Finite Group Databasebeta

← #8529

Subgroups — #8529

Order 256.

324 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 71 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#3553C211yesmaximal, characteristic
2#1991C211yesmaximal, characteristic, complemented
3#1534C211yesmaximal, characteristic, complemented
4#1512C212yesmaximal, complemented
5#1494C212yesmaximal, complemented
6#564C411yescharacteristic, complemented
7#563C412yescomplemented
8#562C411yescharacteristic, complemented
9#554—22no—
10#553—22no—
11#552—22no—
12#541—22no—
13#539—22no—
14#539—22no—
15#529C2212yes—
16C2×#125C2211yescharacteristic
17#432C2212yes—
18C2×C4×C8C2211yescharacteristic
19#382C2211yescharacteristic
20#340—22no—
21#330—22no—
22#328—22no—
23#327—22no—
24Q8⋊C8—22no—
25D4⋊C8—22no—
26C22×C8—21no—
27C4⋊Q8—22no—
28#126—22no—
29#126—22no—
30#126—22no—
31#126—22no—
32#126—22no—
33#125—22no—
34#125—22no—
35#125—22no—
36#125—22no—
37#125—22no—
38#125—22no—
39#125C2×C411yescharacteristic
40#125C2×C411yescharacteristic
41#125C2×C412yes—
42#125D412yes—
43#124C2×C412yes—
44#123—22no—
45#123—22no—
46#123—22no—
47#123—22no—
48#123—22no—
49C22⋊Q8—22no—
50C22⋊Q8—22no—
51C22⋊Q8—22no—
52C22⋊Q8—22no—
53C22⋊Q8—22no—
54C22⋊Q8—22no—
55C22⋊Q8D412yes—
56#121—22no—
57#121—22no—
58#121D412yes—
59C4×Q8—22no—
60C4×Q8D412yes—
61C4×D4—22no—
62C4×D4—22no—
63C4×D4—22no—
64C4×D4—22no—
65C4×D4D412yes—
66#117—22no—
67#117—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4D412yes—
72C2×C42C2311yescharacteristic
73C4⋊C8—22no—
74C22⋊C8—22no—
75#97—22no—
76C4×C8—22no—
77#95—22no—
78C2×Q8—22no—
79C2×Q8C22⋊C412yes—
80C2×D4—22no—
81C2×D4—22no—
82C2×D4C22⋊C412yes—
83C22×C4—22no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—22no—
87C22×C4—41no—
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D412yes—
90C2×C8—22no—
91C2×C8—41no—
92C2×C8—41no—
93C2×C8—42no—
94C4⋊C4—22no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—22no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4—22no—
103C4⋊C4—22no—
104C4⋊C4—22no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C4⋊C4—22no—
109C4⋊C4—42no—
110C4⋊C4C22⋊C412yes—
111C4⋊C4C22⋊C412yes—
112C4⋊C4C22⋊C412yes—
113C4⋊C4C22⋊C412yes—
114C22⋊C4—22no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C22⋊C4—22no—
121C22⋊C4—22no—
122C22⋊C4—22no—
123C42—22no—
124C42—22no—
125C42—22no—
126C42C22×C411yescharacteristic
127C42C2×D412yes—
128C42C2×D411yescharacteristic
129C23—22no—
130C23#12011yescharacteristic
131Q8—42no—
132Q8—82no—
133D4—42no—
134D4—42no—
135D4—42no—
136D4—82no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—42no—
145C2×C4—82no—
146C2×C4—82no—
147C2×C4—82no—
148C2×C4—82no—
149C2×C4—82no—
150C2×C4—81no—
151C2×C4C2×C22⋊C411yescharacteristic
152C2×C4C2×C22⋊C412yes—
153C2×C4#12012yes—
154C2×C4#12011yescharacteristic
155C8—42no—
156C8—82no—
157C22—22no—
158C22—42no—
159C22—42no—
160C22—42no—
161C22#37911yescharacteristic
162C22#45411yescharacteristic
163C22#45611yescharacteristic
164C4—21no—
165C4—21no—
166C4—22no—
167C4—22no—
168C4—42no—
169C4—42no—
170C4—42no—
171C4—42no—
172C4—42no—
173C4—161no—
174C2—21no—
175C2—21no—
176C2—82no—
177C2#179311yescharacteristic
178C2#179412yes—
subgroupG/H
center, socleC22#379
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#3553C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1991C2semidirect
3. #1534C2semidirect
4. #1512C2semidirect
5. #1494C2semidirect
6. #564C4semidirect
7. #563C4semidirect
8. #562C4semidirect

Maximal subgroups

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

subgroupindex
#35532
#19912
#15342
#15122
#14942