Finite Group Databasebeta

← #25276

Subgroups — #25276

Order 256.

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

normal subgroups only

This group has 405 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
3#2546C211yesmaximal, characteristic
4#2365C211yesmaximal, characteristic
5#2066C212yesmaximal
6#2043C212yesmaximal
7#1981C211yesmaximal, characteristic
8#1981C212yesmaximal
9#1939C211yesmaximal, characteristic
10#1927C212yesmaximal
11C4×#103C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4×Q8—21no—
16C2×C4×Q8C2211yescharacteristic
17#487—24no—
18#475—24no—
19C2×#106C2212yes—
20C2×C4⋊C8C2211yescharacteristic
21C2×C4⋊C8C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2212yes—
29C2×C4×C8C2211yescharacteristic
30#398—22no—
31#391—21no—
32#391C2212yes—
33#389—21no—
34#389—21no—
35#389C2212yes—
36#387—22no—
37C4×C4⋊C4C2212yes—
38C4×C4⋊C4C2211yescharacteristic
39C4×C4⋊C4C2211yescharacteristic
40#340C2211yescharacteristic
41#340C2211yescharacteristic
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8C2312yes—
45C22×Q8C2311yescharacteristic
46C22×C8C2311yescharacteristic
47C22×C8C2312yes—
48C22×C8C2311yescharacteristic
49C4⋊Q8—24no—
50C4⋊Q8—22no—
51C4⋊Q8—22no—
52C4⋊Q8—22no—
53C4⋊Q8—42no—
54C4⋊Q8D414yes—
55C4×Q8—22no—
56C4×Q8—22no—
57C4×Q8—44no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4D411yescharacteristic
66C2×C4⋊C4D411yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2311yescharacteristic
71C2×C42—22no—
72C2×C42—21no—
73C2×C42—21no—
74C2×C42C2312yes—
75C2×C42C2311yescharacteristic
76#106—24no—
77C4⋊C8—22no—
78C4⋊C8—42no—
79#103—22no—
80#103—24no—
81#103—24no—
82#103—24no—
83#103—22no—
84#103—24no—
85#103—42no—
86C4×C8—22no—
87#95—21no—
88#95—22no—
89#95—22no—
90#95—22no—
91#95—22no—
92#95D412yes—
93C2×Q8—24no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—22no—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8—21no—
104C2×Q8—22no—
105C2×Q8—42no—
106C2×Q8—42no—
107C2×Q8#4112yes—
108C22×C4—21no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4C2×D412yes—
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D412yes—
117C22×C4C2×D411yescharacteristic
118C22×C4C2411yescharacteristic
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—22no—
122C2×C8—21no—
123C2×C8—22no—
124C2×C8—21no—
125C2×C8—22no—
126C2×C8—21no—
127C2×C8—21no—
128C4⋊C4—24no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—44no—
137C4⋊C4—44no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—44no—
142C4⋊C4SD1614yes—
143C4⋊C4C2×D414yes—
144C4⋊C4#4112yes—
145C42—22no—
146C42—22no—
147C42—42no—
148C42—44no—
149C42—42no—
150C42C2×D412yes—
151C42#4112yes—
152C23C22×D411yescharacteristic
153Q8—22no—
154Q8—22no—
155Q8—41no—
156Q8—41no—
157Q8—48no—
158Q8—42no—
159Q8—42no—
160Q8—42no—
161Q8—42no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—24no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4#12112yes—
188C2×C4#12112yes—
189C2×C4C2×SD1614yes—
190C2×C4C2×SD1612yes—
191C2×C4#13712yes—
192C2×C4C22×D411yescharacteristic
193C2×C4C2×#4111yescharacteristic
194C2×C4C2×#4111yescharacteristic
195C2×C4C2×#4111yescharacteristic
196C2×C4H(5,𝔽2)11yescharacteristic
197C2×C42−1+411yescharacteristic
198C8—41no—
199C8—41no—
200C8—42no—
201C22C2×#12111yescharacteristic
202C22#54611yescharacteristic
203C22#54811yescharacteristic
204C22C22×SD1611yescharacteristic
205C22C2×#13711yescharacteristic
206C22#57811yescharacteristic
207C22#57811yescharacteristic
208C4—24no—
209C4—22no—
210C4—42no—
211C4—42no—
212C4—42no—
213C4—44no—
214C4—82no—
215C4—81no—
216C4—81no—
217C4#46114yes—
218C4#52911yescharacteristic
219C4#54211yescharacteristic
220C4#54811yescharacteristic
221C4#55011yescharacteristic
222C2#242611yescharacteristic
223C2C2×#46111yescharacteristic
224C2#304411yescharacteristic
225C2#329811yescharacteristic
226C2#332811yescharacteristic
227C2#335911yescharacteristic
228C2#337411yescharacteristic
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
#25462
#23652
#20662
#20432
#19812
#19392
#19272
C4×#1032