Finite Group Databasebeta

← #29931

Subgroups — #29931

Order 256.

355 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 111 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#2620C211yesmaximal, characteristic
2#2314C211yesmaximal, characteristic, complemented
3#2095C211yesmaximal, characteristic, complemented
4#2093C211yesmaximal, characteristic, complemented
5#2074C211yesmaximal, characteristic, complemented
6#2073C211yesmaximal, characteristic, complemented
7#2036C212yesmaximal, complemented
8#2033C212yesmaximal
9#1983C212yesmaximal, complemented
10#1947C212yesmaximal, complemented
11#1798C211yesmaximal, characteristic, complemented
12C2×#124C2211yescharacteristic
13C2×C22⋊Q8C2211yescharacteristic
14C2×#121C2211yescharacteristic
15C2×#117C2211yescharacteristic
16C2×#107C2212yes—
17C2×#107C2212yes—
18C2×C4⋊C8C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#102C2212yes—
22C2×#102C2212yes—
23C2×C22⋊C8C2212yes—
24#396—22no—
25#394—22no—
26#391C2211yescharacteristic
27#390C2211yescharacteristic
28#388—22no—
29#387—22no—
30#384C2211yescharacteristic
31#384C2211yescharacteristic
32#382C2211yescharacteristic
33#380C2212yes—
34C4×C4⋊C4C2212yes—
35#340C2212yes—
36#340C2212yes—
37#336C2211yescharacteristic
38#336C2211yescharacteristic
39C22×Q8C2311yescharacteristic
40C22×D4—21no—
41C22×D4C2311yescharacteristic
42C23×C4C2311yescharacteristic
43C22×C8C2312yes—
44C22×C8C2312yes—
45#124—24no—
46C22⋊Q8—42no—
47#121—42no—
48#117—24no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4D412yes—
53C2×C4⋊C4C2311yescharacteristic
54C2×C4⋊C4C2311yescharacteristic
55C2×C4⋊C4C2312yes—
56C2×C4⋊C4C2312yes—
57C2×C22⋊C4—21no—
58C2×C22⋊C4—21no—
59C2×C22⋊C4—22no—
60C2×C22⋊C4—22no—
61C2×C22⋊C4—22no—
62C2×C22⋊C4D412yes—
63C2×C42—22no—
64C2×C42C2311yescharacteristic
65C2×C42C2311yescharacteristic
66#107—42no—
67#107—42no—
68C4⋊C8—24no—
69#103—42no—
70#103—42no—
71#102—42no—
72#102—42no—
73C22⋊C8—42no—
74#95—22no—
75#95—22no—
76#95—22no—
77#95—22no—
78C24—21no—
79C24C2×D411yescharacteristic
80C2×Q8—22no—
81C2×Q8—21no—
82C2×Q8—22no—
83C2×Q8—21no—
84C2×D4—22no—
85C2×D4—21no—
86C2×D4—22no—
87C2×D4—21no—
88C2×D4—42no—
89C2×D4—82no—
90C22×C4—22no—
91C22×C4—21no—
92C22×C4—22no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4C2×D412yes—
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D412yes—
102C22×C4C2411yescharacteristic
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8—22no—
108C2×C8—22no—
109C4⋊C4—24no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—42no—
117C4⋊C4—82no—
118C4⋊C4—82no—
119C22⋊C4—24no—
120C22⋊C4—42no—
121C22⋊C4—42no—
122C22⋊C4—44no—
123C22⋊C4—82no—
124C22⋊C4—82no—
125C42—22no—
126C42—82no—
127C42#4114yes—
128C23—21no—
129C23—21no—
130C23—21no—
131C23—22no—
132C23—22no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—42no—
137C23—42no—
138C23C22×D411yescharacteristic
139Q8—42no—
140Q8—42no—
141D4—42no—
142D4—42no—
143D4—84no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4#12314yes—
168C2×C4C2×#4112yes—
169C2×C4H(5,𝔽2)11yescharacteristic
170C2×C4H(5,𝔽2)12yes—
171C2×C42−1+411yescharacteristic
172C8—42no—
173C8—42no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—41no—
178C22—42no—
179C22—82no—
180C22—81no—
181C22—81no—
182C22—81no—
183C22—82no—
184C22C2×#12311yescharacteristic
185C22#53511yescharacteristic
186C22#53711yescharacteristic
187C22#57612yes—
188C22#57812yes—
189C4—24no—
190C4—42no—
191C4—41no—
192C4—81no—
193C4—81no—
194C4—82no—
195C4—81no—
196C4—82no—
197C4#54012yes—
198C4#54212yes—
199C2—81no—
200C2—161no—
201C2#262411yescharacteristic
202C2#309812yes—
203C2#322011yescharacteristic
204C2#322311yescharacteristic
205C2#326212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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
2. #2314C2semidirect
3. #2095C2semidirect
4. #2093C2semidirect
5. #2074C2semidirect
6. #2073C2semidirect
7. #2036C2semidirect
9. #1983C2semidirect
10. #1947C2semidirect
11. #1798C2semidirect

Maximal subgroups

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

subgroupindex
#26202
#23142
#20952
#20932
#20742
#20732
#20362
#20332
#19832
#19472
#17982