Finite Group Databasebeta

← #41011815

Subgroups — #41011815

Order 27008.

262 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 85 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#27213199C211yesmaximal, characteristic, complemented
2C4×#16809946C211yesmaximal, characteristic, complemented
3D4⋊(C211⋊C8)C214yesmaximal, complemented
4#532×C211C211yesmaximal, characteristic
5C2×#16809946—21no—
6C4⋊(C211⋊C8)C2212yes—
7C22⋊(C211⋊C8)—21no—
8#20269888—21no—
9C4×C211⋊C8—21no—
10C4×C211⋊C8C2212yes—
11C2×#41×C211—21no—
12C4⋊Q8×C211C411yescharacteristic
13#127×C211C411yescharacteristic
14#124×C211—21no—
15#121×C211—21no—
16#121×C211C412yes—
17C4×D4×C211—21no—
18C4×D4×C211C2212yes—
19C2×C42×C211C2211yescharacteristic
20#16809946—24no—
21C2×C211⋊C8—22no—
22C2×C211⋊C8—22no—
23#41×C211—44no—
24C2×Q8×C211—21no—
25C2×D4×C211—22no—
26C2×D4×C211—21no—
27C2×D4×C211—21no—
28C2×D4×C211C2×C412yes—
29C22×C4×C211—22no—
30C22×C4×C211C2×C411yescharacteristic
31C22×C4×C211D411yescharacteristic
32C22×C4×C211D411yescharacteristic
33C4⋊C4×C211—21no—
34C4⋊C4×C211C2×C412yes—
35C22⋊C4×C211—22no—
36C22⋊C4×C211—22no—
37C42×C211C2×C411yescharacteristic
38C42×C211D411yescharacteristic
39C42×C211D411yescharacteristic
40C42×C211C2311yescharacteristic
41C211⋊C8—24no—
42C211⋊C8—42no—
43C23×C211—22no—
44C23×C211C22⋊C411yescharacteristic
45Q8×C211—42no—
46D4×C211—24no—
47D4×C211—42no—
48D4×C211—44no—
49D4×C211—42no—
50C2×C4×C211—22no—
51C2×C4×C211—22no—
52C2×C4×C211—22no—
53C2×C4×C211—22no—
54C2×C4×C211—44no—
55C2×C4×C211C22⋊C411yescharacteristic
56C2×C4×C211C22⋊C411yescharacteristic
57C2×C4×C211C22⋊C411yescharacteristic
58C2×C4×C211C22×C411yescharacteristic
59C2×C4×C211C2×D411yescharacteristic
60C2×C4×C211C2×D411yescharacteristic
61C22×C211—21no—
62C22×C211—21no—
63C22×C211—21no—
64C22×C211—42no—
65C22×C211—42no—
66C22×C211—42no—
67C22×C211C2×C22⋊C411yescharacteristic
68C4×C211—21no—
69C4×C211—21no—
70C4×C211—21no—
71C4×C211—42no—
72C4×C211#10412yes—
73C4×C211#13112yes—
74C4×C211AGL(1,ℤ/8)12yes—
75C2×C211—41no—
76C2×C211—82no—
77C2×C211#40811yescharacteristic
78C2×C211#41811yescharacteristic
79C2×C211C2×#10411yescharacteristic
80C211#148411yescharacteristic, complemented, Sylow
81#1484—2111nomaximal, Sylow
82#532D21111yescharacteristic
83#431—2111no—
84C4×M16—2111no—
85D4⋊C8—2114no—
86C2×#41—21no—
87C2×M16—4221no—
88C4⋊Q8Dic21111yescharacteristic
89#127Dic21111yescharacteristic
90#124—21no—
91#121—21no—
92#121Dic21112yes—
93C4×D4—21no—
94C4×D4C2×D21112yes—
95C2×C42C2×D21111yescharacteristic
96C4⋊C8—2112no—
97C22⋊C8—4221no—
98#97—4221no—
99C4×C8—4221no—
100C4×C8—2112no—
101#41—44no—
102C2×Q8—21no—
103C2×D4—22no—
104C2×D4—21no—
105C2×D4—21no—
106C2×D4C2×Dic21112yes—
107C22×C4—22no—
108C22×C4C2×Dic21111yescharacteristic
109C22×C4C22⋊D21111yescharacteristic
110C22×C4C22⋊D21111yescharacteristic
111M16—4224no—
112C2×C8—4222no—
113C2×C8—4222no—
114C4⋊C4—21no—
115C4⋊C4C2×Dic21112yes—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C42C2×Dic21111yescharacteristic
119C42C22⋊D21111yescharacteristic
120C42C22⋊D21111yescharacteristic
121C42C22×D21111yescharacteristic
122C23—22no—
123C23C22⋊Dic21111yescharacteristic
124Q8—42no—
125D4—24no—
126D4—42no—
127D4—44no—
128D4—42no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—44no—
134C2×C4C22⋊Dic21111yescharacteristic
135C2×C4C22⋊Dic21111yescharacteristic
136C2×C4C22⋊Dic21111yescharacteristic
137C2×C4C22×Dic21111yescharacteristic
138C2×C4C2×C22⋊D21111yescharacteristic
139C2×C4C2×C22⋊D21111yescharacteristic
140C8—8442no—
141C8—4224no—
142C22—21no—
143C22—21no—
144C22—21no—
145C22—42no—
146C22—42no—
147C22—42no—
148C22C2×C22⋊Dic21111yescharacteristic
149C4—21no—
150C4—21no—
151C4—21no—
152C4—42no—
153C4#2026990412yes—
154C4#2026998412yes—
155C4#2027000312yes—
156C2—41no—
157C2—82no—
158C2#2721311311yescharacteristic
159C2#2721315111yescharacteristic
160C2C2×#2026990411yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D211
commutatorC2×C4×C211C22×C4
FrattiniC42C22×D211
Fitting, prime radical#532×C211C2
socleC22×C211C2×C22⋊C4
cosocleC42×C211C23
hypercenter#532D211
hypocenterC211#1484
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #27213199C2semidirect
2. C4×#16809946C2semidirect
3. D4⋊(C211⋊C8)C2semidirect
80. C211#1484semidirect

Maximal subgroups

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

subgroupindex
#272131992
C4×#168099462
D4⋊(C211⋊C8)2
#532×C2112
#1484211