Finite Group Databasebeta

← #29321

Subgroups — #29321

Order 256.

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

normal subgroups only

This group has 329 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
1#2859C211yesmaximal, characteristic
2C4×#125C211yesmaximal, characteristic
3#2085C212yesmaximal
4#2059C212yesmaximal
5#1948C211yesmaximal, characteristic
6#1948C211yesmaximal, characteristic
7#1947C211yesmaximal, characteristic
8#1947C211yesmaximal, characteristic
9#1944C212yesmaximal
10#1943C212yesmaximal
11#1767C211yesmaximal, characteristic
12C2×#125—21no—
13C2×#125C2211yescharacteristic
14C2×#107C2212yes—
15C2×#107C2212yes—
16C2×#106C2212yes—
17C2×#106C2212yes—
18C2×C4⋊C8C2211yescharacteristic
19C2×C4⋊C8C2211yescharacteristic
20C2×#97C2212yes—
21#389—21no—
22#389—21no—
23#389C2211yescharacteristic
24#389C2211yescharacteristic
25#389C2212yes—
26#389C2212yes—
27#387—21no—
28#387—21no—
29#384C2211yescharacteristic
30#384C2211yescharacteristic
31#382—21no—
32#382C2211yescharacteristic
33C4×C4⋊C4—21no—
34C4×C4⋊C4—21no—
35C4×C4⋊C4C2211yescharacteristic
36C4×C4⋊C4C2211yescharacteristic
37C43C2211yescharacteristic
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×C8C2312yes—
44C22×C8C2312yes—
45#125—22no—
46#125—22no—
47#125—44no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4C2311yescharacteristic
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C42—21no—
65C2×C42—21no—
66C2×C42D411yescharacteristic
67C2×C42D411yescharacteristic
68C2×C42D411yescharacteristic
69C2×C42D411yescharacteristic
70C2×C42C2311yescharacteristic
71C2×C42C2311yescharacteristic
72C2×C42C2311yescharacteristic
73#107—24no—
74#107—42no—
75#106—24no—
76#106—42no—
77C4⋊C8Q814yes—
78C4⋊C8Q814yes—
79#97—42no—
80#95—21no—
81#95—21no—
82#95—21no—
83#95—21no—
84#95—21no—
85#95—21no—
86#95—22no—
87#95—22no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2411yescharacteristic
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—22no—
104C2×C8—22no—
105C2×C8C2×Q814yes—
106C2×C8C2×Q814yes—
107C4⋊C4—24no—
108C4⋊C4—24no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4—21no—
114C4⋊C4—22no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—21no—
118C4⋊C4—22no—
119C4⋊C4—42no—
120C4⋊C4—42no—
121C4⋊C4—42no—
122C4⋊C4—42no—
123C4⋊C4—44no—
124C4⋊C4—44no—
125C4⋊C4—42no—
126C4⋊C4—42no—
127C4⋊C4#4112yes—
128C4⋊C4#4112yes—
129C42—22no—
130C42—22no—
131C42—21no—
132C42—21no—
133C42—21no—
134C42—21no—
135C42—41no—
136C42—41no—
137C42—42no—
138C42—42no—
139C42C2×Q812yes—
140C42C2×Q812yes—
141C23C22×D411yescharacteristic
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4C4⋊Q812yes—
175C2×C4C4⋊Q812yes—
176C2×C4AGL(1,ℤ/8)12yes—
177C2×C4#13712yes—
178C2×C4C22×Q811yescharacteristic
179C2×C4C22×Q811yescharacteristic
180C2×C4C2×#4111yescharacteristic
181C2×C4C2×#4111yescharacteristic
182C2×C4H(5,𝔽2)11yescharacteristic
183C2×C42−1+411yescharacteristic
184C8—24no—
185C8—24no—
186C22#52511yescharacteristic
187C22C2×C4⋊Q811yescharacteristic
188C22#53711yescharacteristic
189C22C2×AGL(1,ℤ/8)11yescharacteristic
190C22C2×#13711yescharacteristic
191C22#57511yescharacteristic
192C22#57511yescharacteristic
193C4—22no—
194C4—41no—
195C4—42no—
196C4—41no—
197C4—42no—
198C4—41no—
199C4—41no—
200C4—82no—
201C4—81no—
202C4—82no—
203C4—81no—
204C4#50114yes—
205C4#55411yescharacteristic
206C4#55411yescharacteristic
207C4#55711yescharacteristic
208C4#55711yescharacteristic
209C2#258411yescharacteristic
210C2#311511yescharacteristic
211C2#311611yescharacteristic
212C2C2×#50111yescharacteristic
213C2#316611yescharacteristic
214C2#325111yescharacteristic
215C2#325211yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#28592
C4×#1252
#20852
#20592
#19482
#19472
#19442
#19432
#17672