Finite Group Databasebeta

← #1856×C211 #41009476

Subgroups — #1856×C211 #41009476

Order 27008.

324 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 112 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#518×C211C211yesmaximal, characteristic
2#428×C211C212yesmaximal, complemented
3C2×#100×C211C211yesmaximal, characteristic
4#390×C211C211yesmaximal, characteristic
5#339×C211C212yesmaximal, complemented
6C22×D4×C211C2211yescharacteristic
7C2×M16×C211C2212yes—
8C4×D4×C211—22no—
9C4×D4×C211—22no—
10#117×C211C2212yes—
11C2×C4⋊C4×C211C2211yescharacteristic
12C2×C22⋊C4×C211—22no—
13C2×C22⋊C4×C211—22no—
14C2×C42×C211C2211yescharacteristic
15#107×C211—22no—
16#106×C211—22no—
17#100×C211C414yescomplemented
18C24×C211D412yes—
19C2×D4×C211—22no—
20C2×D4×C211—22no—
21C2×D4×C211C2×C412yes—
22C2×D4×C211Q812yes—
23C22×C4×C211—21no—
24C22×C4×C211—22no—
25C22×C4×C211—22no—
26C22×C4×C211—21no—
27C22×C4×C211C2311yescharacteristic
28M16×C211—22no—
29M16×C211C2×C414yescomplemented
30C2×C8×C211—22no—
31C4⋊C4×C211—22no—
32C4⋊C4×C211D412yes—
33C4⋊C4×C211D412yes—
34C22⋊C4×C211—24no—
35C22⋊C4×C211—22no—
36C22⋊C4×C211—44no—
37C42×C211—22no—
38C42×C211—22no—
39C23×C211—22no—
40C23×C211—22no—
41C23×C211—42no—
42C23×C211—42no—
43C23×C211C4⋊C414yes—
44C23×C211C2×D411yescharacteristic
45D4×C211—24no—
46D4×C211—24no—
47C2×C4×C211—22no—
48C2×C4×C211—22no—
49C2×C4×C211—22no—
50C2×C4×C211—41no—
51C2×C4×C211—41no—
52C2×C4×C211—42no—
53C2×C4×C211—41no—
54C2×C4×C211—82no—
55C2×C4×C211—82no—
56C2×C4×C211C22×C411yescharacteristic
57C2×C4×C211C2×D412yes—
58C2×C4×C211C2×Q811yescharacteristic
59C2×C4×C211#4112yes—
60C8×C211—24no—
61C22×C211—21no—
62C22×C211—21no—
63C22×C211—24no—
64C22×C211—44no—
65C22×C211—42no—
66C22×C211—42no—
67C22×C211—82no—
68C22×C211C2×C4⋊C411yescharacteristic
69C22×C211#12011yescharacteristic
70C22×C211#12311yescharacteristic
71C4×C211—42no—
72C4×C211—42no—
73C4×C211—42no—
74C4×C211—81no—
75C4×C211C4×D412yes—
76C4×C211C22⋊Q812yes—
77C2×C211—21no—
78C2×C211—21no—
79C2×C211—44no—
80C2×C211#38511yescharacteristic
81C2×C211#45311yescharacteristic
82C2×C211#45511yescharacteristic
83C211#185611yescharacteristic, direct product factor, Sylow
84#1856C21111yesmaximal, characteristic, direct product factor, Sylow
85#518C2×C21111yescharacteristic
86#428C2×C21112yescomplemented
87C2×#100C2×C21111yescharacteristic
88#390C2×C21111yescharacteristic
89#339C2×C21112yescomplemented
90C22×D4C22×C21111yescharacteristic
91C2×M16C22×C21112yes—
92C4×D4—22no—
93C4×D4—22no—
94#117C22×C21112yes—
95C2×C4⋊C4C22×C21111yescharacteristic
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C42C22×C21111yescharacteristic
99#107—22no—
100#106—22no—
101#100C4×C21114yescomplemented
102C24D4×C21112yes—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4C2×C4×C21112yes—
106C2×D4Q8×C21112yes—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4C23×C21111yescharacteristic
112M16—22no—
113M16C2×C4×C21114yescomplemented
114C2×C8—22no—
115C4⋊C4—22no—
116C4⋊C4D4×C21112yes—
117C4⋊C4D4×C21112yes—
118C22⋊C4—24no—
119C22⋊C4—22no—
120C22⋊C4—44no—
121C42—22no—
122C42—22no—
123C23—22no—
124C23—22no—
125C23—42no—
126C23—42no—
127C23C4⋊C4×C21114yes—
128C23C2×D4×C21111yescharacteristic
129D4—24no—
130D4—24no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—42no—
137C2×C4—41no—
138C2×C4—82no—
139C2×C4—82no—
140C2×C4C22×C4×C21111yescharacteristic
141C2×C4C2×D4×C21112yes—
142C2×C4C2×Q8×C21111yescharacteristic
143C2×C4#41×C21112yes—
144C8—24no—
145C22—21no—
146C22—21no—
147C22—24no—
148C22—44no—
149C22—42no—
150C22—42no—
151C22—82no—
152C22C2×C4⋊C4×C21111yescharacteristic
153C22#120×C21111yescharacteristic
154C22#123×C21111yescharacteristic
155C4—42no—
156C4—42no—
157C4—42no—
158C4—81no—
159C4C4×D4×C21112yes—
160C4C22⋊Q8×C21112yes—
161C2—21no—
162C2—21no—
163C2—44no—
164C2#385×C21111yescharacteristic
165C2#453×C21111yescharacteristic
166C2#455×C21111yescharacteristic
subgroupG/H
center, socleC22×C211#120
commutatorC2×C4C22×C4×C211
Frattini, cosocleC22×C4C23×C211
prime radicalC22×D4×C211C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C211 ◁ C22×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. #428×C211C2semidirect
5. #339×C211C2semidirect
17. #100×C211C4semidirect
29. M16×C211C2×C4semidirect
83. C211#1856direct (N × H)
84. #1856C211direct (N × H)
86. #428C2×C211semidirect
89. #339C2×C211semidirect
101. #100C4×C211semidirect
113. M16C2×C4×C211semidirect

Maximal subgroups

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

subgroupindex
#518×C2112
#428×C2112
C2×#100×C2112
#390×C2112
#339×C2112
#1856211