Finite Group Databasebeta

← #30870

Subgroups — #30870

Order 256.

427 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 427 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C2×#473C212yesmaximal, complemented
2#3063C212yesmaximal
3#3044C212yesmaximal, complemented
4#2064C212yesmaximal, complemented
5#2016C211yesmaximal, characteristic, complemented
6#2007C211yesmaximal, characteristic
7#2005C212yesmaximal, complemented
8#1866C212yesmaximal, complemented
9#1857C211yesmaximal, characteristic, complemented
10C2×#137C2212yes—
11C2×#137C2212yes—
12C22×Q16C2212yes—
13C2×#131C2212yes—
14#537C2211yescharacteristic
15#536C2212yes—
16C2×C4⋊Q8C2211yescharacteristic
17#519C2211yescharacteristic
18#473—28no—
19#470—22no—
20#468—22no—
21#467—22no—
22#465—22no—
23#464—22no—
24#464—22no—
25#462—22no—
26#462—22no—
27C2×#108C2212yes—
28#428C2212yes—
29#424C2212yes—
30C2×#104C2212yes—
31#419C2212yes—
32#416C2212yes—
33C2×#103C2212yes—
34C2×#101C2212yes—
35C2×#101C2211yescharacteristic
36#391C2211yescharacteristic
37#343C2211yescharacteristic
38#339C2212yes—
39#336C2211yescharacteristic
40C2×#41C2312yes—
41C22×Q8C2312yes—
42C22×Q8C2311yescharacteristic
43#137—24no—
44#137—24no—
45#137—42no—
46#137—42no—
47C2×Q16—24no—
48C2×Q16—22no—
49C2×Q16—22no—
50C2×Q16—22no—
51C2×Q16—22no—
52C2×Q16—42no—
53C2×SD16—22no—
54C2×SD16—22no—
55#131—24no—
56#131—42no—
57C2×M16—22no—
58C2×M16C2312yes—
59C2×M16C2312yes—
60C22×C8—22no—
61C22×C8C2312yes—
62C4⋊Q8—22no—
63C4⋊Q8—42no—
64#124—22no—
65#123—42no—
66C22⋊Q8—22no—
67C22⋊Q8—22no—
68C22⋊Q8—22no—
69#121—22no—
70#121—41no—
71C4×Q8—22no—
72C4×Q8—22no—
73#117—41no—
74#117C2312yes—
75C2×C4⋊C4—41no—
76C2×C4⋊C4C2311yescharacteristic
77C2×C42C2311yescharacteristic
78#108—24no—
79#107—22no—
80#106—22no—
81C4⋊C8—22no—
82C4⋊C8—22no—
83#104—42no—
84#103—22no—
85#103—22no—
86#103—22no—
87#103—22no—
88#103D412yes—
89#103D412yes—
90#102D412yes—
91#102D412yes—
92#101—24no—
93#101D414yescomplemented
94#95—41no—
95#41—22no—
96#41—22no—
97#41—22no—
98#41—22no—
99C2×Q8—21no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8C2×D412yes—
109C2×Q8C2×D412yes—
110C2×D4—22no—
111C2×D4—41no—
112C2×D4C2×D412yes—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4C2411yescharacteristic
119Q16—44no—
120Q16—44no—
121Q16—44no—
122Q16—44no—
123SD16—44no—
124SD16—44no—
125M16—22no—
126M16—22no—
127M16—22no—
128M16—44no—
129M16C2×D414yes—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—44no—
135C2×C8C2×D412yes—
136C2×C8C2×D412yes—
137C4⋊C4—22no—
138C4⋊C4—41no—
139C4⋊C4—41no—
140C4⋊C4—41no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—44no—
144C4⋊C4C2×D412yes—
145C4⋊C4C2×D412yes—
146C22⋊C4—22no—
147C22⋊C4—42no—
148C22⋊C4—42no—
149C22⋊C4—42no—
150C42—22no—
151C42—22no—
152C42—42no—
153C23—22no—
154C23H(5,𝔽2)11yescharacteristic
155Q8—22no—
156Q8—24no—
157Q8—22no—
158Q8—42no—
159Q8—42no—
160Q8—42no—
161Q8—42no—
162Q8—42no—
163D4—22no—
164D4—22no—
165D4—42no—
166D4—42no—
167D4—161no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—81no—
184C2×C4C22×D412yes—
185C2×C4C22×D411yescharacteristic
186C2×C4H(5,𝔽2)12yes—
187C2×C42−1+411yescharacteristic
188C8—22no—
189C8—22no—
190C8—24no—
191C8—42no—
192C22—21no—
193C22—21no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22#53711yescharacteristic
198C22D4211yescharacteristic
199C22#56011yescharacteristic
200C4—42no—
201C4—42no—
202C4—42no—
203C4—42no—
204C4—42no—
205C4—82no—
206C4—81no—
207C4#53712yes—
208C4D4212yes—
209C2—21no—
210C2—21no—
211C2—82no—
212C2#267511yescharacteristic
213C2#328911yescharacteristic
214C2#329311yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radical#537C22
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. C2×#473C2semidirect
3. #3044C2semidirect
4. #2064C2semidirect
5. #2016C2semidirect
7. #2005C2semidirect
8. #1866C2semidirect
9. #1857C2semidirect
93. #101D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4732
#30632
#30442
#20642
#20162
#20072
#20052
#18662
#18572