Finite Group Databasebeta

← #17004

Subgroups — #17004

Order 256.

597 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 597 conjugacy classes of subgroups — too many to draw. Only its 175 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
1C2×#398C211yesmaximal, characteristic
2C2×#395C211yesmaximal, characteristic
3#1446C212yesmaximal
4#1443C212yesmaximal
5#1442C212yesmaximal
6#1441C212yesmaximal
7#1441C212yesmaximal
8#1436C212yesmaximal
9C4⋊#95C211yesmaximal, characteristic
10C23×Q8C2211yescharacteristic
11C22×C4⋊C4C2212yes—
12C22×C4⋊C4C2212yes—
13C22×C4⋊C4C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C42C2211yescharacteristic
17#398—24no—
18#395—24no—
19C2×#95C2212yes—
20C2×#95C2211yescharacteristic
21C2×#95C2211yescharacteristic
22C2×#95C2212yes—
23C2×#95C2212yes—
24C2×#95C2212yes—
25C2×#95C2212yes—
26C2×#95C2212yes—
27C2×#95C2212yes—
28C2×#95C2212yes—
29C2×#95C2212yes—
30C2×#95C2212yes—
31C2×#95C2212yes—
32C2×#95C2212yes—
33C22×Q8—22no—
34C22×Q8—24no—
35C22×Q8—42no—
36C22×Q8—41no—
37C22×Q8—41no—
38C23×C4C2312yes—
39C23×C4C2312yes—
40C23×C4C2311yescharacteristic
41C23×C4C2311yescharacteristic
42C23×C4C2312yes—
43C23×C4C2312yes—
44C23×C4C2312yes—
45C23×C4C2312yes—
46C23×C4C2311yescharacteristic
47C2×C4⋊C4—24no—
48C2×C4⋊C4—22no—
49C2×C4⋊C4—24no—
50C2×C4⋊C4—24no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—24no—
53C2×C4⋊C4—24no—
54C2×C4⋊C4—41no—
55C2×C4⋊C4—41no—
56C2×C4⋊C4—42no—
57C2×C4⋊C4—41no—
58C2×C4⋊C4—41no—
59C2×C4⋊C4—41no—
60C2×C4⋊C4—41no—
61C2×C4⋊C4D414yes—
62C2×C42—22no—
63C2×C42—41no—
64C2×C42—41no—
65#95—22no—
66#95—24no—
67#95—22no—
68#95—24no—
69#95—24no—
70#95—42no—
71#95—42no—
72#95—42no—
73#95—42no—
74#95—42no—
75#95—42no—
76#95—42no—
77#95—42no—
78#95Q818yescomplemented
79C24C2411yescharacteristic
80C2×Q8—48no—
81C2×Q8—44no—
82C2×Q8—44no—
83C2×Q8—48no—
84C2×Q8—44no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4C2×D412yes—
136C22×C4C2×D412yes—
137C22×C4C2×D412yes—
138C22×C4C2×Q814yes—
139C22×C4C2×Q814yes—
140C22×C4C2×Q814yes—
141C22×C4#4114yes—
142C22×C4#4112yes—
143C22×C4#4112yes—
144C4⋊C4—44no—
145C4⋊C4—48no—
146C4⋊C4—44no—
147C4⋊C4—44no—
148C4⋊C4—48no—
149C42—44no—
150C23C22×D411yescharacteristic
151C23C22×Q812yes—
152C23C2×#4111yescharacteristic
153C23C2×#4111yescharacteristic
154C23C2×#4112yes—
155C23H(5,𝔽2)11yescharacteristic
156C23H(5,𝔽2)11yescharacteristic
157C23H(5,𝔽2)11yescharacteristic
158C23H(5,𝔽2)12yes—
159C232−1+411yescharacteristic
160C232−1+412yes—
161Q8—416no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—24no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—24no—
172C2×C4—24no—
173C2×C4—24no—
174C2×C4—24no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—24no—
181C2×C4—24no—
182C2×C4—24no—
183C2×C4—24no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4#12414yes—
221C22C2×#12411yescharacteristic
222C22#53312yes—
223C22#53411yescharacteristic
224C22#53711yescharacteristic
225C22#53811yescharacteristic
226C22#53811yescharacteristic
227C22#53812yes—
228C22#53911yescharacteristic
229C22#53911yescharacteristic
230C22#54112yes—
231C22#54212yes—
232C22#54412yes—
233C22#54412yes—
234C22#54612yes—
235C22D4×Q812yes—
236C22#55111yescharacteristic
237C22#55412yes—
238C22#55412yes—
239C22#55612yes—
240C22Q8211yescharacteristic
241C22#56011yescharacteristic
242C22#56212yes—
243C22#56311yescharacteristic
244C4—24no—
245C4—42no—
246C4—42no—
247C4—44no—
248C4—44no—
249C4—44no—
250C4—44no—
251C4—82no—
252C4—82no—
253C2#249212yes—
254C2#258611yescharacteristic
255C2#263911yescharacteristic
256C2#266612yes—
257C2#269711yescharacteristic
258C2#270412yes—
259C2#273411yescharacteristic
260C2#278812yes—
261C2#278911yescharacteristic
262C2#279211yescharacteristic
263C2#283011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
78. #95Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#3982
C2×#3952
#14462
#14432
#14422
#14412
#14362
C4⋊#952