Finite Group Databasebeta

← #44408215

Subgroups — #44408215

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×#20300335C211yesmaximal, characteristic
2#28409798C212yesmaximal, complemented
3#28409016C211yesmaximal, characteristic, complemented
4#28409013C214yesmaximal, complemented
5C2×#20300206C212yesmaximal, complemented
6#101×D227C214yesmaximal, complemented
7#413×C227C211yesmaximal, characteristic, complemented
8#20300335—24no—
9#20300335—24no—
10C2×#16813884C412yes—
11C2×#16813884C2211yescharacteristic
12C2×#16813883C414yes—
13C2×#16813883C2212yes—
14C22×Dic454C412yes—
15C22×Dic454C2211yescharacteristic
16C2×Q8×D227C2212yes—
17#20300296—24no—
18#20300294—24no—
19C2×#16813868C2212yes—
20M16×D227C2218yes—
21#20300207C2212yes—
22#20300206C2218yes—
23#20300205C2212yes—
24C2×#41×C227C2211yescharacteristic
25C2×M16×C227C2212yes—
26#101×C227C2212yes—
27#100×C227C2212yes—
28#16813884—24no—
29#16813884—24no—
30#16813884D414yes—
31#16813884D414yes—
32#16813883—216no—
33#16813883—28no—
34C2×Dic454—24no—
35C2×Dic454—24no—
36C2×Dic454—22no—
37C2×Dic454—22no—
38C2×Dic454C2×C414yes—
39C2×Dic454C2×C412yes—
40C2×Dic454C2×C411yescharacteristic
41C2×Dic454C2312yes—
42Q8×D227—28no—
43C2×D908C2×C411yescharacteristic
44C2×C22⋊D227C2×C414yes—
45C2×C22⋊D227C2×C412yes—
46C22×Dic227C2×C414yes—
47C22×Dic227C2×C412yes—
48C2×C4×D227C2×C414yes—
49C2×C4×D227C2312yes—
50#16813868—22no—
51#16813868C2314yes—
52#16813867—28no—
53C2×C227⋊C8—22no—
54C8×D227—28no—
55#41×C227—22no—
56#41×C227—22no—
57C2×Q8×C227C2311yescharacteristic
58C2×D4×C227C2×C412yes—
59C2×D4×C227C2311yescharacteristic
60C22×C4×C227C2×C412yes—
61C22×C4×C227C2311yescharacteristic
62M16×C227—22no—
63M16×C227C2314yes—
64C2×C8×C227—22no—
65C22×D227C22×C412yes—
66Dic454—22no—
67Dic454—22no—
68Dic454—24no—
69Dic454—28no—
70Dic454C22⋊C412yes—
71Dic454C22⋊C412yes—
72D908C22⋊C412yes—
73D908C22⋊C412yes—
74C22⋊D227—28no—
75C22⋊D227C22⋊C418yes—
76C2×Dic227—24no—
77C2×Dic227—28no—
78C2×Dic227C22×C414yes—
79C2×Dic227C22×C412yes—
80C4×D227—28no—
81C4×D227C2×D418yes—
82C227⋊C8—24no—
83C23×C227C22×C412yes—
84C23×C227C22×C411yescharacteristic
85Q8×C227—22no—
86D4×C227—24no—
87D4×C227—22no—
88C2×C4×C227—22no—
89C2×C4×C227—22no—
90C2×C4×C227C22×C412yes—
91C2×C4×C227C22×C411yescharacteristic
92C2×C4×C227C2×D412yes—
93C2×C4×C227C2×D412yes—
94C2×C4×C227C2411yescharacteristic
95C8×C227—24no—
96C22×C227—22no—
97C22×C227—41no—
98C22×C227—42no—
99C22×C227C2×C22⋊C412yes—
100C22×C227C23×C411yescharacteristic
101C2×D227—42no—
102C2×D227C2×C22⋊C414yes—
103C4×C227—22no—
104C4×C227C2×C22⋊C411yescharacteristic
105C4×C227C2×C22⋊C411yescharacteristic
106C4×C227C22×D412yes—
107Dic227—24no—
108Dic227C2×C22⋊C414yes—
109C2×C227—21no—
110C2×C227—22no—
111C2×C227—42no—
112C2×C227C22×C22⋊C411yescharacteristic
113D227—24no—
114C227#288911yescharacteristic, complemented, Sylow
115#2889—2271nomaximal, Sylow
116C2×2−1+4—2271no—
117#568—2272no—
118#413—2271no—
119#413—2274no—
120#413D22711yescharacteristic, complemented
121C2×#101—2272no—
122C2×#101—2274no—
1232−1+4—4544no—
1242−1+4—4544no—
125C2×#41—2274no—
126C2×#41—2272no—
127C2×#41—2272no—
128C2×#41—2271no—
129C2×#41C2×D22711yescharacteristic
130C22×Q8—2272no—
131C22×Q8—2272no—
132C22×Q8—2271no—
133#131—4544no—
134#131—4544no—
135C2×M16—2272no—
136C2×M16—2278no—
137C2×M16C2×D22712yes—
138#101—2272no—
139#101—2278no—
140#101C2×D22712yes—
141#100—2272no—
142#100C2×D22712yes—
143#41—22no—
144#41—22no—
145#41—4544no—
146#41—4544no—
147#41—45416no—
148#41—4548no—
149#41—2274no—
150#41—2274no—
151C2×Q8—4544no—
152C2×Q8—4544no—
153C2×Q8—4548no—
154C2×Q8—4542no—
155C2×Q8—4542no—
156C2×Q8—2272no—
157C2×Q8—2274no—
158C2×Q8—2271no—
159C2×Q8—2272no—
160C2×Q8C22×D22711yescharacteristic
161C2×D4—2274no—
162C2×D4—2272no—
163C2×D4—2271no—
164C2×D4C4×D22712yes—
165C2×D4C22×D22711yescharacteristic
166C22×C4—2272no—
167C22×C4—2274no—
168C22×C4—2274no—
169C22×C4—2272no—
170C22×C4C4×D22712yes—
171C22×C4C22×D22711yescharacteristic
172M16—22no—
173M16—4548no—
174M16—4542no—
175M16—2274no—
176M16C22×D22714yes—
177C2×C8—22no—
178C2×C8—4548no—
179C2×C8—4542no—
180C23—2272no—
181C23C2×C4×D22712yes—
182C23C2×C4×D22711yescharacteristic
183Q8—22no—
184Q8—4548no—
185Q8—4544no—
186Q8—4542no—
187Q8—4542no—
188Q8—2272no—
189Q8—2272no—
190D4—24no—
191D4—22no—
192D4—4548no—
193D4—2278no—
194D4—2272no—
195D4—2272no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—4548no—
199C2×C4—4544no—
200C2×C4—4548no—
201C2×C4—2274no—
202C2×C4—2272no—
203C2×C4—2278no—
204C2×C4C2×C4×D22712yes—
205C2×C4C2×C4×D22711yescharacteristic
206C2×C4D4×D22712yes—
207C2×C4D4×D22712yes—
208C2×C4C23×D22711yescharacteristic
209C8—24no—
210C8—4544no—
211C22—22no—
212C22—41no—
213C22—42no—
214C22—9082no—
215C22—2274no—
216C22C22⋊C4×D22712yes—
217C22C22×C4×D22711yescharacteristic
218C4—22no—
219C4—4544no—
220C4—2274no—
221C4C22⋊C4×D22711yescharacteristic
222C4C22⋊C4×D22711yescharacteristic
223C4C2×D4×D22712yes—
224C2—21no—
225C2—22no—
226C2—42no—
227C2—4544no—
228C2C2×C22⋊C4×D22711yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D227
commutatorC22×C227C23×C4
FrattiniC2×C4C23×D227
Fitting#413×C227C2
socleC2×C227C22×C22⋊C4
prime radicalC2×#20300335C2
cosocleC2×C4×C227C24
hypercenter#413D227
hypocenterC227#2889
LCSG ▷ C22×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #28409798C2semidirect
3. #28409016C2semidirect
4. #28409013C2semidirect
5. C2×#20300206C2semidirect
6. #101×D227C2semidirect
7. #413×C227C2semidirect
114. C227#2889semidirect
120. #413D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203003352
#284097982
#284090162
#284090132
C2×#203002062
#101×D2272
#413×C2272
#2889227