Finite Group Databasebeta

← #23807

Subgroups — #23807

Order 256.

503 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 503 conjugacy classes of subgroups — too many to draw. Only its 139 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×#441C212yesmaximal, complemented
2C2×#440C212yesmaximal
3C4×C4⋊Q8C211yesmaximal, characteristic, complemented
4C4×#124C211yesmaximal, characteristic
5#2097C212yesmaximal, complemented
6#2095C211yesmaximal, characteristic, complemented
7#2094C211yesmaximal, characteristic
8#2016C211yesmaximal, characteristic, complemented
9#2015C212yesmaximal
10#2013C211yesmaximal, characteristic
11#1745C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#124C2211yescharacteristic
16C2×C4×Q8C2212yes—
17C2×C4×Q8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19#441—28no—
20#440—28no—
21C2×#107C2212yes—
22C2×#106C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#102C2212yes—
27C2×#97C2212yes—
28#391C2212yes—
29#391C2211yescharacteristic
30#390C2211yescharacteristic
31#389—22no—
32#388—22no—
33#384C2211yescharacteristic
34#382C2211yescharacteristic
35C4×C4⋊C4—22no—
36C4×C22⋊C4—22no—
37C43C2211yescharacteristic
38#336C2211yescharacteristic
39#336C2211yescharacteristic
40#336C2212yes—
41C22×Q8C2311yescharacteristic
42C22×Q8C2312yes—
43C22×D4C2311yescharacteristic
44C23×C4—21no—
45C2×Q16—24no—
46C2×Q16—24no—
47C2×Q16—24no—
48C2×SD16—24no—
49C2×SD16—24no—
50C2×SD16—24no—
51C22×C8C2312yes—
52C22×C8C2312yes—
53C4⋊Q8—24no—
54#124—22no—
55#124—22no—
56C4×Q8—24no—
57C4×Q8—24no—
58C4×Q8—24no—
59C4×D4—24no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2312yes—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—22no—
70C2×C42—21no—
71C2×C42—22no—
72C2×C42D412yes—
73C2×C42D412yes—
74C2×C42C2311yescharacteristic
75C2×C42C2311yescharacteristic
76C2×C42C2311yescharacteristic
77#107—24no—
78#106—24no—
79#103—24no—
80#103—24no—
81#103—24no—
82#102—24no—
83#97D418yes—
84#95—22no—
85#95—22no—
86#95—24no—
87C24—21no—
88C2×Q8—21no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8#4112yes—
97C2×Q8#4114yes—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—21no—
101C2×D4#4112yes—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—44no—
108C22×C4—42no—
109C22×C4C2×D412yes—
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D412yes—
113C22×C4C2411yescharacteristic
114Q16—216no—
115SD16—216no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8C2×D414yes—
121C2×C8C2×D414yes—
122C4⋊C4—28no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—44no—
128C4⋊C4—42no—
129C4⋊C4—44no—
130C22⋊C4—28no—
131C22⋊C4—42no—
132C22⋊C4—44no—
133C42—24no—
134C42—21no—
135C42—21no—
136C42—22no—
137C42—42no—
138C42—44no—
139C42—42no—
140C42C2×D414yes—
141C23—22no—
142C23—41no—
143C23—41no—
144C23—42no—
145C23—42no—
146C23C22×D411yescharacteristic
147Q8—24no—
148Q8—24no—
149Q8—24no—
150Q8—42no—
151Q8—42no—
152Q8—42no—
153D4—24no—
154D4—42no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—24no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—44no—
173C2×C4—42no—
174C2×C4—84no—
175C2×C4#12712yes—
176C2×C4#12712yes—
177C2×C4#13714yes—
178C2×C4C22×D412yes—
179C2×C4C2×#4112yes—
180C2×C4C2×#4111yescharacteristic
181C2×C4C2×#4111yescharacteristic
182C8—24no—
183C8—24no—
184C22—44no—
185C22—42no—
186C22—82no—
187C22—81no—
188C22—81no—
189C22#52511yescharacteristic
190C22#52511yescharacteristic
191C22C2×#12711yescharacteristic
192C22C2×#13712yes—
193C22#57512yes—
194C4—24no—
195C4—22no—
196C4—22no—
197C4—42no—
198C4—42no—
199C4—44no—
200C4—81no—
201C4—82no—
202C4—81no—
203C4#52512yes—
204C4#52512yes—
205C2—82no—
206C2#239711yescharacteristic
207C2#311612yes—
208C2#311712yes—
209C2#315211yescharacteristic
210C2#315311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#441C2semidirect
3. C4×C4⋊Q8C2semidirect
5. #2097C2semidirect
6. #2095C2semidirect
8. #2016C2semidirect
11. #1745C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4412
C2×#4402
C4×C4⋊Q82
C4×#1242
#20972
#20952
#20942
#20162
#20152
#20132
#17452