Finite Group Databasebeta

← #28213

Subgroups — #28213

Order 256.

363 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 123 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×#477C211yesmaximal, characteristic
2C2×#475C211yesmaximal, characteristic
3#2331C211yesmaximal, characteristic
4#2327C211yesmaximal, characteristic
5#2085C212yesmaximal
6#2043C212yesmaximal
7#1928C212yesmaximal
8#1927C212yesmaximal
9#1832C211yesmaximal, characteristic
10#1805C211yesmaximal, characteristic
11#1805C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×C4×Q8C2211yescharacteristic
16#477—24no—
17#475—24no—
18C2×#107C2212yes—
19C2×#106C2212yes—
20C2×C4⋊C8C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26#391—22no—
27#391C2211yescharacteristic
28#391C2211yescharacteristic
29#389C2212yes—
30#387—22no—
31#387—22no—
32#387—22no—
33#382C2211yescharacteristic
34#382C2211yescharacteristic
35C4×C4⋊C4C2212yes—
36#376C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#336C2212yes—
40#336C2212yes—
41C22×Q8—21no—
42C22×Q8C2311yescharacteristic
43C22×Q8C2311yescharacteristic
44C22×C8C2312yes—
45C22×C8C2312yes—
46C4⋊Q8—22no—
47C4⋊Q8—22no—
48C4⋊Q8—22no—
49C4⋊Q8D414yes—
50C4×Q8—22no—
51C4×Q8—22no—
52C4×Q8—22no—
53C4×Q8—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2312yes—
63C2×C42—21no—
64C2×C42—21no—
65C2×C42—22no—
66C2×C42C2311yescharacteristic
67C2×C42C2311yescharacteristic
68C2×C42C2311yescharacteristic
69#107—24no—
70#106—24no—
71C4⋊C8—22no—
72C4⋊C8—22no—
73#103—24no—
74#103—24no—
75#103—24no—
76#103—24no—
77#95—22no—
78#95—22no—
79#95—21no—
80#95—21no—
81#95—22no—
82#95—22no—
83#95—22no—
84#95D412yes—
85#95D412yes—
86C2×Q8—24no—
87C2×Q8—21no—
88C2×Q8—21no—
89C2×Q8—21no—
90C2×Q8—21no—
91C2×Q8—21no—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—82no—
98C2×Q8#4112yes—
99C2×Q8#4112yes—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4C2×D412yes—
107C22×C4C2×D412yes—
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2411yescharacteristic
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C4⋊C4—28no—
118C4⋊C4—24no—
119C4⋊C4—22no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—42no—
127C4⋊C4—42no—
128C4⋊C4—82no—
129C4⋊C4C2×D414yes—
130C42—21no—
131C42—21no—
132C42—21no—
133C42—21no—
134C42—42no—
135C42—42no—
136C42—82no—
137C42C2×D412yes—
138C42#4112yes—
139C23C22×D411yescharacteristic
140Q8—22no—
141Q8—22no—
142Q8—22no—
143Q8—22no—
144Q8—41no—
145Q8—41no—
146Q8—41no—
147Q8—41no—
148Q8—84no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—24no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4#12112yes—
174C2×C4#12112yes—
175C2×C4C22×D411yescharacteristic
176C2×C4C2×#4111yescharacteristic
177C2×C4C2×#4111yescharacteristic
178C2×C4C2×#4111yescharacteristic
179C2×C42−1+411yescharacteristic
180C2×C42−1+411yescharacteristic
181C8—42no—
182C8—42no—
183C22C2×#12111yescharacteristic
184C22#52511yescharacteristic
185C22#53611yescharacteristic
186C22#57712yes—
187C22#57812yes—
188C4—22no—
189C4—22no—
190C4—42no—
191C4—42no—
192C4—44no—
193C4—42no—
194C4—42no—
195C4—82no—
196C4—81no—
197C4—81no—
198C4#54811yescharacteristic
199C4#54811yescharacteristic
200C4#55211yescharacteristic
201C4#55211yescharacteristic
202C2#253611yescharacteristic
203C2#304211yescharacteristic
204C2#304411yescharacteristic
205C2#312112yes—
206C2#324212yes—
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ 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
C2×#4772
C2×#4752
#23312
#23272
#20852
#20432
#19282
#19272
#18322
#18052