Finite Group Databasebeta

← #41014044

Subgroups — #41014044

Order 27008.

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

normal subgroups only

This group has 224 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
1C2×#20269912C211yesmaximal, characteristic, complemented
2C2×#20269905C211yesmaximal, characteristic
3#27213140C211yesmaximal, characteristic
4#27213114C211yesmaximal, characteristic
5#27213102C211yesmaximal, characteristic, complemented
6#27212901C211yesmaximal, characteristic, complemented
7#339×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10C22×C211⋊C8C2211yescharacteristic
11C22×C211⋊C8C2211yescharacteristic
12#20269912—22no—
13#20269905—22no—
14#20269900C411yescharacteristic, complemented
15#20269900C411yescharacteristic, complemented
16#20269897C411yescharacteristic, complemented
17#20269897C411yescharacteristic, complemented
18(C2×D211)⋊C8—21no—
19(C2×D211)⋊C8—21no—
20#20269888—21no—
21C4×C211⋊C8—21no—
22C2×M16×C211C2211yescharacteristic
23#117×C211C2211yescharacteristic
24C2×C42×C211C2211yescharacteristic
25#16809962—21no—
26#16809962—21no—
27#16809962—21no—
28#16809962—21no—
29C2×Dic422C2×C411yescharacteristic
30C2×D844C2×C411yescharacteristic
31C2×C22⋊D211—21no—
32C2×C4×D211—21no—
33#16809946—21no—
34#16809946D412yes—
35C2×C211⋊C8—21no—
36C2×C211⋊C8—21no—
37C2×C211⋊C8—21no—
38C2×C211⋊C8—21no—
39C2×C211⋊C8—21no—
40C2×C211⋊C8C2×C411yescharacteristic
41C2×C211⋊C8C2×C411yescharacteristic
42C2×C211⋊C8D411yescharacteristic
43C2×C211⋊C8D411yescharacteristic
44C22×C4×C211—21no—
45C22×C4×C211C2311yescharacteristic
46M16×C211—21no—
47M16×C211—21no—
48C2×C8×C211—21no—
49C4⋊C4×C211C2×C411yescharacteristic
50C4⋊C4×C211C2×C411yescharacteristic
51C22⋊C4×C211—21no—
52C42×C211—21no—
53C42×C211—21no—
54C42×C211—21no—
55C22×D211—21no—
56Dic422—21no—
57Dic422—21no—
58D844—21no—
59D844—21no—
60C22⋊D211—41no—
61C22⋊D211—41no—
62C2×Dic211—21no—
63C4×D211—41no—
64C4×D211—41no—
65C211⋊C8—21no—
66C211⋊C8—21no—
67C211⋊C8—22no—
68C211⋊C8—21no—
69C211⋊C8—21no—
70C23×C211#4111yescharacteristic
71C2×C4×C211—22no—
72C2×C4×C211—21no—
73C2×C4×C211—21no—
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—41no—
83C22×C211—21no—
84C22×C211—21no—
85C22×C211C4×D411yescharacteristic
86C22×C211#12411yescharacteristic
87C22×C211#12611yescharacteristic
88C2×D211—41no—
89C2×D211—41no—
90C2×D211—41no—
91C4×C211—42no—
92C4×C211—41no—
93C4×C211—41no—
94C4×C211#11711yescharacteristic
95C4×C211C4×D411yescharacteristic
96C4×C211#12111yescharacteristic
97C4×C211#12311yescharacteristic
98Dic211—41no—
99C2×C211—21no—
100C2×C211—21no—
101C2×C211#38811yescharacteristic
102C2×C211#44311yescharacteristic
103C2×C211#44411yescharacteristic
104D211—81no—
105C211#192911yescharacteristic, complemented, Sylow
106#1929—2111nomaximal, Sylow
107C2×#108—2111no—
108C2×#104—2111no—
109#416—2111no—
110#408—2111no—
111#405—2111no—
112#339D21111yescharacteristic, complemented
113#336—2111no—
114C2×#41—2111no—
115C2×M16—2111no—
116C2×M16C2×D21111yescharacteristic
117C22×C8—2111no—
118C22×C8—2111no—
119#117C2×D21111yescharacteristic
120C2×C42C2×D21111yescharacteristic
121#108—4222no—
122#104—4222no—
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—4221no—
132#41—4221no—
133#41—4221no—
134#41—4221no—
135C2×Q8—2111no—
136C2×D4—4221no—
137C2×D4—2111no—
138C22×C4—21no—
139C22×C4—4221no—
140C22×C4C22×D21111yescharacteristic
141M16—21no—
142M16—21no—
143M16—4221no—
144M16—2112no—
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⋊C4C4×D21111yescharacteristic
156C4⋊C4C4×D21111yescharacteristic
157C22⋊C4—21no—
158C42—21no—
159C42—21no—
160C42—21no—
161C23—4221no—
162C23#1680996111yescharacteristic
163Q8—4221no—
164Q8—4221no—
165D4—8441no—
166D4—8441no—
167D4—4221no—
168D4—4221no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—8441no—
175C2×C4—8441no—
176C2×C4—4221no—
177C2×C4C2×C4×D21111yescharacteristic
178C2×C4D4×D21111yescharacteristic
179C2×C4D4×D21111yescharacteristic
180C2×C4#1680996211yescharacteristic
181C2×C4#1680996211yescharacteristic
182C2×C4#1680996311yescharacteristic
183C8—41no—
184C8—4221no—
185C8—4221no—
186C8—4222no—
187C8—4221no—
188C8—4221no—
189C22—21no—
190C22—21no—
191C22—8441no—
192C22—8441no—
193C22—8441no—
194C22(C22⋊D211)⋊C411yescharacteristic
195C22#2026996311yescharacteristic
196C22#2026997211yescharacteristic
197C4—42no—
198C4—41no—
199C4—41no—
200C4—8441no—
201C4#2026993411yescharacteristic
202C4#2026993911yescharacteristic
203C4#2026995011yescharacteristic
204C4#2026996211yescharacteristic
205C2—21no—
206C2—21no—
207C2—16881no—
208C2#2721303511yescharacteristic
209C2#2721326011yescharacteristic
210C2#2721326511yescharacteristic
subgroupG/H
centerC2×C4D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#339×C211C2
socleC22×C211C4×D4
prime radicalC2×#16809962C22
cosocleC22×C4×C211C23
hypercenter#339D211
hypocenterC211#1929
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. C2×#20269912C2semidirect
5. #27213102C2semidirect
6. #27212901C2semidirect
7. #339×C211C2semidirect
14. #20269900C4semidirect
15. #20269900C4semidirect
16. #20269897C4semidirect
17. #20269897C4semidirect
105. C211#1929semidirect
112. #339D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699122
C2×#202699052
#272131402
#272131142
#272131022
#272129012
#339×C2112
#1929211