Finite Group Databasebeta

← #24648

Subgroups — #24648

Order 256.

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

normal subgroups only

This group has 408 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
1C2×#462C211yesmaximal, characteristic
2C2×#441C211yesmaximal, characteristic
3#2546C211yesmaximal, characteristic
4#2356C211yesmaximal, characteristic
5#2056C212yesmaximal
6#2016C212yesmaximal
7#1981C212yesmaximal
8#1960C211yesmaximal, characteristic
9#1940C211yesmaximal, characteristic
10#1927C212yesmaximal
11#1805C211yesmaximal, characteristic
12C22×Q16C2211yescharacteristic
13C22×Q16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2212yes—
16C2×C4×Q8C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18#462—24no—
19#441—24no—
20C2×#106C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2212yes—
23C2×#103C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×#97C2211yescharacteristic
30#398—21no—
31#398—21no—
32#391C2212yes—
33#391C2212yes—
34#389—22no—
35#387—22no—
36#383—21no—
37#382C2211yescharacteristic
38C4×C4⋊C4C2212yes—
39C4×C4⋊C4C2211yescharacteristic
40#376—21no—
41#340C2212yes—
42#336C2212yes—
43C22×Q8C2312yes—
44C22×Q8C2311yescharacteristic
45C22×Q8C2311yescharacteristic
46C2×Q16—22no—
47C2×Q16—22no—
48C2×Q16—24no—
49C2×Q16—22no—
50C2×Q16—42no—
51C2×Q16—42no—
52C22×C8C2312yes—
53C22×C8C2311yescharacteristic
54C22×C8C2311yescharacteristic
55C4⋊Q8—24no—
56C4⋊Q8—22no—
57C4⋊Q8—22no—
58C4⋊Q8—42no—
59C4×Q8—22no—
60C4×Q8—22no—
61C4×Q8—22no—
62C4×Q8D414yes—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4D412yes—
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2312yes—
72C2×C4⋊C4C2311yescharacteristic
73C2×C42—21no—
74C2×C42—21no—
75C2×C42—22no—
76C2×C42C2312yes—
77C2×C42C2311yescharacteristic
78#106—22no—
79C4⋊C8—22no—
80C4⋊C8—42no—
81#103—22no—
82#103—24no—
83#103—22no—
84#103—24no—
85#103—42no—
86#103—42no—
87#97—22no—
88#95—21no—
89#95—22no—
90#95—22no—
91#95—21no—
92#95—22no—
93#95—22no—
94#95—22no—
95#95D411yescharacteristic
96#95D411yescharacteristic
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8—21no—
100C2×Q8—22no—
101C2×Q8—21no—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×Q8—21no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—21no—
109C2×Q8—21no—
110C2×Q8—21no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8C2×D412yes—
114C2×Q8#4112yes—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4C2×D412yes—
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D412yes—
125C22×C4C2411yescharacteristic
126Q16—28no—
127Q16—48no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—22no—
132C2×C8—21no—
133C2×C8—22no—
134C2×C8—21no—
135C2×C8—22no—
136C2×C8#4112yes—
137C4⋊C4—24no—
138C4⋊C4—24no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—22no—
142C4⋊C4—21no—
143C4⋊C4—22no—
144C4⋊C4—44no—
145C4⋊C4—44no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4C2×D412yes—
150C42—24no—
151C42—22no—
152C42—22no—
153C42—42no—
154C42—44no—
155C42C2×D412yes—
156C42#4112yes—
157C23C22×D411yescharacteristic
158Q8—22no—
159Q8—22no—
160Q8—22no—
161Q8—22no—
162Q8—41no—
163Q8—41no—
164Q8—41no—
165Q8—41no—
166Q8—42no—
167Q8—42no—
168Q8—42no—
169Q8—42no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4#12112yes—
194C2×C4#12112yes—
195C2×C4#13714yes—
196C2×C4C22×D411yescharacteristic
197C2×C4C2×#4111yescharacteristic
198C2×C4C2×#4111yescharacteristic
199C2×C4C2×#4111yescharacteristic
200C2×C4H(5,𝔽2)11yescharacteristic
201C2×C4H(5,𝔽2)11yescharacteristic
202C8—22no—
203C8—41no—
204C8—42no—
205C22C2×#12111yescharacteristic
206C22#54611yescharacteristic
207C22#55011yescharacteristic
208C22C2×#13712yes—
209C22#57812yes—
210C4—22no—
211C4—22no—
212C4—24no—
213C4—42no—
214C4—42no—
215C4—42no—
216C4—42no—
217C4—82no—
218C4—82no—
219C4—81no—
220C4#52511yescharacteristic
221C4#53511yescharacteristic
222C4#55311yescharacteristic
223C4#55511yescharacteristic
224C2#241411yescharacteristic
225C2#303911yescharacteristic
226C2#304411yescharacteristic
227C2#331212yes—
228C2#339812yes—
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×#4622
C2×#4412
#25462
#23562
#20562
#20162
#19812
#19602
#19402
#19272
#18052