Finite Group Databasebeta

← #44398816

Subgroups — #44398816

Order 29056.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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
1#28409747C211yesmaximal, characteristic, complemented
2#28409404C211yesmaximal, characteristic, complemented
3#28409224C211yesmaximal, characteristic, complemented
4#28408780C211yesmaximal, characteristic, complemented
5D4⋊(C227⋊C8)C211yesmaximal, characteristic, complemented
6#28408759C211yesmaximal, characteristic, complemented
7#459×C227C211yesmaximal, characteristic, complemented
8C2×#16813872—22no—
9C2×#16813869—21no—
10#20300288C2211yescharacteristic
11#20300274—22no—
12#20300262—21no—
13#20300259—21no—
14C4×D908C2211yescharacteristic
15#20300247—21no—
16#20300246—21no—
17C4⋊(C227⋊C8)C2211yescharacteristic
18#20300209—21no—
19C4×C227⋊C8C2211yescharacteristic
20C2×D8×C227—21no—
21#127×C227C2211yescharacteristic
22C4×D4×C227C2211yescharacteristic
23C4⋊C8×C227C2211yescharacteristic
24#102×C227—21no—
25C2×Dic454D412yes—
26C2×D908D411yescharacteristic
27C2×C22⋊D227—22no—
28C2×C4×D227—21no—
29#16813872—44no—
30#16813869—42no—
31C2×C227⋊C8—21no—
32C2×C227⋊C8—21no—
33C4⋊Dic227—21no—
34C4⋊Dic227—21no—
35C4⋊Dic227D411yescharacteristic
36C22⋊Dic227—22no—
37(C2×D227)⋊C4—22no—
38(C2×D227)⋊C4—21no—
39C4×Dic227—22no—
40C2×D4×C227—21no—
41C2×D4×C227—21no—
42C2×D4×C227—21no—
43C2×D4×C227D411yescharacteristic
44C22×C4×C227—21no—
45D8×C227—42no—
46C2×C8×C227—21no—
47C4⋊C4×C227D411yescharacteristic
48C22⋊C4×C227—21no—
49C42×C227C2311yescharacteristic
50C22×D227—21no—
51Dic454—42no—
52D908—21no—
53D908—21no—
54C22⋊D227—82no—
55C2×Dic227—21no—
56C2×Dic227—22no—
57C4×D227—41no—
58C4×D227—41no—
59C227⋊C8—22no—
60C227⋊C8—41no—
61C23×C227—21no—
62C23×C227—41no—
63D4×C227—21no—
64D4×C227—21no—
65D4×C227—42no—
66D4×C227—41no—
67D4×C227—41no—
68D4×C227—41no—
69D4×C227—41no—
70C2×C4×C227—21no—
71C2×C4×C227—41no—
72C2×C4×C227—41no—
73C2×C4×C227C2×D411yescharacteristic
74C2×C4×C227C2×D411yescharacteristic
75C2×C4×C227C2×D411yescharacteristic
76C8×C227—41no—
77C22×C227—41no—
78C22×C227—41no—
79C22×C227—41no—
80C22×C227—81no—
81C22×C227—81no—
82C22×C227—81no—
83C22×C227#12011yescharacteristic
84C2×D227—41no—
85C2×D227—41no—
86C2×D227—41no—
87C4×C227—21no—
88C4×C227—41no—
89C4×C227#13511yescharacteristic
90C4×C227#13511yescharacteristic
91C4×C227AGL(1,ℤ/8)11yescharacteristic
92C4×C227AGL(1,ℤ/8)11yescharacteristic
93Dic227—41no—
94Dic227—42no—
95C2×C227—81no—
96C2×C227—161no—
97C2×C227#44911yescharacteristic
98C2×C227#44911yescharacteristic
99C2×C227#45311yescharacteristic
100D227—81no—
101C227#162411yescharacteristic, complemented, Sylow
102#1624—2271nomaximal, Sylow
103#555—2271no—
104#492—2271no—
105#459—2271no—
106#459D22711yescharacteristic, complemented
107#333—2271no—
108D4⋊C8—2271no—
109D4⋊C8—2271no—
110C2×SD16—4542no—
111C2×D8—21no—
112C2×D8—4541no—
113C4⋊Q8—2271no—
114#127C2×D22711yescharacteristic
115#124—4542no—
116#121—4541no—
117#121—4541no—
118C4×D4—2271no—
119C4×D4C2×D22711yescharacteristic
120#117—4541no—
121#117—4541no—
122C4⋊C8—2271no—
123C4⋊C8C2×D22711yescharacteristic
124#102—21no—
125#102—4541no—
126C4×C8—2271no—
127C2×Q8—2272no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—4542no—
132C2×D4—2271no—
133C2×D4C22⋊D22711yescharacteristic
134C22×C4—21no—
135C22×C4—4541no—
136SD16—9084no—
137D8—42no—
138D8—9082no—
139C2×C8—21no—
140C2×C8—4541no—
141C2×C8—4541no—
142C4⋊C4—4541no—
143C4⋊C4—4541no—
144C4⋊C4—2271no—
145C4⋊C4C22⋊D22711yescharacteristic
146C22⋊C4—21no—
147C22⋊C4—4542no—
148C22⋊C4—4542no—
149C22⋊C4—4541no—
150C42—4542no—
151C42C22×D22711yescharacteristic
152C23—21no—
153C23—41no—
154C23—4541no—
155Q8—9082no—
156D4—21no—
157D4—21no—
158D4—42no—
159D4—41no—
160D4—41no—
161D4—41no—
162D4—41no—
163D4—18162no—
164D4—4541no—
165D4—4541no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—9081no—
170C2×C4—9081no—
171C2×C4—4541no—
172C2×C4—4542no—
173C2×C4D4×D22711yescharacteristic
174C2×C4D4×D22711yescharacteristic
175C2×C4C2×C22⋊D22711yescharacteristic
176C8—41no—
177C8—9081no—
178C8—4542no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22—81no—
184C22—81no—
185C22—9081no—
186C22—9081no—
187C22—9081no—
188C22#2030025611yescharacteristic
189C4—21no—
190C4—41no—
191C4—9081no—
192C4—9082no—
193C4#2030030711yescharacteristic
194C4#2030030811yescharacteristic
195C4#2030031211yescharacteristic
196C4#2030031411yescharacteristic
197C2—81no—
198C2—161no—
199C2—18161no—
200C2#2840917411yescharacteristic
201C2#2840917511yescharacteristic
202C2#2840919811yescharacteristic
subgroupG/H
centerC22#20300256
commutator, cosocleC42×C227C23
FrattiniC42C22×D227
Fitting#459×C227C2
socleC22×C227#120
hypercenter#459D227
hypocenterC227#1624
LCSG ▷ C42×C227 ▷ C22×C227 ▷ C227
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #28409747C2semidirect
2. #28409404C2semidirect
3. #28409224C2semidirect
4. #28408780C2semidirect
5. D4⋊(C227⋊C8)C2semidirect
6. #28408759C2semidirect
7. #459×C227C2semidirect
101. C227#1624semidirect
106. #459D227semidirect

Maximal subgroups

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

subgroupindex
#284097472
#284094042
#284092242
#284087802
D4⋊(C227⋊C8)2
#284087592
#459×C2272
#1624227