Finite Group Databasebeta

← #8365

Subgroups — #8365

Order 256.

391 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 391 conjugacy classes of subgroups — too many to draw. Only its 75 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×#556C211yesmaximal, characteristic
2#1826C212yesmaximal, complemented
3C4×#99C211yesmaximal, characteristic
4#1508C212yesmaximal, complemented
5#1465C211yesmaximal, characteristic
6#556—24no—
7#556—24no—
8C2×C4⋊Q8C411yescharacteristic, complemented
9C2×#126—21no—
10C2×#126—21no—
11C2×#124—21no—
12C2×C22⋊Q8C411yescharacteristic, complemented
13C2×C22⋊Q8C2211yescharacteristic
14C2×C4×Q8—21no—
15C2×C4×Q8C411yescharacteristic, complemented
16C2×C4×D4C2211yescharacteristic
17C2×#117C411yescharacteristic, complemented
18C2×#117C2211yescharacteristic
19C2×#99—21no—
20#407C2212yes—
21C4×C22⋊C4C2212yes—
22#342—21no—
23#339—24no—
24#328—24no—
25#324—21no—
26#323—21no—
27C22×Q8—21no—
28C22×Q8C2×C411yescharacteristic
29C22×D4D411yescharacteristic
30C23×C4C2×C411yescharacteristic
31C23×C4C2311yescharacteristic
32C2×M16—22no—
33C4⋊Q8—22no—
34C4⋊Q8—22no—
35#126—28no—
36#126—28no—
37#124—28no—
38C22⋊Q8—22no—
39C22⋊Q8—22no—
40C22⋊Q8—22no—
41C22⋊Q8D414yes—
42C4×Q8—28no—
43C4×Q8—22no—
44C4×Q8—22no—
45C4×D4—22no—
46C4×D4—22no—
47#117—22no—
48#117—22no—
49#117—22no—
50#117D414yes—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4C2×C411yescharacteristic
56C2×C4⋊C4C2×C411yescharacteristic
57C2×C4⋊C4C2×C411yescharacteristic
58C2×C4⋊C4D411yescharacteristic
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—21no—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4D411yescharacteristic
65C2×C22⋊C4D411yescharacteristic
66C2×C42—21no—
67C2×C42—21no—
68C2×C42—21no—
69C2×C42—22no—
70C2×C42C2×C411yescharacteristic
71#99—24no—
72C22⋊C8—22no—
73C22⋊C8—22no—
74#95—22no—
75C24C22⋊C411yescharacteristic
76C24C2×D411yescharacteristic
77C2×Q8—24no—
78C2×Q8—21no—
79C2×Q8—42no—
80C2×Q8—42no—
81C2×Q8—42no—
82C2×Q8C22⋊C412yes—
83C2×D4—22no—
84C2×D4—22no—
85C2×D4—41no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—42no—
93C22×C4—42no—
94C22×C4—42no—
95C22×C4C22⋊C411yescharacteristic
96C22×C4C22⋊C411yescharacteristic
97C22×C4C22⋊C411yescharacteristic
98C22×C4C22×C411yescharacteristic
99C22×C4C2×D412yes—
100C22×C4C2×D412yes—
101C22×C4C2×D411yescharacteristic
102M16—44no—
103C2×C8—42no—
104C4⋊C4—24no—
105C4⋊C4—24no—
106C4⋊C4—24no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—24no—
111C4⋊C4—22no—
112C4⋊C4C22⋊C412yes—
113C4⋊C4C22⋊C412yes—
114C4⋊C4C22⋊C412yes—
115C22⋊C4—24no—
116C22⋊C4—24no—
117C22⋊C4—24no—
118C22⋊C4—24no—
119C22⋊C4—22no—
120C22⋊C4—22no—
121C22⋊C4—24no—
122C22⋊C4—42no—
123C42—24no—
124C42—24no—
125C42—21no—
126C42—21no—
127C42—24no—
128C42—44no—
129C23—21no—
130C23—21no—
131C23—21no—
132C23—21no—
133C23—21no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23—41no—
138C23C2×C22⋊C411yescharacteristic
139C23#12012yes—
140Q8—42no—
141Q8—48no—
142Q8—42no—
143D4—44no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—24no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—44no—
167C2×C4—82no—
168C2×C4—82no—
169C2×C4#10412yes—
170C2×C4C2×C22⋊C411yescharacteristic
171C2×C4C2×C22⋊C411yescharacteristic
172C2×C4#12012yes—
173C8—82no—
174C22—21no—
175C22—21no—
176C22—22no—
177C22—22no—
178C22—22no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22—81no—
184C22—81no—
185C22—81no—
186C22#37911yescharacteristic
187C22C2×#10411yescharacteristic
188C22#42111yescharacteristic
189C4—22no—
190C4—21no—
191C4—22no—
192C4—41no—
193C4—42no—
194C4—41no—
195C4—42no—
196C4—42no—
197C4—42no—
198C4—84no—
199C4#41012yes—
200C2—21no—
201C2—21no—
202C2—42no—
203C2—81no—
204C2#178211yescharacteristic
205C2#178911yescharacteristic
206C2#179611yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1826C2semidirect
4. #1508C2semidirect
8. C2×C4⋊Q8C4semidirect
12. C2×C22⋊Q8C4semidirect
15. C2×C4×Q8C4semidirect
17. C2×#117C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5562
#18262
C4×#992
#15082
#14652