Finite Group Databasebeta

← #41014628

Subgroups — #41014628

Order 27008.

290 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 67 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#27213727C211yesmaximal, characteristic
2C2×#20269903C212yesmaximal, complemented
3#27213148C212yesmaximal, complemented
4#27213011C211yesmaximal, characteristic, complemented
5#343×C211C211yesmaximal, characteristic, complemented
6C2×#16809962C2212yes—
7#20269957—22no—
8#20269952—22no—
9#20269951—22no—
10C2×C4⋊Dic211C2211yescharacteristic
11C2×C4×Dic211C2211yescharacteristic
12#20269903—24no—
13#20269902—22no—
14#20269899—22no—
15C211⋊#95—22no—
16C2×M16×C211C2212yes—
17C2×C4⋊C4×C211C2211yescharacteristic
18#16809962—22no—
19#16809962—22no—
20#16809962—22no—
21#16809962—22no—
22C2×Dic422D412yes—
23C2×D844—22no—
24C2×D844D412yes—
25C2×C22⋊D211—22no—
26C22×Dic211—21no—
27C22×Dic211—21no—
28C2×C4×D211—22no—
29C4⋊Dic211—22no—
30C4⋊Dic211—22no—
31(C2×D211)⋊C4—24no—
32C4×Dic211—21no—
33C4×Dic211D412yescomplemented
34C22×C4×C211D412yes—
35C22×C4×C211C2311yescharacteristic
36M16×C211—22no—
37M16×C211—22no—
38C2×C8×C211—22no—
39C4⋊C4×C211—22no—
40C22×D211—22no—
41Dic422—22no—
42Dic422—22no—
43D844—22no—
44D844—22no—
45D844—82no—
46C22⋊D211—42no—
47C22⋊D211—42no—
48C2×Dic211—22no—
49C2×Dic211—22no—
50C2×Dic211—41no—
51C2×Dic211—41no—
52C2×Dic211—42no—
53C2×Dic211—41no—
54C4×D211—42no—
55C4×D211—42no—
56C23×C211C2×D411yescharacteristic
57C2×C4×C211—22no—
58C2×C4×C211—42no—
59C2×C4×C211C2×D411yescharacteristic
60C2×C4×C211C2×D412yes—
61C2×C4×C211#4112yes—
62C2×C4×C211#4111yescharacteristic
63C8×C211—42no—
64C22×C211—21no—
65C22×C211—21no—
66C22×C211#12011yescharacteristic
67C22×C211#12111yescharacteristic
68C22×C211#12411yescharacteristic
69C2×D211—42no—
70C2×D211—42no—
71C2×D211—42no—
72C4×C211—42no—
73C4×C211#12112yes—
74C4×C211#12312yes—
75Dic211—42no—
76Dic211—42no—
77Dic211—81no—
78C2×C211—21no—
79C2×C211—21no—
80C2×C211#39411yescharacteristic
81C2×C211#45411yescharacteristic
82C2×C211#45511yescharacteristic
83D211—82no—
84C211#204011yescharacteristic, complemented, Sylow
85#2040—2111nomaximal, Sylow
86#537—2111no—
87C2×#104—2112no—
88#417—2112no—
89#382—2111no—
90#343D21111yescharacteristic, complemented
91C2×#41—2112no—
92C2×M16C2×D21112yes—
93C22⋊Q8—4222no—
94#121—4222no—
95#121—4222no—
96C2×C4⋊C4—2111no—
97C2×C4⋊C4C2×D21111yescharacteristic
98C2×C42—2111no—
99#104—4224no—
100#103—4222no—
101#102—4222no—
102#95—4222no—
103#41—4222no—
104#41—4222no—
105#41—4222no—
106#41—4222no—
107C2×Q8—2112no—
108C2×D4—4222no—
109C2×D4—4222no—
110C2×D4—2112no—
111C22×C4—4221no—
112C22×C4—4221no—
113C22×C4—4222no—
114C22×C4D84412yes—
115C22×C4C22×D21111yescharacteristic
116M16—22no—
117M16—22no—
118C2×C8—22no—
119C4⋊C4—22no—
120C4⋊C4—4222no—
121C4⋊C4—4222no—
122C22⋊C4—4224no—
123C42—4221no—
124C42—2112no—
125C23—4222no—
126C23C2×D84411yescharacteristic
127Q8—4222no—
128Q8—4222no—
129D4—8442no—
130D4—8442no—
131D4—16882no—
132D4—4222no—
133D4—4222no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—8441no—
137C2×C4—8441no—
138C2×C4—8442no—
139C2×C4—8442no—
140C2×C4—8442no—
141C2×C4—8441no—
142C2×C4—4222no—
143C2×C4—4222no—
144C2×C4D4×D21111yescharacteristic
145C2×C4D4×D21112yes—
146C2×C4#1680996212yes—
147C2×C4#1680996311yescharacteristic
148C8—42no—
149C22—21no—
150C22—21no—
151C22—8442no—
152C22—8442no—
153C22—8442no—
154C22#2026994711yescharacteristic
155C22#2026995211yescharacteristic
156C22#2026996511yescharacteristic
157C4—42no—
158C4—8442no—
159C4—8442no—
160C4—16881no—
161C4#2026995012yes—
162C4#2026996212yes—
163C2—21no—
164C2—21no—
165C2—16882no—
166C2#2721306411yescharacteristic
167C2#2721332111yescharacteristic
168C2#2721332611yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#343×C211C2
socleC22×C211#120
prime radical#27213727C2
hypercenter#343D211
hypocenterC211#2040
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
2. C2×#20269903C2semidirect
3. #27213148C2semidirect
4. #27213011C2semidirect
5. #343×C211C2semidirect
33. C4×Dic211D4semidirect
84. C211#2040semidirect
90. #343D211semidirect

Maximal subgroups

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

subgroupindex
#272137272
C2×#202699032
#272131482
#272130112
#343×C2112
#2040211