Finite Group Databasebeta

← #10029

Subgroups — #10029

Order 256.

245 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 245 conjugacy classes of subgroups — too many to draw. Only its 51 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#3246C211yesmaximal, characteristic
2#3113C211yesmaximal, characteristic, complemented
3#3112C211yesmaximal, characteristic
4#1985C211yesmaximal, characteristic, complemented
5#1531C211yesmaximal, characteristic, complemented
6#1530C211yesmaximal, characteristic
7#1290C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9#532C2211yescharacteristic
10C2×C4×Q8C2211yescharacteristic
11#483—21no—
12#480—21no—
13#477—21no—
14#476—21no—
15#475—21no—
16#475—21no—
17#465—22no—
18#462—21no—
19#461—21no—
20#460—21no—
21#460—21no—
22#451—21no—
23#448—21no—
24#441—22no—
25#440—22no—
26#432C2211yescharacteristic
27#432C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×#97C2211yescharacteristic
30#336—22no—
31#330—22no—
32#329—22no—
33C2×#41—21no—
34C22×Q8D411yescharacteristic
35C2×Q16—22no—
36C2×SD16—22no—
37C2×SD16—22no—
38C22×C8—21no—
39C22×C8D412yes—
40C4⋊Q8D411yescharacteristic
41C4⋊Q8D411yescharacteristic
42C4⋊Q8D411yescharacteristic
43#127D411yescharacteristic
44#125—21no—
45#124—21no—
46C22⋊Q8—21no—
47C22⋊Q8—21no—
48#121—21no—
49#121—21no—
50C4×Q8—21no—
51C4×Q8—21no—
52C4×Q8—21no—
53C4×Q8—21no—
54C4×Q8—21no—
55C4×Q8—21no—
56C4×D4—21no—
57C4×D4—21no—
58#117—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4D411yescharacteristic
61C2×C42—21no—
62C2×C42C2311yescharacteristic
63#107—22no—
64#106—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#103—22no—
74#102—22no—
75#102—22no—
76C22⋊C8—21no—
77C22⋊C8—21no—
78#97—21no—
79#97—21no—
80#97D412yes—
81#97D412yes—
82#41—82no—
83C2×Q8—21no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×Q8—42no—
89C2×D4—21no—
90C2×D4—21no—
91C2×D4—21no—
92C2×D4—41no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—41no—
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4#4111yescharacteristic
99Q16—44no—
100SD16—44no—
101SD16—82no—
102C2×C8—22no—
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—41no—
107C2×C8—41no—
108C2×C8—42no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C4⋊C4—41no—
120C4⋊C4—41no—
121C22⋊C4—41no—
122C22⋊C4—41no—
123C22⋊C4—41no—
124C42—41no—
125C42—41no—
126C42—41no—
127C42C2×D411yescharacteristic
128C42C2×D411yescharacteristic
129C42C2×D411yescharacteristic
130C42C2×D411yescharacteristic
131C23—41no—
132C23#12111yescharacteristic
133Q8—22no—
134Q8—22no—
135Q8—41no—
136Q8—41no—
137Q8—42no—
138Q8—41no—
139Q8—41no—
140Q8—81no—
141D4—42no—
142D4—41no—
143D4—41no—
144D4—81no—
145D4—82no—
146C2×C4—22no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—82no—
159C2×C4#12011yescharacteristic
160C2×C4#12011yescharacteristic
161C2×C4#12011yescharacteristic
162C2×C4#12111yescharacteristic
163C2×C4#12111yescharacteristic
164C2×C4#12711yescharacteristic
165C8—42no—
166C8—42no—
167C8—42no—
168C8—42no—
169C22—21no—
170C22—21no—
171C22—81no—
172C22—81no—
173C22—81no—
174C22#39211yescharacteristic
175C22#47112yes—
176C4—21no—
177C4—21no—
178C4—21no—
179C4—21no—
180C4—21no—
181C4—42no—
182C4—81no—
183C4—81no—
184C4—81no—
185C4—81no—
186C4#45511yescharacteristic
187C4#45611yescharacteristic
188C2—22no—
189C2—161no—
190C2#200711yescharacteristic
191C2#201011yescharacteristic
192C2#201911yescharacteristic
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. #3113C2semidirect
4. #1985C2semidirect
5. #1531C2semidirect
7. #1290C2semidirect

Maximal subgroups

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

subgroupindex
#32462
#31132
#31122
#19852
#15312
#15302
#12902