Finite Group Databasebeta

← #29963

Subgroups — #29963

Order 256.

395 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 395 conjugacy classes of subgroups — too many to draw. Only its 119 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#2629C211yesmaximal, characteristic
2#2310C211yesmaximal, characteristic
3#2096C211yesmaximal, characteristic, complemented
4#2095C211yesmaximal, characteristic, complemented
5#2068C212yesmaximal, complemented
6#2066C212yesmaximal, complemented
7#2026C211yesmaximal, characteristic, complemented
8#2026C211yesmaximal, characteristic, complemented
9#1854C212yesmaximal
10#1798C211yesmaximal, characteristic
11#1787C212yesmaximal
12C23×Q8C2211yescharacteristic
13C2×#125C2211yescharacteristic
14C2×#117C2211yescharacteristic
15C2×#117C2211yescharacteristic
16C2×#107C2212yes—
17C2×#106C2212yes—
18C2×C4⋊C8C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×C22⋊C8C2212yes—
24#398—22no—
25#396—22no—
26#391—22no—
27#391C2211yescharacteristic
28#391C2211yescharacteristic
29#391C2211yescharacteristic
30#391C2211yescharacteristic
31#389C2212yes—
32#384C2211yescharacteristic
33#381—22no—
34#380C2212yes—
35#340C2212yes—
36#340C2212yes—
37#336C2211yescharacteristic
38#336C2211yescharacteristic
39C22×Q8—21no—
40C22×Q8—22no—
41C22×Q8—42no—
42C22×Q8—42no—
43C22×Q8Q814yes—
44C22×Q8C2311yescharacteristic
45C22×Q8C2311yescharacteristic
46C23×C4—21no—
47C23×C4C2311yescharacteristic
48C22×C8C2312yes—
49C22×C8C2312yes—
50#125—42no—
51#117—42no—
52#117—42no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4D412yes—
57C2×C4⋊C4C2311yescharacteristic
58C2×C4⋊C4C2311yescharacteristic
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2312yes—
61C2×C22⋊C4—21no—
62C2×C22⋊C4—22no—
63C2×C22⋊C4D412yes—
64C2×C42—21no—
65C2×C42C2311yescharacteristic
66C2×C42C2311yescharacteristic
67#107—42no—
68#106—42no—
69C4⋊C8—42no—
70#103—42no—
71#103—42no—
72#103—42no—
73#103—42no—
74C22⋊C8—24no—
75#95—22no—
76#95—22no—
77#95—22no—
78#95—22no—
79#95—22no—
80#95—22no—
81C24C2×D411yescharacteristic
82C2×Q8—24no—
83C2×Q8—21no—
84C2×Q8—21no—
85C2×Q8—22no—
86C2×Q8—22no—
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—48no—
90C2×Q8—42no—
91C2×Q8—42no—
92C2×Q8—48no—
93C2×Q8C2×Q812yes—
94C2×Q8C2×Q812yes—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—24no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4C2×D412yes—
107C22×C4C2×D412yes—
108C22×C4C2×D411yescharacteristic
109C22×C4C2×Q812yes—
110C22×C4#4112yes—
111C22×C4C2411yescharacteristic
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—42no—
125C4⋊C4—44no—
126C4⋊C4—82no—
127C4⋊C4—82no—
128C22⋊C4—42no—
129C22⋊C4—42no—
130C22⋊C4—82no—
131C42—22no—
132C42—22no—
133C42—42no—
134C23—21no—
135C23—21no—
136C23—22no—
137C23—22no—
138C23C22⋊Q812yes—
139C23C22×D411yescharacteristic
140Q8—24no—
141Q8—24no—
142Q8—48no—
143Q8—42no—
144Q8—42no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—24no—
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—48no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4C22⋊Q812yes—
173C2×C4C22×Q811yescharacteristic
174C2×C4C2×#4111yescharacteristic
175C2×C4H(5,𝔽2)12yes—
176C2×C4H(5,𝔽2)12yes—
177C8—42no—
178C8—42no—
179C22—24no—
180C22—22no—
181C22—42no—
182C22—41no—
183C22—41no—
184C22C2×C22⋊Q811yescharacteristic
185C22#53411yescharacteristic
186C22#53511yescharacteristic
187C22#57712yes—
188C22#57812yes—
189C4—22no—
190C4—42no—
191C4—42no—
192C4—42no—
193C4—42no—
194C4—82no—
195C4—81no—
196C4—82no—
197C4—81no—
198C4#53812yes—
199C4#54312yes—
200C2—42no—
201C2#262611yescharacteristic
202C2#308212yes—
203C2#320112yes—
204C2#322111yescharacteristic
205C2#322311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2096C2semidirect
4. #2095C2semidirect
5. #2068C2semidirect
6. #2066C2semidirect
7. #2026C2semidirect
8. #2026C2semidirect

Maximal subgroups

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

subgroupindex
#26292
#23102
#20962
#20952
#20682
#20662
#20262
#18542
#17982
#17872