Finite Group Databasebeta

← #18685

Subgroups — #18685

Order 256.

465 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 465 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#3544C211yesmaximal, characteristic
2#3238C212yesmaximal, complemented
3C2×#494C211yesmaximal, characteristic, complemented
4#1634C212yesmaximal, complemented
5#1633C212yesmaximal, complemented
6#1632C214yesmaximal, complemented
7#1558C211yesmaximal, characteristic, complemented
8#1494C212yesmaximal, complemented
9C22×Q16—22no—
10#563—22no—
11Q82C2211yescharacteristic
12#555C2211yescharacteristic
13#554C2212yes—
14#552—22no—
15#547—22no—
16#544—22no—
17#542—22no—
18#536—22no—
19C2×C4⋊Q8C2211yescharacteristic
20C2×C4⋊Q8C2211yescharacteristic
21#529C2212yes—
22#501—22no—
23#494—22no—
24#494C2212yes—
25#494C2212yes—
26#489—22no—
27#483—22no—
28#480—22no—
29#477C2214yes—
30#476C2214yes—
31#432C2212yes—
32C2×C4×C8C2211yescharacteristic
33#333C2212yes—
34#333C2212yes—
35Q8⋊C8C2214yes—
36D4⋊C8C2214yes—
37C2×#41—22no—
38C22×Q8—21no—
39C22×Q8D412yes—
40C2×Q16—24no—
41C2×Q16—24no—
42C2×Q16—48no—
43C22×C8—21no—
44C4⋊Q8—22no—
45C4⋊Q8—22no—
46C4⋊Q8—22no—
47C4⋊Q8—22no—
48C4⋊Q8—22no—
49C4⋊Q8—21no—
50C4⋊Q8D412yes—
51C4⋊Q8C2312yes—
52C4⋊Q8C2311yescharacteristic
53C4⋊Q8C2311yescharacteristic
54#126—24no—
55#126—22no—
56#125—24no—
57#125—22no—
58#125D412yes—
59#124—22no—
60#124—22no—
61#124D412yes—
62#123—24no—
63#123—24no—
64#123—22no—
65C22⋊Q8—24no—
66C22⋊Q8—24no—
67C22⋊Q8—24no—
68C22⋊Q8—22no—
69C22⋊Q8D412yes—
70#121—22no—
71#121D412yes—
72C4×Q8—22no—
73C4×Q8C2312yes—
74C4×D4—22no—
75C4×D4—22no—
76C4×D4C2312yes—
77#117—22no—
78#117—22no—
79#117—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C42C2311yescharacteristic
84#107—24no—
85#106—24no—
86C4⋊C8C2314yes—
87#103—24no—
88#102—24no—
89C22⋊C8—22no—
90#97—22no—
91C4×C8—21no—
92C4×C8C2312yes—
93#41—84no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—44no—
98C2×Q8—44no—
99C2×Q8—41no—
100C2×Q8C2×D412yes—
101C2×Q8C2×D412yes—
102C2×Q8C2×D412yes—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4C2×D412yes—
106C22×C4—22no—
107C22×C4—24no—
108C22×C4—22no—
109C22×C4—41no—
110C22×C4C2×D412yes—
111C22×C4C2×D411yescharacteristic
112Q16—48no—
113Q16—48no—
114C2×C8—24no—
115C2×C8—22no—
116C2×C8—24no—
117C4⋊C4—24no—
118C4⋊C4—24no—
119C4⋊C4—24no—
120C4⋊C4—24no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—24no—
128C4⋊C4—42no—
129C4⋊C4C2×D412yes—
130C4⋊C4C2×D412yes—
131C22⋊C4—24no—
132C22⋊C4—24no—
133C22⋊C4—22no—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C22⋊C4—22no—
137C22⋊C4—22no—
138C42—22no—
139C42—22no—
140C42—22no—
141C42C2×D411yescharacteristic
142C42C2×D412yes—
143C42C2411yescharacteristic
144C23—22no—
145C23#12011yescharacteristic
146Q8—24no—
147Q8—42no—
148Q8—42no—
149Q8—44no—
150Q8—82no—
151Q8—82no—
152Q8—82no—
153D4—24no—
154D4—44no—
155D4—82no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—44no—
164C2×C4—44no—
165C2×C4—42no—
166C2×C4—84no—
167C2×C4—82no—
168C2×C4#12012yes—
169C2×C4#12011yescharacteristic
170C2×C4C22×D411yescharacteristic
171C2×C4C22×D412yes—
172C8—24no—
173C8—44no—
174C22—22no—
175C22—42no—
176C22—42no—
177C22—42no—
178C22#45312yes—
179C22C2×#12011yescharacteristic
180C4—22no—
181C4—21no—
182C4—21no—
183C4—42no—
184C4—42no—
185C4—42no—
186C4—42no—
187C4—42no—
188C4—82no—
189C4#57814yes—
190C2—22no—
191C2—82no—
192C2#301112yes—
193C2C2×#45311yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3544C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3238C2semidirect
3. C2×#494C2semidirect
4. #1634C2semidirect
5. #1633C2semidirect
6. #1632C2semidirect
7. #1558C2semidirect
8. #1494C2semidirect

Maximal subgroups

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

subgroupindex
#35442
#32382
C2×#4942
#16342
#16332
#16322
#15582
#14942