Finite Group Databasebeta

← #30556

Subgroups — #30556

Order 256.

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

normal subgroups only

This group has 423 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#3137C212yesmaximal, complemented
2#3011C212yesmaximal
3#2052C212yesmaximal, complemented
4#2028C211yesmaximal, characteristic, complemented
5#2016C211yesmaximal, characteristic, complemented
6#2005C212yesmaximal, complemented
7#1979C212yesmaximal, complemented
8#1929C212yesmaximal, complemented
9#1793C211yesmaximal, characteristic
10C2×2−1+4C2211yescharacteristic
11C2×#137C2212yes—
12C22×Q16C2212yes—
13#544C2211yescharacteristic
14#536C2212yes—
15C2×C4⋊Q8C2211yescharacteristic
16#491—22no—
17#489—22no—
18#488—22no—
19#487—22no—
20#452—22no—
21#452—22no—
22#451—22no—
23#451—22no—
24C2×#108C2212yes—
25C2×#104C2212yes—
26#419C2212yes—
27#419C2212yes—
28#417C2212yes—
29#416C2212yes—
30C2×#103C2212yes—
31C2×#101C2211yescharacteristic
32C2×#101C2212yes—
33#408C2212yes—
34#405C2212yes—
35#391C2211yescharacteristic
36#343C2211yescharacteristic
37#339C2212yes—
38#336C2211yescharacteristic
392−1+4—22no—
402−1+4—41no—
412−1+4—42no—
42C2×#41—22no—
43C2×#41—41no—
44C2×#41C2312yes—
45C22×Q8—22no—
46C22×Q8C2312yes—
47C22×Q8C2311yescharacteristic
48#137—42no—
49#137—42no—
50C2×Q16—22no—
51C2×Q16—24no—
52C2×Q16—22no—
53C2×Q16—22no—
54C2×Q16—42no—
55C2×SD16—22no—
56C2×M16C2312yes—
57C2×M16C2312yes—
58C22×C8C2312yes—
59C4⋊Q8—22no—
60C4⋊Q8—22no—
61C4⋊Q8—42no—
62#125—22no—
63#124—22no—
64#123—42no—
65C22⋊Q8—22no—
66C22⋊Q8—22no—
67C22⋊Q8—41no—
68#117C2312yes—
69C2×C4⋊C4—41no—
70C2×C4⋊C4C2311yescharacteristic
71C2×C42C2311yescharacteristic
72#108—42no—
73#104—42no—
74#103—22no—
75#103—22no—
76#103—22no—
77#103—22no—
78#103—22no—
79#103—22no—
80#103—22no—
81#103D412yes—
82#103D412yes—
83#102—22no—
84#102D412yes—
85#102D412yes—
86#101—41no—
87#101—42no—
88C22⋊C8—22no—
89C22⋊C8—22no—
90#97—22no—
91C4×C8—22no—
92#95—41no—
93#41—22no—
94#41—22no—
95#41—22no—
96#41—22no—
97#41—44no—
98#41—44no—
99#41—44no—
100#41—44no—
101C2×Q8—21no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—41no—
111C2×Q8—41no—
112C2×Q8—44no—
113C2×Q8—44no—
114C2×Q8C2×D412yes—
115C2×Q8#4112yes—
116C2×D4—22no—
117C2×D4—41no—
118C2×D4C2×D412yes—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—42no—
125C22×C4C2411yescharacteristic
126Q16—44no—
127Q16—44no—
128Q16—44no—
129SD16—44no—
130M16—22no—
131M16—22no—
132M16—22no—
133M16—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8C2×D412yes—
139C2×C8C2×D412yes—
140C4⋊C4—22no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—41no—
144C4⋊C4—42no—
145C4⋊C4—44no—
146C4⋊C4C2×D412yes—
147C4⋊C4C2×D412yes—
148C22⋊C4—22no—
149C22⋊C4—42no—
150C22⋊C4—42no—
151C42—22no—
152C42—22no—
153C23—22no—
154C23H(5,𝔽2)11yescharacteristic
155Q8—22no—
156Q8—24no—
157Q8—22no—
158Q8—44no—
159Q8—42no—
160Q8—42no—
161Q8—42no—
162Q8—42no—
163Q8—42no—
164Q8—161no—
165D4—22no—
166D4—22no—
167D4—44no—
168D4—42no—
169D4—42no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—41no—
176C2×C4—48no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—81no—
187C2×C4C22×D412yes—
188C2×C4C2×#4111yescharacteristic
189C2×C4H(5,𝔽2)11yescharacteristic
190C2×C4H(5,𝔽2)12yes—
191C8—22no—
192C8—22no—
193C8—42no—
194C8—42no—
195C22—21no—
196C22—21no—
197C22—42no—
198C22—42no—
199C22—42no—
200C22#53811yescharacteristic
201C22D4211yescharacteristic
202C22H(3,𝔽4)11yescharacteristic
203C4—42no—
204C4—42no—
205C4—42no—
206C4—42no—
207C4—42no—
208C4—82no—
209C4—81no—
210C4#53512yes—
211C4#54612yes—
212C2—21no—
213C2—21no—
214C2—82no—
215C2#267011yescharacteristic
216C2#328911yescharacteristic
217C2#329311yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×2−1+4C22
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. #3137C2semidirect
3. #2052C2semidirect
4. #2028C2semidirect
5. #2016C2semidirect
6. #2005C2semidirect
7. #1979C2semidirect
8. #1929C2semidirect

Maximal subgroups

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

subgroupindex
#31372
#30112
#20522
#20282
#20162
#20052
#19792
#19292
#17932