Finite Group Databasebeta

← #25850

Subgroups — #25850

Order 256.

528 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 153 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×#554C211yesmaximal, characteristic
2C2×#501C211yesmaximal, characteristic, complemented
3C2×#474C211yesmaximal, characteristic
4#2567C211yesmaximal, characteristic, complemented
5#2059C212yesmaximal, complemented
6#2057C212yesmaximal, complemented
7#1989C211yesmaximal, characteristic, complemented
8#1985C211yesmaximal, characteristic, complemented
9#1864C212yesmaximal
10#1862C212yesmaximal
11#1762C211yesmaximal, characteristic
12#554—24no—
13#554—24no—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×#125C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C2×C22⋊Q8—21no—
19C2×C22⋊Q8—21no—
20C2×C22⋊Q8—21no—
21C2×C4×Q8C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C22×C4⋊C4—21no—
26#501—24no—
27#474—24no—
28C2×#107C2212yes—
29C2×#107C2212yes—
30C2×#106C2212yes—
31C2×C4⋊C8C2211yescharacteristic
32C2×#102C2212yes—
33C2×#102C2212yes—
34C2×#97C2211yescharacteristic
35#395—21no—
36#395—21no—
37#391—21no—
38#389C2212yes—
39#389C2212yes—
40#384—21no—
41C4×C4⋊C4C2211yescharacteristic
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45#336C2212yes—
46#336C2212yes—
47C22×Q8—21no—
48C22×Q8—21no—
49C22×Q8D411yescharacteristic
50C22×D4C2311yescharacteristic
51C23×C4—21no—
52C23×C4—21no—
53C23×C4—21no—
54C22×C8C2312yes—
55C22×C8C2312yes—
56C4⋊Q8—22no—
57C4⋊Q8—22no—
58C4⋊Q8—22no—
59C4⋊Q8—22no—
60#125—22no—
61#125—22no—
62#125—22no—
63#125D414yes—
64C22⋊Q8—28no—
65C22⋊Q8—28no—
66C22⋊Q8—28no—
67C4×Q8—22no—
68C4×Q8—22no—
69C4×D4—22no—
70C4×D4—22no—
71C4×D4—22no—
72C4×D4—22no—
73C4×D4—22no—
74C4×D4D414yes—
75C2×C4⋊C4—28no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4C2312yes—
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2312yes—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C42—21no—
96C2×C42C2311yescharacteristic
97C2×C42C2311yescharacteristic
98C2×C42C2311yescharacteristic
99#107—24no—
100#107—24no—
101#106—24no—
102C4⋊C8—22no—
103#102—24no—
104#102Q818yescomplemented
105#97—22no—
106#95—21no—
107#95—22no—
108#95—22no—
109C24—21no—
110C2×Q8—24no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—41no—
114C2×Q8—42no—
115C2×Q8—82no—
116C2×Q8—82no—
117C2×D4—21no—
118C2×D4—22no—
119C2×D4C2×D412yes—
120C2×D4C2×Q814yes—
121C22×C4—24no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—24no—
127C22×C4—24no—
128C22×C4—22no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—82no—
132C22×C4—82no—
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C22×C4C2411yescharacteristic
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—24no—
144C2×C8C2×Q814yes—
145C4⋊C4—24no—
146C4⋊C4—24no—
147C4⋊C4—24no—
148C4⋊C4—22no—
149C4⋊C4—24no—
150C4⋊C4—24no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—42no—
160C4⋊C4—82no—
161C4⋊C4—82no—
162C4⋊C4C2×D412yes—
163C4⋊C4C2×D412yes—
164C4⋊C4C2×D412yes—
165C4⋊C4C2×Q814yes—
166C22⋊C4—24no—
167C22⋊C4—24no—
168C22⋊C4—24no—
169C42—21no—
170C42—21no—
171C42—21no—
172C42—21no—
173C42—42no—
174C42C2×D412yes—
175C42C2×D412yes—
176C23—22no—
177C23—24no—
178C23—41no—
179C23—41no—
180C23—42no—
181C23C22×D411yescharacteristic
182Q8—44no—
183Q8—84no—
184Q8—84no—
185D4—22no—
186D4—24no—
187D4C4⋊Q814yes—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—24no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—48no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—84no—
215C2×C4—84no—
216C2×C4#12012yes—
217C2×C4#12012yes—
218C2×C4#13712yes—
219C2×C4C22×D411yescharacteristic
220C2×C4C22×D411yescharacteristic
221C2×C4C22×Q812yes—
222C2×C42−1+412yes—
223C8—24no—
224C8—42no—
225C22—24no—
226C22—44no—
227C22—44no—
228C22—42no—
229C22—42no—
230C22C2×#12011yescharacteristic
231C22D4×Q812yes—
232C22C2×#13711yescharacteristic
233C22#57511yescharacteristic
234C22#57612yes—
235C4—22no—
236C4—22no—
237C4—22no—
238C4—44no—
239C4—41no—
240C4—42no—
241C4—41no—
242C4—41no—
243C4—42no—
244C4—42no—
245C4—82no—
246C4C2×C4⋊Q811yescharacteristic
247C4#53611yescharacteristic
248C4D4×Q812yes—
249C2—44no—
250C2#243411yescharacteristic
251C2#300711yescharacteristic
252C2#300811yescharacteristic
253C2#338512yes—
254C2#338912yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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
2. C2×#501C2semidirect
4. #2567C2semidirect
5. #2059C2semidirect
6. #2057C2semidirect
7. #1989C2semidirect
8. #1985C2semidirect
104. #102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#5012
C2×#4742
#25672
#20592
#20572
#19892
#19852
#18642
#18622
#17622