Finite Group Databasebeta

← #2071×C211 #41009691

Subgroups — #2071×C211 #41009691

Order 27008.

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

normal subgroups only

This group has 274 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
1C2×#102×C211C212yesmaximal
2#390×C211C211yesmaximal, characteristic
3#384×C211C211yesmaximal, characteristic, complemented
4#340×C211C212yesmaximal, complemented
5#336×C211C211yesmaximal, characteristic, complemented
6C22×D4×C211C2211yescharacteristic
7C22×C8×C211C2212yes—
8C2×C4⋊C4×C211—21no—
9C2×C4⋊C4×C211—21no—
10C2×C4⋊C4×C211C2211yescharacteristic
11C2×C4⋊C4×C211C2212yes—
12C2×C22⋊C4×C211—21no—
13C2×C22⋊C4×C211—21no—
14C2×C42×C211C2211yescharacteristic
15#102×C211—24no—
16C24×C211—21no—
17C2×D4×C211—22no—
18C2×D4×C211—22no—
19C2×D4×C211—21no—
20C2×D4×C211D412yes—
21C22×C4×C211—21no—
22C22×C4×C211—22no—
23C22×C4×C211D411yescharacteristic
24C22×C4×C211D411yescharacteristic
25C22×C4×C211C2311yescharacteristic
26C2×C8×C211—22no—
27C2×C8×C211—22no—
28C2×C8×C211—22no—
29C4⋊C4×C211—22no—
30C4⋊C4×C211—22no—
31C4⋊C4×C211—22no—
32C4⋊C4×C211—42no—
33C4⋊C4×C211—42no—
34C22⋊C4×C211—42no—
35C22⋊C4×C211—42no—
36C42×C211—22no—
37C23×C211—22no—
38C23×C211—41no—
39C23×C211—41no—
40C23×C211—41no—
41C23×C211—41no—
42C23×C211—41no—
43C23×C211—41no—
44C23×C211C2×D411yescharacteristic
45D4×C211—24no—
46D4×C211—42no—
47C2×C4×C211—21no—
48C2×C4×C211—21no—
49C2×C4×C211—21no—
50C2×C4×C211—21no—
51C2×C4×C211—42no—
52C2×C4×C211—41no—
53C2×C4×C211—41no—
54C2×C4×C211—42no—
55C2×C4×C211—41no—
56C2×C4×C211—42no—
57C2×C4×C211D812yes—
58C2×C4×C211SD1612yes—
59C2×C4×C211C2×D411yescharacteristic
60C2×C4×C211#4112yes—
61C2×C4×C211#4112yes—
62C2×C4×C211#4111yescharacteristic
63C8×C211—42no—
64C22×C211—42no—
65C22×C211—42no—
66C22×C211—42no—
67C22×C211—81no—
68C22×C211—81no—
69C22×C211—81no—
70C22×C211—81no—
71C22×C211#12111yescharacteristic
72C22×C211#12311yescharacteristic
73C22×C211#12411yescharacteristic
74C22×C211C2×D811yescharacteristic
75C22×C211C2×SD1611yescharacteristic
76C22×C211AGL(1,ℤ/8)11yescharacteristic
77C22×C211#13711yescharacteristic
78C4×C211—22no—
79C4×C211—22no—
80C4×C211—82no—
81C4×C211—81no—
82C4×C211#12312yes—
83C4×C211#12612yes—
84C2×C211—82no—
85C2×C211#39711yescharacteristic
86C2×C211#45911yescharacteristic
87C2×C211#46111yescharacteristic
88C2×C211#48011yescharacteristic
89C2×C211#48111yescharacteristic
90C2×C211#48611yescharacteristic
91C2×C211#48911yescharacteristic
92C211#207111yescharacteristic, direct product factor, Sylow
93#2071C21111yesmaximal, characteristic, direct product factor, Sylow
94C2×#102C2×C21112yes—
95#390C2×C21111yescharacteristic
96#384C2×C21111yescharacteristic, complemented
97#340C2×C21112yescomplemented
98#336C2×C21111yescharacteristic, complemented
99C22×D4C22×C21111yescharacteristic
100C22×C8C22×C21112yes—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4C22×C21111yescharacteristic
104C2×C4⋊C4C22×C21112yes—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C42C22×C21111yescharacteristic
108#102—24no—
109C24—21no—
110C2×D4—22no—
111C2×D4—22no—
112C2×D4—21no—
113C2×D4D4×C21112yes—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4D4×C21111yescharacteristic
117C22×C4D4×C21111yescharacteristic
118C22×C4C23×C21111yescharacteristic
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—42no—
126C4⋊C4—42no—
127C22⋊C4—42no—
128C22⋊C4—42no—
129C42—22no—
130C23—22no—
131C23—41no—
132C23—41no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23C2×D4×C21111yescharacteristic
138D4—24no—
139D4—42no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—42no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—42no—
148C2×C4—41no—
149C2×C4—42no—
150C2×C4D8×C21112yes—
151C2×C4SD16×C21112yes—
152C2×C4C2×D4×C21111yescharacteristic
153C2×C4#41×C21112yes—
154C2×C4#41×C21112yes—
155C2×C4#41×C21111yescharacteristic
156C8—42no—
157C22—42no—
158C22—42no—
159C22—42no—
160C22—81no—
161C22—81no—
162C22—81no—
163C22—81no—
164C22#121×C21111yescharacteristic
165C22#123×C21111yescharacteristic
166C22#124×C21111yescharacteristic
167C22C2×D8×C21111yescharacteristic
168C22C2×SD16×C21111yescharacteristic
169C22AGL(1,ℤ/8)×C21111yescharacteristic
170C22#137×C21111yescharacteristic
171C4—22no—
172C4—22no—
173C4—82no—
174C4—81no—
175C4#123×C21112yes—
176C4#126×C21112yes—
177C2—82no—
178C2#397×C21111yescharacteristic
179C2#459×C21111yescharacteristic
180C2#461×C21111yescharacteristic
181C2#480×C21111yescharacteristic
182C2#481×C21111yescharacteristic
183C2#486×C21111yescharacteristic
184C2#489×C21111yescharacteristic
subgroupG/H
center, socleC23×C211C2×D4
commutator, Frattini, cosocleC22×C4C23×C211
prime radicalC22×D4×C211C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C211 ◁ C2×C42×C211 ◁ G

Complements

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

NH ≅ G/Ntype
3. #384×C211C2semidirect
4. #340×C211C2semidirect
5. #336×C211C2semidirect
92. C211#2071direct (N × H)
93. #2071C211direct (N × H)
96. #384C2×C211semidirect
97. #340C2×C211semidirect
98. #336C2×C211semidirect

Maximal subgroups

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

subgroupindex
C2×#102×C2112
#390×C2112
#384×C2112
#340×C2112
#336×C2112
#2071211