Finite Group Databasebeta

← #29676

Subgroups — #29676

Order 256.

537 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 537 conjugacy classes of subgroups — too many to draw. Only its 127 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×#541C211yesmaximal, characteristic, complemented
2C2×#464C212yesmaximal, complemented
3#2620C211yesmaximal, characteristic
4#2026C211yesmaximal, characteristic, complemented
5#2025C211yesmaximal, characteristic, complemented
6#2016C211yesmaximal, characteristic, complemented
7#2013C211yesmaximal, characteristic, complemented
8#2001C212yesmaximal, complemented
9#2000C212yesmaximal
10#1803C212yesmaximal, complemented
11#1798C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14#541—28no—
15C2×C4⋊Q8C2211yescharacteristic
16C2×#126—22no—
17C2×#125—22no—
18C2×#125C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×C22⋊Q8C2212yes—
21C2×C22⋊Q8C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×C4×Q8C2212yes—
24C2×#117C2211yescharacteristic
25#464—28no—
26C2×C4⋊C8C2212yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×#103C2212yes—
30C2×#102C2212yes—
31C2×C22⋊C8C2212yes—
32#396—22no—
33#394—22no—
34#391C2211yescharacteristic
35#390C2211yescharacteristic
36#384C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#336C2211yescharacteristic
40#336C2211yescharacteristic
41C22×Q8C2311yescharacteristic
42C22×Q8C2312yes—
43C22×D4—21no—
44C22×D4C2311yescharacteristic
45C23×C4C2311yescharacteristic
46C2×Q16—24no—
47C2×Q16—42no—
48C2×Q16—42no—
49C2×SD16—24no—
50C2×SD16—42no—
51C2×SD16—42no—
52C22×C8C2312yes—
53C22×C8C2312yes—
54C4⋊Q8—24no—
55#126—216no—
56#125—216no—
57#125—24no—
58#124—24no—
59C22⋊Q8—24no—
60C22⋊Q8—24no—
61C22⋊Q8—42no—
62#121—42no—
63C4×Q8—24no—
64C4×Q8D418yescomplemented
65#117—24no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4D412yes—
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2312yes—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4D412yes—
82C2×C42—22no—
83C2×C42C2311yescharacteristic
84C2×C42C2311yescharacteristic
85C4⋊C8—24no—
86#103—24no—
87#103—42no—
88#103—42no—
89#102—24no—
90C22⋊C8—42no—
91#95—22no—
92C24—21no—
93C24C2×D411yescharacteristic
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—21no—
101C2×Q8—22no—
102C2×Q8—21no—
103C2×Q8C2×D414yes—
104C2×D4—22no—
105C2×D4—22no—
106C2×D4—21no—
107C2×D4—21no—
108C2×D4—42no—
109C2×D4—82no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4C2×D412yes—
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D412yes—
122C22×C4C2411yescharacteristic
123Q16—48no—
124SD16—48no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C4⋊C4—28no—
132C4⋊C4—28no—
133C4⋊C4—28no—
134C4⋊C4—28no—
135C4⋊C4—24no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—42no—
140C4⋊C4—82no—
141C4⋊C4C2×D414yes—
142C22⋊C4—28no—
143C22⋊C4—24no—
144C22⋊C4—42no—
145C22⋊C4—42no—
146C22⋊C4—44no—
147C22⋊C4—82no—
148C42—28no—
149C42—22no—
150C42C2×D414yes—
151C23—21no—
152C23—21no—
153C23—21no—
154C23—22no—
155C23—22no—
156C23—41no—
157C23—41no—
158C23—41no—
159C23—42no—
160C23—42no—
161C23C22×D411yescharacteristic
162Q8—24no—
163Q8—24no—
164Q8—42no—
165Q8—42no—
166Q8—42no—
167Q8—42no—
168D4—42no—
169D4—42no—
170D4—84no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—24no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4#12014yes—
195C2×C4C22×D412yes—
196C2×C4H(5,𝔽2)11yescharacteristic
197C2×C4H(5,𝔽2)12yes—
198C2×C42−1+411yescharacteristic
199C8—42no—
200C8—42no—
201C22—41no—
202C22—41no—
203C22—41no—
204C22—42no—
205C22—42no—
206C22—81no—
207C22—81no—
208C22—81no—
209C22—82no—
210C22—82no—
211C22C2×#12011yescharacteristic
212C22#53511yescharacteristic
213C22#53711yescharacteristic
214C22#57712yes—
215C22#57812yes—
216C4—24no—
217C4—41no—
218C4—44no—
219C4—42no—
220C4—44no—
221C4—81no—
222C4—81no—
223C4—81no—
224C4#53512yes—
225C4#53712yes—
226C2—81no—
227C2—161no—
228C2#261511yescharacteristic
229C2#301312yes—
230C2#322211yescharacteristic
231C2#322311yescharacteristic
232C2#325812yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#541C2semidirect
2. C2×#464C2semidirect
4. #2026C2semidirect
5. #2025C2semidirect
6. #2016C2semidirect
7. #2013C2semidirect
8. #2001C2semidirect
10. #1803C2semidirect
11. #1798C2semidirect
64. C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5412
C2×#4642
#26202
#20262
#20252
#20162
#20132
#20012
#20002
#18032
#17982