Finite Group Databasebeta

← #29371

Subgroups — #29371

Order 256.

373 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 373 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C4×C4⋊Q8C211yesmaximal, characteristic
2#2295C211yesmaximal, characteristic
3#2099C211yesmaximal, characteristic
4#2096C211yesmaximal, characteristic
5#2043C211yesmaximal, characteristic
6#2016C211yesmaximal, characteristic
7#1927C212yesmaximal
8#1878C212yesmaximal
9#1878C212yesmaximal
10#1865C212yesmaximal
11#1839C211yesmaximal, characteristic
12C22×Q16C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×C4×Q8C2212yes—
16C2×#106C2212yes—
17C2×#103C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21#391C2212yes—
22#391C2211yescharacteristic
23#391C2211yescharacteristic
24#389—21no—
25#389—21no—
26#389C2211yescharacteristic
27#389C2211yescharacteristic
28#387—21no—
29#387—21no—
30#384C2211yescharacteristic
31C4×C4⋊C4—21no—
32C4×C4⋊C4—21no—
33C4×C4⋊C4—21no—
34C4×C4⋊C4C2211yescharacteristic
35#376—21no—
36C43C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#336C2211yescharacteristic
40#336C2211yescharacteristic
41#336C2212yes—
42#336C2212yes—
43C22×Q8C2312yes—
44C22×Q8C2311yescharacteristic
45C2×Q16—24no—
46C2×Q16—24no—
47C2×Q16—42no—
48C22×C8C2312yes—
49C22×C8C2312yes—
50C4⋊Q8—24no—
51C4×Q8—22no—
52C4×Q8—22no—
53C4×Q8—42no—
54C4×Q8—42no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C42—21no—
68C2×C42—22no—
69C2×C42—21no—
70C2×C42D411yescharacteristic
71C2×C42D411yescharacteristic
72C2×C42D411yescharacteristic
73C2×C42D411yescharacteristic
74C2×C42C2311yescharacteristic
75C2×C42C2311yescharacteristic
76C2×C42C2311yescharacteristic
77#106—42no—
78#103—24no—
79#103—24no—
80#103—42no—
81#103—42no—
82#95—21no—
83#95—21no—
84#95—21no—
85#95—21no—
86#95—21no—
87#95—21no—
88#95—22no—
89#95—22no—
90#95—21no—
91C2×Q8—21no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8#4114yes—
100C2×Q8#4112yes—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2411yescharacteristic
114Q16—48no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C4⋊C4—24no—
122C4⋊C4—24no—
123C4⋊C4—24no—
124C4⋊C4—21no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—44no—
130C4⋊C4—42no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4#4112yes—
135C42—24no—
136C42—22no—
137C42—22no—
138C42—22no—
139C42—22no—
140C42—21no—
141C42—21no—
142C42—22no—
143C42—22no—
144C42—44no—
145C42—42no—
146C23C22×D411yescharacteristic
147Q8—24no—
148Q8—24no—
149Q8—24no—
150Q8—42no—
151Q8—42no—
152Q8—42no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4#13512yes—
186C2×C4#13712yes—
187C2×C4#13712yes—
188C2×C4#13712yes—
189C2×C4C2×#4111yescharacteristic
190C2×C4C2×#4112yes—
191C2×C4C2×#4111yescharacteristic
192C2×C4H(5,𝔽2)12yes—
193C8—42no—
194C8—42no—
195C22#52511yescharacteristic
196C22#52511yescharacteristic
197C22#53511yescharacteristic
198C22C2×#13511yescharacteristic
199C22C2×#13711yescharacteristic
200C22C2×#13711yescharacteristic
201C22C2×#13711yescharacteristic
202C4—22no—
203C4—22no—
204C4—22no—
205C4—22no—
206C4—41no—
207C4—42no—
208C4—42no—
209C4—44no—
210C4—41no—
211C4—81no—
212C4—82no—
213C4—81no—
214C4#55112yes—
215C4#55312yes—
216C2#259811yescharacteristic
217C2#310711yescharacteristic
218C2#310811yescharacteristic
219C2#311311yescharacteristic
220C2#311311yescharacteristic
221C2#321011yescharacteristic
222C2#321811yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C4×C4⋊Q82
#22952
#20992
#20962
#20432
#20162
#19272
#18782
#18652
#18392