Finite Group Databasebeta

← #41014686

Subgroups — #41014686

Order 27008.

266 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 63 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#27213946C211yesmaximal, characteristic
2#27213169C211yesmaximal, characteristic, complemented
3C2×#20269903C212yesmaximal, complemented
4#27213128C212yesmaximal, complemented
5#338×C211C211yesmaximal, characteristic, complemented
6C2×#16809962C2212yes—
7#20270010—22no—
8#20270005—22no—
9#20269999—22no—
10C2×#16809946C2211yescharacteristic
11C2×C4×Dic211C2211yescharacteristic
12Dic211⋊C8—22no—
13#20269903—22no—
14#20269903—22no—
15#20269895—22no—
16#20269893—22no—
17C2×M16×C211C2212yes—
18C2×M16×C211C2211yescharacteristic
19#16809962—22no—
20#16809962—22no—
21#16809962—22no—
22#16809962—22no—
23C2×Dic422D412yes—
24C2×D844D412yes—
25C2×C22⋊D211—22no—
26C22×Dic211—21no—
27C2×C4×D211—22no—
28Dic844—22no—
29#16809951—24no—
30D1688—22no—
31#16809946—22no—
32C2×C211⋊C8—21no—
33C4×Dic211—21no—
34C4×Dic211D411yescharacteristic
35C4×Dic211D411yescharacteristic
36C22×C4×C211C2311yescharacteristic
37M16×C211—22no—
38M16×C211—22no—
39M16×C211—22no—
40C2×C8×C211—22no—
41C2×C8×C211D412yes—
42C22×D211—22no—
43Dic422—22no—
44Dic422—22no—
45D844—22no—
46D844—22no—
47C22⋊D211—42no—
48C22⋊D211—42no—
49C2×Dic211—21no—
50C2×Dic211—21no—
51C2×Dic211—22no—
52C2×Dic211—42no—
53C4×D211—42no—
54C4×D211—42no—
55C211⋊C8—41no—
56C23×C211#4111yescharacteristic
57C2×C4×C211C2×D412yes—
58C2×C4×C211C2×D411yescharacteristic
59C2×C4×C211C2×D411yescharacteristic
60C2×C4×C211#4112yes—
61C8×C211—22no—
62C8×C211—42no—
63C22×C211—41no—
64C22×C211#12111yescharacteristic
65C22×C211#12312yes—
66C2×D211—42no—
67C2×D211—82no—
68C4×C211#12011yescharacteristic
69C4×C211#12112yes—
70C4×C211#12411yescharacteristic
71Dic211—22no—
72Dic211—41no—
73Dic211—42no—
74C2×C211—21no—
75C2×C211—22no—
76C2×C211#39411yescharacteristic
77D211—82no—
78C211#205011yescharacteristic, complemented, Sylow
79#2050—2111nomaximal, Sylow
80#575—2111no—
81#423—2111no—
82C2×#104—2112no—
83#413—2112no—
84#338D21111yescharacteristic, complemented
85C2×#41—2112no—
86#137—4222no—
87AGL(1,ℤ/8)—4222no—
88#135—4222no—
89C2×M16—2111no—
90C2×M16C2×D21112yes—
91C2×M16C2×D21111yescharacteristic
92C2×C42—2111no—
93C4⋊C8—4222no—
94#104—4222no—
95#104—4222no—
96#101—4222no—
97#100—4222no—
98#41—4222no—
99#41—4222no—
100#41—4222no—
101#41—4222no—
102C2×Q8—2112no—
103C2×D4—4222no—
104C2×D4—2112no—
105C22×C4—4221no—
106C22×C4—4222no—
107C22×C4C22×D21111yescharacteristic
108Q16—4222no—
109SD16—4224no—
110D8—4222no—
111M16—22no—
112M16—22no—
113M16—22no—
114M16—4222no—
115C2×C8—22no—
116C2×C8—4221no—
117C2×C8D84412yes—
118C42—4221no—
119C42—2111no—
120C42—2111no—
121C23—4222no—
122C23#1680996311yescharacteristic
123Q8—4222no—
124Q8—4222no—
125D4—8442no—
126D4—8442no—
127D4—4222no—
128D4—4222no—
129C2×C4—8442no—
130C2×C4—8442no—
131C2×C4—8442no—
132C2×C4—4221no—
133C2×C4—4222no—
134C2×C4—4221no—
135C2×C4D4×D21112yes—
136C2×C4D4×D21111yescharacteristic
137C2×C4C2×D84411yescharacteristic
138C2×C4#1680996212yes—
139C8—22no—
140C8—42no—
141C8—8441no—
142C22—41no—
143C22—8442no—
144C22—16882no—
145C22#2026995211yescharacteristic
146C22#2026996212yes—
147C4—8441no—
148C4—8442no—
149C4—4222no—
150C4#2026994711yescharacteristic
151C4#2026995012yes—
152C4#2026996511yescharacteristic
153C2—21no—
154C2—22no—
155C2—16882no—
156C2#2721306411yescharacteristic
subgroupG/H
centerC4#20269965
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#338×C211C2
socleC2×C211#394
prime radical#27213946C2
hypercenter#338D211
hypocenterC211#2050
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #338

Complements

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

NH ≅ G/Ntype
2. #27213169C2semidirect
3. C2×#20269903C2semidirect
4. #27213128C2semidirect
5. #338×C211C2semidirect
78. C211#2050semidirect
84. #338D211semidirect

Maximal subgroups

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

subgroupindex
#272139462
#272131692
C2×#202699032
#272131282
#338×C2112
#2050211