Finite Group Databasebeta

← #10178

Subgroups — #10178

Order 256.

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

normal subgroups only

This group has 296 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#3208C211yesmaximal, characteristic, complemented
2#3077C211yesmaximal, characteristic, complemented
3C2×#459C211yesmaximal, characteristic
4#1545C211yesmaximal, characteristic, complemented
5#1481C211yesmaximal, characteristic, complemented
6#1460C211yesmaximal, characteristic, complemented
7#1301C211yesmaximal, characteristic, complemented
8C22×D8—21no—
9#533C2211yescharacteristic
10C2×#127C2211yescharacteristic
11C2×C4×D4C2211yescharacteristic
12#492—21no—
13#486—21no—
14#478—22no—
15#476—21no—
16#474—21no—
17#460—22no—
18#459—22no—
19#459—22no—
20#450—22no—
21#431C2211yescharacteristic
22#431C2211yescharacteristic
23#428—21no—
24#423C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26#418—21no—
27C2×#102—21no—
28#331—22no—
29Q8⋊C8—22no—
30#323—22no—
31C22×D4—21no—
32C22×D4—21no—
33C22×D4—21no—
34C22×D4D411yescharacteristic
35C23×C4—21no—
36C2×SD16—24no—
37C2×D8—24no—
38C2×D8—42no—
39C2×D8—42no—
40C2×M16—21no—
41C22×C8—21no—
42C4⋊Q8D411yescharacteristic
43C4⋊Q8D411yescharacteristic
44#127—21no—
45#127—21no—
46#127—21no—
47#127D412yes—
48#125—21no—
49C22⋊Q8—21no—
50C22⋊Q8—21no—
51C4×Q8—21no—
52C4×Q8—21no—
53C4×D4—21no—
54C4×D4—21no—
55C4×D4—21no—
56C4×D4—21no—
57#117—21no—
58C2×C4⋊C4D411yescharacteristic
59C2×C22⋊C4—21no—
60C2×C42C2311yescharacteristic
61#107—22no—
62#106—22no—
63C4⋊C8—21no—
64C4⋊C8—21no—
65C4⋊C8—21no—
66C4⋊C8—21no—
67C4⋊C8D412yes—
68C4⋊C8D412yes—
69#102—24no—
70#102—22no—
71#102—22no—
72#102—24no—
73C22⋊C8—21no—
74C22⋊C8D412yes—
75C4×C8—21no—
76C4×C8—21no—
77C24—21no—
78C24—41no—
79C2×Q8—21no—
80C2×Q8—21no—
81C2×D4—21no—
82C2×D4—21no—
83C2×D4—22no—
84C2×D4—22no—
85C2×D4—22no—
86C2×D4—21no—
87C2×D4—21no—
88C2×D4—21no—
89C2×D4—41no—
90C2×D4—41no—
91C2×D4—41no—
92C2×D4—41no—
93C2×D4—41no—
94C2×D4—41no—
95C2×D4—41no—
96C2×D4—82no—
97C2×D4—82no—
98C2×D4SD1612yescomplemented
99C22×C4—21no—
100C22×C4—41no—
101C22×C4—41no—
102C22×C4—82no—
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4#4111yescharacteristic
106SD16—84no—
107D8—48no—
108M16—41no—
109M16—41no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—41no—
114C2×C8—41no—
115C2×C8—41no—
116C4⋊C4—21no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—41no—
123C4⋊C4—41no—
124C22⋊C4—41no—
125C22⋊C4—41no—
126C22⋊C4—41no—
127C42—41no—
128C42C2×D411yescharacteristic
129C42C2×D411yescharacteristic
130C42C2×D411yescharacteristic
131C42C2×D411yescharacteristic
132C23—22no—
133C23—41no—
134C23—41no—
135C23—42no—
136C23—81no—
137C23—81no—
138C23—82no—
139C23—82no—
140C23—81no—
141C23—81no—
142C23#12111yescharacteristic
143Q8—42no—
144Q8—42no—
145D4—22no—
146D4—22no—
147D4—44no—
148D4—41no—
149D4—42no—
150D4—41no—
151D4—42no—
152D4—82no—
153D4—82no—
154D4—81no—
155D4—81no—
156D4—82no—
157C2×C4—22no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—82no—
163C2×C4—82no—
164C2×C4—81no—
165C2×C4#12011yescharacteristic
166C2×C4#12011yescharacteristic
167C2×C4#12011yescharacteristic
168C2×C4#12111yescharacteristic
169C2×C4#12111yescharacteristic
170C2×C4#12711yescharacteristic
171C2×C4C2×SD1611yescharacteristic
172C2×C4#13511yescharacteristic
173C2×C4AGL(1,ℤ/8)11yescharacteristic
174C2×C4#13711yescharacteristic
175C8—42no—
176C8—42no—
177C8—42no—
178C8—81no—
179C22—21no—
180C22—21no—
181C22—42no—
182C22—82no—
183C22—82no—
184C22—81no—
185C22—81no—
186C22—82no—
187C22—82no—
188C22—82no—
189C22—162no—
190C22—162no—
191C22#39211yescharacteristic
192C22#46511yescharacteristic
193C22#46811yescharacteristic
194C4—21no—
195C4—21no—
196C4—21no—
197C4—81no—
198C4—81no—
199C4—81no—
200C4#44811yescharacteristic
201C4#44911yescharacteristic
202C4#45011yescharacteristic
203C4#45211yescharacteristic
204C4#47212yes—
205C2—21no—
206C2—21no—
207C2—82no—
208C2—162no—
209C2#200011yescharacteristic
210C2#200811yescharacteristic
211C2#201811yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radicalC2×#459C2
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
1. #3208C2semidirect
2. #3077C2semidirect
4. #1545C2semidirect
5. #1481C2semidirect
6. #1460C2semidirect
7. #1301C2semidirect
98. C2×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#32082
#30772
C2×#4592
#15452
#14812
#14602
#13012