Finite Group Databasebeta

← #2195×C211 #41009815

Subgroups — #2195×C211 #41009815

Order 27008.

206 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 64 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#578×C211C211yesmaximal, characteristic
2#510×C211C211yesmaximal, characteristic, complemented
3C2×Q32×C211C211yesmaximal, characteristic, complemented
4#473×C211C211yesmaximal, characteristic, complemented
5#362×C211C211yesmaximal, characteristic, complemented
6#359×C211C211yesmaximal, characteristic, complemented
7#350×C211C211yesmaximal, characteristic, complemented
82−1+4×C211—21no—
9#137×C211—22no—
10#137×C211—21no—
11#137×C211—21no—
12#135×C211—21no—
13#135×C211C2211yescharacteristic
14C2×Q16×C211—21no—
15C2×Q16×C211C2211yescharacteristic
16C2×Q16×C211C2211yescharacteristic
17#131×C211C2211yescharacteristic
18Q32×C211—22no—
19Q32×C211—21no—
20SD32×C211—21no—
21M32×C211C2211yescharacteristic
22C2×C16×C211C2211yescharacteristic
23#108×C211C2211yescharacteristic
24#101×C211—21no—
25#41×C211—22no—
26#41×C211—21no—
27#41×C211—42no—
28#41×C211—41no—
29#41×C211D411yescharacteristic
30C2×Q8×C211—22no—
31C2×Q8×C211—21no—
32C2×Q8×C211—21no—
33C2×Q8×C211—41no—
34Q16×C211—22no—
35Q16×C211—21no—
36Q16×C211—21no—
37Q16×C211—21no—
38Q16×C211—41no—
39Q16×C211D411yescharacteristic
40Q16×C211D412yes—
41SD16×C211—22no—
42SD16×C211—21no—
43SD16×C211—41no—
44D8×C211D411yescharacteristic
45M16×C211—21no—
46M16×C211—21no—
47M16×C211D411yescharacteristic
48C2×C8×C211—21no—
49C2×C8×C211C2311yescharacteristic
50C16×C211—21no—
51C16×C211—21no—
52Q8×C211—22no—
53Q8×C211—21no—
54Q8×C211—42no—
55Q8×C211—41no—
56Q8×C211—41no—
57Q8×C211—41no—
58Q8×C211D811yescharacteristic
59D4×C211—21no—
60D4×C211—21no—
61D4×C211—42no—
62D4×C211—41no—
63D4×C211D811yescharacteristic
64C2×C4×C211—21no—
65C2×C4×C211—42no—
66C2×C4×C211—42no—
67C2×C4×C211—41no—
68C2×C4×C211—41no—
69C2×C4×C211—41no—
70C2×C4×C211C2×D411yescharacteristic
71C8×C211—21no—
72C8×C211—41no—
73C8×C211C2×D411yescharacteristic
74C8×C211C2×D411yescharacteristic
75C22×C211—21no—
76C22×C211—41no—
77C22×C211AGL(1,ℤ/8)11yescharacteristic
78C4×C211—21no—
79C4×C211—42no—
80C4×C211—41no—
81C4×C211—81no—
82C4×C211#12011yescharacteristic
83C4×C211C2×D811yescharacteristic
84C2×C211—21no—
85C2×C211—41no—
86C2×C211—81no—
87C2×C211#44711yescharacteristic
88C211#219511yescharacteristic, direct product factor, Sylow
89#2195C21111yesmaximal, characteristic, direct product factor, Sylow
90#578C2×C21111yescharacteristic
91#510C2×C21111yescharacteristic, complemented
92C2×Q32C2×C21111yescharacteristic, complemented
93#473C2×C21111yescharacteristic, complemented
94#362C2×C21111yescharacteristic, complemented
95#359C2×C21111yescharacteristic, complemented
96#350C2×C21111yescharacteristic, complemented
972−1+4—21no—
98#137—22no—
99#137—21no—
100#137—21no—
101#135—21no—
102#135C22×C21111yescharacteristic
103C2×Q16—21no—
104C2×Q16C22×C21111yescharacteristic
105C2×Q16C22×C21111yescharacteristic
106#131C22×C21111yescharacteristic
107Q32—22no—
108Q32—21no—
109SD32—21no—
110M32C22×C21111yescharacteristic
111C2×C16C22×C21111yescharacteristic
112#108C22×C21111yescharacteristic
113#101—21no—
114#41—22no—
115#41—21no—
116#41—42no—
117#41—41no—
118#41D4×C21111yescharacteristic
119C2×Q8—22no—
120C2×Q8—21no—
121C2×Q8—21no—
122C2×Q8—41no—
123Q16—22no—
124Q16—21no—
125Q16—21no—
126Q16—21no—
127Q16—41no—
128Q16D4×C21111yescharacteristic
129Q16D4×C21112yes—
130SD16—22no—
131SD16—21no—
132SD16—41no—
133D8D4×C21111yescharacteristic
134M16—21no—
135M16—21no—
136M16D4×C21111yescharacteristic
137C2×C8—21no—
138C2×C8C23×C21111yescharacteristic
139C16—21no—
140C16—21no—
141Q8—22no—
142Q8—21no—
143Q8—42no—
144Q8—41no—
145Q8—41no—
146Q8—41no—
147Q8D8×C21111yescharacteristic
148D4—21no—
149D4—21no—
150D4—42no—
151D4—41no—
152D4D8×C21111yescharacteristic
153C2×C4—21no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4C2×D4×C21111yescharacteristic
160C8—21no—
161C8—41no—
162C8C2×D4×C21111yescharacteristic
163C8C2×D4×C21111yescharacteristic
164C22—21no—
165C22—41no—
166C22AGL(1,ℤ/8)×C21111yescharacteristic
167C4—21no—
168C4—42no—
169C4—41no—
170C4—81no—
171C4#120×C21111yescharacteristic
172C4C2×D8×C21111yescharacteristic
173C2—21no—
174C2—41no—
175C2—81no—
176C2#447×C21111yescharacteristic
subgroupG/H
center, socleC2×C211#447
commutator, Frattini, cosocleC2×C8C23×C211
prime radical#578×C211C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C211 ◁ C2×C4×C211 ◁ #131×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. #510×C211C2semidirect
3. C2×Q32×C211C2semidirect
4. #473×C211C2semidirect
5. #362×C211C2semidirect
6. #359×C211C2semidirect
7. #350×C211C2semidirect
88. C211#2195direct (N × H)
89. #2195C211direct (N × H)
91. #510C2×C211semidirect
92. C2×Q32C2×C211semidirect
93. #473C2×C211semidirect
94. #362C2×C211semidirect
95. #359C2×C211semidirect
96. #350C2×C211semidirect

Maximal subgroups

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

subgroupindex
#578×C2112
#510×C2112
C2×Q32×C2112
#473×C2112
#362×C2112
#359×C2112
#350×C2112
#2195211