Finite Group Databasebeta

← #41013470

Subgroups — #41013470

Order 27008.

504 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C22×C22⋊Dic211C211yesmaximal, characteristic
2#27213002C211yesmaximal, characteristic
3#27212912C214yesmaximal, complemented
4#379×C211C211yesmaximal, characteristic
5C23×Dic211C412yescomplemented
6C2×C22⋊Dic211—22no—
7C2×C22⋊Dic211—22no—
8C2×C22⋊Dic211—22no—
9C2×C22⋊Dic211C412yescomplemented
10C2×C22⋊Dic211C2212yes—
11C2×C22⋊Dic211C2212yes—
12C25×C211C2211yescharacteristic
13C2×C22⋊C4×C211—22no—
14C2×C22⋊C4×C211C2212yes—
15C22×Dic211—22no—
16C22×Dic211—22no—
17C22×Dic211—22no—
18C22×Dic211—44no—
19C22×Dic211C2×C414yes—
20C22⋊Dic211—24no—
21C22⋊Dic211—22no—
22C22⋊Dic211—22no—
23C22⋊Dic211—24no—
24C22⋊Dic211—44no—
25C24×C211—24no—
26C24×C211—44no—
27C24×C211C2×C411yescharacteristic
28C24×C211C2×C411yescharacteristic
29C24×C211D412yes—
30C24×C211D412yes—
31C24×C211C2311yescharacteristic
32C22×C4×C211—22no—
33C22⋊C4×C211—24no—
34C22⋊C4×C211—44no—
35C2×Dic211—24no—
36C2×Dic211—44no—
37C2×Dic211—44no—
38C2×Dic211—44no—
39C2×Dic211—44no—
40C2×Dic211—42no—
41C23×C211—24no—
42C23×C211—22no—
43C23×C211—24no—
44C23×C211—24no—
45C23×C211—22no—
46C23×C211—24no—
47C23×C211—24no—
48C23×C211—42no—
49C23×C211—42no—
50C23×C211—42no—
51C23×C211—42no—
52C23×C211—416no—
53C23×C211C22⋊C412yes—
54C23×C211C22⋊C412yes—
55C23×C211C22×C411yescharacteristic
56C23×C211C2×D411yescharacteristic
57C23×C211C2×D411yescharacteristic
58C23×C211#4114yes—
59C2×C4×C211—44no—
60C2×C4×C211—42no—
61C22×C211—28no—
62C22×C211—22no—
63C22×C211—22no—
64C22×C211—22no—
65C22×C211—22no—
66C22×C211—22no—
67C22×C211—22no—
68C22×C211—22no—
69C22×C211—22no—
70C22×C211—44no—
71C22×C211—48no—
72C22×C211—48no—
73C22×C211—44no—
74C22×C211#9914yes—
75C22×C211C2×C22⋊C411yescharacteristic
76C22×C211#11712yes—
77C22×C211#12312yes—
78C22×C211#12312yes—
79C4×C211—82no—
80Dic211—44no—
81Dic211—82no—
82C2×C211—22no—
83C2×C211—22no—
84C2×C211—24no—
85C2×C211—22no—
86C2×C211—42no—
87C2×C211#38111yescharacteristic
88C2×C211C2×#9912yes—
89C211#181911yescharacteristic, complemented, Sylow
90#1819—2111nomaximal, Sylow
91C22×C22⋊C4—2111no—
92#379—2111no—
93#379D21111yescharacteristic
94#342—2114no—
95C25C2×D21111yescharacteristic
96C23×C4—2112no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—4222no—
99C2×C22⋊C4—4222no—
100C2×C22⋊C4—4222no—
101C2×C22⋊C4—2112no—
102C2×C22⋊C4—2112no—
103C2×C22⋊C4—2112no—
104C2×C22⋊C4C2×D21112yes—
105C24—24no—
106C24—44no—
107C24C4×D21111yescharacteristic
108C24C4×D21111yescharacteristic
109C24C22⋊D21112yes—
110C24D84412yes—
111C24C22×D21111yescharacteristic
112C22×C4—22no—
113C22×C4—8444no—
114C22×C4—4222no—
115C22×C4—4222no—
116C22×C4—4222no—
117C22×C4—2114no—
118C22⋊C4—24no—
119C22⋊C4—44no—
120C22⋊C4—8444no—
121C22⋊C4—4224no—
122C22⋊C4—4224no—
123C22⋊C4—4222no—
124C22⋊C4—4222no—
125C23—24no—
126C23—22no—
127C23—24no—
128C23—24no—
129C23—22no—
130C23—24no—
131C23—24no—
132C23—42no—
133C23—42no—
134C23—42no—
135C23—42no—
136C23—416no—
137C23(C2×D211)⋊C412yes—
138C23(C2×D211)⋊C412yes—
139C23C2×C4×D21111yescharacteristic
140C23C2×C22⋊D21111yescharacteristic
141C23C2×D84411yescharacteristic
142C23#1680996114yes—
143C2×C4—44no—
144C2×C4—42no—
145C2×C4—8444no—
146C2×C4—8444no—
147C2×C4—8444no—
148C2×C4—8444no—
149C2×C4—8442no—
150C2×C4—4224no—
151C22—28no—
152C22—22no—
153C22—22no—
154C22—22no—
155C22—22no—
156C22—22no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—44no—
161C22—48no—
162C22—48no—
163C22—44no—
164C22#2026989114yes—
165C22C2×(C2×D211)⋊C411yescharacteristic
166C22#2026993312yes—
167C22#2026995812yes—
168C22#2026996112yes—
169C4—82no—
170C4—8444no—
171C4—16882no—
172C2—22no—
173C2—22no—
174C2—24no—
175C2—22no—
176C2—42no—
177C2#2721300711yescharacteristic
178C2C2×#2026989112yes—
subgroupG/H
centerC22C2×(C2×D211)⋊C4
commutatorC23×C211C22×C4
FrattiniC24C22×D211
Fitting#379×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC25×C211C22
cosocleC24×C211C23
hypercenter#379D211
hypocenterC211#1819
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C25 ◁ #379

Complements

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

NH ≅ G/Ntype
3. #27212912C2semidirect
5. C23×Dic211C4semidirect
9. C2×C22⋊Dic211C4semidirect
89. C211#1819semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Dic2112
#272130022
#272129122
#379×C2112
#1819211