Finite Group Databasebeta

← #1884×C211 #41009504

Subgroups — #1884×C211 #41009504

Order 27008.

332 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 108 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#568×C211C211yesmaximal, characteristic, complemented
2#532×C211C211yesmaximal, characteristic
3#421×C211C212yesmaximal, complemented
4#410×C211C211yesmaximal, characteristic, complemented
5#339×C211C212yesmaximal, complemented
6C2×#41×C211C2211yescharacteristic
7C2×#41×C211C2211yescharacteristic
8#131×C211—24no—
9C2×M16×C211—22no—
10C2×M16×C211C2212yes—
11C4⋊Q8×C211C411yescharacteristic, complemented
12#127×C211C411yescharacteristic, complemented
13#124×C211C411yescharacteristic, complemented
14#124×C211C411yescharacteristic, complemented
15#121×C211—22no—
16C4×D4×C211—22no—
17#117×C211C2212yes—
18C2×C42×C211C2211yescharacteristic
19#104×C211—24no—
20#99×C211—22no—
21#41×C211—22no—
22#41×C211—22no—
23#41×C211—22no—
24#41×C211—22no—
25C2×Q8×C211C2×C411yescharacteristic
26C2×Q8×C211C2×C411yescharacteristic
27C2×D4×C211—22no—
28C2×D4×C211—21no—
29C2×D4×C211C2×C411yescharacteristic
30C2×D4×C211C2×C411yescharacteristic
31C2×D4×C211D412yes—
32C22×C4×C211—21no—
33C22×C4×C211—21no—
34C22×C4×C211D412yes—
35C22×C4×C211C2311yescharacteristic
36M16×C211—24no—
37M16×C211—22no—
38M16×C211D414yescomplemented
39C2×C8×C211—24no—
40C2×C8×C211—22no—
41C4⋊C4×C211—22no—
42C4⋊C4×C211—22no—
43C22⋊C4×C211—22no—
44C22⋊C4×C211—22no—
45C22⋊C4×C211—22no—
46C42×C211—21no—
47C42×C211—22no—
48C42×C211C2×C411yescharacteristic
49C42×C211C2×C411yescharacteristic
50C23×C211—21no—
51C23×C211C22⋊C412yes—
52C23×C211C2×D411yescharacteristic
53Q8×C211—22no—
54Q8×C211—22no—
55D4×C211—24no—
56D4×C211—22no—
57D4×C211—22no—
58D4×C211—44no—
59D4×C211—42no—
60C2×C4×C211—21no—
61C2×C4×C211—22no—
62C2×C4×C211—22no—
63C2×C4×C211—21no—
64C2×C4×C211—21no—
65C2×C4×C211—42no—
66C2×C4×C211—41no—
67C2×C4×C211—41no—
68C2×C4×C211—42no—
69C2×C4×C211C22⋊C412yes—
70C2×C4×C211C22×C411yescharacteristic
71C2×C4×C211C2×D412yes—
72C2×C4×C211C2×D411yescharacteristic
73C2×C4×C211#4112yes—
74C8×C211—24no—
75C22×C211—22no—
76C22×C211—41no—
77C22×C211—42no—
78C22×C211—42no—
79C22×C211—41no—
80C22×C211C2×C22⋊C411yescharacteristic
81C22×C211#12112yes—
82C4×C211—22no—
83C4×C211—22no—
84C4×C211—41no—
85C4×C211—41no—
86C4×C211—82no—
87C4×C211C4×D412yes—
88C4×C211#12011yescharacteristic
89C4×C211#12311yescharacteristic
90C2×C211—21no—
91C2×C211—22no—
92C2×C211—42no—
93C2×C211—81no—
94C2×C211#38611yescharacteristic
95C211#188411yescharacteristic, direct product factor, Sylow
96#1884C21111yesmaximal, characteristic, direct product factor, Sylow
97#568C2×C21111yescharacteristic, complemented
98#532C2×C21111yescharacteristic
99#421C2×C21112yescomplemented
100#410C2×C21111yescharacteristic, complemented
101#339C2×C21112yescomplemented
102C2×#41C22×C21111yescharacteristic
103C2×#41C22×C21111yescharacteristic
104#131—24no—
105C2×M16—22no—
106C2×M16C22×C21112yes—
107C4⋊Q8C4×C21111yescharacteristic, complemented
108#127C4×C21111yescharacteristic, complemented
109#124C4×C21111yescharacteristic, complemented
110#124C4×C21111yescharacteristic, complemented
111#121—22no—
112C4×D4—22no—
113#117C22×C21112yes—
114C2×C42C22×C21111yescharacteristic
115#104—24no—
116#99—22no—
117#41—22no—
118#41—22no—
119#41—22no—
120#41—22no—
121C2×Q8C2×C4×C21111yescharacteristic
122C2×Q8C2×C4×C21111yescharacteristic
123C2×D4—22no—
124C2×D4—21no—
125C2×D4C2×C4×C21111yescharacteristic
126C2×D4C2×C4×C21111yescharacteristic
127C2×D4D4×C21112yes—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4D4×C21112yes—
131C22×C4C23×C21111yescharacteristic
132M16—24no—
133M16—22no—
134M16D4×C21114yescomplemented
135C2×C8—24no—
136C2×C8—22no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C42—21no—
143C42—22no—
144C42C2×C4×C21111yescharacteristic
145C42C2×C4×C21111yescharacteristic
146C23—21no—
147C23C22⋊C4×C21112yes—
148C23C2×D4×C21111yescharacteristic
149Q8—22no—
150Q8—22no—
151D4—24no—
152D4—22no—
153D4—22no—
154D4—44no—
155D4—42no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4C22⋊C4×C21112yes—
166C2×C4C22×C4×C21111yescharacteristic
167C2×C4C2×D4×C21112yes—
168C2×C4C2×D4×C21111yescharacteristic
169C2×C4#41×C21112yes—
170C8—24no—
171C22—22no—
172C22—41no—
173C22—42no—
174C22—42no—
175C22—41no—
176C22C2×C22⋊C4×C21111yescharacteristic
177C22#121×C21112yes—
178C4—22no—
179C4—22no—
180C4—41no—
181C4—41no—
182C4—82no—
183C4C4×D4×C21112yes—
184C4#120×C21111yescharacteristic
185C4#123×C21111yescharacteristic
186C2—21no—
187C2—22no—
188C2—42no—
189C2—81no—
190C2#386×C21111yescharacteristic
subgroupG/H
centerC4×C211#120
commutatorC2×C4C22×C4×C211
Frattini, cosocleC22×C4C23×C211
socleC2×C211#386
prime radical#532×C211C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C211 ◁ C22×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
1. #568×C211C2semidirect
3. #421×C211C2semidirect
4. #410×C211C2semidirect
5. #339×C211C2semidirect
11. C4⋊Q8×C211C4semidirect
12. #127×C211C4semidirect
13. #124×C211C4semidirect
14. #124×C211C4semidirect
38. M16×C211D4semidirect
95. C211#1884direct (N × H)
96. #1884C211direct (N × H)
97. #568C2×C211semidirect
99. #421C2×C211semidirect
100. #410C2×C211semidirect
101. #339C2×C211semidirect
107. C4⋊Q8C4×C211semidirect
108. #127C4×C211semidirect
109. #124C4×C211semidirect
110. #124C4×C211semidirect
134. M16D4×C211semidirect

Maximal subgroups

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

subgroupindex
#568×C2112
#532×C2112
#421×C2112
#410×C2112
#339×C2112
#1884211