—
98C2×Q16—22no—
99C2×Q16—22no—
100C2×Q16—22no—
101C2×Q16—22no—
102C2×Q16—41no—
103C2×Q16—41no—
104C2×Q16—42no—
105C2×Q16—42no—
106C2×Q16—82no—
107C2×Q16C2×D412yes—
108C22×C8C2×D411yescharacteristic
109C22×C8#4111yescharacteristic
110C4⋊Q8—22no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—41no—
114C4⋊Q8—41no—
115C4⋊Q8—41no—
116#125—41no—
117C22⋊Q8—41no—
118C22⋊Q8—41no—
119C4×Q8—22no—
120C4×Q8—22no—
121C4×Q8—41no—
122C4×Q8—41no—
123C4×Q8—41no—
124C4×Q8—41no—
125#117—41no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C42—41no—
131C2×C42C2×D411yescharacteristic
132Q32—48no—
133M32—42no—
134C2×C16—22no—
135C2×C16—22no—
136C2×C16—42no—
137#107—21no—
138#107—21no—
139#107—21no—
140#107—21no—
141#107—21no—
142#107C2×D412yes—
143#106—44no—
144#106—41no—
145#106—41no—
146#106—44no—
147C4⋊C8—21no—
148C4⋊C8—22no—
149C4⋊C8—44no—
150C4⋊C8D812yes—
151#103—22no—
152#103—22no—
153#103—44no—
154#103—44no—
155C22⋊C8—21no—
156C22⋊C8—41no—
157C22⋊C8D812yes—
158#97—44no—
159C4×C8—41no—
160C4×C8C2×D411yescharacteristic
161C4×C8C2×D412yes—
162C4×C8C2×D411yescharacteristic
163C4×C8C2×D412yes—
164C4×C8#4111yescharacteristic
165C4×C8C2411yescharacteristic
166C2×Q8—22no—
167C2×Q8—41no—
168C2×Q8—42no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—81no—
174C2×Q8—82no—
175C2×Q8—82no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—81no—
179C22×C4#12111yescharacteristic
180C22×C4C2×D811yescharacteristic
181C22×C4AGL(1,ℤ/8)11yescharacteristic
182Q16—22no—
183Q16—22no—
184Q16—42no—
185Q16—42no—
186Q16—42no—
187Q16—82no—
188Q16—82no—
189Q16—82no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—42no—
193C2×C8#12111yescharacteristic
194C2×C8#12111yescharacteristic
195C2×C8C2×D814yes—
196C2×C8AGL(1,ℤ/8)12yes—
197C2×C8#13712yes—
198C2×C8C22×D411yescharacteristic
199C2×C8C2×#4111yescharacteristic
200C4⋊C4—22no—
201C4⋊C4—42no—
202C4⋊C4—41no—
203C4⋊C4—42no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—42no—
207C4⋊C4—41no—
208C4⋊C4—42no—
209C4⋊C4—41no—
210C4⋊C4—81no—
211C4⋊C4—81no—
212C4⋊C4—81no—
213C4⋊C4—81no—
214C22⋊C4—81no—
215C42—42no—
216C42—42no—
217C42—81no—
218C42#12111yescharacteristic
219C42C2×D811yescharacteristic
220C42AGL(1,ℤ/8)11yescharacteristic
221C42C22×D411yescharacteristic
222C16—42no—
223C16—42no—
224C23#45911yescharacteristic
225Q8—42no—
226Q8—42no—
227Q8—82no—
228Q8—82no—
229Q8—82no—
230Q8—82no—
231Q8—82no—
232Q8—162no—
233C2×C4—22no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—82no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—82no—
241C2×C4—161no—
242C2×C4—162no—
243C2×C4#45911yescharacteristic
244C2×C4#45911yescharacteristic
245C2×C4#45911yescharacteristic
246C2×C4#46812yes—
247C2×C4#47112yes—
248C2×C4C2×#12111yescharacteristic
249C2×C4C22×D811yescharacteristic
250C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
251C8—24no—
252C8—21no—
253C8—21no—
254C8—42no—
255C8—82no—
256C8#46812yes—
257C8C2×AGL(1,ℤ/8)11yescharacteristic
258C8C2×#13711yescharacteristic
259C22—21no—
260C22—21no—
261C22#168512yes—
262C22C2×#45911yescharacteristic
263C4—21no—
264C4—21no—
265C4—21no—
266C4—82no—
267C4—82no—
268C4—161no—
269C4—161no—
270C4—162no—
271C4#168512yes—
272C4C2×#46811yescharacteristic
273C4C2×#47111yescharacteristic
274C4#341712yes—
275C2—22no—
276C2C2×#168511yescharacteristic
277C2#3358811yescharacteristic
278C2#3359611yescharacteristic
subgroupG/H
center, socleC22C2×#459
commutator, Frattini, cosocleC4×C8C24
prime radicalC23#459
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #12490C2semidirect
6. #12488C2semidirect

Maximal subgroups

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

subgroupindex
#336282
C2×#22092
#196532
#178972
#124902
#124882
#120052
#119392
#119322