Finite Group Databasebeta

← #2028×C211 #41009648

Subgroups — #2028×C211 #41009648

Order 27008.

246 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 84 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#544×C211C211yesmaximal, characteristic
2#419×C211C212yesmaximal
3#391×C211C211yesmaximal, characteristic
4#343×C211C211yesmaximal, characteristic
5#339×C211C212yesmaximal
6C22×Q8×C211C2211yescharacteristic
7C2×M16×C211C2212yes—
8C4⋊Q8×C211—22no—
9#125×C211—22no—
10C22⋊Q8×C211—22no—
11#117×C211C2212yes—
12C2×C4⋊C4×C211C2211yescharacteristic
13C2×C42×C211C2211yescharacteristic
14#103×C211—22no—
15#103×C211—22no—
16#95×C211—22no—
17C2×Q8×C211—22no—
18C2×Q8×C211—21no—
19C2×Q8×C211—22no—
20C2×Q8×C211—22no—
21C2×Q8×C211Q812yescomplemented
22C22×C4×C211—21no—
23C22×C4×C211—21no—
24C22×C4×C211D412yes—
25C22×C4×C211C2311yescharacteristic
26M16×C211—22no—
27M16×C211—22no—
28C2×C8×C211—22no—
29C4⋊C4×C211—22no—
30C4⋊C4×C211—24no—
31C4⋊C4×C211—22no—
32C4⋊C4×C211D412yes—
33C4⋊C4×C211D412yes—
34C22⋊C4×C211—22no—
35C42×C211—22no—
36C42×C211—22no—
37C23×C211C2×D411yescharacteristic
38Q8×C211—24no—
39Q8×C211—42no—
40Q8×C211—82no—
41C2×C4×C211—22no—
42C2×C4×C211—22no—
43C2×C4×C211—22no—
44C2×C4×C211—22no—
45C2×C4×C211—22no—
46C2×C4×C211—41no—
47C2×C4×C211—41no—
48C2×C4×C211—41no—
49C2×C4×C211—41no—
50C2×C4×C211—42no—
51C2×C4×C211C2×D412yes—
52C2×C4×C211C2×Q811yescharacteristic
53C2×C4×C211#4112yes—
54C2×C4×C211#4111yescharacteristic
55C8×C211—42no—
56C22×C211—21no—
57C22×C211—21no—
58C22×C211#12011yescharacteristic
59C22×C211C22⋊Q811yescharacteristic
60C22×C211#12411yescharacteristic
61C4×C211—42no—
62C4×C211—42no—
63C4×C211—42no—
64C4×C211—42no—
65C4×C211—42no—
66C4×C211C22⋊Q812yes—
67C4×C211#12412yes—
68C2×C211—21no—
69C2×C211—21no—
70C2×C211#39311yescharacteristic
71C2×C211#45511yescharacteristic
72C2×C211#45611yescharacteristic
73C211#202811yescharacteristic, direct product factor, Sylow
74#2028C21111yesmaximal, characteristic, direct product factor, Sylow
75#544C2×C21111yescharacteristic
76#419C2×C21112yes—
77#391C2×C21111yescharacteristic
78#343C2×C21111yescharacteristic
79#339C2×C21112yes—
80C22×Q8C22×C21111yescharacteristic
81C2×M16C22×C21112yes—
82C4⋊Q8—22no—
83#125—22no—
84C22⋊Q8—22no—
85#117C22×C21112yes—
86C2×C4⋊C4C22×C21111yescharacteristic
87C2×C42C22×C21111yescharacteristic
88#103—22no—
89#103—22no—
90#95—22no—
91C2×Q8—22no—
92C2×Q8—21no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8Q8×C21112yescomplemented
96C22×C4—21no—
97C22×C4—21no—
98C22×C4D4×C21112yes—
99C22×C4C23×C21111yescharacteristic
100M16—22no—
101M16—22no—
102C2×C8—22no—
103C4⋊C4—22no—
104C4⋊C4—24no—
105C4⋊C4—22no—
106C4⋊C4D4×C21112yes—
107C4⋊C4D4×C21112yes—
108C22⋊C4—22no—
109C42—22no—
110C42—22no—
111C23C2×D4×C21111yescharacteristic
112Q8—24no—
113Q8—42no—
114Q8—82no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—42no—
125C2×C4C2×D4×C21112yes—
126C2×C4C2×Q8×C21111yescharacteristic
127C2×C4#41×C21112yes—
128C2×C4#41×C21111yescharacteristic
129C8—42no—
130C22—21no—
131C22—21no—
132C22#120×C21111yescharacteristic
133C22C22⋊Q8×C21111yescharacteristic
134C22#124×C21111yescharacteristic
135C4—42no—
136C4—42no—
137C4—42no—
138C4—42no—
139C4—42no—
140C4C22⋊Q8×C21112yes—
141C4#124×C21112yes—
142C2—21no—
143C2—21no—
144C2#393×C21111yescharacteristic
145C2#455×C21111yescharacteristic
146C2#456×C21111yescharacteristic
subgroupG/H
center, socleC22×C211#120
commutator, Frattini, cosocleC22×C4C23×C211
prime radicalC23×C211C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C211 ◁ C22×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
21. C2×Q8×C211Q8semidirect
73. C211#2028direct (N × H)
74. #2028C211direct (N × H)
95. C2×Q8Q8×C211semidirect

Maximal subgroups

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

subgroupindex
#544×C2112
#419×C2112
#391×C2112
#343×C2112
#339×C2112
#2028211