Finite Group Databasebeta

← #41013431

Subgroups — #41013431

Order 27008.

316 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C2×#20269911C211yesmaximal, characteristic, complemented
2C2×#20269909C211yesmaximal, characteristic, complemented
3#27213006C212yesmaximal
4#27212909C212yesmaximal, complemented
5C22×M16×C211C211yesmaximal, characteristic
6C2×C4⋊Dic211C2212yes—
7C2×C4⋊Dic211C2212yes—
8C2×C22⋊Dic211—22no—
9#20269911—22no—
10#20269909—22no—
11C211⋊#95—22no—
12C23×C4×C211C2211yescharacteristic
13C2×M16×C211—22no—
14C2×M16×C211—22no—
15C2×M16×C211C414yescomplemented
16C22×C8×C211C2211yescharacteristic
17C22×C8×C211C2211yescharacteristic
18C22×Dic211—22no—
19C22×Dic211—22no—
20C4⋊Dic211—22no—
21C4⋊Dic211—22no—
22C4⋊Dic211—22no—
23C4⋊Dic211—22no—
24C22⋊Dic211—44no—
25C24×C211D411yescharacteristic
26C22×C4×C211—21no—
27C22×C4×C211—21no—
28C22×C4×C211—21no—
29C22×C4×C211—21no—
30C22×C4×C211C2×C412yes—
31C22×C4×C211D411yescharacteristic
32C22×C4×C211Q812yes—
33C22×C4×C211C2311yescharacteristic
34M16×C211—28no—
35C2×C8×C211—21no—
36C2×C8×C211—21no—
37C2×C8×C211C2×C412yes—
38C2×C8×C211C2×C412yes—
39C2×C8×C211D412yes—
40C2×C8×C211D412yes—
41C2×Dic211—42no—
42C2×Dic211—42no—
43C2×Dic211—42no—
44C2×Dic211—42no—
45C2×Dic211—42no—
46C2×Dic211—42no—
47C23×C211—21no—
48C23×C211—21no—
49C23×C211—22no—
50C23×C211—22no—
51C23×C211C4⋊C412yes—
52C23×C211C2×D411yescharacteristic
53C2×C4×C211—21no—
54C2×C4×C211—21no—
55C2×C4×C211—22no—
56C2×C4×C211—22no—
57C2×C4×C211—22no—
58C2×C4×C211—22no—
59C2×C4×C211C4⋊C412yes—
60C2×C4×C211C22×C411yescharacteristic
61C2×C4×C211C2×D411yescharacteristic
62C2×C4×C211C2×D411yescharacteristic
63C2×C4×C211C2×Q811yescharacteristic
64C2×C4×C211#4111yescharacteristic
65C2×C4×C211#4111yescharacteristic
66C8×C211—22no—
67C8×C211C22⋊C414yescomplemented
68C22×C211—22no—
69C22×C211—24no—
70C22×C211—42no—
71C22×C211—41no—
72C22×C211—41no—
73C22×C211C2×C4⋊C411yescharacteristic
74C22×C211#12111yescharacteristic
75C22×C211#12111yescharacteristic
76C22×C211AGL(1,ℤ/8)12yes—
77C22×C211#13712yes—
78C4×C211—22no—
79C4×C211C2×C22⋊C411yescharacteristic
80C4×C211#11711yescharacteristic
81C4×C211C22⋊Q811yescharacteristic
82C4×C211C22⋊Q811yescharacteristic
83Dic211—82no—
84Dic211—82no—
85C2×C211—42no—
86C2×C211#38011yescharacteristic
87C2×C211#42812yes—
88C2×C211#46812yes—
89C2×C211#46911yescharacteristic
90C2×C211#47011yescharacteristic
91C211#180911yescharacteristic, complemented, Sylow
92#1809—2111nomaximal, Sylow
93C22×M16D21111yescharacteristic
94C2×#107—2111no—
95C2×#106—2111no—
96#380—2112no—
97#340—2112no—
98C23×C4C2×D21111yescharacteristic
99C2×M16—22no—
100C2×M16—22no—
101C2×M16Dic21114yescomplemented
102C22×C8C2×D21111yescharacteristic
103C22×C8C2×D21111yescharacteristic
104C2×C4⋊C4—2112no—
105C2×C4⋊C4—2112no—
106C2×C22⋊C4—4222no—
107#107—4222no—
108#106—4222no—
109#95—4222no—
110C24D84411yescharacteristic
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—4222no—
116C22×C4—4222no—
117C22×C4C2×Dic21112yes—
118C22×C4D84411yescharacteristic
119C22×C4Dic42212yes—
120C22×C4C22×D21111yescharacteristic
121M16—28no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8C2×Dic21112yes—
125C2×C8C2×Dic21112yes—
126C2×C8C22⋊D21112yes—
127C2×C8C22⋊D21112yes—
128C4⋊C4—4222no—
129C4⋊C4—4222no—
130C4⋊C4—4222no—
131C4⋊C4—4222no—
132C22⋊C4—8444no—
133C23—21no—
134C23—21no—
135C23—22no—
136C23—22no—
137C23C4⋊Dic21112yes—
138C23C2×D84411yescharacteristic
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—8442no—
146C2×C4—8442no—
147C2×C4—8442no—
148C2×C4—8442no—
149C2×C4—8442no—
150C2×C4—8442no—
151C2×C4C4⋊Dic21112yes—
152C2×C4C22×Dic21111yescharacteristic
153C2×C4C2×C22⋊D21111yescharacteristic
154C2×C4C2×C22⋊D21111yescharacteristic
155C2×C4C2×Dic42211yescharacteristic
156C2×C4#1680996211yescharacteristic
157C2×C4#1680996211yescharacteristic
158C8—22no—
159C8C22⋊Dic21114yescomplemented
160C22—24no—
161C22—22no—
162C22—42no—
163C22—41no—
164C22—41no—
165C22C2×C4⋊Dic21111yescharacteristic
166C22#2026995111yescharacteristic
167C22#2026995111yescharacteristic
168C22#2027000512yes—
169C22#2027001012yes—
170C4—22no—
171C4—16882no—
172C4—16882no—
173C4C2×C22⋊Dic21111yescharacteristic
174C4#2026993611yescharacteristic
175C4#2026995611yescharacteristic
176C4#2026995611yescharacteristic
177C2—42no—
178C2#2721300611yescharacteristic
179C2#2721318812yes—
180C2#2721339212yes—
181C2#2721339911yescharacteristic
182C2#2721340411yescharacteristic
subgroupG/H
centerC23C2×D844
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC22×M16×C211C2
socleC23×C211C2×D4
prime radicalC24×C211D4
cosocleC22×C4×C211C23
hypercenterC22×M16D211
hypocenterC211#1809
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
1. C2×#20269911C2semidirect
2. C2×#20269909C2semidirect
4. #27212909C2semidirect
15. C2×M16×C211C4semidirect
67. C8×C211C22⋊C4semidirect
91. C211#1809semidirect
101. C2×M16Dic211semidirect
159. C8C22⋊Dic211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699112
C2×#202699092
#272130062
#272129092
C22×M16×C2112
#1809211