Finite Group Databasebeta

← #49130758

Subgroups — #49130758

Order 36224.

500 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 133 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×#22583641C211yesmaximal, characteristic, complemented
2C2×#22583638C211yesmaximal, characteristic
3C2×#22583599C211yesmaximal, characteristic, complemented
4#31826933C211yesmaximal, characteristic, complemented
5#31826690C214yesmaximal, complemented
6#31826683C214yesmaximal, complemented
7#31826488C211yesmaximal, characteristic, complemented
8#31826469C211yesmaximal, characteristic, complemented
9#419×C283C211yesmaximal, characteristic, complemented
10C2×#17985392C2211yescharacteristic
11C22×D1132C2211yescharacteristic
12#22583641—24no—
13#22583638—24no—
14C2×#17985382C2212yescomplemented
15C2×#17985381C2212yes—
16C2×#17985379C2212yescomplemented
17C2×D2264C2212yes—
18C2×#17985376C2211yescharacteristic
19#22583599—21no—
20#22583599—21no—
21#22583599C2214yes—
22#22583575C2212yescomplemented
23#22583572C2212yescomplemented
24(C22⋊D283)⋊C4—22no—
25#22583567—21no—
26C2×(C2×D283)⋊C4—22no—
27C2×C4×Dic283C2211yescharacteristic
28C42×D283—21no—
29Dic283⋊C8C2214yescomplemented
30#22583530C2214yes—
31C22×Q8×C283C2211yescharacteristic
32C2×M16×C283C2211yescharacteristic
33#117×C283C2211yescharacteristic
34#103×C283C2214yescomplemented
35C23×D283—21no—
36#17985392—22no—
37#17985392D414yes—
38C2×Dic566C2311yescharacteristic
39C2×D1132—22no—
40C2×D1132—21no—
41C2×D1132—22no—
42C2×D1132C2312yes—
43C2×D1132C2311yescharacteristic
44C2×C22⋊D283—21no—
45C22×Dic283D412yes—
46C2×C4×D283—22no—
47C2×C4×D283—21no—
48#17985382—24no—
49#17985381—24no—
50#17985379—24no—
51D2264—24no—
52#17985376—22no—
53C2×C283⋊C8C2312yes—
54Dic283⋊C4—22no—
55(C2×D283)⋊C4—24no—
56(C2×D283)⋊C4—24no—
57(C2×D283)⋊C4—22no—
58C4×Dic283—22no—
59C4×Dic283D411yescharacteristic
60C4×Dic283D411yescharacteristic
61C4×Dic283C2312yes—
62C2×Q8×C283—21no—
63C2×Q8×C283—22no—
64C2×Q8×C283—22no—
65C2×Q8×C283C2312yes—
66C22×C4×C283—21no—
67C22×C4×C283C2311yescharacteristic
68M16×C283—22no—
69C2×C8×C283C2312yes—
70C4⋊C4×C283C2312yes—
71C22⋊C4×C283—21no—
72C42×C283—21no—
73C22×D283—21no—
74C22×D283—22no—
75C22×D283—44no—
76C22×D283—42no—
77Dic566—21no—
78Dic566C2×D412yes—
79D1132—22no—
80D1132—21no—
81D1132—22no—
82D1132—42no—
83D1132C2×D412yes—
84C22⋊D283—24no—
85C2×Dic283—21no—
86C2×Dic283—42no—
87C2×Dic283C2×D412yes—
88C2×Dic283C2×D412yes—
89C4×D283—24no—
90C4×D283—44no—
91C283⋊C8—22no—
92C23×C283C2×D411yescharacteristic
93Q8×C283—22no—
94Q8×C283—22no—
95Q8×C283—42no—
96C2×C4×C283—22no—
97C2×C4×C283—22no—
98C2×C4×C283—42no—
99C2×C4×C283C2×D411yescharacteristic
100C2×C4×C283C2×D412yes—
101C2×C4×C283#4112yes—
102C2×C4×C283C2411yescharacteristic
103C8×C283—22no—
104C22×C283—22no—
105C22×C283#12112yes—
106C22×C283C22×D411yescharacteristic
107C2×D283—22no—
108C2×D283—42no—
109C2×D283—42no—
110C2×D283—44no—
111C2×D283—84no—
112C4×C283—42no—
113C4×C283—42no—
114C4×C283#12112yes—
115C4×C283C22×D411yescharacteristic
116C4×C283C2×#4111yescharacteristic
117Dic283—22no—
118Dic283—22no—
119Dic283—22no—
120C2×C283—22no—
121C2×C283C2×#12111yescharacteristic
122C2×C283#57512yes—
123D283—42no—
124D283—82no—
125C283#304011yescharacteristic, complemented, Sylow
126#3040—2831nomaximal, Sylow
127C2×#137—2831no—
128C2×AGL(1,ℤ/8)—2831no—
129C2×#124—2831no—
130C4×#41—2831no—
131#464—2834no—
132#463—2834no—
133#423—2831no—
134#419D28311yescharacteristic, complemented
135#418—2831no—
136C2×#41—2831no—
137C22×Q8C2×D28311yescharacteristic
138C22×D4—2831no—
139#137—5664no—
140AGL(1,ℤ/8)—5664no—
141C2×Q16—2832no—
142C2×SD16—2832no—
143C2×SD16—2832no—
144C2×D8—2832no—
145C2×M16—2831no—
146C2×M16C2×D28311yescharacteristic
147#124—2834no—
148#124—5661no—
149#124—5661no—
150C4×Q8—2832no—
151C4×D4—2832no—
152C4×D4—5662no—
153#117—5661no—
154#117C2×D28311yescharacteristic
155C2×C22⋊C4—5662no—
156C2×C42—2831no—
157C2×C42—5661no—
158C4⋊C8—2834no—
159#103C2×D28314yescomplemented
160#102—2834no—
161C24—5661no—
162#41—2834no—
163#41—5662no—
164C2×Q8—21no—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8—2831no—
168C2×Q8C22×D28312yes—
169C2×D4—2831no—
170C2×D4—2832no—
171C2×D4—5661no—
172C2×D4—5662no—
173C2×D4—5661no—
174C2×D4—5662no—
175C22×C4—21no—
176C22×C4—2832no—
177C22×C4—5662no—
178C22×C4—5661no—
179C22×C4C22×D28311yescharacteristic
180Q16—5664no—
181SD16—5664no—
182SD16—5664no—
183D8—5664no—
184M16—22no—
185M16—5662no—
186C2×C8—2832no—
187C2×C8C22×D28312yes—
188C4⋊C4—5662no—
189C4⋊C4C22×D28312yes—
190C22⋊C4—21no—
191C22⋊C4—5664no—
192C22⋊C4—5664no—
193C22⋊C4—5662no—
194C42—21no—
195C42—2831no—
196C42—2831no—
197C42—2832no—
198C42—5662no—
199C23—5661no—
200C23—5662no—
201C23—11324no—
202C23—11322no—
203C23D4×D28311yescharacteristic
204Q8—22no—
205Q8—22no—
206Q8—42no—
207Q8—2832no—
208Q8—5661no—
209D4—2832no—
210D4—5664no—
211D4—5662no—
212D4—5661no—
213D4—5662no—
214D4—11322no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—42no—
218C2×C4—2832no—
219C2×C4—2832no—
220C2×C4—5661no—
221C2×C4—5664no—
222C2×C4—11324no—
223C2×C4—11322no—
224C2×C4D4×D28311yescharacteristic
225C2×C4D4×D28312yes—
226C2×C4#1798539212yes—
227C2×C4C23×D28311yescharacteristic
228C8—22no—
229C8—5662no—
230C22—22no—
231C22—5662no—
232C22—11322no—
233C22—11322no—
234C22—11324no—
235C22—22644no—
236C22#2258358312yes—
237C22C2×D4×D28311yescharacteristic
238C4—42no—
239C4—42no—
240C4—5662no—
241C4—5662no—
242C4—5662no—
243C4#2258358312yes—
244C4C2×D4×D28311yescharacteristic
245C4C2×#1798539211yescharacteristic
246C2—22no—
247C2—11322no—
248C2—22642no—
249C2C2×#2258358311yescharacteristic
250C2#3182726412yes—
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#419×C283C2
socleC22×C283C22×D4
prime radicalC2×#22583638C2
hypercenter#419D283
hypocenterC283#3040
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C2×#22583641C2semidirect
3. C2×#22583599C2semidirect
4. #31826933C2semidirect
5. #31826690C2semidirect
6. #31826683C2semidirect
7. #31826488C2semidirect
8. #31826469C2semidirect
9. #419×C283C2semidirect
14. C2×#17985382C22semidirect
16. C2×#17985379C22semidirect
22. #22583575C22semidirect
23. #22583572C22semidirect
29. Dic283⋊C8C22semidirect
34. #103×C283C22semidirect
125. C283#3040semidirect
134. #419D283semidirect
159. #103C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836412
C2×#225836382
C2×#225835992
#318269332
#318266902
#318266832
#318264882
#318264692
#419×C2832
#3040283