Finite Group Databasebeta

← #44400255

Subgroups — #44400255

Order 29056.

312 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 79 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#28409798C211yesmaximal, characteristic
2#28409608C211yesmaximal, characteristic
3#28409042C211yesmaximal, characteristic, complemented
4#28409040C211yesmaximal, characteristic, complemented
5#28409005C211yesmaximal, characteristic
6#28408788C211yesmaximal, characteristic, complemented
7H(3,ℤ/4)×C227C211yesmaximal, characteristic, complemented
8C2×#16813884C2211yescharacteristic
9C22×Dic454C2211yescharacteristic
10#20300296—22no—
11#20300294—22no—
12C2×#16813868C2211yescharacteristic
13M16×D227—22no—
14#20300288C412yescomplemented
15#20300276C412yescomplemented
16#20300272—22no—
17#20300268—22no—
18#20300247C2211yescharacteristic
19#20300213—22no—
20#20300211—22no—
21#20300202—22no—
22C2×M16×C227C2211yescharacteristic
23#117×C227C2211yescharacteristic
24#117×C227C2211yescharacteristic
25#16813884—21no—
26#16813884—21no—
27#16813884—21no—
28#16813884—21no—
29C2×Dic454—21no—
30C2×Dic454—21no—
31C2×Dic454—21no—
32C2×Dic454—21no—
33C2×Dic454—21no—
34C2×Dic454C2×C412yes—
35C2×Dic454C2×C411yescharacteristic
36C2×D908C2×C411yescharacteristic
37C2×C22⋊D227D412yes—
38C22×Dic227—21no—
39C2×C4×D227D412yes—
40#16813868—21no—
41#16813868—21no—
42#16813867—24no—
43C2×C227⋊C8D412yes—
44C8×D227—24no—
45C4⋊Dic227—24no—
46C4⋊Dic227—21no—
47C22⋊Dic227—21no—
48(C2×D227)⋊C4—24no—
49C4×Dic227—21no—
50C22×C4×C227C2311yescharacteristic
51M16×C227—21no—
52M16×C227—21no—
53C2×C8×C227D412yes—
54C4⋊C4×C227—21no—
55C4⋊C4×C227—21no—
56C22⋊C4×C227—21no—
57C22⋊C4×C227—21no—
58C42×C227—21no—
59C42×C227C2×C412yes—
60C22×D227C22⋊C412yes—
61Dic454—21no—
62Dic454—21no—
63Dic454—22no—
64Dic454—22no—
65Dic454—41no—
66Dic454—41no—
67D908—21no—
68D908—21no—
69C22⋊D227—22no—
70C22⋊D227—22no—
71C2×Dic227—22no—
72C2×Dic227—41no—
73C2×Dic227—41no—
74C2×Dic227—41no—
75C2×Dic227C22⋊C412yes—
76C4×D227—22no—
77C4×D227—22no—
78C227⋊C8—22no—
79C23×C227C2×D411yescharacteristic
80C2×C4×C227—22no—
81C2×C4×C227—41no—
82C2×C4×C227C22×C411yescharacteristic
83C2×C4×C227C2×D411yescharacteristic
84C2×C4×C227C2×D411yescharacteristic
85C2×C4×C227C2×D411yescharacteristic
86C2×C4×C227#4111yescharacteristic
87C2×C4×C227#4111yescharacteristic
88C8×C227—22no—
89C22×C227—41no—
90C22×C227C2×C22⋊C411yescharacteristic
91C22×C227#12111yescharacteristic
92C22×C227#12111yescharacteristic
93C2×D227—22no—
94C2×D227—42no—
95C4×C227—42no—
96C4×C227—81no—
97C4×C227C4×D411yescharacteristic
98C4×C227C4×D411yescharacteristic
99C4×C227#12011yescharacteristic
100C4×C227#12311yescharacteristic
101Dic227—22no—
102Dic227—42no—
103Dic227—81no—
104C2×C227—21no—
105C2×C227—21no—
106C2×C227—21no—
107C2×C227#38611yescharacteristic
108D227—42no—
109C227#190911yescharacteristic, complemented, Sylow
110#1909—2271nomaximal, Sylow
111#568—2271no—
112#536—2271no—
113#419—2271no—
114#419—2271no—
115#410—2271no—
116H(3,ℤ/4)—2271no—
117H(3,ℤ/4)D22711yescharacteristic, complemented
118C2×#41—2271no—
119C22×Q8—2271no—
120#131—4542no—
121#131—4542no—
122C2×M16—4542no—
123C2×M16—2271no—
124C2×M16C2×D22711yescharacteristic
125C4⋊Q8—2272no—
126#124—2272no—
127#123—4542no—
128C22⋊Q8—4542no—
129#117—2271no—
130#117C2×D22711yescharacteristic
131#117C2×D22711yescharacteristic
132#103—4542no—
133#103—4542no—
134#99—4542no—
135#41—4541no—
136#41—4541no—
137#41—4541no—
138#41—4541no—
139C2×Q8—4541no—
140C2×Q8—4541no—
141C2×Q8—4541no—
142C2×Q8—4541no—
143C2×Q8—4541no—
144C2×Q8—2272no—
145C2×Q8—2271no—
146C2×D4—2272no—
147C2×D4—2271no—
148C22×C4—4541no—
149C22×C4—2272no—
150C22×C4C22×D22711yescharacteristic
151M16—21no—
152M16—21no—
153M16—4544no—
154M16—4541no—
155M16—4541no—
156C2×C8—4544no—
157C2×C8—2272no—
158C2×C8D90812yes—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—4544no—
162C4⋊C4—4541no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—4544no—
166C22⋊C4—4541no—
167C42—21no—
168C42—4541no—
169C42C4×D22712yes—
170C23—2272no—
171C23D4×D22711yescharacteristic
172Q8—9081no—
173Q8—9081no—
174Q8—4542no—
175Q8—4542no—
176Q8—4541no—
177Q8—4541no—
178D4—4542no—
179D4—4542no—
180D4—4541no—
181D4—4541no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—9081no—
185C2×C4—9081no—
186C2×C4—9081no—
187C2×C4—4542no—
188C2×C4—4542no—
189C2×C4—4542no—
190C2×C4—2272no—
191C2×C4C2×C4×D22711yescharacteristic
192C2×C4D4×D22711yescharacteristic
193C2×C4D4×D22711yescharacteristic
194C2×C4C2×D90811yescharacteristic
195C2×C4#1681388411yescharacteristic
196C2×C4#1681388511yescharacteristic
197C8—22no—
198C8—4542no—
199C22—41no—
200C22—9082no—
201C22—4542no—
202C22C22⋊C4×D22711yescharacteristic
203C22#2030026111yescharacteristic
204C22#2030026311yescharacteristic
205C4—42no—
206C4—81no—
207C4—18161no—
208C4—9082no—
209C4—4542no—
210C4#2030025011yescharacteristic
211C4C4×D90811yescharacteristic
212C4#2030025811yescharacteristic
213C4#2030027311yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—9082no—
218C2#2840891011yescharacteristic
subgroupG/H
centerC2#28408910
commutatorC2×C4×C227C22×C4
FrattiniC22×C4C22×D227
FittingH(3,ℤ/4)×C227C2
socleC2×C227#386
prime radicalC2×#16813884C22
cosocleC22×C4×C227C23
hypercenterH(3,ℤ/4)D227
hypocenterC227#1909
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22×C4 ◁ H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
3. #28409042C2semidirect
4. #28409040C2semidirect
6. #28408788C2semidirect
7. H(3,ℤ/4)×C227C2semidirect
14. #20300288C4semidirect
15. #20300276C4semidirect
109. C227#1909semidirect
117. H(3,ℤ/4)D227semidirect

Maximal subgroups

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

subgroupindex
#284097982
#284096082
#284090422
#284090402
#284090052
#284087882
H(3,ℤ/4)×C2272
#1909227