Finite Group Databasebeta

← #49129723

Subgroups — #49129723

Order 36224.

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

normal subgroups only

This group has 716 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×#22583657C211yesmaximal, characteristic
2#31826942C212yesmaximal, complemented
3#31826439C211yesmaximal, characteristic, complemented
4#31826438C212yesmaximal, complemented
5#31826437C218yesmaximal, complemented
6#410×C283C211yesmaximal, characteristic, complemented
7#22583657—24no—
8#22583657—24no—
9C2×#17985392C412yescomplemented
10C2×#17985392C2211yescharacteristic
11C2×#17985391C412yescomplemented
12C2×#17985391C2214yes—
13C22×Dic566C412yescomplemented
14C22×Dic566C2211yescharacteristic
15C2×Q8×D283C412yescomplemented
16C4×Dic566—24no—
17#22583575—24no—
18#22583569C2212yes—
19#22583566C2218yes—
20#22583525C2214yes—
21#22583524C2218yes—
22C2×#41×C283C2211yescharacteristic
23#117×C283C2212yes—
24#99×C283C2214yes—
25#17985392—24no—
26#17985392—24no—
27#17985392—22no—
28#17985392—22no—
29#17985391—28no—
30#17985391—216no—
31C2×Dic566—24no—
32C2×Dic566—24no—
33C2×Dic566C2×C414yes—
34C2×Dic566C2×C411yescharacteristic
35C2×Dic566C2×C412yes—
36C2×Dic566C2×C412yes—
37C2×Dic566D414yes—
38C2×Dic566D414yes—
39Q8×D283—28no—
40C2×D1132C2×C411yescharacteristic
41C2×C22⋊D283C2×C414yes—
42C2×C22⋊D283C2312yes—
43C22×Dic283C2×C414yes—
44C22×Dic283C2312yes—
45C2×C4×D283C2×C412yes—
46C2×C4×D283C2×C414yes—
47C4⋊Dic283—22no—
48Dic283⋊C4—28no—
49C22⋊Dic283C2314yes—
50C4×Dic283—28no—
51C4×Dic283—22no—
52#41×C283—22no—
53#41×C283—22no—
54C2×Q8×C283C2×C411yescharacteristic
55C2×D4×C283C2×C411yescharacteristic
56C2×D4×C283C2312yes—
57C22×C4×C283C2×C412yes—
58C22×C4×C283C2311yescharacteristic
59C4⋊C4×C283—22no—
60C22⋊C4×C283C2314yes—
61C42×C283—22no—
62C22×D283C22×C412yes—
63Dic566—21no—
64Dic566—21no—
65Dic566—28no—
66Dic566C22⋊C418yes—
67Dic566C22⋊C414yes—
68Dic566C22⋊C414yes—
69D1132—21no—
70D1132—21no—
71C22⋊D283—24no—
72C22⋊D283—28no—
73C2×Dic283—28no—
74C2×Dic283—24no—
75C2×Dic283C22×C412yes—
76C2×Dic283C22×C414yes—
77C2×Dic283C2×D418yes—
78C4×D283—24no—
79C4×D283—28no—
80C23×C283C22×C412yes—
81C23×C283C2411yescharacteristic
82Q8×C283—22no—
83D4×C283—24no—
84D4×C283—22no—
85C2×C4×C283—22no—
86C2×C4×C283—22no—
87C2×C4×C283—24no—
88C2×C4×C283C22×C412yes—
89C2×C4×C283C22×C411yescharacteristic
90C2×C4×C283C22×C411yescharacteristic
91C2×C4×C283C2×D412yes—
92C2×C4×C283C2×D412yes—
93C22×C283—22no—
94C22×C283—41no—
95C22×C283—42no—
96C22×C283C23×C411yescharacteristic
97C22×C283C22×D412yes—
98C2×D283—22no—
99C2×D283—42no—
100C4×C283—22no—
101C4×C283—44no—
102C4×C283C2×C22⋊C411yescharacteristic
103C4×C283C2×C22⋊C411yescharacteristic
104C4×C283C2×C22⋊C412yes—
105Dic283—22no—
106Dic283—44no—
107Dic283C2×C22⋊C418yes—
108C2×C283—22no—
109C2×C283—21no—
110C2×C283—42no—
111C2×C283C22×C22⋊C411yescharacteristic
112D283—42no—
113C283#288411yescharacteristic, complemented, Sylow
114#2884—2831nomaximal, Sylow
115C2×2−1+4—2831no—
116#519—2832no—
117#410—2831no—
118#410—2832no—
119#410—2838no—
120#410D28311yescharacteristic, complemented
1212−1+4—5664no—
1222−1+4—5664no—
123C2×#41—2832no—
124C2×#41—2834no—
125C2×#41—2832no—
126C2×#41—2831no—
127C2×#41C2×D28311yescharacteristic
128C22×Q8—2832no—
129C22×Q8—2832no—
130C22×Q8—2831no—
131C4×Q8—5664no—
132C4×Q8—5664no—
133#117—2832no—
134#117—2838no—
135#117C2×D28312yes—
136#99—2834no—
137#99—2838no—
138#99C2×D28314yes—
139#41—22no—
140#41—22no—
141#41—5664no—
142#41—5664no—
143#41—56616no—
144#41—5668no—
145#41—5662no—
146#41—5662no—
147C2×Q8—2832no—
148C2×Q8—2832no—
149C2×Q8—2834no—
150C2×Q8—2831no—
151C2×Q8—2834no—
152C2×Q8—2834no—
153C2×Q8—5664no—
154C2×Q8—5664no—
155C2×Q8—5668no—
156C2×Q8C4×D28311yescharacteristic
157C2×D4—2834no—
158C2×D4—2832no—
159C2×D4—2831no—
160C2×D4C4×D28311yescharacteristic
161C2×D4C22×D28312yes—
162C22×C4—2832no—
163C22×C4—2834no—
164C22×C4—2834no—
165C22×C4—2832no—
166C22×C4C4×D28312yes—
167C22×C4C22×D28311yescharacteristic
168C4⋊C4—22no—
169C4⋊C4—5668no—
170C4⋊C4—5662no—
171C22⋊C4—2834no—
172C22⋊C4C22×D28314yes—
173C42—22no—
174C42—5668no—
175C42—5662no—
176C23—2832no—
177C23C2×C4×D28312yes—
178C23C23×D28311yescharacteristic
179Q8—22no—
180Q8—2834no—
181Q8—2834no—
182Q8—2838no—
183Q8—5668no—
184Q8—5661no—
185Q8—5661no—
186D4—24no—
187D4—22no—
188D4—5668no—
189D4—5664no—
190D4—5661no—
191D4—5661no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—2834no—
196C2×C4—2838no—
197C2×C4—2832no—
198C2×C4—5668no—
199C2×C4—5668no—
200C2×C4—5664no—
201C2×C4—5664no—
202C2×C4C2×C4×D28312yes—
203C2×C4C2×C4×D28311yescharacteristic
204C2×C4C2×C4×D28311yescharacteristic
205C2×C4D4×D28312yes—
206C2×C4D4×D28312yes—
207C22—22no—
208C22—41no—
209C22—42no—
210C22—5662no—
211C22—11322no—
212C22C22×C4×D28311yescharacteristic
213C22C2×D4×D28312yes—
214C4—22no—
215C4—44no—
216C4—2838no—
217C4—5662no—
218C4—11324no—
219C4C22⋊C4×D28311yescharacteristic
220C4C22⋊C4×D28311yescharacteristic
221C4C22⋊C4×D28312yes—
222C2—22no—
223C2—21no—
224C2—42no—
225C2—11322no—
226C2C2×C22⋊C4×D28311yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D283
commutatorC22×C283C23×C4
FrattiniC23C23×D283
Fitting#410×C283C2
socleC2×C283C22×C22⋊C4
prime radicalC2×#22583657C2
cosocleC23×C283C24
hypercenter#410D283
hypocenterC283#2884
LCSG ▷ C22×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #31826942C2semidirect
3. #31826439C2semidirect
4. #31826438C2semidirect
5. #31826437C2semidirect
6. #410×C283C2semidirect
9. C2×#17985392C4semidirect
11. C2×#17985391C4semidirect
13. C22×Dic566C4semidirect
15. C2×Q8×D283C4semidirect
113. C283#2884semidirect
120. #410D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836572
#318269422
#318264392
#318264382
#318264372
#410×C2832
#2884283