Finite Group Databasebeta

← #10243

Subgroups — #10243

Order 256.

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

normal subgroups only

This group has 289 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#3226C211yesmaximal, characteristic
2#3037C211yesmaximal, characteristic
3#2546C211yesmaximal, characteristic, complemented
4#1546C211yesmaximal, characteristic, complemented
5#1502C211yesmaximal, characteristic
6#1479C211yesmaximal, characteristic, complemented
7#1303C211yesmaximal, characteristic, complemented
8C2×#137—21no—
9#532C2211yescharacteristic
10C2×C4⋊Q8—21no—
11C2×C4⋊Q8C2211yescharacteristic
12C2×C4×Q8C2211yescharacteristic
13#500—21no—
14#486—21no—
15#480—22no—
16#474—22no—
17#464—22no—
18#462—21no—
19#460—21no—
20#431C2211yescharacteristic
21#431C2211yescharacteristic
22#423C2211yescharacteristic
23#417—21no—
24#398—22no—
25#391—21no—
26#389—22no—
27C4×C4⋊C4C2211yescharacteristic
28#332—22no—
29#327—22no—
30Q8⋊C8—22no—
31C2×#41—21no—
32C22×Q8—21no—
33C22×Q8D411yescharacteristic
34#137—42no—
35#137—42no—
36C2×Q16—22no—
37C2×SD16—22no—
38C2×M16—21no—
39C4⋊Q8—21no—
40C4⋊Q8—21no—
41C4⋊Q8—21no—
42C4⋊Q8—21no—
43C4⋊Q8—42no—
44C4⋊Q8—42no—
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—24no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4D411yescharacteristic
61C2×C4⋊C4D412yes—
62C2×C42—21no—
63C2×C42—21no—
64C2×C42C2311yescharacteristic
65#107—24no—
66C4⋊C8—21no—
67C4⋊C8—21no—
68C4⋊C8D412yes—
69C4⋊C8D412yes—
70#103—22no—
71#102—22no—
72#102—24no—
73C22⋊C8—21no—
74C22⋊C8—21no—
75C4×C8—21no—
76C4×C8—21no—
77#95—24no—
78#95D412yes—
79#41—41no—
80#41—41no—
81#41—41no—
82#41—41no—
83C2×Q8—21no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×Q8—21no—
89C2×Q8—41no—
90C2×Q8—41no—
91C2×Q8—41no—
92C2×Q8—41no—
93C2×Q8—41no—
94C2×Q8—41no—
95C2×Q8D812yes—
96C2×D4—21no—
97C2×D4—21no—
98C2×D4—21no—
99C2×D4—41no—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—41no—
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108Q16—44no—
109SD16—44no—
110M16—41no—
111M16—41no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—41no—
115C4⋊C4—24no—
116C4⋊C4—21no—
117C4⋊C4—22no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—22no—
121C4⋊C4—21no—
122C4⋊C4—48no—
123C4⋊C4—41no—
124C4⋊C4—41no—
125C4⋊C4—41no—
126C4⋊C4—41no—
127C22⋊C4—41no—
128C22⋊C4—41no—
129C42—41no—
130C42—41no—
131C42—41no—
132C42—41no—
133C42C2×D411yescharacteristic
134C42C2×D411yescharacteristic
135C42C2×D411yescharacteristic
136C42#4111yescharacteristic
137C23—41no—
138C23#12011yescharacteristic
139Q8—22no—
140Q8—22no—
141Q8—41no—
142Q8—41no—
143Q8—41no—
144Q8—41no—
145Q8—81no—
146Q8—81no—
147Q8—81no—
148Q8—81no—
149D4—41no—
150D4—41no—
151D4—42no—
152D4—81no—
153D4—81no—
154D4—81no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—22no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4#12011yescharacteristic
172C2×C4#12011yescharacteristic
173C2×C4#12111yescharacteristic
174C2×C4#12111yescharacteristic
175C2×C4#12111yescharacteristic
176C2×C4#12711yescharacteristic
177C2×C4C2×D811yescharacteristic
178C2×C4#13511yescharacteristic
179C2×C4AGL(1,ℤ/8)11yescharacteristic
180C2×C4AGL(1,ℤ/8)11yescharacteristic
181C8—42no—
182C8—42no—
183C8—81no—
184C22—21no—
185C22—21no—
186C22—81no—
187C22—81no—
188C22—81no—
189C22#39211yescharacteristic
190C22#44711yescharacteristic
191C22#44911yescharacteristic
192C4—21no—
193C4—21no—
194C4—21no—
195C4—44no—
196C4—42no—
197C4—82no—
198C4—81no—
199C4—81no—
200C4#44711yescharacteristic
201C4#44911yescharacteristic
202C4#45612yes—
203C4#46611yescharacteristic
204C4#46911yescharacteristic
205C2—21no—
206C2—21no—
207C2—161no—
208C2#199911yescharacteristic
209C2#200511yescharacteristic
210C2#200911yescharacteristic
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
3. #2546C2semidirect
4. #1546C2semidirect
6. #1479C2semidirect
7. #1303C2semidirect

Maximal subgroups

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

subgroupindex
#32262
#30372
#25462
#15462
#15022
#14792
#13032