Finite Group Databasebeta

← #10016

Subgroups — #10016

Order 256.

280 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 57 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#3104C211yesmaximal, characteristic, complemented
2#3101C211yesmaximal, characteristic
3#2398C211yesmaximal, characteristic
4#1995C211yesmaximal, characteristic, complemented
5#1487C211yesmaximal, characteristic, complemented
6#1486C211yesmaximal, characteristic
7#1274C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×C4×Q8C2211yescharacteristic
10C2×C4×D4C2211yescharacteristic
11#484—21no—
12#483—21no—
13#477—21no—
14#476—21no—
15#462—21no—
16#461—21no—
17#451—21no—
18#450—21no—
19C4×Q16—22no—
20C4×SD16—22no—
21#431C2211yescharacteristic
22#431C2211yescharacteristic
23#423C2211yescharacteristic
24#400—21no—
25#397—21no—
26#394—21no—
27#393—21no—
28C4×C22⋊C4—22no—
29#376C2211yescharacteristic
30#339—22no—
31Q8⋊C8—22no—
32D4⋊C8—22no—
33C22×Q8D411yescharacteristic
34C22×D4D411yescharacteristic
35C23×C4—21no—
36C2×Q16—22no—
37C2×SD16—22no—
38C2×M16—21no—
39C4⋊Q8D411yescharacteristic
40C4⋊Q8D411yescharacteristic
41#125—21no—
42C22⋊Q8—21no—
43C22⋊Q8—21no—
44C4×Q8—21no—
45C4×Q8—21no—
46C4×Q8—21no—
47C4×Q8—21no—
48C4×Q8—21no—
49C4×Q8—21no—
50C4×D4—21no—
51C4×D4—21no—
52C4×D4—21no—
53C4×D4—21no—
54#117—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4D411yescharacteristic
57C2×C4⋊C4D411yescharacteristic
58C2×C22⋊C4—22no—
59C2×C22⋊C4—22no—
60C2×C22⋊C4—21no—
61C2×C42—21no—
62C2×C42D412yes—
63C2×C42C2311yescharacteristic
64#107—22no—
65#106—22no—
66C4⋊C8—21no—
67C4⋊C8—21no—
68C4⋊C8—21no—
69C4⋊C8—21no—
70#103—22no—
71#103—22no—
72#103—22no—
73#102—22no—
74C22⋊C8—21no—
75C22⋊C8—21no—
76C4×C8D412yes—
77C4×C8D412yes—
78#95—22no—
79#95—22no—
80#95—21no—
81#95—21no—
82C24—21no—
83C2×Q8—21no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—21no—
87C2×Q8—42no—
88C2×D4—21no—
89C2×D4—21no—
90C2×D4—42no—
91C22×C4—21no—
92C22×C4—22no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4—42no—
98C22×C4—42no—
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102Q16—44no—
103SD16—44no—
104M16—42no—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8—41no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—41no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C4⋊C4—41no—
120C22⋊C4—44no—
121C22⋊C4—44no—
122C22⋊C4—41no—
123C22⋊C4—41no—
124C22⋊C4—41no—
125C42—22no—
126C42—22no—
127C42—41no—
128C42—41no—
129C42—41no—
130C42C2×D411yescharacteristic
131C42C2×D411yescharacteristic
132C42C2×D411yescharacteristic
133C42#4111yescharacteristic
134C23—22no—
135C23—42no—
136C23—41no—
137C23—41no—
138C23—42no—
139C23#12011yescharacteristic
140Q8—22no—
141Q8—22no—
142Q8—41no—
143Q8—41no—
144Q8—42no—
145Q8—42no—
146D4—22no—
147D4—22no—
148D4—41no—
149D4—41no—
150C2×C4—22no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—82no—
165C2×C4—81no—
166C2×C4—81no—
167C2×C4#12011yescharacteristic
168C2×C4#12011yescharacteristic
169C2×C4#12111yescharacteristic
170C2×C4#12111yescharacteristic
171C2×C4#12111yescharacteristic
172C2×C4#12711yescharacteristic
173C2×C4#13512yes—
174C2×C4#13712yes—
175C8—42no—
176C8—42no—
177C8—81no—
178C22—21no—
179C22—21no—
180C22—44no—
181C22—42no—
182C22—82no—
183C22—81no—
184C22—81no—
185C22#39211yescharacteristic
186C22#45212yes—
187C4—21no—
188C4—21no—
189C4—21no—
190C4—21no—
191C4—42no—
192C4—82no—
193C4—81no—
194C4—81no—
195C4—81no—
196C4—81no—
197C4#46412yes—
198C4#49511yescharacteristic
199C4#49711yescharacteristic
200C2—22no—
201C2—82no—
202C2#201011yescharacteristic
203C2#201011yescharacteristic
204C2#201511yescharacteristic
subgroupG/H
centerC2×C4#127
commutator, Frattini, cosocleC2×C42C23
socleC22#392
prime radicalC22×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3104C2semidirect
4. #1995C2semidirect
5. #1487C2semidirect
7. #1274C2semidirect

Maximal subgroups

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

subgroupindex
#31042
#31012
#23982
#19952
#14872
#14862
#12742