Finite Group Databasebeta

← #41014801

Subgroups — #41014801

Order 27008.

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

normal subgroups only

This group has 274 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
1C2×#20269898C212yesmaximal
2#27213016C211yesmaximal, characteristic, complemented
3#27212908C212yesmaximal, complemented
4#27212901C211yesmaximal, characteristic, complemented
5#390×C211C211yesmaximal, characteristic
6C22×C211⋊C8C2212yes—
7C2×C4⋊Dic211—21no—
8C2×C4⋊Dic211—21no—
9C2×C4⋊Dic211C2212yes—
10#20269898—24no—
11C22×D4×C211C2211yescharacteristic
12C2×C4⋊C4×C211C2211yescharacteristic
13C2×C22⋊C4×C211—21no—
14C2×C22⋊C4×C211—21no—
15C2×C42×C211C2211yescharacteristic
16C22×Dic211—22no—
17C2×C211⋊C8—22no—
18C2×C211⋊C8—22no—
19C2×C211⋊C8—22no—
20C4⋊Dic211—22no—
21C4⋊Dic211—22no—
22C4⋊Dic211—42no—
23C4⋊Dic211—42no—
24C24×C211—21no—
25C2×D4×C211—22no—
26C2×D4×C211—22no—
27C2×D4×C211—21no—
28C2×D4×C211D412yes—
29C22×C4×C211—21no—
30C22×C4×C211D411yescharacteristic
31C22×C4×C211D411yescharacteristic
32C22×C4×C211C2311yescharacteristic
33C4⋊C4×C211—22no—
34C22⋊C4×C211—42no—
35C22⋊C4×C211—42no—
36C42×C211—22no—
37C2×Dic211—42no—
38C2×Dic211—42no—
39C2×Dic211—42no—
40C211⋊C8—42no—
41C23×C211—22no—
42C23×C211—41no—
43C23×C211—41no—
44C23×C211—41no—
45C23×C211—41no—
46C23×C211—41no—
47C23×C211—41no—
48C23×C211C2×D411yescharacteristic
49D4×C211—24no—
50D4×C211—42no—
51C2×C4×C211—21no—
52C2×C4×C211—21no—
53C2×C4×C211—21no—
54C2×C4×C211—21no—
55C2×C4×C211—41no—
56C2×C4×C211—41no—
57C2×C4×C211—41no—
58C2×C4×C211D812yes—
59C2×C4×C211SD1612yes—
60C2×C4×C211C2×D411yescharacteristic
61C2×C4×C211#4112yes—
62C2×C4×C211#4112yes—
63C2×C4×C211#4111yescharacteristic
64C22×C211—42no—
65C22×C211—42no—
66C22×C211—42no—
67C22×C211—81no—
68C22×C211—81no—
69C22×C211—81no—
70C22×C211—81no—
71C22×C211#12111yescharacteristic
72C22×C211#12311yescharacteristic
73C22×C211#12411yescharacteristic
74C22×C211C2×D811yescharacteristic
75C22×C211C2×SD1611yescharacteristic
76C22×C211AGL(1,ℤ/8)11yescharacteristic
77C22×C211#13711yescharacteristic
78C4×C211—22no—
79C4×C211—22no—
80C4×C211—81no—
81C4×C211#12312yes—
82C4×C211#12612yes—
83Dic211—82no—
84C2×C211—82no—
85C2×C211#39711yescharacteristic
86C2×C211#45911yescharacteristic
87C2×C211#46111yescharacteristic
88C2×C211#48011yescharacteristic
89C2×C211#48111yescharacteristic
90C2×C211#48611yescharacteristic
91C2×C211#48911yescharacteristic
92C211#207111yescharacteristic, complemented, Sylow
93#2071—2111nomaximal, Sylow
94C2×#102—2112no—
95#390D21111yescharacteristic
96#384—2111no—
97#340—2112no—
98#336—2111no—
99C22×D4C2×D21111yescharacteristic
100C22×C8—2112no—
101C2×C4⋊C4—4221no—
102C2×C4⋊C4—4221no—
103C2×C4⋊C4—2112no—
104C2×C4⋊C4C2×D21111yescharacteristic
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C42C2×D21111yescharacteristic
108#102—4224no—
109C24—21no—
110C2×D4—22no—
111C2×D4—22no—
112C2×D4—21no—
113C2×D4D84412yes—
114C22×C4—21no—
115C22×C4—4222no—
116C22×C4C22⋊D21111yescharacteristic
117C22×C4C22⋊D21111yescharacteristic
118C22×C4C22×D21111yescharacteristic
119C2×C8—4222no—
120C2×C8—4222no—
121C2×C8—4222no—
122C4⋊C4—22no—
123C4⋊C4—8442no—
124C4⋊C4—8442no—
125C4⋊C4—4222no—
126C4⋊C4—4222no—
127C22⋊C4—42no—
128C22⋊C4—42no—
129C42—22no—
130C23—22no—
131C23—41no—
132C23—41no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23C2×C22⋊D21111yescharacteristic
138D4—24no—
139D4—42no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—8442no—
148C2×C4—8442no—
149C2×C4—8442no—
150C2×C4#1680994712yes—
151C2×C4#1680995012yes—
152C2×C4C2×D84411yescharacteristic
153C2×C4#1680996112yes—
154C2×C4#1680996112yes—
155C2×C4#1680996211yescharacteristic
156C8—8442no—
157C22—42no—
158C22—42no—
159C22—42no—
160C22—81no—
161C22—81no—
162C22—81no—
163C22—81no—
164C22#2026995111yescharacteristic
165C22#2026995811yescharacteristic
166C22#2026996411yescharacteristic
167C22C2×#1680994711yescharacteristic
168C22C2×#1680995011yescharacteristic
169C22#2027000211yescharacteristic
170C22#2027000711yescharacteristic
171C4—22no—
172C4—22no—
173C4—81no—
174C4—16882no—
175C4#2026996112yes—
176C4#2026997012yes—
177C2—82no—
178C2#2721307111yescharacteristic
179C2#2721334111yescharacteristic
180C2#2721335111yescharacteristic
181C2#2721345711yescharacteristic
182C2#2721346211yescharacteristic
183C2#2721349211yescharacteristic
184C2#2721350511yescharacteristic
subgroupG/H
centerC23C2×C22⋊D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#390×C211C2
socleC23×C211C2×D4
prime radicalC22×D4×C211C22
hypercenter#390D211
hypocenterC211#2071
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #27213016C2semidirect
3. #27212908C2semidirect
4. #27212901C2semidirect
92. C211#2071semidirect

Maximal subgroups

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

subgroupindex
C2×#202698982
#272130162
#272129082
#272129012
#390×C2112
#2071211