Finite Group Databasebeta

← #26320

Subgroups — #26320

Order 256.

349 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 349 conjugacy classes of subgroups — too many to draw. Only its 131 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#2568C211yesmaximal, characteristic
2Q8×C4⋊C4C211yesmaximal, characteristic
3#2096C212yesmaximal
4#2072C212yesmaximal
5#1984C211yesmaximal, characteristic
6#1981C211yesmaximal, characteristic
7#1956C211yesmaximal, characteristic
8#1948C211yesmaximal, characteristic
9#1927C212yesmaximal
10#1927C212yesmaximal
11C4×#106C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4×Q8—21no—
14C2×C4×Q8C2211yescharacteristic
15C2×#106C2211yescharacteristic
16C2×#106C2211yescharacteristic
17C2×#106C2212yes—
18C2×C4⋊C8C2211yescharacteristic
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×C4×C8C2211yescharacteristic
24#400—22no—
25#391—21no—
26#391C2212yes—
27#391C2212yes—
28#389—21no—
29#389—22no—
30#384C2212yes—
31#384C2211yescharacteristic
32#382—21no—
33C4×C4⋊C4C2211yescharacteristic
34C4×C4⋊C4C2212yes—
35C4×C4⋊C4C2211yescharacteristic
36#340C2211yescharacteristic
37#340C2212yes—
38#340C2211yescharacteristic
39#336C2212yes—
40#336C2212yes—
41C22×Q8D411yescharacteristic
42C22×Q8C2312yes—
43C22×C8C2311yescharacteristic
44C22×C8C2312yes—
45C22×C8C2311yescharacteristic
46C4⋊Q8—22no—
47C4⋊Q8—22no—
48C4×Q8—22no—
49C4×Q8—22no—
50C4×Q8—44no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4D411yescharacteristic
58C2×C4⋊C4D412yes—
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C42—22no—
65C2×C42—21no—
66C2×C42—21no—
67C2×C42C2312yes—
68C2×C42C2311yescharacteristic
69#106—22no—
70#106—22no—
71#106—42no—
72C4⋊C8—22no—
73#103—24no—
74#103—24no—
75#103—42no—
76#103—42no—
77C4×C8—22no—
78#95—22no—
79#95—21no—
80#95—21no—
81#95—22no—
82#95—22no—
83#95—22no—
84#95—22no—
85C2×Q8—22no—
86C2×Q8—22no—
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8SD1614yes—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—21no—
100C22×C4C2×D412yes—
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D412yes—
103C22×C4C2×D411yescharacteristic
104C22×C4C2411yescharacteristic
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—21no—
111C2×C8—22no—
112C2×C8—21no—
113C2×C8#4112yes—
114C4⋊C4—24no—
115C4⋊C4—22no—
116C4⋊C4—21no—
117C4⋊C4—22no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—22no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—42no—
126C4⋊C4—42no—
127C4⋊C4—44no—
128C4⋊C4—42no—
129C4⋊C4—81no—
130C4⋊C4—82no—
131C4⋊C4#4112yes—
132C42—22no—
133C42—22no—
134C42—44no—
135C42—42no—
136C42—42no—
137C42#4112yes—
138C42#4112yes—
139C23C22×D411yescharacteristic
140Q8—28no—
141Q8—42no—
142Q8—42no—
143Q8—42no—
144Q8—42no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4#12312yes—
171C2×C4#12312yes—
172C2×C4C2×SD1614yes—
173C2×C4C2×SD1612yes—
174C2×C4#13712yes—
175C2×C4C2×#4111yescharacteristic
176C2×C4C2×#4111yescharacteristic
177C2×C4C2×#4111yescharacteristic
178C2×C4C2×#4111yescharacteristic
179C2×C4H(5,𝔽2)11yescharacteristic
180C2×C42−1+411yescharacteristic
181C8—22no—
182C8—41no—
183C8—42no—
184C22C2×#12311yescharacteristic
185C22#54811yescharacteristic
186C22#55011yescharacteristic
187C22C22×SD1611yescharacteristic
188C22C2×#13711yescharacteristic
189C22#57711yescharacteristic
190C22#57811yescharacteristic
191C4—24no—
192C4—22no—
193C4—42no—
194C4—42no—
195C4—81no—
196C4—82no—
197C4—81no—
198C4—81no—
199C4—82no—
200C4#48314yes—
201C4#52911yescharacteristic
202C4#54211yescharacteristic
203C4#55611yescharacteristic
204C4#55911yescharacteristic
205C2#244811yescharacteristic
206C2C2×#48311yescharacteristic
207C2#309711yescharacteristic
208C2#335911yescharacteristic
209C2#336911yescharacteristic
210C2#339211yescharacteristic
211C2#339811yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ 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
#25682
Q8×C4⋊C42
#20962
#20722
#19842
#19812
#19562
#19482
#19272
C4×#1062