Finite Group Databasebeta

← #1638×C211 #41009258

Subgroups — #1638×C211 #41009258

Order 27008.

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

normal subgroups only

This group has 194 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#556×C211C211yesmaximal, characteristic
2#487×C211C211yesmaximal, characteristic, complemented
3#477×C211C211yesmaximal, characteristic, complemented
4#476×C211C211yesmaximal, characteristic
5#332×C211C211yesmaximal, characteristic, complemented
6Q8⋊C8×C211C211yesmaximal, characteristic, complemented
7D4⋊C8×C211C211yesmaximal, characteristic
8C4⋊Q8×C211C2211yescharacteristic
9C4⋊Q8×C211C2211yescharacteristic
10#126×C211—22no—
11#124×C211—21no—
12C22⋊Q8×C211—21no—
13C4×Q8×C211—21no—
14C4×Q8×C211C2211yescharacteristic
15C4×D4×C211C2211yescharacteristic
16#117×C211—21no—
17#107×C211—21no—
18#106×C211—21no—
19C4⋊C8×C211C2211yescharacteristic
20C4⋊C8×C211C2211yescharacteristic
21#103×C211—21no—
22#103×C211—22no—
23#102×C211—21no—
24C4×C8×C211C2211yescharacteristic
25C2×Q8×C211—21no—
26C2×Q8×C211—21no—
27C2×Q8×C211D411yescharacteristic
28C2×D4×C211D411yescharacteristic
29C22×C4×C211—21no—
30C2×C8×C211—21no—
31C2×C8×C211—21no—
32C2×C8×C211—21no—
33C4⋊C4×C211—22no—
34C4⋊C4×C211—21no—
35C4⋊C4×C211—21no—
36C4⋊C4×C211—21no—
37C4⋊C4×C211—21no—
38C4⋊C4×C211D412yes—
39C4⋊C4×C211D411yescharacteristic
40C4⋊C4×C211D411yescharacteristic
41C22⋊C4×C211—22no—
42C22⋊C4×C211—22no—
43C22⋊C4×C211—21no—
44C42×C211—22no—
45C42×C211—21no—
46C42×C211C2311yescharacteristic
47C23×C211—21no—
48Q8×C211—21no—
49Q8×C211—21no—
50Q8×C211—42no—
51Q8×C211—41no—
52Q8×C211—41no—
53D4×C211—21no—
54D4×C211—21no—
55C2×C4×C211—21no—
56C2×C4×C211—21no—
57C2×C4×C211—21no—
58C2×C4×C211—22no—
59C2×C4×C211—41no—
60C2×C4×C211—41no—
61C2×C4×C211—41no—
62C2×C4×C211C2×D411yescharacteristic
63C2×C4×C211C2×D411yescharacteristic
64C2×C4×C211C2×D411yescharacteristic
65C8×C211—41no—
66C8×C211—41no—
67C8×C211—41no—
68C22×C211—41no—
69C22×C211—41no—
70C22×C211—41no—
71C22×C211#12011yescharacteristic
72C4×C211—21no—
73C4×C211—41no—
74C4×C211—41no—
75C4×C211—42no—
76C4×C211—41no—
77C4×C211—81no—
78C4×C211#13511yescharacteristic
79C4×C211#13511yescharacteristic
80C4×C211#13711yescharacteristic
81C4×C211#13711yescharacteristic
82C2×C211—81no—
83C2×C211#45211yescharacteristic
84C2×C211#45211yescharacteristic
85C2×C211#45411yescharacteristic
86C211#163811yescharacteristic, direct product factor, Sylow
87#1638C21111yesmaximal, characteristic, direct product factor, Sylow
88#556C2×C21111yescharacteristic
89#487C2×C21111yescharacteristic, complemented
90#477C2×C21111yescharacteristic, complemented
91#476C2×C21111yescharacteristic
92#332C2×C21111yescharacteristic, complemented
93Q8⋊C8C2×C21111yescharacteristic, complemented
94D4⋊C8C2×C21111yescharacteristic
95C4⋊Q8C22×C21111yescharacteristic
96C4⋊Q8C22×C21111yescharacteristic
97#126—22no—
98#124—21no—
99C22⋊Q8—21no—
100C4×Q8—21no—
101C4×Q8C22×C21111yescharacteristic
102C4×D4C22×C21111yescharacteristic
103#117—21no—
104#107—21no—
105#106—21no—
106C4⋊C8C22×C21111yescharacteristic
107C4⋊C8C22×C21111yescharacteristic
108#103—21no—
109#103—22no—
110#102—21no—
111C4×C8C22×C21111yescharacteristic
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8D4×C21111yescharacteristic
115C2×D4D4×C21111yescharacteristic
116C22×C4—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C4⋊C4—22no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4D4×C21112yes—
126C4⋊C4D4×C21111yescharacteristic
127C4⋊C4D4×C21111yescharacteristic
128C22⋊C4—22no—
129C22⋊C4—22no—
130C22⋊C4—21no—
131C42—22no—
132C42—21no—
133C42C23×C21111yescharacteristic
134C23—21no—
135Q8—21no—
136Q8—21no—
137Q8—42no—
138Q8—41no—
139Q8—41no—
140D4—21no—
141D4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—22no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4C2×D4×C21111yescharacteristic
150C2×C4C2×D4×C21111yescharacteristic
151C2×C4C2×D4×C21111yescharacteristic
152C8—41no—
153C8—41no—
154C8—41no—
155C22—41no—
156C22—41no—
157C22—41no—
158C22#120×C21111yescharacteristic
159C4—21no—
160C4—41no—
161C4—41no—
162C4—42no—
163C4—41no—
164C4—81no—
165C4#135×C21111yescharacteristic
166C4#135×C21111yescharacteristic
167C4#137×C21111yescharacteristic
168C4#137×C21111yescharacteristic
169C2—81no—
170C2#452×C21111yescharacteristic
171C2#452×C21111yescharacteristic
172C2#454×C21111yescharacteristic
subgroupG/H
center, socleC22×C211#120
commutator, Frattini, cosocleC42C23×C211
prime radicalC2×D4×C211D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C211 ◁ C42×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. #487×C211C2semidirect
3. #477×C211C2semidirect
5. #332×C211C2semidirect
6. Q8⋊C8×C211C2semidirect
86. C211#1638direct (N × H)
87. #1638C211direct (N × H)
89. #487C2×C211semidirect
90. #477C2×C211semidirect
92. #332C2×C211semidirect
93. Q8⋊C8C2×C211semidirect

Maximal subgroups

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

subgroupindex
#556×C2112
#487×C2112
#477×C2112
#476×C2112
#332×C2112
Q8⋊C8×C2112
D4⋊C8×C2112
#1638211