Finite Group Databasebeta

← #10181

Subgroups — #10181

Order 256.

265 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 265 conjugacy classes of subgroups — too many to draw. Only its 61 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#3206C211yesmaximal, characteristic
2C2×#477C211yesmaximal, characteristic, complemented
3#3037C211yesmaximal, characteristic
4#1549C211yesmaximal, characteristic, complemented
5#1480C211yesmaximal, characteristic
6#1461C211yesmaximal, characteristic, complemented
7#1301C211yesmaximal, characteristic, complemented
8C2×#137—21no—
9#532C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11C2×C4×Q8C2211yescharacteristic
12#492—21no—
13#487—21no—
14#477—22no—
15#477—22no—
16#464—22no—
17#462—21no—
18#461—22no—
19#460—21no—
20#448—22no—
21#431C2211yescharacteristic
22#431C2211yescharacteristic
23C2×#107—21no—
24#423C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26#417—21no—
27C2×#103—21no—
28#332—22no—
29D4⋊C8—22no—
30#324—22no—
31C2×#41—21no—
32C22×Q8—21no—
33C22×Q8D411yescharacteristic
34#137—42no—
35#137—42no—
36C2×Q16—22no—
37C2×SD16—22no—
38C2×SD16—24no—
39C2×M16—21no—
40C22×C8—21no—
41C4⋊Q8—21no—
42C4⋊Q8—21no—
43C4⋊Q8—21no—
44C4⋊Q8D412yes—
45C4⋊Q8D411yescharacteristic
46#127D411yescharacteristic
47#124—21no—
48#121—21no—
49#121—21no—
50C4×Q8—21no—
51C4×Q8—21no—
52C4×Q8—21no—
53C4×Q8—21no—
54C4×D4—21no—
55C4×D4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4D411yescharacteristic
60C2×C42—21no—
61C2×C42C2311yescharacteristic
62#107—24no—
63C4⋊C8—21no—
64C4⋊C8—21no—
65C4⋊C8—21no—
66C4⋊C8—21no—
67C4⋊C8D412yes—
68C4⋊C8D412yes—
69#103—24no—
70#103—22no—
71#103—24no—
72#102—22no—
73C22⋊C8—21no—
74C22⋊C8D412yes—
75C4×C8—21no—
76C4×C8—21no—
77#41—41no—
78#41—41no—
79#41—41no—
80#41—41no—
81C2×Q8—21no—
82C2×Q8—21no—
83C2×Q8—21no—
84C2×Q8—21no—
85C2×Q8—22no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×Q8—41no—
89C2×Q8—82no—
90C2×Q8SD1612yescomplemented
91C2×D4—21no—
92C2×D4—21no—
93C2×D4—21no—
94C2×D4—41no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—41no—
98C22×C4—41no—
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4#4111yescharacteristic
102Q16—44no—
103SD16—44no—
104SD16—84no—
105M16—41no—
106M16—41no—
107C2×C8—22no—
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—41no—
111C2×C8—41no—
112C2×C8—41no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—22no—
116C4⋊C4—21no—
117C4⋊C4—22no—
118C4⋊C4—21no—
119C4⋊C4—41no—
120C4⋊C4—41no—
121C4⋊C4—41no—
122C4⋊C4—41no—
123C22⋊C4—41no—
124C22⋊C4—41no—
125C42—41no—
126C42—41no—
127C42C2×D411yescharacteristic
128C42C2×D411yescharacteristic
129C42C2×D411yescharacteristic
130C42C2×D411yescharacteristic
131C23—41no—
132C23#12111yescharacteristic
133Q8—22no—
134Q8—22no—
135Q8—44no—
136Q8—41no—
137Q8—41no—
138Q8—41no—
139Q8—41no—
140Q8—82no—
141D4—42no—
142D4—41no—
143D4—41no—
144D4—42no—
145D4—81no—
146D4—81no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—42no—
155C2×C4—82no—
156C2×C4—81no—
157C2×C4—81no—
158C2×C4—81no—
159C2×C4#12011yescharacteristic
160C2×C4#12011yescharacteristic
161C2×C4#12011yescharacteristic
162C2×C4#12111yescharacteristic
163C2×C4#12111yescharacteristic
164C2×C4#12711yescharacteristic
165C2×C4C2×SD1611yescharacteristic
166C2×C4#13511yescharacteristic
167C2×C4AGL(1,ℤ/8)11yescharacteristic
168C2×C4#13711yescharacteristic
169C8—42no—
170C8—42no—
171C8—42no—
172C8—81no—
173C22—21no—
174C22—21no—
175C22—81no—
176C22—81no—
177C22—81no—
178C22#39211yescharacteristic
179C22#46511yescharacteristic
180C22#46811yescharacteristic
181C4—21no—
182C4—21no—
183C4—21no—
184C4—42no—
185C4—81no—
186C4—81no—
187C4—82no—
188C4#44811yescharacteristic
189C4#44911yescharacteristic
190C4#45011yescharacteristic
191C4#45211yescharacteristic
192C4#47312yes—
193C2—21no—
194C2—21no—
195C2—161no—
196C2#200011yescharacteristic
197C2#200811yescharacteristic
198C2#202011yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#532C22
LCSG ▷ C2×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
2. C2×#477C2semidirect
4. #1549C2semidirect
6. #1461C2semidirect
7. #1301C2semidirect
90. C2×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#32062
C2×#4772
#30372
#15492
#14802
#14612
#13012