Finite Group Databasebeta

← #32813

Subgroups — #32813

Order 256.

352 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 103 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#2642C211yesmaximal, characteristic, complemented
2#2640C211yesmaximal, characteristic
3#2056C212yesmaximal, complemented
4#2055C212yesmaximal
5#2026C211yesmaximal, characteristic, complemented
6#2025C211yesmaximal, characteristic
7#2024C212yesmaximal, complemented
8#2016C211yesmaximal, characteristic, complemented
9#2015C212yesmaximal
10#2013C211yesmaximal, characteristic
11#1832C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×#125C2211yescharacteristic
17C2×#124C2211yescharacteristic
18C2×C4⋊C8C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#102C2212yes—
23#400—22no—
24#400—22no—
25#397—22no—
26#393—22no—
27#391C2212yes—
28#391C2211yescharacteristic
29#391C2211yescharacteristic
30#391C2212yes—
31#390C2211yescharacteristic
32#390C2211yescharacteristic
33#376C2211yescharacteristic
34#340C2212yes—
35#340C2212yes—
36#336C2211yescharacteristic
37#336C2212yes—
38#336C2211yescharacteristic
39C22×Q8—21no—
40C22×Q8C2311yescharacteristic
41C22×Q8C2312yes—
42C22×D4C2311yescharacteristic
43C2×Q16—42no—
44C2×Q16—42no—
45C2×Q16—42no—
46C2×SD16—42no—
47C2×SD16—42no—
48C2×SD16—42no—
49C22×C8C2312yes—
50C22×C8C2312yes—
51C4⋊Q8—42no—
52C4⋊Q8—42no—
53#125—42no—
54#124—42no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C22⋊C4—21no—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—22no—
65C2×C42C2311yescharacteristic
66C2×C42C2311yescharacteristic
67C2×C42C2311yescharacteristic
68C4⋊C8—42no—
69#103—42no—
70#103—42no—
71#103—42no—
72#102—42no—
73#95—22no—
74#95—22no—
75#95—24no—
76#95—22no—
77#95—22no—
78#95D412yes—
79#95D412yes—
80C24—21no—
81C2×Q8—22no—
82C2×Q8—22no—
83C2×Q8—22no—
84C2×Q8—22no—
85C2×Q8—22no—
86C2×Q8—22no—
87C2×Q8—22no—
88C2×Q8—21no—
89C2×Q8—21no—
90C2×Q8—22no—
91C2×Q8—42no—
92C2×Q8—82no—
93C2×D4—22no—
94C2×D4—22no—
95C2×D4—21no—
96C2×D4—21no—
97C22×C4—21no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4C2×D412yes—
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D412yes—
106C22×C4C2411yescharacteristic
107Q16—48no—
108SD16—48no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—44no—
120C4⋊C4—42no—
121C4⋊C4—44no—
122C22⋊C4—42no—
123C22⋊C4—42no—
124C22⋊C4—82no—
125C22⋊C4—82no—
126C42—21no—
127C42—22no—
128C42—22no—
129C42—21no—
130C23—41no—
131C23—41no—
132C23—41no—
133C23—42no—
134C23—42no—
135C23C22×D411yescharacteristic
136Q8—42no—
137Q8—42no—
138Q8—42no—
139Q8—42no—
140Q8—42no—
141Q8—42no—
142Q8—84no—
143D4—42no—
144D4—42no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4H(5,𝔽2)11yescharacteristic
169C2×C4H(5,𝔽2)12yes—
170C2×C4H(5,𝔽2)12yes—
171C2×C42−1+411yescharacteristic
172C8—42no—
173C8—42no—
174C22—81no—
175C22—81no—
176C22—81no—
177C22—82no—
178C22—82no—
179C22#53511yescharacteristic
180C22#53511yescharacteristic
181C22#53711yescharacteristic
182C22#57712yes—
183C22#57812yes—
184C4—41no—
185C4—41no—
186C4—42no—
187C4—42no—
188C4—81no—
189C4—82no—
190C4—81no—
191C4—81no—
192C4—82no—
193C4H(3,𝔽4)12yes—
194C4#56212yes—
195C2—161no—
196C2#281911yescharacteristic
197C2#322211yescharacteristic
198C2#322311yescharacteristic
199C2#322512yes—
200C2#325812yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2642C2semidirect
3. #2056C2semidirect
5. #2026C2semidirect
7. #2024C2semidirect
8. #2016C2semidirect
11. #1832C2semidirect

Maximal subgroups

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

subgroupindex
#26422
#26402
#20562
#20552
#20262
#20252
#20242
#20162
#20152
#20132
#18322