Finite Group Databasebeta

← #41014049

Subgroups — #41014049

Order 27008.

224 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 75 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×#20269912C211yesmaximal, characteristic, complemented
2#27213142C211yesmaximal, characteristic
3#27213113C211yesmaximal, characteristic
4#27213104C211yesmaximal, characteristic, complemented
5#27212906C211yesmaximal, characteristic, complemented
6#27212901C211yesmaximal, characteristic, complemented
7C2×#104×C211C211yesmaximal, characteristic
8C2×#16809946C2211yescharacteristic
9C22×C211⋊C8C2211yescharacteristic
10C22×C211⋊C8C2211yescharacteristic
11#20269936C2211yescharacteristic
12#20269912—22no—
13#20269901C411yescharacteristic, complemented
14#20269901C411yescharacteristic, complemented
15#20269898C411yescharacteristic, complemented
16#20269898C411yescharacteristic, complemented
17C22⋊(C211⋊C8)—21no—
18C22⋊(C211⋊C8)—21no—
19#20269887—21no—
20C8×Dic211—21no—
21C2×#41×C211C2211yescharacteristic
22C2×M16×C211C2211yescharacteristic
23C2×C42×C211C2211yescharacteristic
24#104×C211—22no—
25#16809946—21no—
26#16809946—21no—
27C2×C211⋊C8—21no—
28C2×C211⋊C8—21no—
29C2×C211⋊C8—21no—
30C2×C211⋊C8—21no—
31C2×C211⋊C8—21no—
32C2×C211⋊C8C2×C411yescharacteristic
33C2×C211⋊C8C2×C411yescharacteristic
34C2×C211⋊C8D411yescharacteristic
35C2×C211⋊C8D411yescharacteristic
36C4⋊Dic211C2×C411yescharacteristic
37C4⋊Dic211C2×C411yescharacteristic
38C22⋊Dic211—21no—
39C4×Dic211—21no—
40#41×C211—21no—
41#41×C211—21no—
42#41×C211—21no—
43#41×C211—21no—
44C2×Q8×C211C2×C411yescharacteristic
45C2×D4×C211—21no—
46C2×D4×C211C2×C411yescharacteristic
47C22×C4×C211—21no—
48C22×C4×C211—21no—
49C22×C4×C211C2311yescharacteristic
50M16×C211—21no—
51M16×C211D412yes—
52C2×C8×C211—21no—
53C42×C211—21no—
54C42×C211—21no—
55C2×Dic211—21no—
56C2×Dic211—21no—
57C211⋊C8—21no—
58C211⋊C8—21no—
59C211⋊C8—21no—
60C211⋊C8—21no—
61C211⋊C8—41no—
62C23×C211—21no—
63C23×C211#4111yescharacteristic
64Q8×C211—21no—
65Q8×C211—21no—
66D4×C211—21no—
67D4×C211—21no—
68D4×C211—41no—
69D4×C211—41no—
70C2×C4×C211—22no—
71C2×C4×C211—21no—
72C2×C4×C211—41no—
73C2×C4×C211—41no—
74C2×C4×C211—41no—
75C2×C4×C211—41no—
76C2×C4×C211C22×C411yescharacteristic
77C2×C4×C211C2×D411yescharacteristic
78C2×C4×C211C2×D411yescharacteristic
79C2×C4×C211#4111yescharacteristic
80C2×C4×C211#4111yescharacteristic
81C2×C4×C211#4111yescharacteristic
82C8×C211—22no—
83C22×C211—21no—
84C22×C211—21no—
85C22×C211—41no—
86C22×C211—41no—
87C22×C211—41no—
88C22×C211C4×D411yescharacteristic
89C22×C211#12411yescharacteristic
90C22×C211#12611yescharacteristic
91C4×C211—42no—
92C4×C211—41no—
93C4×C211#11711yescharacteristic
94C4×C211C4×D411yescharacteristic
95C4×C211#12111yescharacteristic
96C4×C211#12311yescharacteristic
97Dic211—41no—
98Dic211—41no—
99C2×C211—21no—
100C2×C211—21no—
101C2×C211—81no—
102C2×C211#38811yescharacteristic
103C2×C211#44311yescharacteristic
104C2×C211#44411yescharacteristic
105C211#192911yescharacteristic, complemented, Sylow
106#1929—2111nomaximal, Sylow
107C2×#108—2111no—
108C2×#104D21111yescharacteristic
109#416—2111no—
110#408—2111no—
111#405—2111no—
112#339—2111no—
113#336—2111no—
114C2×#41C2×D21111yescharacteristic
115C2×M16—2111no—
116C2×M16C2×D21111yescharacteristic
117C22×C8—2111no—
118C22×C8—2111no—
119#117—2111no—
120C2×C42C2×D21111yescharacteristic
121#108—4222no—
122#104—22no—
123#103—2111no—
124#103—2111no—
125#102—2111no—
126#102—2111no—
127C22⋊C8—4221no—
128C22⋊C8—4221no—
129#97—4221no—
130C4×C8—4221no—
131#41—21no—
132#41—21no—
133#41—21no—
134#41—21no—
135C2×Q8C4×D21111yescharacteristic
136C2×D4—21no—
137C2×D4C4×D21111yescharacteristic
138C22×C4—21no—
139C22×C4—21no—
140C22×C4C22×D21111yescharacteristic
141M16—21no—
142M16—4221no—
143M16—4221no—
144M16C22⋊D21112yes—
145C2×C8—21no—
146C2×C8—4221no—
147C2×C8—4221no—
148C2×C8—4221no—
149C2×C8—4221no—
150C2×C8—4221no—
151C2×C8—2111no—
152C2×C8—2111no—
153C2×C8—2111no—
154C2×C8—2111no—
155C4⋊C4—2111no—
156C4⋊C4—2111no—
157C22⋊C4—4221no—
158C42—21no—
159C42—21no—
160C42—4221no—
161C23—21no—
162C23#1680996111yescharacteristic
163Q8—21no—
164Q8—21no—
165D4—21no—
166D4—21no—
167D4—41no—
168D4—41no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—4221no—
176C2×C4—4221no—
177C2×C4C2×C4×D21111yescharacteristic
178C2×C4D4×D21111yescharacteristic
179C2×C4C2×C22⋊D21111yescharacteristic
180C2×C4#1680996111yescharacteristic
181C2×C4#1680996111yescharacteristic
182C2×C4#1680996211yescharacteristic
183C8—22no—
184C8—8441no—
185C8—4221no—
186C8—4221no—
187C8—4221no—
188C8—4221no—
189C22—21no—
190C22—21no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22(C22⋊D211)⋊C411yescharacteristic
195C22#2026996411yescharacteristic
196C22#2026997011yescharacteristic
197C4—42no—
198C4—41no—
199C4—8441no—
200C4—8441no—
201C4#2026993311yescharacteristic
202C4C4×C22⋊D21111yescharacteristic
203C4#2026994911yescharacteristic
204C4#2026995911yescharacteristic
205C2—21no—
206C2—21no—
207C2—81no—
208C2#2721304011yescharacteristic
209C2#2721326011yescharacteristic
210C2#2721326511yescharacteristic
subgroupG/H
centerC2×C4D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#104×C211C2
socleC22×C211C4×D4
prime radicalC2×#41×C211C22
cosocleC22×C4×C211C23
hypercenterC2×#104D211
hypocenterC211#1929
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
1. C2×#20269912C2semidirect
4. #27213104C2semidirect
5. #27212906C2semidirect
6. #27212901C2semidirect
13. #20269901C4semidirect
14. #20269901C4semidirect
15. #20269898C4semidirect
16. #20269898C4semidirect
105. C211#1929semidirect

Maximal subgroups

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

subgroupindex
C2×#202699122
#272131422
#272131132
#272131042
#272129062
#272129012
C2×#104×C2112
#1929211