Finite Group Databasebeta

← #29152

Subgroups — #29152

Order 256.

399 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 399 conjugacy classes of subgroups — too many to draw. Only its 127 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#2660C211yesmaximal, characteristic, complemented
2#2328C211yesmaximal, characteristic
3#2025C212yesmaximal, complemented
4#2023C212yesmaximal, complemented
5#1961C211yesmaximal, characteristic
6#1957C211yesmaximal, characteristic
7#1836C211yesmaximal, characteristic, complemented
8#1835C211yesmaximal, characteristic, complemented
9#1804C212yesmaximal, complemented
10#1802C212yesmaximal, complemented
11#1768C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×#127C2211yescharacteristic
14C2×#125C2211yescharacteristic
15C2×C4×D4C2212yes—
16C2×C4⋊C8C2212yes—
17C2×#102C2212yes—
18C2×#102C2212yes—
19C2×#102C2212yes—
20C2×#102C2212yes—
21C2×#97C2211yescharacteristic
22C2×C4×C8C2211yescharacteristic
23#398—22no—
24#395—22no—
25#390C2212yes—
26#386—22no—
27#386—22no—
28#384C2211yescharacteristic
29#384C2211yescharacteristic
30#382C2211yescharacteristic
31#382C2211yescharacteristic
32#382C2211yescharacteristic
33#376C2211yescharacteristic
34#340C2212yes—
35#340C2212yes—
36#340C2212yes—
37#340C2212yes—
38#336C2212yes—
39C22×Q8—21no—
40C22×D4—22no—
41C22×D4C2312yes—
42C23×C4—22no—
43C22×C8C2312yes—
44C22×C8C2312yes—
45C4⋊Q8—42no—
46#127—22no—
47#127Q814yescomplemented
48#125—42no—
49C4×D4—24no—
50C4×D4—24no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4C2312yes—
57C2×C4⋊C4C2311yescharacteristic
58C2×C4⋊C4C2311yescharacteristic
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C22⋊C4—22no—
62C2×C22⋊C4—22no—
63C2×C22⋊C4—22no—
64C2×C42C2311yescharacteristic
65C2×C42C2311yescharacteristic
66C2×C42C2311yescharacteristic
67C4⋊C8—24no—
68#102—42no—
69#102—42no—
70#102—42no—
71#102—42no—
72#97—22no—
73C4×C8—22no—
74#95—22no—
75#95—22no—
76#95—22no—
77#95D412yes—
78#95D412yes—
79C24—22no—
80C2×Q8—42no—
81C2×Q8—82no—
82C2×D4—28no—
83C2×D4—22no—
84C2×D4—22no—
85C2×D4—22no—
86C2×D4—22no—
87C2×D4—22no—
88C2×D4—82no—
89C2×D4—82no—
90C2×D4C2×Q814yes—
91C22×C4—21no—
92C22×C4—22no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—44no—
97C22×C4—82no—
98C22×C4—82no—
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D412yes—
102C22×C4C2×D412yes—
103C22×C4C2411yescharacteristic
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8—22no—
108C2×C8—22no—
109C2×C8—22no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—42no—
117C4⋊C4—42no—
118C4⋊C4—42no—
119C4⋊C4—82no—
120C4⋊C4—82no—
121C22⋊C4—44no—
122C22⋊C4—82no—
123C22⋊C4—82no—
124C42—22no—
125C42C2×Q812yes—
126C42#4112yes—
127C42#4114yes—
128C23—24no—
129C23—42no—
130C23—42no—
131C23—42no—
132C23—42no—
133C23—42no—
134C23—42no—
135C23C22×D411yescharacteristic
136Q8—84no—
137D4—24no—
138D4—24no—
139D4—42no—
140D4—42no—
141D4—88no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—42no—
151C2×C4—41no—
152C2×C4—41no—
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—88no—
162C2×C4C22⋊Q814yes—
163C2×C4#12414yes—
164C2×C4C22×Q811yescharacteristic
165C2×C4C2×#4112yes—
166C2×C4C2×#4111yescharacteristic
167C2×C4H(5,𝔽2)12yes—
168C8—42no—
169C8—42no—
170C22—44no—
171C22—44no—
172C22—44no—
173C22—82no—
174C22—82no—
175C22—82no—
176C22—82no—
177C22C2×C22⋊Q811yescharacteristic
178C22C2×#12411yescharacteristic
179C22#53511yescharacteristic
180C22#57612yes—
181C22#57712yes—
182C4—24no—
183C4—24no—
184C4—41no—
185C4—41no—
186C4—82no—
187C4—81no—
188C4—81no—
189C4—81no—
190C4—81no—
191C4#55412yes—
192C4#55512yes—
193C2—84no—
194C2#256311yescharacteristic
195C2#308112yes—
196C2#313612yes—
197C2#322011yescharacteristic
198C2#322111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
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. #2660C2semidirect
3. #2025C2semidirect
4. #2023C2semidirect
7. #1836C2semidirect
8. #1835C2semidirect
9. #1804C2semidirect
10. #1802C2semidirect
11. #1768C2semidirect
47. #127Q8semidirect

Maximal subgroups

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

subgroupindex
#26602
#23282
#20252
#20232
#19612
#19572
#18362
#18352
#18042
#18022
#17682