Finite Group Databasebeta

← #25478

Subgroups — #25478

Order 256.

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

normal subgroups only

This group has 409 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×#474C211yesmaximal, characteristic
2C2×#446C211yesmaximal, characteristic, complemented
3#2546C211yesmaximal, characteristic, complemented
4#2538C211yesmaximal, characteristic
5#2087C212yesmaximal, complemented
6#2065C212yesmaximal
7#1990C211yesmaximal, characteristic, complemented
8#1948C211yesmaximal, characteristic, complemented
9#1944C212yesmaximal, complemented
10#1936C212yesmaximal
11C4×#102C211yesmaximal, characteristic
12C2×C4⋊Q8—21no—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C22⋊Q8—21no—
15C2×C4×Q8C2211yescharacteristic
16C2×C4×D4C2211yescharacteristic
17#474—24no—
18#446—24no—
19C2×#107C2212yes—
20C2×#106C2212yes—
21C2×C4⋊C8C2212yes—
22C2×C4⋊C8C2211yescharacteristic
23C2×#102C2211yescharacteristic
24C2×#102C2212yes—
25C2×#102C2211yescharacteristic
26C2×#97C2212yes—
27C2×C4×C8C2211yescharacteristic
28#398—22no—
29#396—22no—
30#391—21no—
31#390C2212yes—
32#389C2212yes—
33#389C2212yes—
34#389C2212yes—
35#384C2211yescharacteristic
36#380—21no—
37C4×C4⋊C4C2211yescharacteristic
38#340C2212yes—
39#340C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#340C2211yescharacteristic
43C22×Q8—21no—
44C22×Q8D411yescharacteristic
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C22×C8C2311yescharacteristic
48C22×C8C2312yes—
49C22×C8C2311yescharacteristic
50C4⋊Q8—22no—
51C4⋊Q8—22no—
52C4⋊Q8—44no—
53C22⋊Q8—44no—
54C4×Q8—22no—
55C4×Q8—22no—
56C4×D4—22no—
57C4×D4—22no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4D411yescharacteristic
65C2×C4⋊C4D412yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2311yescharacteristic
71C2×C22⋊C4—21no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C42—21no—
75C2×C42C2312yes—
76C2×C42C2311yescharacteristic
77#107—24no—
78#106—24no—
79C4⋊C8—24no—
80C4⋊C8D414yes—
81#102—24no—
82#102—22no—
83#102Q814yescomplemented
84#97—24no—
85C4×C8—22no—
86#95—21no—
87#95—22no—
88#95—22no—
89#95—22no—
90C24—21no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—42no—
95C2×Q8—42no—
96C2×Q8—42no—
97C2×D4—21no—
98C2×D4—21no—
99C2×D4—21no—
100C2×D4—21no—
101C2×D4—21no—
102C2×D4C2×Q812yes—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4C2×D412yes—
112C22×C4C2×D412yes—
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2411yescharacteristic
116C2×C8—21no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8C2×D414yes—
124C2×C8C2×Q812yes—
125C4⋊C4—22no—
126C4⋊C4—24no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—22no—
130C4⋊C4—21no—
131C4⋊C4—22no—
132C4⋊C4—21no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—44no—
136C4⋊C4—44no—
137C4⋊C4—44no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—81no—
141C4⋊C4C2×Q812yes—
142C4⋊C4#4112yes—
143C22⋊C4—44no—
144C22⋊C4—42no—
145C22⋊C4—82no—
146C42—22no—
147C42—22no—
148C42—42no—
149C42C2×D412yes—
150C42#4112yes—
151C23—22no—
152C23—41no—
153C23—41no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23—41no—
158C23C22×D411yescharacteristic
159Q8—44no—
160Q8—84no—
161D4—22no—
162D4—22no—
163D4—41no—
164D4—41no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—84no—
189C2×C4#12112yes—
190C2×C4#12112yes—
191C2×C4AGL(1,ℤ/8)12yes—
192C2×C4AGL(1,ℤ/8)12yes—
193C2×C4C22×D411yescharacteristic
194C2×C4C22×Q811yescharacteristic
195C2×C4C2×#4111yescharacteristic
196C2×C4C2×#4111yescharacteristic
197C2×C42−1+411yescharacteristic
198C2×C42−1+411yescharacteristic
199C8—24no—
200C8—22no—
201C8—41no—
202C22—42no—
203C22—42no—
204C22—42no—
205C22—81no—
206C22—81no—
207C22—81no—
208C22—81no—
209C22C2×#12111yescharacteristic
210C22#54811yescharacteristic
211C22D4×Q811yescharacteristic
212C22C2×AGL(1,ℤ/8)11yescharacteristic
213C22C2×AGL(1,ℤ/8)11yescharacteristic
214C22#57711yescharacteristic
215C22#57811yescharacteristic
216C4—22no—
217C4—42no—
218C4—42no—
219C4—42no—
220C4—42no—
221C4—82no—
222C4—81no—
223C4—82no—
224C4#46914yes—
225C4#53311yescharacteristic
226C4#54111yescharacteristic
227C4#54711yescharacteristic
228C4D4×Q811yescharacteristic
229C2—82no—
230C2#242911yescharacteristic
231C2C2×#46911yescharacteristic
232C2#306311yescharacteristic
233C2#336711yescharacteristic
234C2#336811yescharacteristic
235C2#338411yescharacteristic
236C2#338611yescharacteristic
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×#446C2semidirect
3. #2546C2semidirect
5. #2087C2semidirect
7. #1990C2semidirect
8. #1948C2semidirect
9. #1944C2semidirect
83. #102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#4742
C2×#4462
#25462
#25382
#20872
#20652
#19902
#19482
#19442
#19362
C4×#1022