Finite Group Databasebeta

← #25699

Subgroups — #25699

Order 256.

649 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 649 conjugacy classes of subgroups — too many to draw. Only its 147 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×#547C211yesmaximal, characteristic
2C2×#498C211yesmaximal, characteristic, complemented
3C2×#476C211yesmaximal, characteristic, complemented
4#2563C211yesmaximal, characteristic, complemented
5#2034C214yesmaximal, complemented
6#1988C211yesmaximal, characteristic, complemented
7#1954C211yesmaximal, characteristic, complemented
8#1853C214yesmaximal, complemented
9C4×#102C211yesmaximal, characteristic, complemented
10C22×#41—21no—
11#547—24no—
12#547—24no—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#123—22no—
16C2×C22⋊Q8—21no—
17C2×#121C2212yes—
18C2×C4×D4C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C22×C4⋊C4C2212yes—
21#498—24no—
22#476—24no—
23C2×#106C2212yes—
24C2×#106C2214yes—
25C2×C4⋊C8C2211yescharacteristic
26C2×#102C2212yes—
27C2×#102C2212yes—
28C2×C22⋊C8C2214yes—
29C2×C4×C8C2211yescharacteristic
30#393—22no—
31#390—21no—
32#382—21no—
33#380C2214yes—
34C4×C4⋊C4C2211yescharacteristic
35#340C2212yes—
36#340C2214yes—
37C2×#41—28no—
38C2×#41—44no—
39C2×#41—44no—
40C22×Q8—21no—
41C22×Q8—21no—
42C22×D4—22no—
43C22×D4D411yescharacteristic
44C22×D4C2311yescharacteristic
45C23×C4—22no—
46C23×C4—21no—
47C23×C4C2312yes—
48C22×C8C2312yes—
49C22×C8C2312yes—
50C4⋊Q8—22no—
51C4⋊Q8—22no—
52C4⋊Q8—22no—
53C4⋊Q8D414yes—
54#123—216no—
55C22⋊Q8—28no—
56#121—24no—
57#121—24no—
58C4×D4—22no—
59C4×D4—22no—
60C4×D4—22no—
61C4×D4D414yes—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—24no—
65C2×C4⋊C4—24no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4D411yescharacteristic
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2312yes—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4D412yes—
76C2×C42—21no—
77C2×C42C2311yescharacteristic
78#106—24no—
79#106—28no—
80C4⋊C8—22no—
81#102—24no—
82#102—24no—
83C22⋊C8—28no—
84C4×C8—22no—
85#95—21no—
86#95—24no—
87C24—21no—
88C24C2×D412yes—
89#41—432no—
90C2×Q8—24no—
91C2×Q8—42no—
92C2×Q8—42no—
93C2×Q8—42no—
94C2×Q8—82no—
95C2×D4—28no—
96C2×D4—22no—
97C2×D4—22no—
98C2×D4—21no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—44no—
102C2×D4—44no—
103C2×D4SD1614yes—
104C2×D4C2×D412yes—
105C22×C4—28no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—24no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—24no—
112C22×C4—24no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—44no—
116C22×C4—44no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4C2×D412yes—
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2411yescharacteristic
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—24no—
127C2×C8#4114yescomplemented
128C4⋊C4—28no—
129C4⋊C4—24no—
130C4⋊C4—22no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—42no—
138C4⋊C4C2×D412yes—
139C4⋊C4C2×D414yes—
140C22⋊C4—28no—
141C22⋊C4—24no—
142C22⋊C4—24no—
143C22⋊C4—84no—
144C42—42no—
145C42C2×D412yes—
146C42C2×D412yes—
147C23—22no—
148C23—24no—
149C23—22no—
150C23—24no—
151C23—22no—
152C23—42no—
153C23—41no—
154C23—41no—
155C23—42no—
156C23C2×SD1614yes—
157C23C22×D411yescharacteristic
158Q8—48no—
159Q8—84no—
160D4—22no—
161D4—22no—
162D4—28no—
163D4—416no—
164D4—42no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—24no—
169C2×C4—24no—
170C2×C4—416no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—44no—
178C2×C4—48no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—44no—
184C2×C4#12012yes—
185C2×C4#12012yes—
186C2×C4C2×SD1612yes—
187C2×C4AGL(1,ℤ/8)12yes—
188C2×C4C22×D411yescharacteristic
189C2×C4C22×D411yescharacteristic
190C2×C4C2×#4112yes—
191C2×C42−1+412yes—
192C8—24no—
193C8—42no—
194C22—28no—
195C22—24no—
196C22—44no—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—42no—
201C22—42no—
202C22—82no—
203C22—82no—
204C22C2×#12011yescharacteristic
205C22#54712yes—
206C22C22×SD1611yescharacteristic
207C22C2×AGL(1,ℤ/8)11yescharacteristic
208C22#57712yes—
209C4—22no—
210C4—42no—
211C4—42no—
212C4—44no—
213C4—82no—
214C4—82no—
215C4#45014yes—
216C4#53211yescharacteristic
217C4#53611yescharacteristic
218C4#54812yes—
219C2—44no—
220C2—82no—
221C2#243211yescharacteristic
222C2C2×#45011yescharacteristic
223C2#301011yescharacteristic
224C2#333512yes—
225C2#335412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#547C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#498C2semidirect
3. C2×#476C2semidirect
4. #2563C2semidirect
5. #2034C2semidirect
6. #1988C2semidirect
7. #1954C2semidirect
8. #1853C2semidirect
9. C4×#102C2semidirect
127. C2×C8#41semidirect

Maximal subgroups

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

subgroupindex
C2×#5472
C2×#4982
C2×#4762
#25632
#20342
#19882
#19542
#18532
C4×#1022