Finite Group Databasebeta

← #506330

Subgroups — #506330

Order 512.

447 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 447 conjugacy classes of subgroups — too many to draw. Only its 128 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#33474C211yesmaximal, characteristic, complemented
2#33464C211yesmaximal, characteristic
3#29270C212yesmaximal, complemented
4#13520C211yesmaximal, characteristic, complemented
5#13519C211yesmaximal, characteristic, complemented
6#13447C211yesmaximal, characteristic
7#13444C211yesmaximal, characteristic
8#13151C212yesmaximal, complemented
9#13146C212yesmaximal, complemented
10#12893C211yesmaximal, characteristic, complemented
11#12829C212yesmaximal, complemented
12#3414C2211yescharacteristic
13#2960C2211yescharacteristic
14#2957C2212yes—
15#2183—22no—
16Q8×C16—22no—
17#2179—22no—
18#2178—22no—
19#2161C2211yescharacteristic
20#2151C2211yescharacteristic
21#2151C2211yescharacteristic
22#2150C2211yescharacteristic
23#2148C2212yes—
24C2×#359C2212yes—
25#2138C2212yes—
26#2108C2211yescharacteristic
27#2108C2211yescharacteristic
28C2×C4×C16C2211yescharacteristic
29#2098—22no—
30#2031—22no—
31#1976C2212yes—
32#1974C2212yes—
33#1953C2212yes—
34#1945C2212yes—
35#1868—22no—
36#1861—22no—
37#1833C2212yes—
38#1389C2212yes—
39#1385C2211yescharacteristic
40#1384C2211yescharacteristic
41#1382C2212yes—
42#1381C2212yes—
43C2×#135C2312yes—
44#568—22no—
45#533D411yescharacteristic
46#510—22no—
47#509—22no—
48#508—22no—
49C2×M32C2311yescharacteristic
50C2×M32C2311yescharacteristic
51C22×C16C2311yescharacteristic
52C22×C16C2311yescharacteristic
53#446—22no—
54C8×Q8—22no—
55#444—24no—
56#443—24no—
57#430—22no—
58C2×#108C2312yes—
59C2×#108C2312yes—
60#427C2312yes—
61#424—22no—
62#424D411yescharacteristic
63#424D411yescharacteristic
64#423D411yescharacteristic
65#421—22no—
66#421—22no—
67C2×#104—22no—
68#410—22no—
69#408—22no—
70#405C2311yescharacteristic
71#405C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73#368—22no—
74C4⋊C16—22no—
75C4⋊C16—22no—
76C4⋊C16—22no—
77#362—22no—
78#361—22no—
79#359—22no—
80#359—22no—
81#358—22no—
82#357—22no—
83#346—21no—
84#346D412yes—
85C4×C16—21no—
86C4×C16—21no—
87C4×C16—21no—
88C4×C16D412yes—
89#344—22no—
90#344—22no—
91#339—22no—
92H(3,ℤ/4)—22no—
93C2×#41—22no—
94#135—22no—
95#135—22no—
96#135—22no—
97#135—22no—
98C2×Q16#4112yes—
99C2×SD16—22no—
100C2×D8#4112yes—
101#131—44no—
102#131—82no—
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—82no—
106C2×M16C2×D411yescharacteristic
107C2×M16C2×D411yescharacteristic
108C22×C8C2×D411yescharacteristic
109C22×C8C2411yescharacteristic
110C4⋊Q8—21no—
111#125—21no—
112C22⋊Q8—22no—
113C4×Q8—22no—
114#117—22no—
115#117C2×D411yescharacteristic
116#117C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118Q32—22no—
119SD32—24no—
120D16—22no—
121M32—22no—
122M32—22no—
123C2×C16—22no—
124C2×C16—21no—
125C2×C16—21no—
126C2×C16—22no—
127C2×C16C2×D411yescharacteristic
128C2×C16C2×D411yescharacteristic
129C2×C16C2×D412yes—
130#108—22no—
131#108—22no—
132#108—22no—
133#108—22no—
134#108—82no—
135#107—22no—
136#106—22no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8—82no—
141#104—44no—
142#104—42no—
143#104—42no—
144#104—82no—
145#99—44no—
146C22⋊C8—82no—
147#97—21no—
148#97#4112yes—
149C4×C8—21no—
150C4×C8—21no—
151C4×C8—21no—
152C4×C8C2×D412yes—
153#41—42no—
154#41—42no—
155#41—42no—
156#41—42no—
157C2×Q8—22no—
158C2×Q8—22no—
159C2×D4—22no—
160C2×D4—42no—
161C22×C4—21no—
162C22×C4—42no—
163C22×C4C22×D411yescharacteristic
164Q16—22no—
165Q16—22no—
166SD16—42no—
167SD16—42no—
168D8—22no—
169D8—22no—
170M16—22no—
171M16—22no—
172M16—42no—
173M16—42no—
174M16—42no—
175M16—42no—
176M16—84no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—22no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—84no—
183C2×C8#12112yes—
184C2×C8C22×D411yescharacteristic
185C2×C8C2×#4112yes—
186C2×C8C2×#4111yescharacteristic
187C2×C82−1+412yes—
188C4⋊C4—24no—
189C4⋊C4—22no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—42no—
193C22⋊C4—21no—
194C22⋊C4—42no—
195C22⋊C4#13512yes—
196C42—21no—
197C42—21no—
198C42—42no—
199C42#12112yes—
200C42#13511yescharacteristic
201C42#13511yescharacteristic
202C16—22no—
203C16—21no—
204C16—21no—
205C16—41no—
206C16—41no—
207C23—42no—
208C23C2×#13511yescharacteristic
209Q8—44no—
210Q8—42no—
211Q8—42no—
212D4—42no—
213D4—42no—
214D4—82no—
215D4—82no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4C2×#12111yescharacteristic
226C2×C4#54812yes—
227C2×C4C2×#13511yescharacteristic
228C2×C4#57612yes—
229C8—22no—
230C8—41no—
231C8—41no—
232C8—82no—
233C8—82no—
234C8#52511yescharacteristic
235C8#53611yescharacteristic
236C8#55212yes—
237C22—41no—
238C22—82no—
239C22—162no—
240C22#305011yescharacteristic
241C22#336112yes—
242C4—22no—
243C4—41no—
244C4—42no—
245C4—42no—
246C4—82no—
247C4—162no—
248C4#253611yescharacteristic
249C4#305811yescharacteristic
250C4#336112yes—
251C2—21no—
252C2—22no—
253C2—162no—
254C2#2818811yescharacteristic
subgroupG/H
centerC4#3058
commutator, Frattini, cosocleC22×C8C24
socleC2#28188
prime radical#3414C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2960 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33474C2semidirect
3. #29270C2semidirect
4. #13520C2semidirect
5. #13519C2semidirect
8. #13151C2semidirect
9. #13146C2semidirect
10. #12893C2semidirect
11. #12829C2semidirect

Maximal subgroups

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

subgroupindex
#334742
#334642
#292702
#135202
#135192
#134472
#134442
#131512
#131462
#128932
#128292