Finite Group Databasebeta

← #1827×C211 #41009447

Subgroups — #1827×C211 #41009447

Order 27008.

262 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 100 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
1C4×#41×C211C211yesmaximal, characteristic
2#413×C211C211yesmaximal, characteristic
3#410×C211C211yesmaximal, characteristic
4#339×C211C212yesmaximal, complemented
5H(3,ℤ/4)×C211C212yesmaximal, complemented
6C2×#41×C211C2211yescharacteristic
7C2×M16×C211C2212yes—
8C4×Q8×C211—21no—
9C4×D4×C211—22no—
10C4×D4×C211—21no—
11#117×C211C412yescomplemented
12#117×C211C2212yes—
13#117×C211C2211yescharacteristic
14C2×C42×C211C412yescomplemented
15C2×C42×C211C2211yescharacteristic
16#101×C211—21no—
17#100×C211—21no—
18#99×C211—22no—
19#41×C211—22no—
20#41×C211—22no—
21C2×Q8×C211D411yescharacteristic
22C2×D4×C211D412yes—
23C2×D4×C211D411yescharacteristic
24C22×C4×C211—22no—
25C22×C4×C211—21no—
26C22×C4×C211C2×C412yes—
27C22×C4×C211C2311yescharacteristic
28M16×C211—22no—
29M16×C211—22no—
30C2×C8×C211—22no—
31C4⋊C4×C211—22no—
32C4⋊C4×C211—21no—
33C4⋊C4×C211—22no—
34C22⋊C4×C211—22no—
35C22⋊C4×C211—21no—
36C22⋊C4×C211—22no—
37C42×C211—22no—
38C42×C211—21no—
39C42×C211—22no—
40C42×C211C2×C411yescharacteristic
41C42×C211C2×C412yes—
42C42×C211C2×C411yescharacteristic
43C23×C211C22⋊C412yes—
44C23×C211C2×D411yescharacteristic
45Q8×C211—22no—
46D4×C211—24no—
47D4×C211—22no—
48C2×C4×C211—21no—
49C2×C4×C211—22no—
50C2×C4×C211—22no—
51C2×C4×C211—22no—
52C2×C4×C211—21no—
53C2×C4×C211—42no—
54C2×C4×C211—44no—
55C2×C4×C211—42no—
56C2×C4×C211C22⋊C412yes—
57C2×C4×C211C22×C411yescharacteristic
58C2×C4×C211C2×D411yescharacteristic
59C2×C4×C211#4112yes—
60C2×C4×C211#4112yes—
61C8×C211—42no—
62C22×C211—22no—
63C22×C211—41no—
64C22×C211—42no—
65C22×C211C2×C22⋊C411yescharacteristic
66C22×C211#12312yes—
67C4×C211—22no—
68C4×C211—22no—
69C4×C211—42no—
70C4×C211—41no—
71C4×C211—82no—
72C4×C211#11711yescharacteristic
73C4×C211#11711yescharacteristic
74C4×C211#12312yes—
75C2×C211—21no—
76C2×C211—22no—
77C2×C211—42no—
78C2×C211#38111yescharacteristic
79C211#182711yescharacteristic, direct product factor, Sylow
80#1827C21111yesmaximal, characteristic, direct product factor, Sylow
81C4×#41C2×C21111yescharacteristic
82#413C2×C21111yescharacteristic
83#410C2×C21111yescharacteristic
84#339C2×C21112yescomplemented
85H(3,ℤ/4)C2×C21112yescomplemented
86C2×#41C22×C21111yescharacteristic
87C2×M16C22×C21112yes—
88C4×Q8—21no—
89C4×D4—22no—
90C4×D4—21no—
91#117C4×C21112yescomplemented
92#117C22×C21112yes—
93#117C22×C21111yescharacteristic
94C2×C42C4×C21112yescomplemented
95C2×C42C22×C21111yescharacteristic
96#101—21no—
97#100—21no—
98#99—22no—
99#41—22no—
100#41—22no—
101C2×Q8D4×C21111yescharacteristic
102C2×D4D4×C21112yes—
103C2×D4D4×C21111yescharacteristic
104C22×C4—22no—
105C22×C4—21no—
106C22×C4C2×C4×C21112yes—
107C22×C4C23×C21111yescharacteristic
108M16—22no—
109M16—22no—
110C2×C8—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C22⋊C4—22no—
115C22⋊C4—21no—
116C22⋊C4—22no—
117C42—22no—
118C42—21no—
119C42—22no—
120C42C2×C4×C21111yescharacteristic
121C42C2×C4×C21112yes—
122C42C2×C4×C21111yescharacteristic
123C23C22⋊C4×C21112yes—
124C23C2×D4×C21111yescharacteristic
125Q8—22no—
126D4—24no—
127D4—22no—
128C2×C4—21no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—21no—
133C2×C4—42no—
134C2×C4—44no—
135C2×C4—42no—
136C2×C4C22⋊C4×C21112yes—
137C2×C4C22×C4×C21111yescharacteristic
138C2×C4C2×D4×C21111yescharacteristic
139C2×C4#41×C21112yes—
140C2×C4#41×C21112yes—
141C8—42no—
142C22—22no—
143C22—41no—
144C22—42no—
145C22C2×C22⋊C4×C21111yescharacteristic
146C22#123×C21112yes—
147C4—22no—
148C4—22no—
149C4—42no—
150C4—41no—
151C4—82no—
152C4#117×C21111yescharacteristic
153C4#117×C21111yescharacteristic
154C4#123×C21112yes—
155C2—21no—
156C2—22no—
157C2—42no—
158C2#381×C21111yescharacteristic
subgroupG/H
centerC4×C211#117
commutatorC2×C4C22×C4×C211
Frattini, cosocleC22×C4C23×C211
socleC2×C211#381
prime radicalC2×#41×C211C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C211 ◁ #117×C211 ◁ G

Complements

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

NH ≅ G/Ntype
4. #339×C211C2semidirect
5. H(3,ℤ/4)×C211C2semidirect
11. #117×C211C4semidirect
14. C2×C42×C211C4semidirect
79. C211#1827direct (N × H)
80. #1827C211direct (N × H)
84. #339C2×C211semidirect
85. H(3,ℤ/4)C2×C211semidirect
91. #117C4×C211semidirect
94. C2×C42C4×C211semidirect

Maximal subgroups

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

subgroupindex
C4×#41×C2112
#413×C2112
#410×C2112
#339×C2112
H(3,ℤ/4)×C2112
#1827211