Finite Group Databasebeta

← #41013504

Subgroups — #41013504

Order 27008.

296 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 81 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#27213621C211yesmaximal, characteristic
2C2×#20269892C211yesmaximal, characteristic
3#27212913C212yesmaximal, complemented
4#27212912C212yesmaximal, complemented
5C2×#99×C211C211yesmaximal, characteristic
6D4×Dic211—22no—
7D4×Dic211—22no—
8#20269936C412yescomplemented
9C2×C22⋊Dic211C412yescomplemented
10C2×C22⋊Dic211C2212yes—
11C2×C22⋊Dic211C2211yescharacteristic
12C2×C22⋊Dic211C2211yescharacteristic
13#20269892—22no—
14C22×D4×C211C2211yescharacteristic
15C2×C22⋊C4×C211C2212yes—
16#99×C211—22no—
17C22×Dic211—22no—
18C22×Dic211—21no—
19C22×Dic211—21no—
20C22×Dic211—22no—
21C4⋊Dic211—22no—
22C22⋊Dic211—22no—
23C22⋊Dic211—21no—
24C22⋊Dic211—21no—
25C22⋊Dic211—22no—
26C22⋊Dic211—22no—
27C22⋊Dic211C2×C412yes—
28C22⋊Dic211C2×C412yes—
29C4×Dic211—22no—
30C24×C211C2×C411yescharacteristic
31C24×C211C2311yescharacteristic
32C2×D4×C211—22no—
33C2×D4×C211—22no—
34C2×D4×C211D412yes—
35C2×D4×C211D412yes—
36C22×C4×C211—22no—
37C22×C4×C211C2×C411yescharacteristic
38C22⋊C4×C211—22no—
39C22⋊C4×C211—22no—
40C2×Dic211—22no—
41C2×Dic211—22no—
42C2×Dic211—42no—
43C2×Dic211—81no—
44C2×Dic211—81no—
45C2×Dic211—82no—
46C2×Dic211—82no—
47C23×C211—21no—
48C23×C211—21no—
49C23×C211—81no—
50C23×C211—81no—
51C23×C211C22⋊C412yes—
52C23×C211C22×C411yescharacteristic
53C23×C211C2×D411yescharacteristic
54C23×C211C2×D411yescharacteristic
55C23×C211#4112yes—
56C23×C211#4112yes—
57D4×C211—24no—
58D4×C211—24no—
59C2×C4×C211—21no—
60C2×C4×C211—21no—
61C2×C4×C211—42no—
62C2×C4×C211—82no—
63C2×C4×C211C22⋊C412yes—
64C22×C211—22no—
65C22×C211—42no—
66C22×C211—41no—
67C22×C211—41no—
68C22×C211—41no—
69C22×C211—42no—
70C22×C211—42no—
71C22×C211—81no—
72C22×C211—81no—
73C22×C211C2×C22⋊C411yescharacteristic
74C22×C211#11711yescharacteristic
75C22×C211#11711yescharacteristic
76C22×C211#12312yes—
77C22×C211#12312yes—
78C4×C211—22no—
79C4×C211—82no—
80Dic211—42no—
81Dic211—42no—
82Dic211—82no—
83C2×C211—21no—
84C2×C211—22no—
85C2×C211—22no—
86C2×C211—21no—
87C2×C211—21no—
88C2×C211—42no—
89C2×C211#38111yescharacteristic
90C211#182811yescharacteristic, complemented, Sylow
91#1828—2111nomaximal, Sylow
92#518—2111no—
93C2×#99—2111no—
94C2×#99D21111yescharacteristic
95#342—2112no—
96#342—2112no—
97C22×D4C2×D21111yescharacteristic
98C4×D4—4222no—
99C4×D4—4222no—
100#117—2112no—
101C2×C22⋊C4—2112no—
102C2×C22⋊C4—2112no—
103C2×C22⋊C4—2111no—
104C2×C22⋊C4—2111no—
105C2×C22⋊C4C2×D21112yes—
106#99—22no—
107#99—4222no—
108C24C4×D21111yescharacteristic
109C24C22×D21111yescharacteristic
110C2×D4—22no—
111C2×D4—22no—
112C2×D4C22⋊D21112yes—
113C2×D4D84412yes—
114C22×C4—22no—
115C22×C4—4222no—
116C22×C4—4222no—
117C22×C4—4221no—
118C22×C4—4221no—
119C22×C4C4×D21111yescharacteristic
120C4⋊C4—4222no—
121C22⋊C4—22no—
122C22⋊C4—22no—
123C22⋊C4—4222no—
124C22⋊C4—4222no—
125C22⋊C4—4222no—
126C22⋊C4—4221no—
127C22⋊C4—4221no—
128C22⋊C4—2112no—
129C22⋊C4—2112no—
130C42—4222no—
131C23—21no—
132C23—21no—
133C23—81no—
134C23—81no—
135C23(C2×D211)⋊C412yes—
136C23C2×C4×D21111yescharacteristic
137C23C2×C22⋊D21111yescharacteristic
138C23C2×D84411yescharacteristic
139C23#1680996112yes—
140C23#1680996112yes—
141D4—24no—
142D4—24no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—42no—
146C2×C4—82no—
147C2×C4—8442no—
148C2×C4—16881no—
149C2×C4—16881no—
150C2×C4—16882no—
151C2×C4—16882no—
152C2×C4—4222no—
153C2×C4—4222no—
154C2×C4(C2×D211)⋊C412yes—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—42no—
160C22—41no—
161C22—42no—
162C22—81no—
163C22—81no—
164C22C2×(C2×D211)⋊C411yescharacteristic
165C22#2026993311yescharacteristic
166C22#2026993311yescharacteristic
167C22#2026995812yes—
168C22#2026996112yes—
169C4—22no—
170C4—82no—
171C4—8442no—
172C4—8442no—
173C4—16882no—
174C2—22no—
175C2—21no—
176C2—22no—
177C2—21no—
178C2—21no—
179C2—42no—
180C2#2721300711yescharacteristic
subgroupG/H
centerC2#27213007
commutatorC23×C211C22×C4
FrattiniC24C22×D211
FittingC2×#99×C211C2
socleC2×C211#381
prime radicalC22×D4×C211C22
cosocleC24×C211C23
hypercenterC2×#99D211
hypocenterC211#1828
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
3. #27212913C2semidirect
4. #27212912C2semidirect
8. #20269936C4semidirect
9. C2×C22⋊Dic211C4semidirect
90. C211#1828semidirect

Maximal subgroups

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

subgroupindex
#272136212
C2×#202698922
#272129132
#272129122
C2×#99×C2112
#1828211