Finite Group Databasebeta

← #8892

Subgroups — #8892

Order 256.

249 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 249 conjugacy classes of subgroups — too many to draw. Only its 93 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
1Q8×C4⋊C4C211yesmaximal, characteristic
2#1947C211yesmaximal, characteristic
3#1834C211yesmaximal, characteristic
4#1803C211yesmaximal, characteristic
5C2×Q8⋊C8C211yesmaximal, characteristic
6#1296C211yesmaximal, characteristic
7#1279C211yesmaximal, characteristic
8C2×C4×Q8—21no—
9C2×C4×Q8C2211yescharacteristic
10C2×#107—21no—
11C2×C4⋊C8C2211yescharacteristic
12C2×C4⋊C8C2211yescharacteristic
13C2×#103—21no—
14C2×C4×C8C2211yescharacteristic
15#389—21no—
16#389—22no—
17#384—21no—
18#384C2211yescharacteristic
19#384C2211yescharacteristic
20C4×C4⋊C4—21no—
21C4×C4⋊C4C2211yescharacteristic
22#340—21no—
23#340—22no—
24#340—21no—
25Q8⋊C8C414yescomplemented
26C22×Q8D411yescharacteristic
27C22×C8—21no—
28C22×C8—21no—
29C22×C8—21no—
30C4×Q8—21no—
31C4×Q8—21no—
32C4×Q8—42no—
33C4×Q8—42no—
34C4×Q8C2×C412yes—
35C4×Q8Q812yes—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—22no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—22no—
44C2×C4⋊C4D411yescharacteristic
45C2×C4⋊C4D411yescharacteristic
46C2×C4⋊C4D411yescharacteristic
47C2×C4⋊C4D412yes—
48C2×C42—21no—
49C2×C42—22no—
50C2×C42—21no—
51C2×C42C2311yescharacteristic
52#107—42no—
53C4⋊C8—21no—
54C4⋊C8—21no—
55C4⋊C8—21no—
56C4⋊C8C2×C412yes—
57#103—42no—
58C4×C8—21no—
59C4×C8C2×C412yes—
60#95—22no—
61#95—21no—
62C2×Q8—21no—
63C2×Q8—21no—
64C2×Q8—21no—
65C2×Q8C4⋊C412yes—
66C2×Q8D812yes—
67C2×Q8Q1612yes—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—22no—
73C22×C4—41no—
74C22×C4C2×D411yescharacteristic
75C22×C4C2×D411yescharacteristic
76C22×C4C2×D411yescharacteristic
77C2×C8—22no—
78C2×C8—22no—
79C2×C8—41no—
80C2×C8—41no—
81C2×C8—41no—
82C2×C8—41no—
83C2×C8—41no—
84C2×C8—41no—
85C2×C8—41no—
86C4⋊C4—22no—
87C4⋊C4—22no—
88C4⋊C4—21no—
89C4⋊C4—21no—
90C4⋊C4—21no—
91C4⋊C4—21no—
92C4⋊C4—42no—
93C4⋊C4—44no—
94C4⋊C4—42no—
95C4⋊C4—41no—
96C4⋊C4—41no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—82no—
104C4⋊C4C4⋊C412yes—
105C4⋊C4SD1612yes—
106C42—22no—
107C42—22no—
108C42—44no—
109C42—41no—
110C42—41no—
111C42C22×C411yescharacteristic
112C42C2×Q811yescharacteristic
113C42#4111yescharacteristic
114C42#4111yescharacteristic
115C23#12011yescharacteristic
116Q8—22no—
117Q8—22no—
118Q8—22no—
119Q8#10714yescomplemented
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—22no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—22no—
126C2×C4—22no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—42no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—41no—
137C2×C4—81no—
138C2×C4—81no—
139C2×C4—81no—
140C2×C4C2×C4⋊C411yescharacteristic
141C2×C4C4×D411yescharacteristic
142C2×C4C4×D411yescharacteristic
143C2×C4C22⋊Q811yescharacteristic
144C2×C4C22⋊Q811yescharacteristic
145C2×C4#12311yescharacteristic
146C2×C4C2×D811yescharacteristic
147C2×C4C2×SD1611yescharacteristic
148C2×C4C2×Q1611yescharacteristic
149C2×C4#13511yescharacteristic
150C2×C4AGL(1,ℤ/8)11yescharacteristic
151C2×C4#13711yescharacteristic
152C2×C4#13711yescharacteristic
153C2×C4#13711yescharacteristic
154C8—42no—
155C8—42no—
156C8—81no—
157C22#38511yescharacteristic
158C22#44711yescharacteristic
159C22#44811yescharacteristic
160C22#45111yescharacteristic
161C22#45211yescharacteristic
162C22#45411yescharacteristic
163C22#45511yescharacteristic
164C4—24no—
165C4—21no—
166C4—21no—
167C4—42no—
168C4—42no—
169C4—81no—
170C4—82no—
171C4—161no—
172C4C2×#10711yescharacteristic
173C4#42811yescharacteristic
174C4C4×SD1611yescharacteristic
175C4#44111yescharacteristic
176C4#47511yescharacteristic
177C4#47911yescharacteristic
178C4#48011yescharacteristic
179C4#48411yescharacteristic
180C2#162711yescharacteristic
181C2#163611yescharacteristic
182C2#164111yescharacteristic
183C2#164511yescharacteristic
184C2#185211yescharacteristic
185C2#186011yescharacteristic
186C2#186511yescharacteristic
subgroupG/H
center, socle, prime radicalC23#120
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
25. Q8⋊C8C4semidirect
119. Q8#107semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊C42
#19472
#18342
#18032
C2×Q8⋊C82
#12962
#12792