Finite Group Databasebeta

← #26532

Subgroups — #26532

Order 256.

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

normal subgroups only

This group has 359 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#2552C211yesmaximal, characteristic
2#2365C211yesmaximal, characteristic
3#2095C212yesmaximal
4#2068C212yesmaximal
5#1981C211yesmaximal, characteristic
6#1956C211yesmaximal, characteristic
7#1947C211yesmaximal, characteristic
8#1939C211yesmaximal, characteristic
9#1863C212yesmaximal
10#1803C212yesmaximal
11C4×#107C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×#125—21no—
14C2×C4×Q8C2212yes—
15C2×#107C2211yescharacteristic
16C2×#107C2211yescharacteristic
17C2×#107C2212yes—
18C2×C4⋊C8C2211yescharacteristic
19C2×C4⋊C8C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×C4×C8C2211yescharacteristic
25#398—22no—
26#391C2212yes—
27#389—21no—
28#389—21no—
29#389C2212yes—
30#387—22no—
31#384C2212yes—
32#384C2211yescharacteristic
33C4×C4⋊C4—21no—
34C4×C4⋊C4C2211yescharacteristic
35C4×C4⋊C4C2211yescharacteristic
36C4×C4⋊C4C2211yescharacteristic
37#340C2211yescharacteristic
38#340C2212yes—
39#340C2211yescharacteristic
40#336C2212yes—
41C22×Q8C2312yes—
42C22×C8C2312yes—
43C22×C8C2311yescharacteristic
44C22×C8C2311yescharacteristic
45C4⋊Q8—22no—
46C4⋊Q8Q814yes—
47#125—44no—
48C4×Q8—24no—
49C4×Q8—42no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4D411yescharacteristic
59C2×C4⋊C4D411yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2311yescharacteristic
65C2×C42—22no—
66C2×C42—21no—
67C2×C42—21no—
68C2×C42C2312yes—
69C2×C42C2311yescharacteristic
70#107—22no—
71#107—22no—
72#107—42no—
73C4⋊C8—22no—
74C4⋊C8—42no—
75#103—24no—
76#103—42no—
77#103—42no—
78#103—42no—
79C4×C8—22no—
80#95—22no—
81#95—21no—
82#95—21no—
83#95—22no—
84#95—22no—
85#95D412yes—
86C2×Q8—22no—
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8C2×Q814yes—
92C22×C4—22no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—22no—
98C22×C4C2×D412yes—
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D412yes—
101C22×C4C2×D411yescharacteristic
102C22×C4C2411yescharacteristic
103C2×C8—21no—
104C2×C8—21no—
105C2×C8—21no—
106C2×C8—22no—
107C2×C8—21no—
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—21no—
111C2×C8#4112yes—
112C4⋊C4—24no—
113C4⋊C4—24no—
114C4⋊C4—22no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—22no—
123C4⋊C4—21no—
124C4⋊C4—44no—
125C4⋊C4—44no—
126C4⋊C4—44no—
127C4⋊C4—42no—
128C4⋊C4—82no—
129C4⋊C4—81no—
130C4⋊C4Q1614yes—
131C4⋊C4#4112yes—
132C42—22no—
133C42—22no—
134C42—44no—
135C42—42no—
136C42—42no—
137C42C2×Q812yes—
138C42#4112yes—
139C23C22×D411yescharacteristic
140Q8—24no—
141Q8—24no—
142Q8—42no—
143Q8—42no—
144C2×C4—22no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—21no—
148C2×C4—24no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4C22⋊Q812yes—
170C2×C4C22⋊Q812yes—
171C2×C4C2×Q1614yes—
172C2×C4C2×Q1612yes—
173C2×C4AGL(1,ℤ/8)12yes—
174C2×C4C22×Q811yescharacteristic
175C2×C4C2×#4111yescharacteristic
176C2×C4C2×#4111yescharacteristic
177C2×C4C2×#4111yescharacteristic
178C2×C4H(5,𝔽2)11yescharacteristic
179C2×C4H(5,𝔽2)11yescharacteristic
180C8—22no—
181C8—41no—
182C8—42no—
183C22C2×C22⋊Q811yescharacteristic
184C22#54611yescharacteristic
185C22#55011yescharacteristic
186C22C22×Q1611yescharacteristic
187C22C2×AGL(1,ℤ/8)11yescharacteristic
188C22#57711yescharacteristic
189C22#57811yescharacteristic
190C4—22no—
191C4—24no—
192C4—42no—
193C4—42no—
194C4—44no—
195C4—81no—
196C4—81no—
197C4—81no—
198C4—82no—
199C4#47714yes—
200C4#52911yescharacteristic
201C4#54011yescharacteristic
202C4#55411yescharacteristic
203C4#55411yescharacteristic
204C2#245411yescharacteristic
205C2C2×#47711yescharacteristic
206C2#308211yescharacteristic
207C2#329911yescharacteristic
208C2#333011yescharacteristic
209C2#339511yescharacteristic
210C2#339711yescharacteristic
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

Maximal subgroups

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

subgroupindex
#25522
#23652
#20952
#20682
#19812
#19562
#19472
#19392
#18632
#18032
C4×#1072