Finite Group Databasebeta

← #29824

Subgroups — #29824

Order 256.

483 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 483 conjugacy classes of subgroups — too many to draw. Only its 135 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, complemented
2C2×#459C211yesmaximal, characteristic
3#2858C211yesmaximal, characteristic, complemented
4#2293C211yesmaximal, characteristic, complemented
5#2067C214yesmaximal, complemented
6#2054C214yesmaximal, complemented
7#1988C211yesmaximal, characteristic, complemented
8#1982C211yesmaximal, characteristic, complemented
9#1842C211yesmaximal, characteristic, complemented
10C22×D8C2212yes—
11C2×C4⋊Q8C2211yescharacteristic
12C2×#127C2211yescharacteristic
13C2×#124—21no—
14C2×#124—21no—
15C2×C4×D4C2211yescharacteristic
16C2×#117—21no—
17#474—24no—
18#459—24no—
19C2×#107C2212yes—
20C2×C4⋊C8C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2214yes—
23C2×#102C2212yes—
24C2×#102C2212yes—
25C2×#102C2214yes—
26#390—22no—
27#390C2212yes—
28#390C2214yes—
29#389C2212yes—
30#388—22no—
31C4×C4⋊C4C2211yescharacteristic
32C4×C22⋊C4—21no—
33C43C2211yescharacteristic
34#340C2214yes—
35C22×Q8—21no—
36C22×D4—21no—
37C22×D4—21no—
38C22×D4C2312yes—
39C22×D4C2311yescharacteristic
40C23×C4—21no—
41C2×D8—24no—
42C2×D8—44no—
43C22×C8C2312yes—
44C22×C8C2312yes—
45C4⋊Q8—22no—
46C4⋊Q8—22no—
47#127—22no—
48#127—22no—
49#124—44no—
50#124—44no—
51C4×D4—22no—
52C4×D4D414yes—
53#117—44no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4C2312yes—
57C2×C4⋊C4C2312yes—
58C2×C4⋊C4C2311yescharacteristic
59C2×C22⋊C4—21no—
60C2×C22⋊C4—22no—
61C2×C22⋊C4—24no—
62C2×C22⋊C4—24no—
63C2×C22⋊C4—22no—
64C2×C42—21no—
65C2×C42D411yescharacteristic
66C2×C42D412yes—
67C2×C42D411yescharacteristic
68C2×C42C2312yes—
69C2×C42C2311yescharacteristic
70#107—24no—
71C4⋊C8—22no—
72C4⋊C8—22no—
73C4⋊C8—44no—
74#102—24no—
75#102—24no—
76#102—44no—
77#95—22no—
78#95—21no—
79C24—21no—
80C24—22no—
81C2×Q8—42no—
82C2×Q8—42no—
83C2×Q8—42no—
84C2×D4—22no—
85C2×D4—21no—
86C2×D4—22no—
87C2×D4—22no—
88C2×D4—24no—
89C2×D4—22no—
90C2×D4—24no—
91C2×D4—44no—
92C2×D4—42no—
93C2×D4—42no—
94C2×D4—42no—
95C2×D4—42no—
96C2×D4C2×D412yes—
97C22×C4—21no—
98C22×C4—24no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—42no—
102C22×C4—42no—
103C22×C4C2×D412yes—
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D412yes—
106C22×C4C2×D411yescharacteristic
107C22×C4C2411yescharacteristic
108D8—48no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—24no—
112C2×C8—24no—
113C4⋊C4—21no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—42no—
119C4⋊C4—44no—
120C4⋊C4C2×D412yes—
121C22⋊C4—24no—
122C22⋊C4—44no—
123C22⋊C4—48no—
124C22⋊C4—44no—
125C22⋊C4—48no—
126C42—22no—
127C42—22no—
128C42—22no—
129C42—42no—
130C42—42no—
131C42D814yescomplemented
132C42C2×D412yes—
133C42#4112yes—
134C23—22no—
135C23—42no—
136C23—41no—
137C23—41no—
138C23—42no—
139C23—42no—
140C23—44no—
141C23—42no—
142C23—42no—
143C23—44no—
144C23C22×D411yescharacteristic
145Q8—84no—
146D4—22no—
147D4—22no—
148D4—42no—
149D4—42no—
150D4—42no—
151D4—44no—
152D4—84no—
153D4—84no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—48no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4#12112yes—
171C2×C4#12112yes—
172C2×C4C2×D814yes—
173C2×C4C2×D812yes—
174C2×C4AGL(1,ℤ/8)12yes—
175C2×C4C22×D411yescharacteristic
176C2×C4C2×#4111yescharacteristic
177C2×C4H(5,𝔽2)12yes—
178C2×C42−1+412yes—
179C8—42no—
180C8—42no—
181C22—42no—
182C22—42no—
183C22—42no—
184C22—82no—
185C22—82no—
186C22—82no—
187C22—84no—
188C22—84no—
189C22—82no—
190C22—82no—
191C22C2×#12111yescharacteristic
192C22#53712yes—
193C22C22×D811yescharacteristic
194C22C2×AGL(1,ℤ/8)11yescharacteristic
195C22#57512yes—
196C4—24no—
197C4—22no—
198C4—42no—
199C4—42no—
200C4—82no—
201C4—82no—
202C4#45914yes—
203C4#53511yescharacteristic
204C4#53611yescharacteristic
205C4#54212yes—
206C2—82no—
207C2—162no—
208C2#262011yescharacteristic
209C2C2×#45911yescharacteristic
210C2#304011yescharacteristic
211C2#324512yes—
212C2#325112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#459C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#474C2semidirect
3. #2858C2semidirect
4. #2293C2semidirect
5. #2067C2semidirect
6. #2054C2semidirect
7. #1988C2semidirect
8. #1982C2semidirect
9. #1842C2semidirect
131. C42D8semidirect

Maximal subgroups

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

subgroupindex
C2×#4742
C2×#4592
#28582
#22932
#20672
#20542
#19882
#19822
#18422