Finite Group Databasebeta

← #10106

Subgroups — #10106

Order 256.

287 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 53 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#3253C211yesmaximal, characteristic
2#3206C211yesmaximal, characteristic, complemented
3#3152C211yesmaximal, characteristic, complemented
4#1997C211yesmaximal, characteristic, complemented
5#1525C211yesmaximal, characteristic, complemented
6#1484C211yesmaximal, characteristic, complemented
7#1299C211yesmaximal, characteristic, complemented
8C2×#137—22no—
9#532C2211yescharacteristic
10#532C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic
12#497—22no—
13#494—21no—
14#492—21no—
15#492—21no—
16#487—21no—
17#476—21no—
18#476—21no—
19#469—22no—
20#461—22no—
21#459—21no—
22#459—21no—
23#448—22no—
24#431C2211yescharacteristic
25#423C2211yescharacteristic
26C4×M16C2211yescharacteristic
27C4×M16C2211yescharacteristic
28#343—22no—
29#329—22no—
30D4⋊C8—22no—
31C2×#41—21no—
32C2×#41—21no—
33C22×Q8D412yes—
34#137—48no—
35C2×Q16—24no—
36C2×SD16—24no—
37C2×SD16—24no—
38C2×D8—22no—
39C2×M16—21no—
40C2×M16—21no—
41C2×M16D412yes—
42C4⋊Q8—21no—
43C4⋊Q8—21no—
44C4⋊Q8—21no—
45C4⋊Q8—21no—
46C4⋊Q8D411yescharacteristic
47C4⋊Q8D411yescharacteristic
48#127D411yescharacteristic
49#127D411yescharacteristic
50#124—21no—
51#124—21no—
52#121—21no—
53#121—21no—
54#121—21no—
55#121—21no—
56C4×D4—21no—
57C4×D4—21no—
58C4×D4—21no—
59C4×D4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C42C2311yescharacteristic
63#106—22no—
64C4⋊C8—21no—
65C4⋊C8—21no—
66C4⋊C8D412yes—
67#103—24no—
68#102—22no—
69#102—22no—
70C22⋊C8—21no—
71#97—21no—
72#97D412yes—
73C4×C8—21no—
74C4×C8—21no—
75C4×C8—21no—
76#41—44no—
77#41—82no—
78C2×Q8—22no—
79C2×Q8—21no—
80C2×Q8—22no—
81C2×Q8—21no—
82C2×Q8—44no—
83C2×D4—21no—
84C2×D4—21no—
85C2×D4—21no—
86C2×D4—21no—
87C2×D4—21no—
88C2×D4—21no—
89C2×D4—41no—
90C2×D4—41no—
91C22×C4—22no—
92C22×C4—41no—
93C22×C4—41no—
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4#4111yescharacteristic
97Q16—48no—
98SD16—48no—
99SD16—84no—
100D8—82no—
101M16—24no—
102M16—22no—
103M16—41no—
104M16—42no—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8—22no—
108C2×C8—41no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—41no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C22⋊C4—41no—
118C22⋊C4—41no—
119C22⋊C4—41no—
120C22⋊C4—41no—
121C42C2×D411yescharacteristic
122C42C2×D411yescharacteristic
123C42C2×D411yescharacteristic
124C42C2×D411yescharacteristic
125C23—41no—
126C23—41no—
127C23#12111yescharacteristic
128Q8—44no—
129Q8—42no—
130Q8—44no—
131Q8—81no—
132D4—42no—
133D4—42no—
134D4—41no—
135D4—41no—
136D4—41no—
137D4—41no—
138D4—42no—
139D4—81no—
140D4—82no—
141D4—82no—
142C2×C4—22no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—22no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4—82no—
150C2×C4—82no—
151C2×C4—82no—
152C2×C4#12011yescharacteristic
153C2×C4#12011yescharacteristic
154C2×C4#12011yescharacteristic
155C2×C4#12111yescharacteristic
156C2×C4#12111yescharacteristic
157C2×C4#12711yescharacteristic
158C8—24no—
159C8—42no—
160C8—42no—
161C8—42no—
162C22—21no—
163C22—21no—
164C22—21no—
165C22—81no—
166C22—81no—
167C22—81no—
168C22—81no—
169C22—81no—
170C22—81no—
171C22#39211yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—81no—
176C4—82no—
177C4—81no—
178C4#45312yes—
179C4#45511yescharacteristic
180C4#45611yescharacteristic
181C4#47112yes—
182C2—41no—
183C2—161no—
184C2—161no—
185C2#200211yescharacteristic
186C2#200711yescharacteristic
187C2#201911yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#3253C2
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. #3206C2semidirect
3. #3152C2semidirect
4. #1997C2semidirect
5. #1525C2semidirect
6. #1484C2semidirect
7. #1299C2semidirect

Maximal subgroups

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

subgroupindex
#32532
#32062
#31522
#19972
#15252
#14842
#12992