Finite Group Databasebeta

← #44408214

Subgroups — #44408214

Order 29056.

704 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 209 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×#20300336C211yesmaximal, characteristic
2#28409798C212yesmaximal, complemented
3#28409015C214yesmaximal, complemented
4#28409014C212yesmaximal, complemented
5C2×#20300207C211yesmaximal, characteristic, complemented
6#101×D227C214yesmaximal, complemented
7C2×#101×C227C211yesmaximal, characteristic, complemented
8#20300336—24no—
9#20300336—24no—
10C2×#16813885C412yes—
11C2×#16813885C2212yes—
12C2×#16813884C414yes—
13C2×#16813884C2211yescharacteristic
14C2×#16813884C2211yescharacteristic
15C2×Q8×D227C412yes—
16C2×Q8×D227C2212yes—
17#20300296—24no—
18#20300294—24no—
19C2×#16813868C2212yes—
20M16×D227C2218yes—
21#20300207C2214yes—
22#20300206C2214yes—
23#20300204C2214yes—
24C22×Q8×C227C2211yescharacteristic
25C2×M16×C227C2212yes—
26#101×C227C2214yes—
27#16813885—28no—
28#16813885—28no—
29#16813884—28no—
30#16813884—28no—
31#16813884—22no—
32#16813884—22no—
33#16813884D414yes—
34#16813884D414yes—
35C2×Dic454C2×C414yes—
36C2×Dic454C2×C411yescharacteristic
37C2×Dic454C2311yescharacteristic
38Q8×D227—28no—
39Q8×D227—28no—
40C2×D908C2×C414yes—
41C2×D908C2×C411yescharacteristic
42C2×D908C2311yescharacteristic
43C2×C22⋊D227C2×C412yes—
44C2×C22⋊D227C2×C412yes—
45C2×C4×D227C2×C414yes—
46C2×C4×D227C2×C412yes—
47C2×C4×D227C2×C414yes—
48C2×C4×D227C2312yes—
49#16813868—22no—
50#16813868C2314yes—
51#16813867—28no—
52C2×C227⋊C8—22no—
53C8×D227—28no—
54C2×Q8×C227—22no—
55C2×Q8×C227—22no—
56C2×Q8×C227C2×C412yes—
57C2×Q8×C227C2312yes—
58C22×C4×C227C2×C412yes—
59C22×C4×C227C2311yescharacteristic
60M16×C227—22no—
61M16×C227C2314yes—
62C2×C8×C227—22no—
63C22×D227C22×C412yes—
64C22×D227C22×C412yes—
65Dic454—22no—
66Dic454—28no—
67Dic454C22⋊C414yes—
68D908—28no—
69D908—22no—
70D908C22⋊C414yes—
71C22⋊D227—22no—
72C22⋊D227—22no—
73C22⋊D227C22⋊C414yes—
74C22⋊D227C22⋊C414yes—
75C2×Dic227C22×C412yes—
76C2×Dic227C22×C412yes—
77C4×D227—28no—
78C4×D227—24no—
79C4×D227—28no—
80C4×D227C2×D418yes—
81C227⋊C8—24no—
82C23×C227C22×C411yescharacteristic
83Q8×C227—24no—
84Q8×C227—24no—
85C2×C4×C227—22no—
86C2×C4×C227—22no—
87C2×C4×C227C22×C414yes—
88C2×C4×C227C22×C411yescharacteristic
89C2×C4×C227C2×D412yes—
90C2×C4×C227C2×D412yes—
91C2×C4×C227C2411yescharacteristic
92C8×C227—24no—
93C22×C227—41no—
94C22×C227C2×C22⋊C411yescharacteristic
95C22×C227C2×C22⋊C411yescharacteristic
96C22×C227C23×C411yescharacteristic
97C2×D227—22no—
98C2×D227—42no—
99C2×D227—42no—
100C2×D227C2×C22⋊C414yes—
101C4×C227—24no—
102C4×C227C2×C22⋊C412yes—
103C4×C227C22×D412yes—
104Dic227—22no—
105Dic227C2×C22⋊C414yes—
106C2×C227—21no—
107C2×C227—21no—
108C2×C227—21no—
109C2×C227C22×C22⋊C411yescharacteristic
110D227—24no—
111D227—42no—
112C227#288911yescharacteristic, complemented, Sylow
113#2889—2271nomaximal, Sylow
114C2×2−1+4—2271no—
115#568—2272no—
116#413—2272no—
117#413—2274no—
118C2×#101—2271no—
119C2×#101—2274no—
120C2×#101D22711yescharacteristic, complemented
1212−1+4—4544no—
1222−1+4—4544no—
123C2×#41—2272no—
124C2×#41—2272no—
125C2×#41—2274no—
126C2×#41—2271no—
127C2×#41—2271no—
128C22×Q8—2272no—
129C22×Q8—2272no—
130C22×Q8C2×D22711yescharacteristic
131#131—4544no—
132#131—4544no—
133C2×M16—2272no—
134C2×M16—2278no—
135C2×M16C2×D22712yes—
136#101—2274no—
137#101—2274no—
138#101C2×D22714yes—
139#100—2274no—
140#41—4548no—
141#41—4548no—
142#41—4548no—
143#41—4548no—
144#41—4542no—
145#41—4542no—
146#41—2274no—
147#41—2274no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—4548no—
151C2×Q8—4548no—
152C2×Q8—2274no—
153C2×Q8—2271no—
154C2×Q8—2271no—
155C2×Q8C4×D22712yes—
156C2×Q8C22×D22712yes—
157C2×D4—2272no—
158C2×D4—2272no—
159C2×D4—2274no—
160C2×D4—2271no—
161C2×D4—2271no—
162C22×C4—2274no—
163C22×C4—2274no—
164C22×C4—2272no—
165C22×C4—2272no—
166C22×C4C4×D22712yes—
167C22×C4C22×D22711yescharacteristic
168M16—22no—
169M16—4548no—
170M16—4542no—
171M16—2274no—
172M16C22×D22714yes—
173C2×C8—22no—
174C2×C8—4548no—
175C2×C8—4542no—
176C23—2272no—
177C23—2272no—
178C23C2×C4×D22711yescharacteristic
179Q8—24no—
180Q8—24no—
181Q8—4548no—
182Q8—4542no—
183Q8—2274no—
184D4—4548no—
185D4—4542no—
186D4—4542no—
187D4—4542no—
188D4—2274no—
189D4—2274no—
190D4—2274no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—4548no—
194C2×C4—4548no—
195C2×C4—4544no—
196C2×C4—2272no—
197C2×C4—2272no—
198C2×C4—2278no—
199C2×C4C2×C4×D22714yes—
200C2×C4C2×C4×D22711yescharacteristic
201C2×C4D4×D22712yes—
202C2×C4D4×D22712yes—
203C2×C4C23×D22711yescharacteristic
204C8—24no—
205C8—4544no—
206C22—41no—
207C22—9082no—
208C22—9082no—
209C22—4542no—
210C22—2274no—
211C22C22⋊C4×D22711yescharacteristic
212C22C22⋊C4×D22711yescharacteristic
213C22C22×C4×D22711yescharacteristic
214C4—24no—
215C4—4542no—
216C4—2274no—
217C4C22⋊C4×D22712yes—
218C4C2×D4×D22712yes—
219C2—21no—
220C2—21no—
221C2—21no—
222C2—9082no—
223C2—4544no—
224C2C2×C22⋊C4×D22711yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D227
commutatorC22×C227C23×C4
FrattiniC2×C4C23×D227
FittingC2×#101×C227C2
socleC2×C227C22×C22⋊C4
prime radicalC2×#20300336C2
cosocleC2×C4×C227C24
hypercenterC2×#101D227
hypocenterC227#2889
LCSG ▷ C22×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. #28409798C2semidirect
3. #28409015C2semidirect
4. #28409014C2semidirect
5. C2×#20300207C2semidirect
6. #101×D227C2semidirect
7. C2×#101×C227C2semidirect
112. C227#2889semidirect
120. C2×#101D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203003362
#284097982
#284090152
#284090142
C2×#203002072
#101×D2272
C2×#101×C2272
#2889227