Finite Group Databasebeta

← #1608×C211 #41009228

Subgroups — #1608×C211 #41009228

Order 27008.

228 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 68 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#543×C211C211yesmaximal, characteristic
2#483×C211C212yesmaximal, complemented
3#397×C211C211yesmaximal, characteristic
4#328×C211C212yesmaximal, complemented
5#323×C211C211yesmaximal, characteristic
6C22⋊Q8×C211—22no—
7C22⋊Q8×C211—21no—
8C22⋊Q8×C211—22no—
9C22⋊Q8×C211C2212yes—
10C2×C4⋊C4×C211C2211yescharacteristic
11C2×C22⋊C4×C211—21no—
12C2×C22⋊C4×C211—21no—
13C2×C22⋊C4×C211C2211yescharacteristic
14#106×C211—22no—
15#103×C211—22no—
16C22⋊C8×C211C2212yes—
17#95×C211—21no—
18#95×C211C2211yescharacteristic
19C24×C211D411yescharacteristic
20C2×Q8×C211—21no—
21C2×Q8×C211D412yes—
22C22×C4×C211—21no—
23C22×C4×C211—21no—
24C22×C4×C211—21no—
25C22×C4×C211—21no—
26C22×C4×C211D411yescharacteristic
27C22×C4×C211C2311yescharacteristic
28C2×C8×C211—22no—
29C4⋊C4×C211—21no—
30C4⋊C4×C211—22no—
31C4⋊C4×C211—22no—
32C4⋊C4×C211—22no—
33C4⋊C4×C211D412yes—
34C22⋊C4×C211—22no—
35C22⋊C4×C211—22no—
36C22⋊C4×C211—22no—
37C22⋊C4×C211—42no—
38C23×C211—21no—
39C23×C211—21no—
40C23×C211—41no—
41C23×C211—41no—
42C23×C211SD1612yes—
43C23×C211C2×D411yescharacteristic
44Q8×C211—42no—
45Q8×C211—81no—
46C2×C4×C211—22no—
47C2×C4×C211—21no—
48C2×C4×C211—22no—
49C2×C4×C211—41no—
50C2×C4×C211—41no—
51C2×C4×C211—41no—
52C2×C4×C211—41no—
53C2×C4×C211—41no—
54C2×C4×C211—41no—
55C2×C4×C211—41no—
56C2×C4×C211—41no—
57C2×C4×C211—81no—
58C2×C4×C211—81no—
59C2×C4×C211C2×D412yes—
60C8×C211—42no—
61C22×C211—21no—
62C22×C211—21no—
63C22×C211—22no—
64C22×C211—42no—
65C22×C211—41no—
66C22×C211—41no—
67C22×C211—81no—
68C22×C211#12011yescharacteristic
69C22×C211C2×SD1611yescharacteristic
70C22×C211#13711yescharacteristic
71C4×C211—22no—
72C4×C211—41no—
73C4×C211—42no—
74C4×C211—42no—
75C4×C211—81no—
76C4×C211—81no—
77C2×C211—21no—
78C2×C211—21no—
79C2×C211—42no—
80C2×C211#44811yescharacteristic
81C2×C211#45511yescharacteristic
82C2×C211#45811yescharacteristic
83C211#160811yescharacteristic, direct product factor, Sylow
84#1608C21111yesmaximal, characteristic, direct product factor, Sylow
85#543C2×C21111yescharacteristic
86#483C2×C21112yescomplemented
87#397C2×C21111yescharacteristic
88#328C2×C21112yescomplemented
89#323C2×C21111yescharacteristic
90C22⋊Q8—22no—
91C22⋊Q8—21no—
92C22⋊Q8—22no—
93C22⋊Q8C22×C21112yes—
94C2×C4⋊C4C22×C21111yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4C22×C21111yescharacteristic
98#106—22no—
99#103—22no—
100C22⋊C8C22×C21112yes—
101#95—21no—
102#95C22×C21111yescharacteristic
103C24D4×C21111yescharacteristic
104C2×Q8—21no—
105C2×Q8D4×C21112yes—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4D4×C21111yescharacteristic
111C22×C4C23×C21111yescharacteristic
112C2×C8—22no—
113C4⋊C4—21no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4D4×C21112yes—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C22⋊C4—22no—
121C22⋊C4—42no—
122C23—21no—
123C23—21no—
124C23—41no—
125C23—41no—
126C23SD16×C21112yes—
127C23C2×D4×C21111yescharacteristic
128Q8—42no—
129Q8—81no—
130C2×C4—22no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—81no—
142C2×C4—81no—
143C2×C4C2×D4×C21112yes—
144C8—42no—
145C22—21no—
146C22—21no—
147C22—22no—
148C22—42no—
149C22—41no—
150C22—41no—
151C22—81no—
152C22#120×C21111yescharacteristic
153C22C2×SD16×C21111yescharacteristic
154C22#137×C21111yescharacteristic
155C4—22no—
156C4—41no—
157C4—42no—
158C4—42no—
159C4—81no—
160C4—81no—
161C2—21no—
162C2—21no—
163C2—42no—
164C2#448×C21111yescharacteristic
165C2#455×C21111yescharacteristic
166C2#458×C21111yescharacteristic
subgroupG/H
center, socleC22×C211#120
commutator, Frattini, cosocleC22×C4C23×C211
prime radicalC24×C211D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C211 ◁ C22×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. #483×C211C2semidirect
4. #328×C211C2semidirect
83. C211#1608direct (N × H)
84. #1608C211direct (N × H)
86. #483C2×C211semidirect
88. #328C2×C211semidirect

Maximal subgroups

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

subgroupindex
#543×C2112
#483×C2112
#397×C2112
#328×C2112
#323×C2112
#1608211