Finite Group Databasebeta

← #28417

Subgroups — #28417

Order 256.

439 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 439 conjugacy classes of subgroups — too many to draw. Only its 139 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×#487C211yesmaximal, characteristic
2C2×#475C211yesmaximal, characteristic
3C2×#462C211yesmaximal, characteristic
4C2×C4×Q16C211yesmaximal, characteristic
5#2568C211yesmaximal, characteristic
6#2546C211yesmaximal, characteristic
7#2097C212yesmaximal
8#2016C212yesmaximal
9#1971C212yesmaximal
10#1960C212yesmaximal
11#1770C211yesmaximal, characteristic
12C22×Q16C2211yescharacteristic
13C22×Q16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2212yes—
16C2×C4⋊Q8C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18C2×C4×Q8C2211yescharacteristic
19#487—24no—
20#475—24no—
21#462—24no—
22C4×Q16—24no—
23C2×#107C2211yescharacteristic
24C2×#106C2212yes—
25C2×C4⋊C8C2211yescharacteristic
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×#103C2212yes—
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#97C2212yes—
33C2×C4×C8C2211yescharacteristic
34#400—22no—
35#398—22no—
36#391—22no—
37#391C2212yes—
38#391C2212yes—
39#389—22no—
40#382C2212yes—
41C4×C4⋊C4C2211yescharacteristic
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8C2312yes—
45C22×Q8C2311yescharacteristic
46C22×Q8C2311yescharacteristic
47C2×Q16—22no—
48C2×Q16—22no—
49C2×Q16—24no—
50C2×Q16—22no—
51C2×Q16—24no—
52C2×Q16—42no—
53C22×C8C2311yescharacteristic
54C22×C8C2312yes—
55C22×C8C2311yescharacteristic
56C4⋊Q8—24no—
57C4⋊Q8—22no—
58C4⋊Q8—22no—
59C4⋊Q8—22no—
60C4⋊Q8—22no—
61C4⋊Q8—42no—
62C4×Q8—22no—
63C4×Q8—22no—
64C4×Q8—22no—
65C4×Q8—22no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—24no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4D412yes—
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2312yes—
76C2×C42—21no—
77C2×C42—21no—
78C2×C42C2312yes—
79C2×C42C2311yescharacteristic
80#107—22no—
81#106—24no—
82C4⋊C8D414yes—
83#103—22no—
84#103—22no—
85#103—24no—
86#103—24no—
87#103—24no—
88#103—24no—
89#97—42no—
90C4×C8D414yes—
91#95—24no—
92#95—24no—
93#95—24no—
94#95D412yes—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8—21no—
100C2×Q8—21no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8—21no—
104C2×Q8—21no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—21no—
108C2×Q8—22no—
109C2×Q8—21no—
110C2×Q8—22no—
111C2×Q8—42no—
112C2×Q8—44no—
113C2×Q8#4112yes—
114C2×Q8#4112yes—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4C2×D414yes—
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2411yescharacteristic
125Q16—28no—
126Q16—48no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—22no—
130C2×C8—21no—
131C2×C8—22no—
132C2×C8—21no—
133C2×C8C2×D412yes—
134C2×C8C2×D412yes—
135C2×C8C2×D414yes—
136C4⋊C4—24no—
137C4⋊C4—21no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—48no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C42—22no—
150C42—22no—
151C42—42no—
152C42—42no—
153C42C2×D412yes—
154C42C2×D412yes—
155C23C22×D411yescharacteristic
156Q8—22no—
157Q8—22no—
158Q8—22no—
159Q8—22no—
160Q8—41no—
161Q8—41no—
162Q8—41no—
163Q8—41no—
164Q8—42no—
165Q8—42no—
166Q8—42no—
167Q8—42no—
168Q8—84no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—24no—
174C2×C4—24no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—44no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4#12712yes—
189C2×C4#12712yes—
190C2×C4#13714yes—
191C2×C4C22×D411yescharacteristic
192C2×C4C22×D411yescharacteristic
193C2×C4C2×#4111yescharacteristic
194C2×C4C2×#4111yescharacteristic
195C2×C4H(5,𝔽2)11yescharacteristic
196C2×C42−1+411yescharacteristic
197C8—22no—
198C8—22no—
199C8—24no—
200C22C2×#12711yescharacteristic
201C22#54611yescharacteristic
202C22#54811yescharacteristic
203C22C2×#13712yes—
204C22#57812yes—
205C4—22no—
206C4—44no—
207C4—42no—
208C4—42no—
209C4—82no—
210C4—81no—
211C4—81no—
212C4—82no—
213C4#49714yes—
214C4#52511yescharacteristic
215C4#53711yescharacteristic
216C4#54611yescharacteristic
217C4#54811yescharacteristic
218C2#254011yescharacteristic
219C2C2×#49711yescharacteristic
220C2#315611yescharacteristic
221C2#332812yes—
222C2#337412yes—
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ 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
C2×#4872
C2×#4752
C2×#4622
C2×C4×Q162
#25682
#25462
#20972
#20162
#19712
#19602
#17702