Finite Group Databasebeta

← #138682

Subgroups — #138682

Order 512.

321 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 321 conjugacy classes of subgroups — too many to draw. Only its 96 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#33475C211yesmaximal, characteristic
2#19204C211yesmaximal, characteristic
3#13207C211yesmaximal, characteristic
4#11932C211yesmaximal, characteristic
5#8677C211yesmaximal, characteristic
6#7279C211yesmaximal, characteristic
7#7157C211yesmaximal, characteristic
8#3229—21no—
9C2×#494C2211yescharacteristic
10#2960C2211yescharacteristic
11#2183—22no—
12#2183—22no—
13#2151—21no—
14#2150C2211yescharacteristic
15C2×#358—21no—
16C2×#358—21no—
17#2108—21no—
18C2×#346C2211yescharacteristic
19#1846C2211yescharacteristic
20#1690C414yescomplemented
21#1670—22no—
22#1564C2211yescharacteristic
23#1564C2211yescharacteristic
24#1546—21no—
25#1518—21no—
26#1362—22no—
27#1286—21no—
28C22×Q16—21no—
29C22×Q16—21no—
30#533D411yescharacteristic
31C2×C4⋊Q8—21no—
32C2×M32—21no—
33C22×C16D411yescharacteristic
34C22×C16D411yescharacteristic
35#498—41no—
36#494—21no—
37#494—21no—
38#494—21no—
39#494C2×C412yes—
40#487—41no—
41#483—42no—
42#475—24no—
43#446—21no—
44#446C2×C412yes—
45C8×Q8—21no—
46C8×Q8D412yes—
47#431—21no—
48#431—21no—
49C2×#107—21no—
50C2×#106—21no—
51#424—21no—
52#423D411yescharacteristic
53#405—21no—
54C2×C4×C8C2311yescharacteristic
55#382—21no—
56C4⋊C16—24no—
57C4⋊C16—21no—
58C4⋊C16D412yes—
59#358—42no—
60#358—42no—
61#346—22no—
62#346—21no—
63#346—21no—
64C4×C16—22no—
65#335—21no—
66#335—21no—
67#334—21no—
68#334C2×C412yes—
69#332—24no—
70#328—42no—
71C22×Q8—41no—
72C2×Q16—22no—
73C2×Q16—22no—
74C2×Q16—41no—
75C2×Q16—41no—
76C2×Q16—41no—
77C2×Q16—41no—
78C2×Q16—41no—
79C2×Q16—41no—
80C2×Q16—81no—
81C2×Q16—81no—
82C2×M16—21no—
83C22×C8C2×D411yescharacteristic
84C22×C8#4111yescharacteristic
85C4⋊Q8—22no—
86C4⋊Q8—41no—
87C4⋊Q8—41no—
88C4⋊Q8—41no—
89C4⋊Q8SD1611yescharacteristic
90C4⋊Q8SD1611yescharacteristic
91#125—21no—
92C22⋊Q8—21no—
93C22⋊Q8D812yes—
94C4×Q8—21no—
95C4×Q8C22⋊C412yes—
96#117—21no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C42C2×D411yescharacteristic
101M32—41no—
102M32—41no—
103C2×C16—21no—
104C2×C16—21no—
105C2×C16—22no—
106C2×C16—21no—
107C2×C16—21no—
108C2×C16—21no—
109C2×C16SD1612yes—
110#107—41no—
111#107—41no—
112#106—44no—
113#106—81no—
114C4⋊C8—22no—
115C4⋊C8—22no—
116C4⋊C8—22no—
117C4⋊C8—21no—
118C4⋊C8—41no—
119C4⋊C8C22⋊C412yes—
120#103—44no—
121C22⋊C8—41no—
122C22⋊C8—41no—
123#97—22no—
124C4×C8—22no—
125C4×C8—41no—
126C4×C8—41no—
127C4×C8C22×C411yescharacteristic
128C4×C8C2×D411yescharacteristic
129C4×C8C2×D411yescharacteristic
130C4×C8#4111yescharacteristic
131#95—41no—
132#95—41no—
133C2×Q8—21no—
134C2×Q8—42no—
135C2×Q8—81no—
136C2×Q8—81no—
137C2×Q8—81no—
138C2×Q8—81no—
139C2×Q8—81no—
140C2×Q8#10212yes—
141C22×C4—81no—
142C22×C4—81no—
143C22×C4#12111yescharacteristic
144C22×C4C2×D811yescharacteristic
145C22×C4#13511yescharacteristic
146Q16—42no—
147Q16—42no—
148Q16—82no—
149Q16—82no—
150Q16—81no—
151Q16—81no—
152M16—41no—
153M16—41no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—22no—
157C2×C8—42no—
158C2×C8—81no—
159C2×C8C4×D411yescharacteristic
160C2×C8C4×D411yescharacteristic
161C2×C8#12011yescharacteristic
162C2×C8#12311yescharacteristic
163C2×C8C2×SD1611yescharacteristic
164C2×C8#13511yescharacteristic
165C2×C8AGL(1,ℤ/8)11yescharacteristic
166C2×C8#13711yescharacteristic
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—81no—
175C4⋊C4—81no—
176C4⋊C4—81no—
177C4⋊C4—81no—
178C4⋊C4#10212yes—
179C22⋊C4—22no—
180C42—22no—
181C42C2×C22⋊C411yescharacteristic
182C42#12111yescharacteristic
183C42C2×SD1611yescharacteristic
184C42#13511yescharacteristic
185C16—22no—
186C16—41no—
187C16—42no—
188C23#46611yescharacteristic
189Q8—42no—
190Q8—81no—
191Q8—82no—
192Q8—82no—
193Q8—161no—
194Q8—161no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—82no—
200C2×C4—161no—
201C2×C4—161no—
202C2×C4—161no—
203C2×C4—161no—
204C2×C4—161no—
205C2×C4#38611yescharacteristic
206C2×C4C2×#10211yescharacteristic
207C2×C4#41611yescharacteristic
208C2×C4#46011yescharacteristic
209C2×C4#46411yescharacteristic
210C2×C4#46511yescharacteristic
211C2×C4#46511yescharacteristic
212C2×C4#46611yescharacteristic
213C2×C4#47111yescharacteristic
214C2×C4#47211yescharacteristic
215C8—21no—
216C8—21no—
217C8—42no—
218C8—82no—
219C8—161no—
220C8C4×SD1611yescharacteristic
221C8#44011yescharacteristic
222C8#44811yescharacteristic
223C8#44911yescharacteristic
224C22—21no—
225C22—21no—
226C22#166811yescharacteristic
227C22#168211yescharacteristic
228C22#189311yescharacteristic
229C4—21no—
230C4—21no—
231C4—21no—
232C4—21no—
233C4—42no—
234C4—42no—
235C4—162no—
236C4—321no—
237C4#166611yescharacteristic
238C4#167811yescharacteristic
239C4#188011yescharacteristic
240C4#190711yescharacteristic
241C2—21no—
242C2—21no—
243C2#922211yescharacteristic
244C2#1306911yescharacteristic
245C2#1307511yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23#466
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
20. #1690C4semidirect

Maximal subgroups

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

subgroupindex
#334752
#192042
#132072
#119322
#86772
#72792
#71572