Finite Group Databasebeta

← #23704

Subgroups — #23704

Order 256.

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

normal subgroups only

This group has 410 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
1C2×C4×Q16C211yesmaximal, characteristic, complemented
2C2×#433C211yesmaximal, characteristic
3#2323C212yesmaximal
4#2087C211yesmaximal, characteristic, complemented
5#2076C211yesmaximal, characteristic
6#2066C212yesmaximal, complemented
7#2056C211yesmaximal, characteristic, complemented
8#2035C212yesmaximal
9#2001C211yesmaximal, characteristic
10C4×#106C211yesmaximal, characteristic, complemented
11#1763C212yesmaximal, complemented
12C22×Q16C2211yescharacteristic
13C2×C22⋊Q8C2212yes—
14C2×C4×Q8C2212yes—
15C2×#117C2211yescharacteristic
16C4×Q16—24no—
17#433—24no—
18C2×#107C2211yescharacteristic
19C2×#106C2212yes—
20C2×C4⋊C8C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×C22⋊C8C2211yescharacteristic
26C2×C22⋊C8C2211yescharacteristic
27C2×#97C2211yescharacteristic
28C2×C4×C8C2211yescharacteristic
29#391C2212yes—
30#389—22no—
31#389C2212yes—
32#388—22no—
33#388—22no—
34#387—22no—
35#380C2212yes—
36C4×C4⋊C4C2212yes—
37#340C2211yescharacteristic
38#340C2212yes—
39#340C2211yescharacteristic
40#340C2212yes—
41C22×Q8C2312yes—
42C23×C4C2311yescharacteristic
43C2×Q16—24no—
44C2×Q16—22no—
45C22×C8C2311yescharacteristic
46C22×C8C2311yescharacteristic
47C22×C8C2311yescharacteristic
48C22×C8C2311yescharacteristic
49C22⋊Q8—24no—
50C22⋊Q8—24no—
51C4×Q8—24no—
52C4×Q8—24no—
53#117—22no—
54#117—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4D411yescharacteristic
59C2×C4⋊C4D411yescharacteristic
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2312yes—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4D411yescharacteristic
66C2×C22⋊C4D411yescharacteristic
67C2×C42—22no—
68C2×C42—22no—
69C2×C42C2311yescharacteristic
70C2×C42C2311yescharacteristic
71#107—22no—
72#106—24no—
73C4⋊C8—22no—
74C4⋊C8—22no—
75#103—24no—
76#103—24no—
77#103—24no—
78C22⋊C8—22no—
79C22⋊C8—22no—
80#97—22no—
81C4×C8D414yes—
82#95—22no—
83#95—22no—
84#95—22no—
85#95—22no—
86#95—22no—
87C24C2×D411yescharacteristic
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8#4114yes—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2411yescharacteristic
108Q16—28no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—21no—
112C2×C8—21no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8C2×D412yes—
116C2×C8C2×D412yes—
117C4⋊C4—28no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—44no—
126C4⋊C4—82no—
127C4⋊C4#4114yes—
128C22⋊C4—28no—
129C22⋊C4—22no—
130C22⋊C4—22no—
131C22⋊C4—82no—
132C42—21no—
133C42—21no—
134C42—44no—
135C42—82no—
136C42C2×D412yes—
137C42#4112yes—
138C23—21no—
139C23—22no—
140C23—22no—
141C23—21no—
142C23—21no—
143C23C22×D411yescharacteristic
144Q8—24no—
145Q8—24no—
146Q8—42no—
147Q8—42no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—24no—
155C2×C4—24no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4#12112yes—
174C2×C4#12112yes—
175C2×C4C22×D411yescharacteristic
176C2×C4C2×#4112yes—
177C2×C4C2×#4111yescharacteristic
178C2×C4C2×#4112yes—
179C8—22no—
180C8—22no—
181C8—41no—
182C8—41no—
183C22—42no—
184C22—42no—
185C22—41no—
186C22—41no—
187C22—41no—
188C22C2×#12111yescharacteristic
189C22#52512yes—
190C22#57712yes—
191C22#57811yescharacteristic
192C22#57811yescharacteristic
193C4—22no—
194C4—22no—
195C4—44no—
196C4—41no—
197C4—44no—
198C4—41no—
199C4—41no—
200C4—82no—
201C4—82no—
202C4#52511yescharacteristic
203C4#52912yes—
204C4#53211yescharacteristic
205C2—81no—
206C2#239211yescharacteristic
207C2#305911yescharacteristic
208C2#306011yescharacteristic
209C2#312012yes—
210C2#312112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×C4×Q16C2semidirect
4. #2087C2semidirect
6. #2066C2semidirect
7. #2056C2semidirect
10. C4×#106C2semidirect
11. #1763C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q162
C2×#4332
#23232
#20872
#20762
#20662
#20562
#20352
#20012
C4×#1062
#17632