Finite Group Databasebeta

← #29200

Subgroups — #29200

Order 256.

353 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 123 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#2647C211yesmaximal, characteristic, complemented
2#2326C211yesmaximal, characteristic
3#2068C212yesmaximal, complemented
4#2067C212yesmaximal
5#1959C211yesmaximal, characteristic
6#1958C211yesmaximal, characteristic
7#1863C212yesmaximal, complemented
8#1862C212yesmaximal
9#1845C211yesmaximal, characteristic, complemented
10#1836C211yesmaximal, characteristic, complemented
11#1768C211yesmaximal, characteristic, complemented
12C2×#124C2211yescharacteristic
13C2×C4×Q8C2211yescharacteristic
14C2×C4×D4C2211yescharacteristic
15C2×#107C2212yes—
16C2×#107C2212yes—
17C2×C4⋊C8C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#102C2212yes—
21C2×#102C2212yes—
22C2×#97C2211yescharacteristic
23C2×C4×C8C2211yescharacteristic
24#400—22no—
25#398—22no—
26#391C2211yescharacteristic
27#390C2211yescharacteristic
28#389C2212yes—
29#389C2212yes—
30#388—22no—
31#385—22no—
32#384C2211yescharacteristic
33#384C2211yescharacteristic
34#382C2211yescharacteristic
35#376C2211yescharacteristic
36#340C2212yes—
37#340C2212yes—
38#336C2212yes—
39C22×Q8C2311yescharacteristic
40C22×D4C2311yescharacteristic
41C23×C4—21no—
42C22×C8C2312yes—
43C22×C8C2312yes—
44#124—22no—
45#124Q814yes—
46C4×Q8—42no—
47C4×D4—42no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4C2311yescharacteristic
54C2×C4⋊C4C2311yescharacteristic
55C2×C4⋊C4C2312yes—
56C2×C4⋊C4C2312yes—
57C2×C22⋊C4—22no—
58C2×C22⋊C4—21no—
59C2×C22⋊C4—22no—
60C2×C42—21no—
61C2×C42C2311yescharacteristic
62C2×C42C2311yescharacteristic
63C2×C42C2311yescharacteristic
64#107—42no—
65#107—42no—
66C4⋊C8—24no—
67#103—24no—
68#103—42no—
69#102—24no—
70#102—42no—
71#97—22no—
72C4×C8—22no—
73#95—22no—
74#95—22no—
75#95—22no—
76#95—22no—
77#95D412yes—
78#95D412yes—
79C24—21no—
80C2×Q8—21no—
81C2×Q8—22no—
82C2×Q8—22no—
83C2×Q8C2×Q812yes—
84C2×D4—22no—
85C2×D4—21no—
86C2×D4—22no—
87C2×D4C2×Q812yes—
88C22×C4—21no—
89C22×C4—22no—
90C22×C4—21no—
91C22×C4—22no—
92C22×C4—21no—
93C22×C4—42no—
94C22×C4—82no—
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D412yes—
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D412yes—
99C22×C4C2411yescharacteristic
100C2×C8—22no—
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—22no—
104C2×C8—22no—
105C2×C8#4114yes—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C4⋊C4—22no—
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—42no—
113C4⋊C4—82no—
114C4⋊C4—82no—
115C4⋊C4—82no—
116C4⋊C4—82no—
117C22⋊C4—28no—
118C22⋊C4—42no—
119C22⋊C4—82no—
120C42—22no—
121C42—22no—
122C42—42no—
123C42C2×Q812yes—
124C42#4112yes—
125C23—22no—
126C23—41no—
127C23—41no—
128C23—42no—
129C23—42no—
130C23C22×D411yescharacteristic
131Q8—24no—
132Q8—42no—
133D4—24no—
134D4—42no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—42no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—84no—
158C2×C4C22⋊Q814yes—
159C2×C4C22×Q811yescharacteristic
160C2×C4C2×#4111yescharacteristic
161C2×C4C2×#4112yes—
162C2×C4H(5,𝔽2)12yes—
163C8—24no—
164C8—42no—
165C22—42no—
166C22—44no—
167C22—82no—
168C22—81no—
169C22—81no—
170C22C2×C22⋊Q811yescharacteristic
171C22#53211yescharacteristic
172C22#53511yescharacteristic
173C22#57612yes—
174C22#57812yes—
175C4—24no—
176C4—42no—
177C4—42no—
178C4—41no—
179C4—41no—
180C4—82no—
181C4—81no—
182C4—82no—
183C4—81no—
184C4#55312yes—
185C4#55412yes—
186C2—82no—
187C2#256611yescharacteristic
188C2#308312yes—
189C2#317512yes—
190C2#322011yescharacteristic
191C2#322311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2647C2semidirect
3. #2068C2semidirect
7. #1863C2semidirect
9. #1845C2semidirect
10. #1836C2semidirect
11. #1768C2semidirect

Maximal subgroups

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

subgroupindex
#26472
#23262
#20682
#20672
#19592
#19582
#18632
#18622
#18452
#18362
#17682