Finite Group Databasebeta

← #29934

Subgroups — #29934

Order 256.

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

normal subgroups only

This group has 334 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#2621C211yesmaximal, characteristic
2#2314C211yesmaximal, characteristic
3#2096C211yesmaximal, characteristic, complemented
4#2095C211yesmaximal, characteristic, complemented
5#2074C211yesmaximal, characteristic, complemented
6#2074C211yesmaximal, characteristic, complemented
7#2036C212yesmaximal
8#2035C212yesmaximal
9#1981C212yesmaximal, complemented
10#1949C212yesmaximal, complemented
11#1798C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C22⋊Q8C2211yescharacteristic
14C2×C22⋊Q8C2211yescharacteristic
15C2×#117C2211yescharacteristic
16C2×#107C2212yes—
17C2×#106C2212yes—
18C2×C4⋊C8C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×C22⋊C8C2212yes—
24#398—22no—
25#397—22no—
26#391C2211yescharacteristic
27#391C2211yescharacteristic
28#388—22no—
29#387—22no—
30#384C2211yescharacteristic
31#382C2211yescharacteristic
32#382C2211yescharacteristic
33#380C2212yes—
34C4×C4⋊C4C2212yes—
35#340C2212yes—
36#340C2212yes—
37#336C2211yescharacteristic
38#336C2211yescharacteristic
39C22×Q8C2311yescharacteristic
40C22×Q8C2311yescharacteristic
41C23×C4C2311yescharacteristic
42C22×C8C2312yes—
43C22×C8C2312yes—
44C4⋊Q8—24no—
45C22⋊Q8—42no—
46C22⋊Q8—42no—
47#117—24no—
48C2×C4⋊C4—21no—
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⋊C4D412yes—
61C2×C42—22no—
62C2×C42C2311yescharacteristic
63C2×C42C2311yescharacteristic
64#107—42no—
65#106—42no—
66C4⋊C8—24no—
67#103—42no—
68#103—42no—
69#103—42no—
70#103—42no—
71C22⋊C8—42no—
72#95—22no—
73#95—22no—
74#95—22no—
75#95—22no—
76#95—22no—
77#95—22no—
78C24C2×D411yescharacteristic
79C2×Q8—22no—
80C2×Q8—22no—
81C2×Q8—21no—
82C2×Q8—21no—
83C2×Q8—22no—
84C2×Q8—22no—
85C2×Q8—21no—
86C2×Q8—21no—
87C22×C4—22no—
88C22×C4—21no—
89C22×C4—22no—
90C22×C4—21no—
91C22×C4—22no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4C2×D412yes—
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D412yes—
100C22×C4C2411yescharacteristic
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—22no—
107C4⋊C4—24no—
108C4⋊C4—22no—
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—42no—
115C4⋊C4—42no—
116C4⋊C4—44no—
117C4⋊C4—82no—
118C22⋊C4—24no—
119C22⋊C4—42no—
120C22⋊C4—42no—
121C22⋊C4—82no—
122C42—22no—
123C42—82no—
124C42#4114yes—
125C23—21no—
126C23—21no—
127C23—21no—
128C23—22no—
129C23—22no—
130C23C22×D411yescharacteristic
131Q8—42no—
132Q8—42no—
133Q8—42no—
134Q8—42no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4#12314yes—
162C2×C4C2×#4112yes—
163C2×C4H(5,𝔽2)11yescharacteristic
164C2×C4H(5,𝔽2)12yes—
165C2×C42−1+411yescharacteristic
166C8—42no—
167C8—42no—
168C22—42no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22—42no—
173C22C2×#12311yescharacteristic
174C22#53511yescharacteristic
175C22#53711yescharacteristic
176C22#57712yes—
177C22#57812yes—
178C4—24no—
179C4—42no—
180C4—41no—
181C4—81no—
182C4—81no—
183C4—81no—
184C4—82no—
185C4—81no—
186C4—82no—
187C4#54012yes—
188C4#54212yes—
189C2—81no—
190C2#262411yescharacteristic
191C2#309712yes—
192C2#322111yescharacteristic
193C2#322311yescharacteristic
194C2#325812yes—
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. #2074C2semidirect
6. #2074C2semidirect
9. #1981C2semidirect
10. #1949C2semidirect

Maximal subgroups

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

subgroupindex
#26212
#23142
#20962
#20952
#20742
#20362
#20352
#19812
#19492
#17982