Finite Group Databasebeta

← #41013502

Subgroups — #41013502

Order 27008.

262 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 89 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#27213130C211yesmaximal, characteristic
2#27213120C211yesmaximal, characteristic
3#27212906C212yesmaximal, complemented
4#27212902C212yesmaximal, complemented
5C4×#41×C211C211yesmaximal, characteristic
6C2×#16809946C2212yes—
7#20269936C2212yes—
8#20269896—21no—
9#20269894—21no—
10#20269892—22no—
11C2×#41×C211C2211yescharacteristic
12C4×Q8×C211—21no—
13C4×D4×C211—22no—
14C4×D4×C211—21no—
15#117×C211C412yescomplemented
16#117×C211C2211yescharacteristic
17C2×C42×C211C412yescomplemented
18C2×C42×C211C2211yescharacteristic
19#16809946—22no—
20#16809946—22no—
21C2×C211⋊C8—22no—
22C4⋊Dic211—22no—
23C22⋊Dic211—22no—
24C4×Dic211—22no—
25#41×C211—22no—
26#41×C211—22no—
27C2×Q8×C211D411yescharacteristic
28C2×D4×C211D412yes—
29C2×D4×C211D411yescharacteristic
30C22×C4×C211—22no—
31C22×C4×C211—21no—
32C22×C4×C211C2×C412yes—
33C22×C4×C211C2311yescharacteristic
34C4⋊C4×C211—22no—
35C4⋊C4×C211—21no—
36C22⋊C4×C211—22no—
37C22⋊C4×C211—21no—
38C42×C211—22no—
39C42×C211—21no—
40C42×C211C2×C411yescharacteristic
41C42×C211C2×C412yes—
42C42×C211C2×C411yescharacteristic
43C2×Dic211—42no—
44C211⋊C8—42no—
45C23×C211C22⋊C412yes—
46C23×C211C2×D411yescharacteristic
47Q8×C211—22no—
48D4×C211—24no—
49D4×C211—22no—
50C2×C4×C211—21no—
51C2×C4×C211—22no—
52C2×C4×C211—22no—
53C2×C4×C211—22no—
54C2×C4×C211—21no—
55C2×C4×C211—42no—
56C2×C4×C211—44no—
57C2×C4×C211C22⋊C412yes—
58C2×C4×C211C22×C411yescharacteristic
59C2×C4×C211C2×D411yescharacteristic
60C2×C4×C211#4112yes—
61C2×C4×C211#4112yes—
62C22×C211—22no—
63C22×C211—41no—
64C22×C211—42no—
65C22×C211C2×C22⋊C411yescharacteristic
66C22×C211#12312yes—
67C4×C211—22no—
68C4×C211—22no—
69C4×C211—42no—
70C4×C211—41no—
71C4×C211#11711yescharacteristic
72C4×C211#11711yescharacteristic
73C4×C211#12312yes—
74Dic211—82no—
75C2×C211—22no—
76C2×C211—21no—
77C2×C211—42no—
78C2×C211#38111yescharacteristic
79C211#182711yescharacteristic, complemented, Sylow
80#1827—2111nomaximal, Sylow
81C4×#41D21111yescharacteristic
82#413—2111no—
83#410—2111no—
84#339—2112no—
85H(3,ℤ/4)—2112no—
86C2×#41C2×D21111yescharacteristic
87C2×M16—2112no—
88C4×Q8—21no—
89C4×D4—22no—
90C4×D4—21no—
91#117—2112no—
92#117Dic21112yescomplemented
93#117C2×D21111yescharacteristic
94C2×C42Dic21112yescomplemented
95C2×C42C2×D21111yescharacteristic
96#101—4221no—
97#100—4221no—
98#99—4222no—
99#41—22no—
100#41—22no—
101C2×Q8C22⋊D21111yescharacteristic
102C2×D4C22⋊D21112yes—
103C2×D4C22⋊D21111yescharacteristic
104C22×C4—22no—
105C22×C4—21no—
106C22×C4C2×Dic21112yes—
107C22×C4C22×D21111yescharacteristic
108M16—4222no—
109M16—4222no—
110C2×C8—4222no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—4222no—
114C22⋊C4—22no—
115C22⋊C4—21no—
116C22⋊C4—4222no—
117C42—22no—
118C42—21no—
119C42—4222no—
120C42C2×Dic21111yescharacteristic
121C42C2×Dic21112yes—
122C42C2×Dic21111yescharacteristic
123C23C22⋊Dic21112yes—
124C23C2×C22⋊D21111yescharacteristic
125Q8—22no—
126D4—24no—
127D4—22no—
128C2×C4—21no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—21no—
133C2×C4—42no—
134C2×C4—44no—
135C2×C4—8442no—
136C2×C4C22⋊Dic21112yes—
137C2×C4C22×Dic21111yescharacteristic
138C2×C4C2×C22⋊D21111yescharacteristic
139C2×C4#1680996212yes—
140C2×C4#1680996212yes—
141C8—8442no—
142C22—22no—
143C22—41no—
144C22—42no—
145C22C2×C22⋊Dic21111yescharacteristic
146C22#2026996012yes—
147C4—22no—
148C4—22no—
149C4—42no—
150C4—41no—
151C4—16882no—
152C4#2026993611yescharacteristic
153C4#2026993611yescharacteristic
154C4#2026996012yes—
155C2—21no—
156C2—22no—
157C2—42no—
158C2#2721300911yescharacteristic
subgroupG/H
centerC4#20269936
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC4×#41×C211C2
socleC2×C211#381
prime radicalC2×#41×C211C22
cosocleC22×C4×C211C23
hypercenterC4×#41D211
hypocenterC211#1827
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ #117 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
3. #27212906C2semidirect
4. #27212902C2semidirect
15. #117×C211C4semidirect
17. C2×C42×C211C4semidirect
79. C211#1827semidirect
92. #117Dic211semidirect
94. C2×C42Dic211semidirect

Maximal subgroups

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

subgroupindex
#272131302
#272131202
#272129062
#272129022
C4×#41×C2112
#1827211