Finite Group Databasebeta

← #41014761

Subgroups — #41014761

Order 27008.

208 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 69 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×#20269912C211yesmaximal, characteristic
2#27213188C211yesmaximal, characteristic
3#27213174C211yesmaximal, characteristic
4#27213155C211yesmaximal, characteristic
5C2×#20269901C211yesmaximal, characteristic
6#27212914C211yesmaximal, characteristic
7C2×#101×C211C211yesmaximal, characteristic
8C2×#16809946C2211yescharacteristic
9C22×C211⋊C8C2211yescharacteristic
10#20269936C2211yescharacteristic
11C2×C4⋊Dic211C2211yescharacteristic
12#20269912—22no—
13#20269911—21no—
14#20269909—21no—
15Dic211⋊C8—22no—
16#20269901—22no—
17#20269901—21no—
18#20269901—21no—
19C22×Q8×C211C2211yescharacteristic
20C2×M16×C211C2211yescharacteristic
21C2×M16×C211C2211yescharacteristic
22#101×C211—22no—
23C22×Dic211—21no—
24#16809946—21no—
25#16809946—21no—
26C2×C211⋊C8—21no—
27C2×C211⋊C8—21no—
28C2×C211⋊C8—21no—
29C2×C211⋊C8D412yes—
30C4⋊Dic211—21no—
31C4⋊Dic211—21no—
32C4⋊Dic211D411yescharacteristic
33C4⋊Dic211D411yescharacteristic
34C22⋊Dic211—21no—
35C4×Dic211—21no—
36C2×Q8×C211—21no—
37C2×Q8×C211—21no—
38C2×Q8×C211—21no—
39C2×Q8×C211—21no—
40C2×Q8×C211—21no—
41C2×Q8×C211Q812yes—
42C22×C4×C211—21no—
43C22×C4×C211C2311yescharacteristic
44M16×C211—21no—
45M16×C211—21no—
46M16×C211—21no—
47M16×C211D412yes—
48C2×C8×C211—21no—
49C2×C8×C211—21no—
50C2×Dic211—21no—
51C2×Dic211—21no—
52C2×Dic211—41no—
53C2×Dic211—81no—
54C211⋊C8—22no—
55C211⋊C8—41no—
56C23×C211#4111yescharacteristic
57Q8×C211—22no—
58Q8×C211—22no—
59Q8×C211—41no—
60Q8×C211—41no—
61C2×C4×C211—22no—
62C2×C4×C211—41no—
63C2×C4×C211—41no—
64C2×C4×C211C2×D411yescharacteristic
65C2×C4×C211C2×D411yescharacteristic
66C2×C4×C211C2×D411yescharacteristic
67C2×C4×C211C2×Q811yescharacteristic
68C2×C4×C211#4111yescharacteristic
69C2×C4×C211#4111yescharacteristic
70C8×C211—22no—
71C8×C211—41no—
72C22×C211—21no—
73C22×C211—21no—
74C22×C211#12111yescharacteristic
75C22×C211C22⋊Q811yescharacteristic
76C22×C211#12611yescharacteristic
77C4×C211—42no—
78C4×C211#12111yescharacteristic
79C4×C211#12111yescharacteristic
80C4×C211C22⋊Q811yescharacteristic
81C4×C211C22⋊Q811yescharacteristic
82Dic211—41no—
83Dic211—41no—
84Dic211—81no—
85C2×C211—21no—
86C2×C211—21no—
87C2×C211#39611yescharacteristic
88C2×C211#47111yescharacteristic
89C2×C211#47311yescharacteristic
90C211#206411yescharacteristic, complemented, Sylow
91#2064—2111nomaximal, Sylow
92C2×#108—2111no—
93#428—2111no—
94#424—2111no—
95#419—2111no—
96C2×#103—2111no—
97C2×#101D21111yescharacteristic
98#343—2111no—
99C22×Q8C2×D21111yescharacteristic
100C2×M16—2111no—
101C2×M16C2×D21111yescharacteristic
102C2×M16C2×D21111yescharacteristic
103C22×C8—2111no—
104#117—2111no—
105C2×C4⋊C4—2111no—
106#108—4222no—
107#107—4221no—
108#106—4221no—
109C4⋊C8—4222no—
110#103—4222no—
111#103—4221no—
112#103—4221no—
113#101—22no—
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—21no—
118C2×Q8—21no—
119C2×Q8Dic42212yes—
120C22×C4—21no—
121C22×C4—4221no—
122C22×C4C22×D21111yescharacteristic
123M16—21no—
124M16—21no—
125M16—21no—
126M16—4221no—
127M16—4221no—
128M16C22⋊D21112yes—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—4221no—
132C2×C8—4221no—
133C2×C8—4221no—
134C2×C8—2112no—
135C4⋊C4—4221no—
136C4⋊C4—4221no—
137C4⋊C4—2111no—
138C4⋊C4—2111no—
139C22⋊C4—4221no—
140C42—4221no—
141C23#1680996211yescharacteristic
142Q8—22no—
143Q8—22no—
144Q8—41no—
145Q8—41no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—8441no—
150C2×C4—16881no—
151C2×C4—4221no—
152C2×C4—4221no—
153C2×C4D4×D21111yescharacteristic
154C2×C4D4×D21111yescharacteristic
155C2×C4C2×C22⋊D21111yescharacteristic
156C2×C4C2×Dic42211yescharacteristic
157C2×C4#1680996111yescharacteristic
158C2×C4#1680996111yescharacteristic
159C8—22no—
160C8—41no—
161C8—8441no—
162C8—4222no—
163C22—21no—
164C22—21no—
165C22#2026995011yescharacteristic
166C22#2026995611yescharacteristic
167C22#2026997011yescharacteristic
168C4—42no—
169C4—8441no—
170C4—8441no—
171C4—16881no—
172C4#2026994811yescharacteristic
173C4#2026994911yescharacteristic
174C4#2026995411yescharacteristic
175C4#2026995411yescharacteristic
176C2—21no—
177C2—21no—
178C2#2721306911yescharacteristic
179C2#2721341211yescharacteristic
180C2#2721342011yescharacteristic
subgroupG/H
centerC22#20269950
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#101×C211C2
socleC22×C211#121
prime radicalC23×C211#41
hypercenterC2×#101D211
hypocenterC211#2064
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
90. C211#2064semidirect

Maximal subgroups

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

subgroupindex
C2×#202699122
#272131882
#272131742
#272131552
C2×#202699012
#272129142
C2×#101×C2112
#2064211