Finite Group Databasebeta

← #41013501

Subgroups — #41013501

Order 27008.

262 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C4×#16809962C211yesmaximal, characteristic
2#27213128C211yesmaximal, characteristic
3#27213119C211yesmaximal, characteristic
4#27212906C211yesmaximal, characteristic, complemented
5#27212903C211yesmaximal, characteristic, complemented
6#27212902C211yesmaximal, characteristic, complemented
7#339×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10C4×Dic422—21no—
11C4×D844—21no—
12C4×C22⋊D211—22no—
13#20269936C2211yescharacteristic
14#20269936C2211yescharacteristic
15#20269934C412yescomplemented
16C42×D211C412yescomplemented
17#20269895—21no—
18#20269893—21no—
19#20269891—22no—
20C2×M16×C211C2211yescharacteristic
21#117×C211C2211yescharacteristic
22C2×C42×C211C2211yescharacteristic
23#16809962—21no—
24#16809962—21no—
25#16809962—21no—
26#16809962—21no—
27C2×Dic422D411yescharacteristic
28C2×D844D411yescharacteristic
29C2×C22⋊D211D412yes—
30C2×C4×D211—22no—
31C2×C4×D211C2×C412yes—
32#16809946—21no—
33#16809946—21no—
34C2×C211⋊C8—21no—
35C4⋊Dic211—21no—
36C4⋊Dic211—21no—
37Dic211⋊C4—22no—
38C22⋊Dic211—21no—
39C22⋊Dic211—21no—
40(C2×D211)⋊C4—22no—
41C4×Dic211—22no—
42C4×Dic211—21no—
43C4×Dic211C2×C412yes—
44C22×C4×C211—21no—
45C22×C4×C211C2311yescharacteristic
46M16×C211—21no—
47M16×C211—21no—
48C2×C8×C211—21no—
49C4⋊C4×C211—21no—
50C22⋊C4×C211—21no—
51C42×C211—21no—
52C42×C211—21no—
53C42×C211C2×C411yescharacteristic
54C42×C211C2×C411yescharacteristic
55C22×D211C22⋊C412yes—
56Dic422—21no—
57Dic422—21no—
58D844—21no—
59D844—21no—
60C22⋊D211—22no—
61C22⋊D211—22no—
62C2×Dic211—22no—
63C2×Dic211—41no—
64C2×Dic211C22⋊C412yes—
65C4×D211—22no—
66C4×D211—22no—
67C4×D211—44no—
68C211⋊C8—41no—
69C23×C211C2×D411yescharacteristic
70C2×C4×C211—21no—
71C2×C4×C211—21no—
72C2×C4×C211—42no—
73C2×C4×C211—41no—
74C2×C4×C211C22×C411yescharacteristic
75C2×C4×C211C2×D411yescharacteristic
76C2×C4×C211#4111yescharacteristic
77C2×C4×C211#4111yescharacteristic
78C2×C4×C211#4111yescharacteristic
79C2×C4×C211#4111yescharacteristic
80C8×C211—41no—
81C22×C211—41no—
82C22×C211C2×C22⋊C411yescharacteristic
83C22×C211#12311yescharacteristic
84C22×C211#12311yescharacteristic
85C2×D211—22no—
86C2×D211—42no—
87C4×C211—22no—
88C4×C211—41no—
89C4×C211—81no—
90C4×C211#11711yescharacteristic
91C4×C211#11711yescharacteristic
92C4×C211#12311yescharacteristic
93C4×C211#12311yescharacteristic
94Dic211—22no—
95Dic211—42no—
96Dic211—81no—
97C2×C211—21no—
98C2×C211—21no—
99C2×C211—21no—
100C2×C211#38111yescharacteristic
101D211—42no—
102C211#182711yescharacteristic, complemented, Sylow
103#1827—2111nomaximal, Sylow
104C4×#41—2111no—
105#413—2111no—
106#410—2111no—
107#339—2111no—
108#339D21111yescharacteristic, complemented
109H(3,ℤ/4)—2111no—
110H(3,ℤ/4)—2111no—
111C2×#41—2111no—
112C2×M16—2111no—
113C2×M16C2×D21111yescharacteristic
114C4×Q8—4221no—
115C4×D4—4221no—
116C4×D4—4222no—
117#117—2111no—
118#117—2111no—
119#117—2112no—
120#117C2×D21111yescharacteristic
121C2×C42—2112no—
122C2×C42C2×D21111yescharacteristic
123#101—4221no—
124#100—4221no—
125#99—4222no—
126#41—4221no—
127#41—4221no—
128#41—4221no—
129#41—4221no—
130C2×Q8—2111no—
131C2×D4—2112no—
132C2×D4—2111no—
133C22×C4—21no—
134C22×C4—4222no—
135C22×C4—2112no—
136C22×C4C22×D21111yescharacteristic
137M16—21no—
138M16—21no—
139M16—4221no—
140M16—4221no—
141C2×C8—21no—
142C2×C8—4221no—
143C4⋊C4—21no—
144C4⋊C4—4222no—
145C4⋊C4—4221no—
146C4⋊C4—4221no—
147C22⋊C4—21no—
148C22⋊C4—4222no—
149C22⋊C4—4221no—
150C22⋊C4—4221no—
151C42—21no—
152C42—21no—
153C42—4222no—
154C42—4221no—
155C42—2112no—
156C42C4×D21111yescharacteristic
157C42C4×D21111yescharacteristic
158C23—2112no—
159C23D4×D21111yescharacteristic
160Q8—4221no—
161Q8—4221no—
162D4—4222no—
163D4—4222no—
164D4—4221no—
165D4—4221no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—8444no—
171C2×C4—8441no—
172C2×C4—4222no—
173C2×C4—4222no—
174C2×C4—4222no—
175C2×C4—2112no—
176C2×C4C2×C4×D21111yescharacteristic
177C2×C4D4×D21111yescharacteristic
178C2×C4#1680996111yescharacteristic
179C2×C4#1680996211yescharacteristic
180C2×C4#1680996211yescharacteristic
181C2×C4#1680996311yescharacteristic
182C8—41no—
183C8—8441no—
184C22—41no—
185C22—8442no—
186C22—4222no—
187C22C22⋊C4×D21111yescharacteristic
188C22#2026995911yescharacteristic
189C22#2026996211yescharacteristic
190C4—22no—
191C4—41no—
192C4—81no—
193C4—8442no—
194C4—16881no—
195C4—4222no—
196C4#2026993411yescharacteristic
197C4#2026993511yescharacteristic
198C4#2026995911yescharacteristic
199C4#2026996211yescharacteristic
200C2—21no—
201C2—21no—
202C2—21no—
203C2—8442no—
204C2#2721300811yescharacteristic
subgroupG/H
centerC4#20269935
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#339×C211C2
socleC2×C211#381
prime radicalC2×#16809962C22
cosocleC22×C4×C211C23
hypercenter#339D211
hypocenterC211#1827
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
4. #27212906C2semidirect
5. #27212903C2semidirect
6. #27212902C2semidirect
7. #339×C211C2semidirect
15. #20269934C4semidirect
16. C42×D211C4semidirect
102. C211#1827semidirect
108. #339D211semidirect

Maximal subgroups

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

subgroupindex
C4×#168099622
#272131282
#272131192
#272129062
#272129032
#272129022
#339×C2112
#1827211