Finite Group Databasebeta

← #9172

Subgroups — #9172

Order 256.

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

normal subgroups only

This group has 232 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#3039C211yesmaximal, characteristic
2#2925C211yesmaximal, characteristic
3#1546C211yesmaximal, characteristic, complemented
4C2×#332C211yesmaximal, characteristic, complemented
5#1477C211yesmaximal, characteristic
6#1301C211yesmaximal, characteristic, complemented
7#1292C211yesmaximal, characteristic, complemented
8C2×#137—21no—
9C2×C4⋊Q8C2211yescharacteristic
10C4×#41C2211yescharacteristic
11#462—21no—
12#462C412yes—
13#461—21no—
14#461C412yes—
15#433—22no—
16#432—21no—
17#431—21no—
18#431C2211yescharacteristic
19#423C2211yescharacteristic
20#423C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23#419—21no—
24#408—21no—
25#405—21no—
26#332—21no—
27#332—21no—
28#332—22no—
29Q8⋊C8—21no—
30D4⋊C8—21no—
31C2×#41D411yescharacteristic
32C22×Q8—21no—
33#137—42no—
34#137—42no—
35C2×Q16—22no—
36C2×SD16—22no—
37C2×M16—21no—
38C2×M16—21no—
39C22×C8—21no—
40C22×C8—21no—
41C4⋊Q8—21no—
42C4⋊Q8—21no—
43C4⋊Q8—21no—
44C4⋊Q8C2×C412yes—
45C4×Q8C2×C411yescharacteristic
46C4×Q8D411yescharacteristic
47C4×D4—21no—
48C4×D4—21no—
49C4×D4C2×C411yescharacteristic
50C4×D4D411yescharacteristic
51#117—21no—
52#117D411yescharacteristic
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C42—21no—
56C2×C42C2311yescharacteristic
57C4⋊C8—21no—
58C4⋊C8—21no—
59C4⋊C8—21no—
60C4⋊C8—21no—
61C4⋊C8—21no—
62C4⋊C8—21no—
63C4⋊C8C2×C412yes—
64C4⋊C8D411yescharacteristic
65C4⋊C8D411yescharacteristic
66C4⋊C8D412yes—
67#103—22no—
68#103—22no—
69C22⋊C8—22no—
70C22⋊C8—22no—
71C22⋊C8—21no—
72#97—22no—
73C4×C8—22no—
74C4×C8—21no—
75#41—21no—
76#41—21no—
77#41—21no—
78#41—21no—
79C2×Q8—22no—
80C2×Q8—41no—
81C2×Q8—41no—
82C2×Q8—41no—
83C2×Q8—41no—
84C2×Q8—41no—
85C2×Q8C22⋊C411yescharacteristic
86C2×D4—21no—
87C2×D4C22⋊C411yescharacteristic
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—41no—
92C22×C4C2×D411yescharacteristic
93C22×C4#4111yescharacteristic
94C22×C4#4111yescharacteristic
95Q16—44no—
96SD16—44no—
97M16—41no—
98M16—41no—
99M16—41no—
100M16—41no—
101C2×C8—22no—
102C2×C8—21no—
103C2×C8—22no—
104C2×C8—21no—
105C2×C8—41no—
106C2×C8—41no—
107C2×C8—41no—
108C2×C8—41no—
109C2×C8—81no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4C22⋊C411yescharacteristic
117C4⋊C4C22⋊C411yescharacteristic
118C22⋊C4—21no—
119C22⋊C4—21no—
120C22⋊C4—21no—
121C42—21no—
122C42—21no—
123C42—21no—
124C42C22×C411yescharacteristic
125C42C2×D411yescharacteristic
126C42C2×D411yescharacteristic
127C42C2×D411yescharacteristic
128C23—21no—
129C23#12311yescharacteristic
130Q8—21no—
131Q8—21no—
132Q8—42no—
133Q8—42no—
134Q8—81no—
135Q8—81no—
136D4—21no—
137D4—21no—
138D4—41no—
139D4—41no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—42no—
151C2×C4—81no—
152C2×C4—81no—
153C2×C4—81no—
154C2×C4C2×C22⋊C411yescharacteristic
155C2×C4C4×D411yescharacteristic
156C2×C4C4×D411yescharacteristic
157C2×C4#12011yescharacteristic
158C2×C4#12111yescharacteristic
159C2×C4#12111yescharacteristic
160C2×C4#13511yescharacteristic
161C2×C4#13511yescharacteristic
162C2×C4AGL(1,ℤ/8)11yescharacteristic
163C2×C4AGL(1,ℤ/8)11yescharacteristic
164C8—42no—
165C8—41no—
166C8—41no—
167C8—42no—
168C8—81no—
169C8—81no—
170C22—21no—
171C22—21no—
172C22—41no—
173C22—41no—
174C22—41no—
175C22#38611yescharacteristic
176C22#48211yescharacteristic
177C22#48211yescharacteristic
178C4—21no—
179C4—21no—
180C4—21no—
181C4—41no—
182C4—41no—
183C4—41no—
184C4—82no—
185C4#41611yescharacteristic
186C4#41811yescharacteristic
187C4#44911yescharacteristic
188C4#44911yescharacteristic
189C4#47111yescharacteristic
190C4#47311yescharacteristic
191C2—21no—
192C2—21no—
193C2—81no—
194C2#187511yescharacteristic
195C2#190211yescharacteristic
196C2#190811yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC2×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1546C2semidirect
4. C2×#332C2semidirect
6. #1301C2semidirect
7. #1292C2semidirect

Maximal subgroups

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

subgroupindex
#30392
#29252
#15462
C2×#3322
#14772
#13012
#12922