Finite Group Databasebeta

← #13232

Subgroups — #13232

Order 256.

224 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 59 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#3057C211yesmaximal, characteristic
2#2135C211yesmaximal, characteristic, complemented
3#2134C211yesmaximal, characteristic
4#2123C211yesmaximal, characteristic, complemented
5#2122C211yesmaximal, characteristic
6#1813C211yesmaximal, characteristic, complemented
7#1757C211yesmaximal, characteristic, complemented
8C22×SD16C411yescharacteristic, complemented
9C2×#131C2211yescharacteristic
10#536C2211yescharacteristic
11#535C2211yescharacteristic
12#468—21no—
13#468—21no—
14#465—21no—
15#465—21no—
16C2×#108—21no—
17#427C411yescharacteristic, complemented
18#419C411yescharacteristic, complemented
19#418C411yescharacteristic, complemented
20#413C2211yescharacteristic
21#413C2211yescharacteristic
22#413C2211yescharacteristic
23#410C2211yescharacteristic
24#405—21no—
25#356—22no—
26#355—22no—
27#352—22no—
28AΣL(1,𝔽16)—22no—
29#344—21no—
30#338—21no—
31C2×#41C2311yescharacteristic
32C22×Q8C2×C411yescharacteristic
33C22×D4C2×C411yescharacteristic
34C2×SD16—21no—
35C2×SD16—21no—
36C2×SD16—41no—
37C2×SD16—41no—
38#131—41no—
39#131—41no—
40C2×M16—21no—
41C2×M16—21no—
42C2×M16—21no—
43C2×M16C2×C411yescharacteristic
44C2×M16D411yescharacteristic
45C2×M16D411yescharacteristic
46C22×C8C2×C411yescharacteristic
47C22×C8D411yescharacteristic
48C22×C8D411yescharacteristic
49C4⋊Q8—21no—
50#127—21no—
51#124—21no—
52#124—21no—
53#123—21no—
54#123—21no—
55C22⋊Q8—21no—
56C22⋊Q8—21no—
57#121—21no—
58#121—21no—
59#120—21no—
60#120—21no—
61#117—21no—
62#117C2×C411yescharacteristic
63#117C2×C411yescharacteristic
64#108—41no—
65#108—41no—
66#107—21no—
67#106—21no—
68#103—21no—
69#103—21no—
70#102—21no—
71#102—21no—
72#101—21no—
73#101—21no—
74#101D412yes—
75#100—21no—
76#100—21no—
77#100—21no—
78#99—21no—
79#99D412yes—
80#97—41no—
81C4×C8—41no—
82C24—21no—
83#41—41no—
84#41—41no—
85C2×Q8—21no—
86C2×Q8—21no—
87C2×Q8—41no—
88C2×Q8—41no—
89C2×Q8C2×D411yescharacteristic
90C2×D4—21no—
91C2×D4—21no—
92C2×D4—41no—
93C2×D4—41no—
94C2×D4—41no—
95C2×D4—41no—
96C2×D4C2×D411yescharacteristic
97C2×D4#4111yescharacteristic
98C2×D4#4111yescharacteristic
99C22×C4—21no—
100C22×C4C22×C411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103SD16—42no—
104SD16—41no—
105SD16—41no—
106M16—22no—
107M16—41no—
108M16—41no—
109M16—41no—
110M16—41no—
111M16—41no—
112M16—41no—
113M16—41no—
114M16—41no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—41no—
122C2×C8—41no—
123C2×C8—41no—
124C2×C8C22⋊C411yescharacteristic
125C2×C8C22⋊C411yescharacteristic
126C2×C8C22⋊C411yescharacteristic
127C2×C8C22⋊C411yescharacteristic
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—41no—
131C4⋊C4—41no—
132C4⋊C4—41no—
133C22⋊C4—21no—
134C22⋊C4—21no—
135C22⋊C4—22no—
136C22⋊C4—41no—
137C22⋊C4—41no—
138C22⋊C4—41no—
139C22⋊C4—41no—
140C42—21no—
141C42—21no—
142C42—41no—
143C23—41no—
144C23—81no—
145C23—161no—
146C23C4×D411yescharacteristic
147C23#12111yescharacteristic
148C23#12411yescharacteristic
149Q8—41no—
150Q8—41no—
151Q8—41no—
152Q8—81no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4—81no—
159D4—161no—
160D4—161no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—81no—
172C2×C4C2×C22⋊C411yescharacteristic
173C2×C4C4×D411yescharacteristic
174C2×C4#12111yescharacteristic
175C2×C4#12711yescharacteristic
176C8—21no—
177C8—21no—
178C8—21no—
179C8—42no—
180C8—81no—
181C8—81no—
182C8#9911yescharacteristic
183C8#9911yescharacteristic
184C22—21no—
185C22—21no—
186C22—21no—
187C22—41no—
188C22—41no—
189C22—41no—
190C22—81no—
191C22—81no—
192C22—81no—
193C22—162no—
194C22#39011yescharacteristic
195C4—21no—
196C4—21no—
197C4—21no—
198C4—81no—
199C4—81no—
200C4—82no—
201C4—81no—
202C4C2×#9911yescharacteristic
203C4#41011yescharacteristic
204C2—21no—
205C2—41no—
206C2—41no—
207C2—41no—
208C2—161no—
209C2#196411yescharacteristic
subgroupG/H
center, socleC2#1964
commutatorC22×C4C22×C4
Frattini, cosocleC2×#41C23
prime radical#535C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G
LECSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2135C2semidirect
4. #2123C2semidirect
6. #1813C2semidirect
7. #1757C2semidirect
8. C22×SD16C4semidirect
17. #427C4semidirect
18. #419C4semidirect
19. #418C4semidirect

Maximal subgroups

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

subgroupindex
#30572
#21352
#21342
#21232
#21222
#18132
#17572