Finite Group Databasebeta

← #41014682

Subgroups — #41014682

Order 27008.

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

normal subgroups only

This group has 266 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#27213169C211yesmaximal, characteristic, complemented
2C2×#20269904C212yesmaximal, complemented
3#27213130C212yesmaximal, complemented
4C211⋊#338C211yesmaximal, characteristic, complemented
5#575×C211C211yesmaximal, characteristic
6C2×#16809946C2211yescharacteristic
7C2×#16809946C2212yes—
8C2×C4×Dic211C2211yescharacteristic
9Dic211⋊C8—22no—
10#20269904—22no—
11#20269904—22no—
12#20269896—22no—
13#20269894—22no—
14C2×#41×C211C2212yes—
15#137×C211—22no—
16AGL(1,ℤ/8)×C211—22no—
17#135×C211—22no—
18C2×M16×C211C2211yescharacteristic
19C22×Dic211—21no—
20#16809946—22no—
21#16809946—22no—
22#16809946—22no—
23C2×C211⋊C8—21no—
24C2×C211⋊C8—22no—
25C4×Dic211—21no—
26C4×Dic211D411yescharacteristic
27C4×Dic211D411yescharacteristic
28#41×C211—22no—
29#41×C211—22no—
30#41×C211—22no—
31#41×C211—22no—
32C2×Q8×C211D412yes—
33C2×D4×C211—22no—
34C2×D4×C211D412yes—
35C22×C4×C211—22no—
36C22×C4×C211C2311yescharacteristic
37Q16×C211—22no—
38SD16×C211—24no—
39D8×C211—22no—
40M16×C211—22no—
41C2×C8×C211D412yes—
42C2×Dic211—21no—
43C2×Dic211—21no—
44C2×Dic211—42no—
45C211⋊C8—41no—
46C211⋊C8—42no—
47C23×C211—22no—
48C23×C211#4111yescharacteristic
49Q8×C211—22no—
50Q8×C211—22no—
51D4×C211—22no—
52D4×C211—22no—
53D4×C211—42no—
54D4×C211—42no—
55C2×C4×C211—22no—
56C2×C4×C211—42no—
57C2×C4×C211—42no—
58C2×C4×C211C2×D411yescharacteristic
59C2×C4×C211C2×D412yes—
60C2×C4×C211C2×D411yescharacteristic
61C2×C4×C211#4112yes—
62C8×C211—22no—
63C22×C211—41no—
64C22×C211—42no—
65C22×C211—82no—
66C22×C211#12111yescharacteristic
67C22×C211#12312yes—
68C4×C211—42no—
69C4×C211#12011yescharacteristic
70C4×C211#12112yes—
71C4×C211#12411yescharacteristic
72Dic211—22no—
73Dic211—41no—
74C2×C211—21no—
75C2×C211—22no—
76C2×C211—82no—
77C2×C211#39411yescharacteristic
78C211#205011yescharacteristic, complemented, Sylow
79#2050—2111nomaximal, Sylow
80#575D21111yescharacteristic
81#423—2111no—
82C2×#104—2112no—
83#413—2112no—
84#338—2111no—
85C2×#41C2×D21112yes—
86#137—22no—
87AGL(1,ℤ/8)—22no—
88#135—22no—
89C2×M16—2112no—
90C2×M16—2111no—
91C2×M16C2×D21111yescharacteristic
92C2×C42—2111no—
93C4⋊C8—4222no—
94#104—4222no—
95#104—4222no—
96#101—4222no—
97#100—4222no—
98#41—22no—
99#41—22no—
100#41—22no—
101#41—22no—
102C2×Q8C22⋊D21112yes—
103C2×D4—22no—
104C2×D4C22⋊D21112yes—
105C22×C4—22no—
106C22×C4—4221no—
107C22×C4C22×D21111yescharacteristic
108Q16—22no—
109SD16—24no—
110D8—22no—
111M16—22no—
112M16—4222no—
113M16—4222no—
114M16—4222no—
115C2×C8—4221no—
116C2×C8—4222no—
117C2×C8C22⋊D21112yes—
118C42—4221no—
119C42—2111no—
120C42—2111no—
121C23—22no—
122C23#1680996111yescharacteristic
123Q8—22no—
124Q8—22no—
125D4—22no—
126D4—22no—
127D4—42no—
128D4—42no—
129C2×C4—22no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—8442no—
133C2×C4—4221no—
134C2×C4—4221no—
135C2×C4D4×D21111yescharacteristic
136C2×C4C2×C22⋊D21112yes—
137C2×C4C2×C22⋊D21111yescharacteristic
138C2×C4#1680996112yes—
139C8—22no—
140C8—8441no—
141C8—8442no—
142C22—41no—
143C22—42no—
144C22—82no—
145C22#2026994811yescharacteristic
146C22#2026995812yes—
147C4—42no—
148C4—8441no—
149C4—4222no—
150C4#2026994611yescharacteristic
151C4#2026994912yes—
152C4#2026996411yescharacteristic
153C2—21no—
154C2—22no—
155C2—82no—
156C2#2721306011yescharacteristic
subgroupG/H
centerC4#20269964
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting, prime radical#575×C211C2
socleC2×C211#394
hypercenter#575D211
hypocenterC211#2050
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #27213169C2semidirect
2. C2×#20269904C2semidirect
3. #27213130C2semidirect
4. C211⋊#338C2semidirect
78. C211#2050semidirect

Maximal subgroups

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

subgroupindex
#272131692
C2×#202699042
#272131302
C211⋊#3382
#575×C2112
#2050211