Finite Group Databasebeta

← #8955

Subgroups — #8955

Order 256.

244 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 73 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#2989C211yesmaximal, characteristic
2C4×#137C211yesmaximal, characteristic
3#1984C211yesmaximal, characteristic, complemented
4#1479C211yesmaximal, characteristic, complemented
5#1476C211yesmaximal, characteristic
6#1283C211yesmaximal, characteristic, complemented
7#1278C211yesmaximal, characteristic, complemented
8C2×#137C411yescharacteristic
9C4×#41C2211yescharacteristic
10C2×C4×Q8C2211yescharacteristic
11#446—21no—
12#441—21no—
13#440—21no—
14C4×Q16—21no—
15C4×SD16—21no—
16#436—21no—
17#435—22no—
18#432—21no—
19#431C2211yescharacteristic
20#431C2211yescharacteristic
21#428C411yescharacteristic
22#423—21no—
23C2×C4⋊C8C2211yescharacteristic
24#419C411yescharacteristic
25#417C411yescharacteristic
26C4×M16C2211yescharacteristic
27C2×C4×C8C2211yescharacteristic
28#336—22no—
29Q8⋊C8—21no—
30Q8⋊C8—21no—
31Q8⋊C8—21no—
32D4⋊C8—21no—
33C2×#41C2×C411yescharacteristic
34C22×Q8C2×C411yescharacteristic
35#137—22no—
36#137—22no—
37C2×Q16—21no—
38C2×SD16—21no—
39C2×M16—22no—
40C2×M16C2×C411yescharacteristic
41C2×M16C2×C411yescharacteristic
42C22×C8—21no—
43C22×C8D412yes—
44C4×Q8—21no—
45C4×Q8—21no—
46C4×Q8—21no—
47C4×Q8—21no—
48C4×Q8D411yescharacteristic
49C4×Q8D411yescharacteristic
50C4×D4—21no—
51C4×D4—21no—
52C4×D4D411yescharacteristic
53C4×D4D411yescharacteristic
54#117—21no—
55#117C2×C411yescharacteristic
56C2×C4⋊C4—21no—
57C2×C4⋊C4C2×C411yescharacteristic
58C2×C42—21no—
59C2×C42—21no—
60C2×C42C2311yescharacteristic
61#107—21no—
62#106—21no—
63C4⋊C8—22no—
64C4⋊C8—21no—
65C4⋊C8—21no—
66C4⋊C8—21no—
67C4⋊C8D411yescharacteristic
68C4⋊C8D411yescharacteristic
69#103—21no—
70#103—21no—
71#103—21no—
72#102—21no—
73C22⋊C8—22no—
74C22⋊C8—21no—
75C22⋊C8—21no—
76#97—22no—
77#97—21no—
78C4×C8—21no—
79C4×C8—21no—
80C4×C8—21no—
81C4×C8—21no—
82C4×C8—21no—
83#41—21no—
84#41—21no—
85#41—21no—
86#41—21no—
87C2×Q8—21no—
88C2×Q8—21no—
89C2×Q8—21no—
90C2×Q8—21no—
91C2×Q8—21no—
92C2×Q8C22⋊C411yescharacteristic
93C2×Q8M1612yescomplemented
94C2×D4—21no—
95C2×D4C22⋊C411yescharacteristic
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4C22×C411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4#4111yescharacteristic
104Q16—24no—
105SD16—24no—
106M16—21no—
107M16—21no—
108M16—21no—
109M16—82no—
110M16M1612yescomplemented
111C2×C8—21no—
112C2×C8—22no—
113C2×C8—21no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—22no—
117C2×C8—41no—
118C2×C8—41no—
119C2×C8—42no—
120C2×C8—41no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—41no—
126C4⋊C4—41no—
127C4⋊C4C22⋊C411yescharacteristic
128C4⋊C4C22⋊C411yescharacteristic
129C22⋊C4—21no—
130C22⋊C4—21no—
131C22⋊C4—21no—
132C42—21no—
133C42—21no—
134C42—21no—
135C42—41no—
136C42—41no—
137C42C2×D411yescharacteristic
138C42C2×D411yescharacteristic
139C42C2×D411yescharacteristic
140C42#4111yescharacteristic
141C23—21no—
142C23C4×D411yescharacteristic
143Q8—22no—
144Q8—21no—
145Q8—21no—
146Q8—22no—
147Q8—41no—
148Q8—41no—
149D4—21no—
150D4—21no—
151D4—41no—
152D4—41no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—81no—
169C2×C4C2×C22⋊C411yescharacteristic
170C2×C4C4×D411yescharacteristic
171C2×C4#12011yescharacteristic
172C2×C4#12111yescharacteristic
173C2×C4#12111yescharacteristic
174C2×C4#12311yescharacteristic
175C2×C4C2×M1611yescharacteristic
176C2×C4C2×M1611yescharacteristic
177C2×C4#13111yescharacteristic
178C2×C4#13111yescharacteristic
179C8—22no—
180C8—42no—
181C8—41no—
182C8—42no—
183C8—41no—
184C8—41no—
185C8—81no—
186C22—21no—
187C22—21no—
188C22—41no—
189C22—41no—
190C22—41no—
191C22#38611yescharacteristic
192C22#43511yescharacteristic
193C22#43611yescharacteristic
194C4—21no—
195C4—21no—
196C4—21no—
197C4—41no—
198C4—42no—
199C4—41no—
200C4—41no—
201C4—81no—
202C4#40711yescharacteristic
203C4#40811yescharacteristic
204C4#43511yescharacteristic
205C4#43611yescharacteristic
206C4#45511yescharacteristic
207C4#45611yescharacteristic
208C2—21no—
209C2—21no—
210C2—81no—
211C2#187011yescharacteristic
212C2#188111yescharacteristic
213C2#188211yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
socleC22#386
prime radicalC2×#41C2×C4
LCSG ▷ C22×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
3. #1984C2semidirect
4. #1479C2semidirect
6. #1283C2semidirect
7. #1278C2semidirect
93. C2×Q8M16semidirect
110. M16M16semidirect

Maximal subgroups

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

subgroupindex
#29892
C4×#1372
#19842
#14792
#14762
#12832
#12782