Finite Group Databasebeta

← #41013585

Subgroups — #41013585

Order 27008.

360 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 121 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×C4⋊Dic211C211yesmaximal, characteristic
2C2×#20269910C212yesmaximal, complemented
3C2×C22⋊(C211⋊C8)C211yesmaximal, characteristic
4#27212908C212yesmaximal, complemented
5#380×C211C211yesmaximal, characteristic
6C23×Dic211—21no—
7C22×C211⋊C8C2212yes—
8C2×C4⋊Dic211—24no—
9C2×C4⋊Dic211—21no—
10C2×C4⋊Dic211C2212yes—
11#20269910—24no—
12C22⋊(C211⋊C8)C414yescomplemented
13C23×C4×C211C2211yescharacteristic
14C2×C4⋊C4×C211C2211yescharacteristic
15C2×C4⋊C4×C211C2211yescharacteristic
16C2×C22⋊C4×C211—21no—
17#95×C211—21no—
18C22×Dic211—24no—
19C22×Dic211—22no—
20C22×Dic211—42no—
21C22×Dic211—42no—
22C2×C211⋊C8—22no—
23C2×C211⋊C8—22no—
24C2×C211⋊C8C2×C414yescomplemented
25C4⋊Dic211—24no—
26C4⋊Dic211—22no—
27C4⋊Dic211D414yes—
28C24×C211D411yescharacteristic
29C22×C4×C211—22no—
30C22×C4×C211—22no—
31C22×C4×C211—21no—
32C22×C4×C211—21no—
33C22×C4×C211C2×C412yes—
34C22×C4×C211D411yescharacteristic
35C22×C4×C211Q812yes—
36C22×C4×C211C2311yescharacteristic
37C4⋊C4×C211—22no—
38C4⋊C4×C211—22no—
39C22⋊C4×C211—42no—
40C2×Dic211—24no—
41C2×Dic211—48no—
42C2×Dic211—42no—
43C2×Dic211—42no—
44C211⋊C8—24no—
45C23×C211—21no—
46C23×C211—21no—
47C23×C211—21no—
48C23×C211—21no—
49C23×C211C4⋊C412yes—
50C23×C211D812yes—
51C23×C211Q1612yes—
52C23×C211C2×D411yescharacteristic
53C2×C4×C211—21no—
54C2×C4×C211—21no—
55C2×C4×C211—24no—
56C2×C4×C211—24no—
57C2×C4×C211—41no—
58C2×C4×C211—41no—
59C2×C4×C211—41no—
60C2×C4×C211—41no—
61C2×C4×C211—41no—
62C2×C4×C211—41no—
63C2×C4×C211C4⋊C412yes—
64C2×C4×C211C22×C411yescharacteristic
65C2×C4×C211C2×D412yes—
66C2×C4×C211C2×Q811yescharacteristic
67C2×C4×C211#4112yes—
68C22×C211—22no—
69C22×C211—22no—
70C22×C211—22no—
71C22×C211—22no—
72C22×C211—22no—
73C22×C211—22no—
74C22×C211#10714yescomplemented
75C22×C211C2×C4⋊C411yescharacteristic
76C22×C211#12011yescharacteristic
77C22×C211#12311yescharacteristic
78C22×C211C2×D811yescharacteristic
79C22×C211C2×Q1611yescharacteristic
80C22×C211AGL(1,ℤ/8)11yescharacteristic
81C22×C211#13711yescharacteristic
82C4×C211—22no—
83C4×C211—81no—
84C4×C211—81no—
85C4×C211C4×D412yes—
86C4×C211C22⋊Q812yes—
87Dic211—44no—
88C2×C211—24no—
89C2×C211#38511yescharacteristic
90C2×C211C2×#10711yescharacteristic
91C2×C211#42811yescharacteristic
92C2×C211#44711yescharacteristic
93C2×C211#45111yescharacteristic
94C2×C211#48011yescharacteristic
95C2×C211#48411yescharacteristic
96C211#185211yescharacteristic, complemented, Sylow
97#1852—2111nomaximal, Sylow
98C22×C4⋊C4—2111no—
99C2×#107—2112no—
100C2×C22⋊C8—2111no—
101#380D21111yescharacteristic
102#340—2112no—
103C23×C4—4221no—
104C23×C4C2×D21111yescharacteristic
105C22×C8—2112no—
106C2×C4⋊C4—4221no—
107C2×C4⋊C4—4224no—
108C2×C4⋊C4—2112no—
109C2×C4⋊C4C2×D21111yescharacteristic
110C2×C4⋊C4C2×D21111yescharacteristic
111C2×C22⋊C4—21no—
112#107—4224no—
113C22⋊C8—2114no—
114#95—21no—
115C24C22⋊D21111yescharacteristic
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—8442no—
121C22×C4—8442no—
122C22×C4—4224no—
123C22×C4—4222no—
124C22×C4C4×D21112yes—
125C22×C4C22⋊D21111yescharacteristic
126C22×C4Dic42212yes—
127C22×C4C22×D21111yescharacteristic
128C2×C8—4222no—
129C2×C8—4222no—
130C2×C8—2114no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—4224no—
134C4⋊C4—4222no—
135C4⋊C4—2114no—
136C22⋊C4—42no—
137C23—21no—
138C23—21no—
139C23—21no—
140C23—21no—
141C23Dic211⋊C412yes—
142C23#1680994712yes—
143C23#1680995212yes—
144C23C2×C22⋊D21111yescharacteristic
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—24no—
148C2×C4—24no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—8448no—
156C2×C4—8442no—
157C2×C4—8442no—
158C2×C4—4224no—
159C2×C4Dic211⋊C412yes—
160C2×C4C2×C4×D21111yescharacteristic
161C2×C4D4×D21112yes—
162C2×C4C2×Dic42211yescharacteristic
163C2×C4#1680996112yes—
164C8—4224no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22#2026991014yescomplemented
172C22C2×Dic211⋊C411yescharacteristic
173C22#2026994511yescharacteristic
174C22#2026995811yescharacteristic
175C22C2×#1680994711yescharacteristic
176C22C2×#1680995211yescharacteristic
177C22#2027000211yescharacteristic
178C22#2027000711yescharacteristic
179C4—22no—
180C4—81no—
181C4—81no—
182C4—8444no—
183C4(C22⋊D211)⋊C412yes—
184C4#2026995412yes—
185C2—24no—
186C2#2721301811yescharacteristic
187C2C2×#2026991011yescharacteristic
188C2#2721318911yescharacteristic
189C2#2721327911yescharacteristic
190C2#2721330111yescharacteristic
191C2#2721345711yescharacteristic
192C2#2721347911yescharacteristic
subgroupG/H
centerC23C2×C22⋊D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#380×C211C2
socleC23×C211C2×D4
prime radicalC24×C211D4
cosocleC22×C4×C211C23
hypercenter#380D211
hypocenterC211#1852
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#20269910C2semidirect
4. #27212908C2semidirect
12. C22⋊(C211⋊C8)C4semidirect
24. C2×C211⋊C8C2×C4semidirect
74. C22×C211#107semidirect
96. C211#1852semidirect
171. C22#20269910semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Dic2112
C2×#202699102
C2×C22⋊(C211⋊C8)2
#272129082
#380×C2112
#1852211