Finite Group Databasebeta

← #27957

Subgroups — #27957

Order 256.

351 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 135 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#2629C211yesmaximal, characteristic
2#2317C211yesmaximal, characteristic
3#2085C212yesmaximal, complemented
4#2084C212yesmaximal, complemented
5#1992C211yesmaximal, characteristic, complemented
6#1991C211yesmaximal, characteristic, complemented
7#1854C212yesmaximal
8#1853C212yesmaximal
9#1847C211yesmaximal, characteristic, complemented
10#1844C211yesmaximal, characteristic, complemented
11#1750C211yesmaximal, characteristic
12C2×#125C2211yescharacteristic
13C2×#117C2211yescharacteristic
14C2×#117C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C2×#107C2212yes—
17C2×#106C2212yes—
18C2×#106C2212yes—
19C2×#106C2212yes—
20C2×C22⋊C8C2212yes—
21C2×#97C2211yescharacteristic
22C2×C4×C8C2211yescharacteristic
23#398—22no—
24#396—22no—
25#389—22no—
26#389C2212yes—
27#385—22no—
28#384C2211yescharacteristic
29#384C2211yescharacteristic
30#384C2211yescharacteristic
31#382C2211yescharacteristic
32#382C2211yescharacteristic
33#380C2212yes—
34#340C2212yes—
35#340C2212yes—
36#340C2212yes—
37#340C2212yes—
38#336C2212yes—
39C23×C4—21no—
40C23×C4C2311yescharacteristic
41C22×C8C2312yes—
42C22×C8C2312yes—
43#125—22no—
44#125—22no—
45#117—22no—
46#117—22no—
47#117—42no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—42no—
55C2×C4⋊C4D412yes—
56C2×C4⋊C4C2311yescharacteristic
57C2×C4⋊C4C2311yescharacteristic
58C2×C4⋊C4C2311yescharacteristic
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4D412yes—
65C2×C42—21no—
66C2×C42C2311yescharacteristic
67C2×C42C2311yescharacteristic
68#107—42no—
69#106—42no—
70#106—42no—
71#106—42no—
72C22⋊C8Q818yes—
73#97—22no—
74C4×C8—22no—
75#95—22no—
76#95—22no—
77#95—22no—
78#95—22no—
79C24C2×D411yescharacteristic
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—22no—
83C22×C4—21no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—42no—
90C22×C4—82no—
91C22×C4C2×D412yes—
92C22×C4C2×D412yes—
93C22×C4C2×D411yescharacteristic
94C22×C4C2×Q814yes—
95C22×C4C2411yescharacteristic
96C2×C8—22no—
97C2×C8—22no—
98C2×C8—22no—
99C2×C8—22no—
100C2×C8C2×Q814yes—
101C2×C8C2×Q814yes—
102C4⋊C4—24no—
103C4⋊C4—22no—
104C4⋊C4—22no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C4⋊C4—22no—
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4—42no—
112C4⋊C4—44no—
113C4⋊C4—82no—
114C4⋊C4—82no—
115C4⋊C4—82no—
116C4⋊C4—82no—
117C22⋊C4—24no—
118C22⋊C4—42no—
119C22⋊C4—82no—
120C42—21no—
121C42—21no—
122C42—42no—
123C42#4112yes—
124C42#4112yes—
125C23—21no—
126C23—21no—
127C23—22no—
128C23—22no—
129C23C4⋊Q812yes—
130C23C22×D411yescharacteristic
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—24no—
136C2×C4—24no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—42no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—84no—
155C2×C4#12314yes—
156C2×C4C4⋊Q812yes—
157C2×C4C22×Q812yes—
158C2×C4C2×#4111yescharacteristic
159C2×C4C2×#4111yescharacteristic
160C2×C4H(5,𝔽2)12yes—
161C8—24no—
162C8—24no—
163C22—24no—
164C22—22no—
165C22—42no—
166C22—41no—
167C22—41no—
168C22C2×#12311yescharacteristic
169C22C2×C4⋊Q811yescharacteristic
170C22#53511yescharacteristic
171C22#57712yes—
172C22#57712yes—
173C4—24no—
174C4—22no—
175C4—42no—
176C4—82no—
177C4—81no—
178C4—82no—
179C4—81no—
180C4—81no—
181C4—81no—
182C4#55412yes—
183C4#55412yes—
184C2—42no—
185C2#250211yescharacteristic
186C2#309912yes—
187C2#316312yes—
188C2#322111yescharacteristic
189C2#322211yescharacteristic
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
3. #2085C2semidirect
4. #2084C2semidirect
5. #1992C2semidirect
6. #1991C2semidirect
9. #1847C2semidirect
10. #1844C2semidirect

Maximal subgroups

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

subgroupindex
#26292
#23172
#20852
#20842
#19922
#19912
#18542
#18532
#18472
#18442
#17502