Finite Group Databasebeta

← #31230

Subgroups — #31230

Order 256.

367 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 367 conjugacy classes of subgroups — too many to draw. Only its 107 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#3097C212yesmaximal, complemented
2#3046C212yesmaximal
3#2064C212yesmaximal, complemented
4#2052C212yesmaximal, complemented
5#2005C212yesmaximal, complemented
6#1974C211yesmaximal, characteristic
7#1956C211yesmaximal, characteristic, complemented
8#1945C211yesmaximal, characteristic, complemented
9#1859C212yesmaximal, complemented
10C2×#137C2212yes—
11C2×#135C2211yescharacteristic
12#536C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14#520C2212yes—
15#485—22no—
16#485—22no—
17#484—22no—
18#483—22no—
19#464—22no—
20#463—22no—
21#462—22no—
22#461—22no—
23C2×#108C2211yescharacteristic
24C2×#108C2211yescharacteristic
25#428C2212yes—
26#427C2211yescharacteristic
27#424C2212yes—
28C2×#104C2212yes—
29#419C2212yes—
30#417C2212yes—
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#101C2212yes—
34#408C2212yes—
35C2×C4×C8C2211yescharacteristic
36#384C2211yescharacteristic
37#343C2212yes—
38#339C2212yes—
39C2×#41C2312yes—
40C22×Q8C2312yes—
41#137—42no—
42#137—42no—
43#135—41no—
44#135—41no—
45C2×Q16—22no—
46C2×Q16D411yescharacteristic
47C2×SD16—22no—
48C2×SD16D412yes—
49C2×D8D411yescharacteristic
50C2×M16C2312yes—
51C2×M16C2312yes—
52C22×C8C2311yescharacteristic
53C22×C8C2311yescharacteristic
54C4⋊Q8—22no—
55C4⋊Q8—22no—
56C4⋊Q8—41no—
57#124—22no—
58#123—42no—
59C22⋊Q8—22no—
60C22⋊Q8—22no—
61C4×Q8—22no—
62C4×D4—22no—
63C4×D4—42no—
64#117—22no—
65#117C2312yes—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4C2312yes—
70C2×C42C2311yescharacteristic
71#108—41no—
72#108—41no—
73#107—22no—
74#107D411yescharacteristic
75#107D411yescharacteristic
76#106—22no—
77#106D412yes—
78C4⋊C8—22no—
79C4⋊C8—22no—
80#104—24no—
81#103—22no—
82#103—22no—
83#103—22no—
84#103—22no—
85#103—22no—
86#103—22no—
87#103—22no—
88#102—22no—
89#101—42no—
90C22⋊C8—22no—
91C22⋊C8—22no—
92C4×C8—41no—
93#41—22no—
94#41—22no—
95#41—22no—
96#41—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8C2×D412yes—
104C2×D4—22no—
105C2×D4C2×D412yes—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—42no—
111C22×C4C2411yescharacteristic
112Q16—41no—
113Q16—44no—
114SD16—42no—
115SD16—44no—
116D8—41no—
117M16—22no—
118M16—22no—
119M16—22no—
120M16—22no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8C2×D411yescharacteristic
128C2×C8C2×D411yescharacteristic
129C2×C8C2×D411yescharacteristic
130C2×C8C2×D411yescharacteristic
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—44no—
137C4⋊C4—82no—
138C4⋊C4C2×D412yes—
139C4⋊C4C2×D412yes—
140C22⋊C4—22no—
141C22⋊C4—42no—
142C22⋊C4—42no—
143C22⋊C4—42no—
144C42—22no—
145C42—21no—
146C42—42no—
147C42#4112yes—
148C23—22no—
149C23H(5,𝔽2)11yescharacteristic
150Q8—22no—
151Q8—22no—
152Q8—42no—
153Q8—42no—
154Q8—42no—
155Q8—42no—
156D4—22no—
157D4—22no—
158D4—42no—
159D4—42no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—82no—
173C2×C4—162no—
174C2×C4C22×D411yescharacteristic
175C2×C4C22×D411yescharacteristic
176C2×C4C2×#4111yescharacteristic
177C2×C4H(5,𝔽2)11yescharacteristic
178C2×C4H(5,𝔽2)11yescharacteristic
179C2×C42−1+411yescharacteristic
180C8—21no—
181C8—21no—
182C8—21no—
183C8—21no—
184C8—42no—
185C8—42no—
186C22—21no—
187C22—21no—
188C22—42no—
189C22—42no—
190C22—42no—
191C22#53811yescharacteristic
192C22D4211yescharacteristic
193C22#56211yescharacteristic
194C4—24no—
195C4—42no—
196C4—42no—
197C4—42no—
198C4—82no—
199C4—82no—
200C4#53411yescharacteristic
201C4#53711yescharacteristic
202C4#54711yescharacteristic
203C4#55011yescharacteristic
204C2—21no—
205C2—21no—
206C2—82no—
207C2#268211yescharacteristic
208C2#329111yescharacteristic
209C2#329311yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#135C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3097C2semidirect
3. #2064C2semidirect
4. #2052C2semidirect
5. #2005C2semidirect
7. #1956C2semidirect
8. #1945C2semidirect
9. #1859C2semidirect

Maximal subgroups

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

subgroupindex
#30972
#30462
#20642
#20522
#20052
#19742
#19562
#19452
#18592