Finite Group Databasebeta

← #41013797

Subgroups — #41013797

Order 27008.

444 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 97 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#27213621C211yesmaximal, characteristic, complemented
2#27213151C212yesmaximal, complemented
3C2×#20269894C211yesmaximal, characteristic, complemented
4#27212906C212yesmaximal, complemented
5C2×#127×C211C211yesmaximal, characteristic
6C2×#16809946C2212yes—
7D4×Dic211—24no—
8#20269936C2212yes—
9C2×C22⋊Dic211—22no—
10#20269898—24no—
11#20269894—22no—
12C22×D4×C211—22no—
13C22×D4×C211C2211yescharacteristic
14C22×D4×C211C2211yescharacteristic
15#127×C211—22no—
16#127×C211C414yescomplemented
17C2×C42×C211C2211yescharacteristic
18C22×Dic211—24no—
19#16809946—22no—
20#16809946—22no—
21C2×C211⋊C8—22no—
22C4⋊Dic211D414yescomplemented
23C22⋊Dic211—24no—
24C22⋊Dic211—22no—
25C4×Dic211—22no—
26C24×C211—21no—
27C24×C211D412yes—
28C2×D4×C211—28no—
29C2×D4×C211—22no—
30C2×D4×C211—22no—
31C2×D4×C211—21no—
32C2×D4×C211—22no—
33C2×D4×C211—22no—
34C2×D4×C211—48no—
35C2×D4×C211C2×C412yes—
36C2×D4×C211C2×C412yescomplemented
37C2×D4×C211D412yes—
38C22×C4×C211—21no—
39C22×C4×C211C2311yescharacteristic
40C42×C211—21no—
41C42×C211C2×C412yescomplemented
42C2×Dic211—24no—
43C2×Dic211—84no—
44C211⋊C8—42no—
45C23×C211—22no—
46C23×C211—22no—
47C23×C211—22no—
48C23×C211—44no—
49C23×C211—44no—
50C23×C211—41no—
51C23×C211—41no—
52C23×C211C22⋊C414yes—
53C23×C211C2×D411yescharacteristic
54D4×C211—24no—
55D4×C211—24no—
56D4×C211—24no—
57D4×C211—416no—
58D4×C211—42no—
59C2×C4×C211—24no—
60C2×C4×C211—22no—
61C2×C4×C211C22×C411yescharacteristic
62C2×C4×C211C2×D412yes—
63C2×C4×C211C2×D411yescharacteristic
64C2×C4×C211#4112yes—
65C22×C211—22no—
66C22×C211—24no—
67C22×C211—44no—
68C22×C211—44no—
69C22×C211—44no—
70C22×C211—42no—
71C22×C211—82no—
72C22×C211—82no—
73C22×C211—82no—
74C22×C211C2×C22⋊C411yescharacteristic
75C22×C211#12011yescharacteristic
76C22×C211#12311yescharacteristic
77C4×C211—24no—
78C4×C211C4×D412yes—
79C4×C211#12112yes—
80Dic211—44no—
81C2×C211—21no—
82C2×C211—21no—
83C2×C211—44no—
84C2×C211—82no—
85C2×C211#38611yescharacteristic
86C2×C211#45312yes—
87C211#188711yescharacteristic, complemented, Sylow
88#1887—2111nomaximal, Sylow
89C2×#127D21111yescharacteristic
90#518—2111no—
91#418—2112no—
92C2×#100—2111no—
93#339—2112no—
94C22×D4—22no—
95C22×D4C2×D21111yescharacteristic
96C22×D4C2×D21111yescharacteristic
97C2×M16—2112no—
98#127—22no—
99#127Dic21114yescomplemented
100C4×D4—4224no—
101#117—2112no—
102C2×C22⋊C4—4222no—
103C2×C42C2×D21111yescharacteristic
104#102—4224no—
105#100—4222no—
106C24—21no—
107C24C22⋊D21112yes—
108C2×D4—28no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—21no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—48no—
115C2×D4C2×Dic21112yes—
116C2×D4C2×Dic21112yescomplemented
117C2×D4C22⋊D21112yes—
118C22×C4—21no—
119C22×C4—4224no—
120C22×C4C22×D21111yescharacteristic
121M16—4222no—
122M16—4222no—
123C2×C8—4222no—
124C4⋊C4—2114no—
125C22⋊C4—4224no—
126C22⋊C4—4222no—
127C42—21no—
128C42—4222no—
129C42C2×Dic21112yescomplemented
130C23—22no—
131C23—22no—
132C23—22no—
133C23—44no—
134C23—44no—
135C23—41no—
136C23—41no—
137C23C22⋊Dic21114yes—
138C23C2×C22⋊D21111yescharacteristic
139D4—24no—
140D4—24no—
141D4—24no—
142D4—416no—
143D4—42no—
144C2×C4—24no—
145C2×C4—22no—
146C2×C4—16884no—
147C2×C4—4224no—
148C2×C4C22×Dic21111yescharacteristic
149C2×C4D4×D21112yes—
150C2×C4C2×C22⋊D21111yescharacteristic
151C2×C4#1680996112yes—
152C8—8442no—
153C22—22no—
154C22—24no—
155C22—44no—
156C22—44no—
157C22—44no—
158C22—42no—
159C22—82no—
160C22—82no—
161C22—82no—
162C22C2×C22⋊Dic21111yescharacteristic
163C22#2026994511yescharacteristic
164C22#2026995811yescharacteristic
165C4—24no—
166C4—8444no—
167C4D4×Dic21112yes—
168C4#2026994912yes—
169C2—21no—
170C2—21no—
171C2—44no—
172C2—82no—
173C2#2721302611yescharacteristic
174C2#2721331312yes—
subgroupG/H
centerC22#20269945
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting, prime radicalC2×#127×C211C2
socleC22×C211#120
cosocleC22×C4×C211C23
hypercenterC2×#127D211
hypocenterC211#1887
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #27213621C2semidirect
2. #27213151C2semidirect
3. C2×#20269894C2semidirect
4. #27212906C2semidirect
16. #127×C211C4semidirect
22. C4⋊Dic211D4semidirect
36. C2×D4×C211C2×C4semidirect
41. C42×C211C2×C4semidirect
87. C211#1887semidirect
99. #127Dic211semidirect
116. C2×D4C2×Dic211semidirect
129. C42C2×Dic211semidirect

Maximal subgroups

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

subgroupindex
#272136212
#272131512
C2×#202698942
#272129062
C2×#127×C2112
#1887211