Finite Group Databasebeta

← #2127×C211 #41009747

Subgroups — #2127×C211 #41009747

Order 27008.

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

normal subgroups only

This group has 286 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#535×C211C211yesmaximal, characteristic
2C2×#99×C211C212yesmaximal, complemented
3#354×C211C212yesmaximal, complemented
4#353×C211C212yesmaximal, complemented
5C2×#41×C211C411yescharacteristic, complemented
6C22×D4×C211C2211yescharacteristic
7#127×C211C411yescharacteristic, complemented
8#127×C211C2211yescharacteristic
9#124×C211C411yescharacteristic, complemented
10#124×C211C2211yescharacteristic
11#121×C211—22no—
12#120×C211—22no—
13#117×C211C411yescharacteristic, complemented
14C2×C22⋊C4×C211—22no—
15#99×C211—22no—
16#99×C211C2214yes—
17C24×C211D412yes—
18#41×C211—42no—
19C2×Q8×C211C2×C411yescharacteristic, complemented
20C2×D4×C211—21no—
21C2×D4×C211—22no—
22C2×D4×C211—22no—
23C2×D4×C211—42no—
24C2×D4×C211C2×C411yescharacteristic
25C2×D4×C211C2×C411yescharacteristic, complemented
26C2×D4×C211D412yes—
27C2×D4×C211C2311yescharacteristic
28C22×C4×C211—21no—
29C22×C4×C211—42no—
30C22×C4×C211C2×C411yescharacteristic
31C4⋊C4×C211—21no—
32C22⋊C4×C211—22no—
33C22⋊C4×C211—21no—
34C22⋊C4×C211—22no—
35C22⋊C4×C211—24no—
36C22⋊C4×C211—42no—
37C42×C211C2×C411yescharacteristic, complemented
38C42×C211C2×C411yescharacteristic, complemented
39C23×C211—22no—
40C23×C211—22no—
41C23×C211—21no—
42C23×C211—82no—
43C23×C211C22⋊C412yes—
44C23×C211C22⋊C411yescharacteristic
45C23×C211C2×D412yes—
46Q8×C211—41no—
47D4×C211—41no—
48D4×C211—42no—
49D4×C211—41no—
50D4×C211—41no—
51D4×C211—42no—
52D4×C211—82no—
53D4×C211—82no—
54C2×C4×C211—22no—
55C2×C4×C211—21no—
56C2×C4×C211—21no—
57C2×C4×C211—21no—
58C2×C4×C211—42no—
59C2×C4×C211—44no—
60C2×C4×C211—84no—
61C2×C4×C211C22⋊C411yescharacteristic
62C2×C4×C211C22×C411yescharacteristic
63C22×C211—22no—
64C22×C211—22no—
65C22×C211—41no—
66C22×C211—42no—
67C22×C211—42no—
68C22×C211—44no—
69C22×C211—42no—
70C22×C211—41no—
71C22×C211—82no—
72C22×C211#9912yes—
73C22×C211C2×C22⋊C411yescharacteristic
74C4×C211—21no—
75C4×C211—21no—
76C4×C211—41no—
77C4×C211—41no—
78C4×C211—41no—
79C4×C211—84no—
80C2×C211—21no—
81C2×C211—22no—
82C2×C211—42no—
83C2×C211—42no—
84C2×C211—81no—
85C2×C211C2×#9911yescharacteristic
86C211#212711yescharacteristic, direct product factor, Sylow
87#2127C21111yesmaximal, characteristic, direct product factor, Sylow
88#535C2×C21111yescharacteristic
89C2×#99C2×C21112yescomplemented
90#354C2×C21112yescomplemented
91#353C2×C21112yescomplemented
92C2×#41C4×C21111yescharacteristic, complemented
93C22×D4C22×C21111yescharacteristic
94#127C4×C21111yescharacteristic, complemented
95#127C22×C21111yescharacteristic
96#124C4×C21111yescharacteristic, complemented
97#124C22×C21111yescharacteristic
98#121—22no—
99#120—22no—
100#117C4×C21111yescharacteristic, complemented
101C2×C22⋊C4—22no—
102#99—22no—
103#99C22×C21114yes—
104C24D4×C21112yes—
105#41—42no—
106C2×Q8C2×C4×C21111yescharacteristic, complemented
107C2×D4—21no—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—42no—
111C2×D4C2×C4×C21111yescharacteristic
112C2×D4C2×C4×C21111yescharacteristic, complemented
113C2×D4D4×C21112yes—
114C2×D4C23×C21111yescharacteristic
115C22×C4—21no—
116C22×C4—42no—
117C22×C4C2×C4×C21111yescharacteristic
118C4⋊C4—21no—
119C22⋊C4—21no—
120C22⋊C4—22no—
121C22⋊C4—22no—
122C22⋊C4—24no—
123C22⋊C4—42no—
124C42C2×C4×C21111yescharacteristic, complemented
125C42C2×C4×C21111yescharacteristic, complemented
126C23—22no—
127C23—22no—
128C23—21no—
129C23—82no—
130C23C22⋊C4×C21112yes—
131C23C22⋊C4×C21111yescharacteristic
132C23C2×D4×C21112yes—
133Q8—41no—
134D4—41no—
135D4—42no—
136D4—41no—
137D4—41no—
138D4—42no—
139D4—82no—
140D4—82no—
141C2×C4—22no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—42no—
146C2×C4—44no—
147C2×C4—84no—
148C2×C4C22⋊C4×C21111yescharacteristic
149C2×C4C22×C4×C21111yescharacteristic
150C22—22no—
151C22—22no—
152C22—41no—
153C22—42no—
154C22—42no—
155C22—44no—
156C22—42no—
157C22—41no—
158C22—82no—
159C22#99×C21112yes—
160C22C2×C22⋊C4×C21111yescharacteristic
161C4—21no—
162C4—21no—
163C4—41no—
164C4—41no—
165C4—41no—
166C4—84no—
167C2—21no—
168C2—22no—
169C2—42no—
170C2—42no—
171C2—81no—
172C2C2×#99×C21111yescharacteristic
subgroupG/H
center, socleC2×C211C2×#99
commutatorC2×C4C22×C4×C211
Frattini, cosocleC2×D4C23×C211
prime radical#535×C211C2
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C211 ◁ C23×C211 ◁ C22×D4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#99×C211C2semidirect
3. #354×C211C2semidirect
4. #353×C211C2semidirect
5. C2×#41×C211C4semidirect
7. #127×C211C4semidirect
9. #124×C211C4semidirect
13. #117×C211C4semidirect
19. C2×Q8×C211C2×C4semidirect
25. C2×D4×C211C2×C4semidirect
37. C42×C211C2×C4semidirect
38. C42×C211C2×C4semidirect
86. C211#2127direct (N × H)
87. #2127C211direct (N × H)
89. C2×#99C2×C211semidirect
90. #354C2×C211semidirect
91. #353C2×C211semidirect
92. C2×#41C4×C211semidirect
94. #127C4×C211semidirect
96. #124C4×C211semidirect
100. #117C4×C211semidirect
106. C2×Q8C2×C4×C211semidirect
112. C2×D4C2×C4×C211semidirect
124. C42C2×C4×C211semidirect
125. C42C2×C4×C211semidirect

Maximal subgroups

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

subgroupindex
#535×C2112
C2×#99×C2112
#354×C2112
#353×C2112
#2127211