Finite Group Databasebeta

← #12727

Subgroups — #12727

Order 256.

405 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 405 conjugacy classes of subgroups — too many to draw. Only its 69 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#3508C211yesmaximal, characteristic
2#2127C212yesmaximal, complemented
3C2×#353C212yesmaximal, complemented
4#1828C212yesmaximal, complemented
5#559—22no—
6#550—22no—
7#546—24no—
8#540—22no—
9#535C411yescharacteristic, complemented
10#535C2211yescharacteristic
11#533C411yescharacteristic, complemented
12#532—22no—
13C2×#127C2211yescharacteristic
14#518C411yescharacteristic, complemented
15#518C2211yescharacteristic
16C4×#41C411yescharacteristic, complemented
17C2×#99C2214yes—
18#354—22no—
19#353—22no—
20#353—22no—
21#353—22no—
22#342—24no—
23C2×#41—24no—
24C2×#41C2×C411yescharacteristic
25C22×D4—22no—
26C22×D4C2×C411yescharacteristic
27C22×D4C2311yescharacteristic
28C4⋊Q8—21no—
29#127—21no—
30#127—22no—
31#127D412yes—
32#127D412yes—
33#127D412yes—
34#125—21no—
35#124—24no—
36#124—21no—
37#124D412yes—
38#123—24no—
39#123—24no—
40C22⋊Q8—22no—
41#121—24no—
42#121—24no—
43#121—24no—
44#121—22no—
45#121—22no—
46#120—22no—
47#120—22no—
48C4×Q8—21no—
49C4×Q8—22no—
50C4×D4—24no—
51C4×D4—21no—
52C4×D4—22no—
53C4×D4—22no—
54C4×D4—22no—
55C4×D4—22no—
56C4×D4—22no—
57#117—21no—
58#117C2×C411yescharacteristic
59#117C2×C411yescharacteristic
60#117C2×C411yescharacteristic
61C2×C22⋊C4—22no—
62C2×C22⋊C4—24no—
63C2×C22⋊C4D414yes—
64C2×C42—21no—
65C2×C42C2×C411yescharacteristic
66#99—24no—
67#99—24no—
68C24—21no—
69C24C2×D412yes—
70#41—42no—
71#41—88no—
72C2×Q8—22no—
73C2×Q8C22⋊C411yescharacteristic
74C2×D4—21no—
75C2×D4—24no—
76C2×D4—24no—
77C2×D4—24no—
78C2×D4—24no—
79C2×D4—21no—
80C2×D4—42no—
81C2×D4—42no—
82C2×D4—84no—
83C2×D4C22⋊C412yes—
84C2×D4C22⋊C411yescharacteristic
85C2×D4C2×D412yes—
86C2×D4C2×D412yes—
87C22×C4—22no—
88C22×C4—21no—
89C22×C4—24no—
90C22×C4—24no—
91C22×C4—21no—
92C22×C4—44no—
93C22×C4C22×C411yescharacteristic
94C4⋊C4—22no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—21no—
99C4⋊C4—21no—
100C4⋊C4—21no—
101C22⋊C4—24no—
102C22⋊C4—22no—
103C22⋊C4—24no—
104C22⋊C4—21no—
105C22⋊C4—24no—
106C22⋊C4—24no—
107C22⋊C4—44no—
108C22⋊C4—44no—
109C22⋊C4C22⋊C412yes—
110C22⋊C4C22⋊C412yes—
111C42—22no—
112C42—21no—
113C42—21no—
114C42—21no—
115C42C22⋊C411yescharacteristic
116C42C22⋊C411yescharacteristic
117C42C22⋊C412yes—
118C23—24no—
119C23—22no—
120C23—21no—
121C23—42no—
122C23—82no—
123C23—82no—
124C23C2×C22⋊C411yescharacteristic
125C23#12014yes—
126Q8—41no—
127Q8—44no—
128D4—42no—
129D4—42no—
130D4—42no—
131D4—41no—
132D4—42no—
133D4—48no—
134D4—42no—
135D4—84no—
136D4—84no—
137D4—84no—
138C2×C4—21no—
139C2×C4—22no—
140C2×C4—21no—
141C2×C4—22no—
142C2×C4—21no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—44no—
148C2×C4—84no—
149C2×C4—84no—
150C2×C4—84no—
151C2×C4—164no—
152C2×C4C2×C22⋊C411yescharacteristic
153C2×C4C2×C22⋊C411yescharacteristic
154C22—24no—
155C22—41no—
156C22—44no—
157C22—42no—
158C22—44no—
159C22—41no—
160C22—42no—
161C22—84no—
162C22—162no—
163C22#37911yescharacteristic
164C22#41012yes—
165C4—22no—
166C4—21no—
167C4—21no—
168C4—42no—
169C4—41no—
170C4—42no—
171C4—42no—
172C4—41no—
173C4—164no—
174C2—21no—
175C2—22no—
176C2—44no—
177C2—81no—
178C2—82no—
179C2#178211yescharacteristic
subgroupG/H
center, socleC2#1782
commutatorC22×C4C22×C4
Frattini, cosocleC22×D4C23
prime radical#3508C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ #518 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2127C2semidirect
3. C2×#353C2semidirect
4. #1828C2semidirect
9. #535C4semidirect
11. #533C4semidirect
14. #518C4semidirect
16. C4×#41C4semidirect

Maximal subgroups

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

subgroupindex
#35082
#21272
C2×#3532
#18282