Finite Group Databasebeta

← #8319

Subgroups — #8319

Order 256.

477 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 477 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C4×2−1+4C211yesmaximal, characteristic
2#1984C211yesmaximal, characteristic, complemented
3#1477C212yesmaximal, complemented
4#1476C212yesmaximal, complemented
5#1461C211yesmaximal, characteristic, complemented
6C2×2−1+4C411yescharacteristic
7#520—22no—
8#520—22no—
9#520C412yes—
10#519—22no—
11#519C411yescharacteristic
12C4×#41—22no—
13C4×#41—22no—
14C4×#41C2212yes—
15C2×C4×Q8—22no—
16C2×C4×Q8C2211yescharacteristic
17#431C2212yes—
18C2×C4⋊C8C2211yescharacteristic
19C2×C4×C8C2211yescharacteristic
20#336—22no—
21Q8⋊C8—22no—
22Q8⋊C8—22no—
23D4⋊C8—22no—
24D4⋊C8—22no—
25#324—22no—
262−1+4—24no—
272−1+4—22no—
282−1+4C814yescomplemented
29C2×#41—22no—
30C2×#41—22no—
31C2×#41C2×C412yes—
32C22×Q8—22no—
33C22×Q8C2×C411yescharacteristic
34C22×C8—21no—
35C22×C8—21no—
36C4×Q8—22no—
37C4×Q8—24no—
38C4×Q8—22no—
39C4×Q8—22no—
40C4×Q8—22no—
41C4×Q8—22no—
42C4×Q8—22no—
43C4×Q8—22no—
44C4×Q8D412yes—
45C4×Q8D412yes—
46C4×D4—22no—
47C4×D4—24no—
48C4×D4—22no—
49C4×D4—22no—
50C4×D4—24no—
51C4×D4—22no—
52C4×D4—22no—
53C4×D4D412yes—
54C4×D4D412yes—
55#117—24no—
56#117—24no—
57#117—22no—
58#117—22no—
59#117—22no—
60#117C2×C412yes—
61C2×C4⋊C4—24no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4C2×C411yescharacteristic
64C2×C4⋊C4D412yes—
65C2×C42—24no—
66C2×C42—22no—
67C2×C42D412yes—
68C2×C42C2311yescharacteristic
69C4⋊C8—22no—
70C4⋊C8—22no—
71C22⋊C8—22no—
72C4×C8—22no—
73C4×C8—22no—
74#41—28no—
75#41—28no—
76#41—28no—
77#41—28no—
78#41—22no—
79#41—22no—
80#41C2×C814yes—
81#41M1614yes—
82C2×Q8—22no—
83C2×Q8—28no—
84C2×Q8—28no—
85C2×Q8—22no—
86C2×Q8—22no—
87C2×Q8C22⋊C412yes—
88C2×Q8C2×C812yes—
89C2×Q8M1612yes—
90C2×D4—22no—
91C2×D4—22no—
92C2×D4C22⋊C412yes—
93C22×C4—24no—
94C22×C4—22no—
95C22×C4—24no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4C22⋊C412yes—
99C22×C4C22⋊C412yes—
100C22×C4C22×C411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C2×C8—22no—
104C2×C8—41no—
105C2×C8—41no—
106C2×C8—42no—
107C2×C8—42no—
108C2×C8—41no—
109C4⋊C4—24no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—24no—
115C4⋊C4—22no—
116C4⋊C4C22⋊C412yes—
117C4⋊C4C22⋊C412yes—
118C22⋊C4—24no—
119C22⋊C4—22no—
120C22⋊C4—22no—
121C22⋊C4—22no—
122C42—24no—
123C42—24no—
124C42—22no—
125C42—22no—
126C42—22no—
127C42—22no—
128C42—22no—
129C42C2×D412yes—
130C42C2×D412yes—
131C23—22no—
132C23C2×C22⋊C411yescharacteristic
133Q8—28no—
134Q8—24no—
135Q8—22no—
136Q8—24no—
137Q8C22⋊C814yes—
138D4—28no—
139D4—24no—
140D4—24no—
141D4—22no—
142D4C22⋊C814yes—
143C2×C4—216no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—24no—
148C2×C4—24no—
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—82no—
159C2×C4—84no—
160C2×C4C22⋊C814yes—
161C2×C4C2×C22⋊C412yes—
162C2×C4#12011yescharacteristic
163C2×C4#12012yes—
164C2×C4#12011yescharacteristic
165C2×C4C22×C811yescharacteristic
166C2×C4C2×M1612yes—
167C2×C4C2×M1611yescharacteristic
168C8—42no—
169C8—82no—
170C8—81no—
171C22—21no—
172C22—21no—
173C22—24no—
174C22—42no—
175C22—42no—
176C22#37911yescharacteristic
177C22C2×C22⋊C811yescharacteristic
178C22#40711yescharacteristic
179C4—21no—
180C4—22no—
181C4—24no—
182C4—24no—
183C4—42no—
184C4—42no—
185C4—42no—
186C4C2×C22⋊C812yes—
187C4#40712yes—
188C4#45411yescharacteristic
189C4#45611yescharacteristic
190C2—21no—
191C2—21no—
192C2—44no—
193C2#177911yescharacteristic
194C2#179311yescharacteristic
195C2#179411yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC2×C4C22×C8
Frattini, cosocleC2×C42C23
socleC22#379
prime radicalC2×2−1+4C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1984C2semidirect
3. #1477C2semidirect
4. #1476C2semidirect
5. #1461C2semidirect
28. 2−1+4C8semidirect

Maximal subgroups

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

subgroupindex
C4×2−1+42
#19842
#14772
#14762
#14612