Finite Group Databasebeta

← #28248

Subgroups — #28248

Order 256.

431 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 431 conjugacy classes of subgroups — too many to draw. Only its 127 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#3061C212yesmaximal, complemented
2#3060C212yesmaximal, complemented
3C2×#444C212yesmaximal, complemented
4#2040C211yesmaximal, characteristic, complemented
5#2040C211yesmaximal, characteristic, complemented
6#2039C212yesmaximal, complemented
7#1847C211yesmaximal, characteristic, complemented
8#1816C212yesmaximal, complemented
9#1815C212yesmaximal, complemented
10C2×#135C2212yes—
11C2×#131C2212yes—
12C2×#131C2212yes—
13#537C2211yescharacteristic
14#537C2212yes—
15#537C2211yescharacteristic
16#470—22no—
17#469—22no—
18#468—22no—
19#468—22no—
20#467—22no—
21#466—22no—
22#465—22no—
23#465—22no—
24#444—28no—
25C2×#108C2212yes—
26C2×#107C2212yes—
27C2×#106C2212yes—
28C2×#104C2212yes—
29C2×#104C2212yes—
30#417C2212yes—
31#417C2212yes—
32#417C2212yes—
33#417C2212yes—
34C2×#97C2211yescharacteristic
35#384C2211yescharacteristic
36#382C2211yescharacteristic
37#343C2211yescharacteristic
38#343C2212yes—
39#343C2211yescharacteristic
40C2×#41C2312yes—
41C2×#41C2312yes—
42#135—24no—
43#135—24no—
44C2×Q16D412yes—
45C2×SD16D412yes—
46C2×SD16D412yes—
47C2×D8D412yes—
48#131—24no—
49#131—24no—
50#131—24no—
51#131—24no—
52C2×M16—22no—
53C2×M16—22no—
54C2×M16C2312yes—
55C2×M16C2312yes—
56C22×C8—22no—
57C22×C8—22no—
58C22×C8C2312yes—
59C22⋊Q8—22no—
60C22⋊Q8—22no—
61C22⋊Q8—22no—
62C22⋊Q8—22no—
63#121—22no—
64#121—22no—
65#121—22no—
66#121—22no—
67#121—41no—
68#121—41no—
69#121—42no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2311yescharacteristic
73C2×C4⋊C4C2312yes—
74C2×C42C2311yescharacteristic
75#108—24no—
76#107—22no—
77#107—22no—
78#106—22no—
79#106—22no—
80#104—24no—
81#104—24no—
82#103—22no—
83#103—22no—
84#103—22no—
85#103—22no—
86#102—22no—
87#102—22no—
88#102—22no—
89#102—22no—
90#97D414yescomplemented
91#95—41no—
92#41—22no—
93#41—22no—
94#41—22no—
95#41—22no—
96#41—22no—
97#41—22no—
98#41—22no—
99#41—22no—
100C2×Q8C2×D412yes—
101C2×Q8C2×D412yes—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—41no—
105C2×D4—42no—
106C2×D4—41no—
107C2×D4C2×D412yes—
108C2×D4C2×D412yes—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4C2411yescharacteristic
116Q16—24no—
117SD16—24no—
118SD16—24no—
119D8—24no—
120M16—22no—
121M16—22no—
122M16—22no—
123M16—22no—
124M16—44no—
125M16—44no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—44no—
129C2×C8—44no—
130C2×C8C2×D414yes—
131C2×C8C2×D412yes—
132C2×C8C2×D412yes—
133C2×C8#4114yes—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—41no—
139C4⋊C4—42no—
140C4⋊C4—41no—
141C4⋊C4—82no—
142C22⋊C4—42no—
143C22⋊C4—42no—
144C22⋊C4—42no—
145C22⋊C4—42no—
146C42—21no—
147C42C2×D412yes—
148C23—22no—
149C23—22no—
150C23H(5,𝔽2)11yescharacteristic
151Q8—22no—
152Q8—22no—
153Q8—22no—
154Q8—22no—
155D4—22no—
156D4—22no—
157D4—22no—
158D4—22no—
159D4—42no—
160D4—42no—
161D4—42no—
162D4—42no—
163D4—161no—
164D4—162no—
165D4—161no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—82no—
179C2×C4—81no—
180C2×C4—81no—
181C2×C4C22×D411yescharacteristic
182C2×C4C22×D412yes—
183C2×C4C2×#4112yes—
184C2×C42−1+411yescharacteristic
185C8—42no—
186C8—42no—
187C8#12114yes—
188C8#12114yes—
189C22—21no—
190C22—21no—
191C22—42no—
192C22—42no—
193C22—42no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22#53711yescharacteristic
198C22D4211yescharacteristic
199C22#55511yescharacteristic
200C4—42no—
201C4—42no—
202C4—42no—
203C4—82no—
204C4—81no—
205C4—81no—
206C4C2×#12112yes—
207C4#54712yes—
208C2—21no—
209C2—21no—
210C2—82no—
211C2—82no—
212C2#253711yescharacteristic
213C2#328811yescharacteristic
214C2#328911yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
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. #3061C2semidirect
2. #3060C2semidirect
3. C2×#444C2semidirect
4. #2040C2semidirect
5. #2040C2semidirect
6. #2039C2semidirect
7. #1847C2semidirect
8. #1816C2semidirect
9. #1815C2semidirect
90. #97D4semidirect

Maximal subgroups

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

subgroupindex
#30612
#30602
C2×#4442
#20402
#20392
#18472
#18162
#18152