Finite Group Databasebeta

← #25834

Subgroups — #25834

Order 256.

623 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 623 conjugacy classes of subgroups — too many to draw. Only its 167 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×D4×Q8C211yesmaximal, characteristic
2C2×#498C211yesmaximal, characteristic, complemented
3C2×#476C211yesmaximal, characteristic
4#2567C211yesmaximal, characteristic, complemented
5#2058C214yesmaximal, complemented
6#1988C211yesmaximal, characteristic, complemented
7#1984C211yesmaximal, characteristic, complemented
8#1864C214yesmaximal
9C4×#102C211yesmaximal, characteristic
10C23×Q8—21no—
11D4×Q8—24no—
12D4×Q8—24no—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2212yes—
15C2×C4⋊Q8C2211yescharacteristic
16C2×C22⋊Q8—21no—
17C2×C22⋊Q8—22no—
18C2×C4×Q8C2211yescharacteristic
19C2×C4×D4C2212yes—
20C2×C4×D4C2211yescharacteristic
21#498—24no—
22#476—24no—
23C2×#106C2212yes—
24C2×#106C2214yes—
25C2×C4⋊C8C2211yescharacteristic
26C2×#102C2212yes—
27C2×#102C2212yes—
28C2×C4×C8C2211yescharacteristic
29#395—22no—
30#391—21no—
31#389C2214yes—
32#384—21no—
33C4×C4⋊C4C2211yescharacteristic
34#340C2212yes—
35#340C2214yes—
36#336C2214yes—
37C22×Q8—28no—
38C22×Q8—22no—
39C22×Q8—21no—
40C22×Q8—21no—
41C22×Q8—44no—
42C22×Q8—44no—
43C22×Q8D411yescharacteristic
44C22×D4C2311yescharacteristic
45C23×C4—22no—
46C23×C4—21no—
47C22×C8C2312yes—
48C22×C8C2312yes—
49C4⋊Q8—24no—
50C4⋊Q8—24no—
51C4⋊Q8—22no—
52C4⋊Q8—22no—
53C4⋊Q8—22no—
54C4⋊Q8D414yes—
55C22⋊Q8—28no—
56C22⋊Q8—216no—
57C4×Q8—22no—
58C4×Q8—22no—
59C4×D4—24no—
60C4×D4—24no—
61C4×D4—22no—
62C4×D4D414yes—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—24no—
67C2×C4⋊C4D411yescharacteristic
68C2×C4⋊C4D412yes—
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2312yes—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—21no—
75C2×C42—21no—
76C2×C42C2312yes—
77C2×C42C2311yescharacteristic
78#106—24no—
79#106—28no—
80C4⋊C8—22no—
81#102—24no—
82#102Q818yescomplemented
83C4×C8—22no—
84#95—21no—
85#95—24no—
86C24—21no—
87C2×Q8—28no—
88C2×Q8—22no—
89C2×Q8—24no—
90C2×Q8—22no—
91C2×Q8—44no—
92C2×Q8—432no—
93C2×Q8—42no—
94C2×Q8—44no—
95C2×Q8—42no—
96C2×Q8—42no—
97C2×Q8—82no—
98C2×Q8SD1614yes—
99C2×D4—21no—
100C2×D4—22no—
101C2×D4C2×D412yes—
102C2×D4C2×Q814yes—
103C22×C4—28no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—21no—
107C22×C4—24no—
108C22×C4—22no—
109C22×C4—44no—
110C22×C4—44no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4C2×D412yes—
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D412yes—
116C22×C4C2×D411yescharacteristic
117C22×C4C2411yescharacteristic
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—24no—
122C2×C8C2×Q814yescomplemented
123C4⋊C4—28no—
124C4⋊C4—24no—
125C4⋊C4—24no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—21no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—42no—
133C4⋊C4—84no—
134C4⋊C4C2×D414yes—
135C4⋊C4C2×D412yes—
136C4⋊C4C2×Q814yescomplemented
137C22⋊C4—28no—
138C22⋊C4—24no—
139C42—22no—
140C42—22no—
141C42—42no—
142C42C2×D412yes—
143C42C2×D412yes—
144C23—22no—
145C23—24no—
146C23—41no—
147C23—41no—
148C23—42no—
149C23C22×D411yescharacteristic
150Q8—28no—
151Q8—416no—
152Q8—48no—
153Q8—84no—
154D4—22no—
155D4—24no—
156D4C4⋊Q814yes—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—24no—
162C2×C4—22no—
163C2×C4—24no—
164C2×C4—24no—
165C2×C4—416no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—48no—
173C2×C4—42no—
174C2×C4—44no—
175C2×C4#12012yes—
176C2×C4#12012yes—
177C2×C4C2×SD1614yes—
178C2×C4C2×SD1612yes—
179C2×C4AGL(1,ℤ/8)12yes—
180C2×C4C22×D411yescharacteristic
181C2×C4C22×D411yescharacteristic
182C2×C4C22×Q812yes—
183C2×C42−1+412yes—
184C8—24no—
185C8—42no—
186C22—24no—
187C22—44no—
188C22—44no—
189C22—42no—
190C22—42no—
191C22C2×#12011yescharacteristic
192C22D4×Q812yes—
193C22C22×SD1611yescharacteristic
194C22C2×AGL(1,ℤ/8)11yescharacteristic
195C22#57712yes—
196C4—24no—
197C4—22no—
198C4—44no—
199C4—44no—
200C4—42no—
201C4—42no—
202C4—82no—
203C4#45014yes—
204C4C2×C4⋊Q811yescharacteristic
205C4#53611yescharacteristic
206C4D4×Q812yes—
207C2—44no—
208C2#243411yescharacteristic
209C2C2×#45011yescharacteristic
210C2#301011yescharacteristic
211C2Q8×SD1612yes—
212C2#338412yes—
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×#498C2semidirect
4. #2567C2semidirect
5. #2058C2semidirect
6. #1988C2semidirect
7. #1984C2semidirect
82. #102Q8semidirect
122. C2×C8C2×Q8semidirect
136. C4⋊C4C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Q82
C2×#4982
C2×#4762
#25672
#20582
#19882
#19842
#18642
C4×#1022