Finite Group Databasebeta

← #26465

Subgroups — #26465

Order 256.

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

normal subgroups only

This group has 379 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×#479C212yesmaximal
2#2552C211yesmaximal, characteristic
3#2365C211yesmaximal, characteristic
4#2066C212yesmaximal
5#2026C212yesmaximal
6#1981C211yesmaximal, characteristic
7#1956C211yesmaximal, characteristic
8#1947C211yesmaximal, characteristic
9#1939C211yesmaximal, characteristic
10#1863C212yesmaximal
11C4×#106C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×#125C2212yes—
14C2×C4×Q8C2212yes—
15#479—28no—
16C2×#107C2212yes—
17C2×#106C2211yescharacteristic
18C2×#106C2211yescharacteristic
19C2×C4⋊C8C2212yes—
20C2×C4⋊C8C2211yescharacteristic
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×C4×C8C2211yescharacteristic
26#398—22no—
27#391C2212yes—
28#389—21no—
29#389—21no—
30#389C2212yes—
31#387—22no—
32#384—21no—
33#384C2211yescharacteristic
34C4×C4⋊C4—21no—
35C4×C4⋊C4C2211yescharacteristic
36C4×C4⋊C4C2211yescharacteristic
37C4×C4⋊C4C2211yescharacteristic
38#340C2212yes—
39#340C2211yescharacteristic
40#340C2211yescharacteristic
41#336C2212yes—
42C22×Q8C2312yes—
43C22×C8C2311yescharacteristic
44C22×C8C2312yes—
45C22×C8C2311yescharacteristic
46C4⋊Q8—22no—
47C4⋊Q8Q814yes—
48#125—24no—
49#125—42no—
50C4×Q8—24no—
51C4×Q8—42no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4D411yescharacteristic
61C2×C4⋊C4D411yescharacteristic
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C42—22no—
68C2×C42—21no—
69C2×C42—21no—
70C2×C42C2312yes—
71C2×C42C2311yescharacteristic
72#107—24no—
73#106—22no—
74#106D414yes—
75C4⋊C8—24no—
76C4⋊C8—22no—
77#103—24no—
78#103—24no—
79#103—24no—
80#103—42no—
81C4×C8—22no—
82#95—22no—
83#95—21no—
84#95—21no—
85#95—22no—
86#95—22no—
87#95D412yes—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8C2×Q814yes—
94C22×C4—21no—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—22no—
100C22×C4C2×D412yes—
101C22×C4C2×D412yes—
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2411yescharacteristic
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—22no—
108C2×C8—21no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—21no—
112C2×C8—21no—
113C2×C8C2×D412yes—
114C4⋊C4—24no—
115C4⋊C4—24no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—22no—
122C4⋊C4—21no—
123C4⋊C4—22no—
124C4⋊C4—21no—
125C4⋊C4—44no—
126C4⋊C4—44no—
127C4⋊C4—44no—
128C4⋊C4—42no—
129C4⋊C4—81no—
130C4⋊C4—82no—
131C4⋊C4C2×D412yes—
132C4⋊C4C2×D412yes—
133C4⋊C4#4112yes—
134C42—22no—
135C42—22no—
136C42—44no—
137C42—42no—
138C42—42no—
139C42C2×Q812yes—
140C42#4112yes—
141C23C22×D411yescharacteristic
142Q8—24no—
143Q8—24no—
144Q8—42no—
145Q8—42no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—24no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4C22⋊Q812yes—
173C2×C4C22⋊Q812yes—
174C2×C4#13512yes—
175C2×C4#13712yes—
176C2×C4C22×D411yescharacteristic
177C2×C4C22×Q811yescharacteristic
178C2×C4C2×#4111yescharacteristic
179C2×C4C2×#4111yescharacteristic
180C2×C4H(5,𝔽2)11yescharacteristic
181C2×C42−1+411yescharacteristic
182C8—22no—
183C8—42no—
184C8—41no—
185C22C2×C22⋊Q811yescharacteristic
186C22D4211yescharacteristic
187C22#54811yescharacteristic
188C22C2×#13511yescharacteristic
189C22C2×#13711yescharacteristic
190C22#57711yescharacteristic
191C22#57811yescharacteristic
192C4—22no—
193C4—42no—
194C4—44no—
195C4—42no—
196C4—42no—
197C4—42no—
198C4—42no—
199C4—81no—
200C4—82no—
201C4#47914yescomplemented
202C4#53211yescharacteristic
203C4#54211yescharacteristic
204C4D4×Q811yescharacteristic
205C4#55411yescharacteristic
206C2#245211yescharacteristic
207C2C2×#47911yescharacteristic
208C2#308211yescharacteristic
209C2#327611yescharacteristic
210C2#328411yescharacteristic
211C2#336411yescharacteristic
212C2#337211yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
201. C4#479semidirect

Maximal subgroups

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

subgroupindex
C2×#4792
#25522
#23652
#20662
#20262
#19812
#19562
#19472
#19392
#18632
C4×#1062