Finite Group Databasebeta

← #25472

Subgroups — #25472

Order 256.

385 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 385 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C2×#475C211yesmaximal, characteristic
2C2×C8×Q8C211yesmaximal, characteristic
3#2582C211yesmaximal, characteristic
4#2546C211yesmaximal, characteristic
5#2088C212yesmaximal
6#2066C212yesmaximal
7#1990C211yesmaximal, characteristic
8#1948C211yesmaximal, characteristic
9#1941C212yesmaximal
10#1933C212yesmaximal
11C4×#103C211yesmaximal, characteristic
12C2×C4⋊Q8—21no—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×C4×Q8C2211yescharacteristic
16#475—24no—
17C8×Q8—24no—
18C2×#106C2212yes—
19C2×#106C2212yes—
20C2×C4⋊C8C2212yes—
21C2×C4⋊C8C2211yescharacteristic
22C2×#103C2212yes—
23C2×#103C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×C4×C8C2212yes—
26C2×C4×C8C2211yescharacteristic
27#398—22no—
28#398—22no—
29#391—21no—
30#391—21no—
31#391C2212yes—
32#389—21no—
33#389C2212yes—
34#389C2212yes—
35#389C2212yes—
36#384C2211yescharacteristic
37C4×C4⋊C4C2211yescharacteristic
38#340C2212yes—
39#340C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#340C2211yescharacteristic
43C22×Q8—21no—
44C22×Q8D411yescharacteristic
45C22×Q8C2311yescharacteristic
46C22×C8C2311yescharacteristic
47C22×C8C2312yes—
48C22×C8C2311yescharacteristic
49C4⋊Q8—22no—
50C4⋊Q8—22no—
51C4⋊Q8—44no—
52C4×Q8—22no—
53C4×Q8—22no—
54C4×Q8—22no—
55C4×Q8—22no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4D411yescharacteristic
64C2×C4⋊C4D412yes—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C42—21no—
71C2×C42—21no—
72C2×C42C2312yes—
73C2×C42C2311yescharacteristic
74#106—24no—
75#106—24no—
76C4⋊C8—24no—
77C4⋊C8D414yes—
78#103—24no—
79#103—22no—
80#103Q814yescomplemented
81C4×C8—24no—
82C4×C8—22no—
83#95—21no—
84#95—22no—
85#95—22no—
86#95—22no—
87#95—22no—
88#95—22no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—21no—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—42no—
97C2×Q8—42no—
98C2×Q8—42no—
99C2×Q8SD1614yes—
100C2×Q8C2×Q812yes—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4C2×D412yes—
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D412yes—
110C22×C4C2×D411yescharacteristic
111C22×C4C2411yescharacteristic
112C2×C8—21no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8C2×D414yes—
120C2×C8C2×Q812yes—
121C4⋊C4—24no—
122C4⋊C4—22no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—22no—
126C4⋊C4—21no—
127C4⋊C4—22no—
128C4⋊C4—21no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4—44no—
134C4⋊C4—44no—
135C4⋊C4—42no—
136C4⋊C4—81no—
137C4⋊C4—82no—
138C4⋊C4C2×Q812yes—
139C4⋊C4#4112yes—
140C42—22no—
141C42—22no—
142C42—42no—
143C42—42no—
144C42C2×D412yes—
145C42#4112yes—
146C23C22×D411yescharacteristic
147Q8—28no—
148Q8—22no—
149Q8—22no—
150Q8—41no—
151Q8—41no—
152Q8—84no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4#12112yes—
179C2×C4#12112yes—
180C2×C4C2×SD1614yes—
181C2×C4C2×SD1612yes—
182C2×C4#13512yes—
183C2×C4C22×D411yescharacteristic
184C2×C4C22×Q811yescharacteristic
185C2×C4C2×#4111yescharacteristic
186C2×C4C2×#4111yescharacteristic
187C2×C42−1+411yescharacteristic
188C2×C42−1+411yescharacteristic
189C8—24no—
190C8—22no—
191C8—41no—
192C22C2×#12111yescharacteristic
193C22#54811yescharacteristic
194C22D4×Q811yescharacteristic
195C22C22×SD1611yescharacteristic
196C22C2×#13511yescharacteristic
197C22#57711yescharacteristic
198C22#57811yescharacteristic
199C4—24no—
200C4—22no—
201C4—42no—
202C4—42no—
203C4—42no—
204C4—42no—
205C4—82no—
206C4—82no—
207C4—81no—
208C4#46514yes—
209C4#53311yescharacteristic
210C4#54111yescharacteristic
211C4#54711yescharacteristic
212C4D4×Q811yescharacteristic
213C2#242911yescharacteristic
214C2C2×#46511yescharacteristic
215C2#306311yescharacteristic
216C2#335911yescharacteristic
217C2#336511yescharacteristic
218C2Q8×SD1611yescharacteristic
219C2#338311yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
80. #103Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#4752
C2×C8×Q82
#25822
#25462
#20882
#20662
#19902
#19482
#19412
#19332
C4×#1032