Finite Group Databasebeta

← #49130625

Subgroups — #49130625

Order 36224.

698 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 161 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#31827263C212yesmaximal, complemented
2#31827039C211yesmaximal, characteristic, complemented
3#31826648C214yesmaximal, complemented
4#31826638C214yesmaximal, complemented
5#31826482C212yesmaximal, complemented
6C2×#22583529C211yesmaximal, characteristic, complemented
7C2×2−1+4×C283C211yesmaximal, characteristic, complemented
8C2×#17985392C2211yescharacteristic
9#22583641—22no—
10#22583640C2214yescomplemented
11#22583636—22no—
12#22583635C2214yescomplemented
13#22583630—24no—
14C2×#17985376C2212yes—
15#22583608C2212yescomplemented
16#22583599C2212yescomplemented
17#22583593—22no—
18#22583586—22no—
19#22583569C2211yescharacteristic
20#22583537C2218yescomplemented
21#22583529C2214yescomplemented
222−1+4×C283—24no—
232−1+4×C283—22no—
242−1+4×C283C2214yes—
25C2×#41×C283—22no—
26C2×#41×C283—22no—
27C2×#41×C283C2212yes—
28C22×Q8×C283—22no—
29C22×Q8×C283C2211yescharacteristic
30#17985392—22no—
31#17985392—22no—
32C2×Dic566C2311yescharacteristic
33C2×D1132C2311yescharacteristic
34C2×C22⋊D283—21no—
35C2×C4×D283—21no—
36#17985382—24no—
37#17985380—24no—
38#17985379—24no—
39#17985377—24no—
40#17985376—22no—
41#17985376C2314yes—
42C2×C283⋊C8—22no—
43C4⋊Dic283—21no—
44Dic283⋊C4—24no—
45C22⋊Dic283—21no—
46(C2×D283)⋊C4—24no—
47C4×Dic283C2312yes—
48#41×C283—28no—
49#41×C283—28no—
50#41×C283—28no—
51#41×C283—28no—
52#41×C283—22no—
53#41×C283—22no—
54#41×C283D414yes—
55#41×C283C2314yes—
56C2×Q8×C283—22no—
57C2×Q8×C283—28no—
58C2×Q8×C283—28no—
59C2×Q8×C283—22no—
60C2×Q8×C283—22no—
61C2×Q8×C283D412yes—
62C2×Q8×C283D412yes—
63C2×Q8×C283C2312yes—
64C2×D4×C283—22no—
65C2×D4×C283—22no—
66C2×D4×C283D412yes—
67C22×C4×C283—24no—
68C22×C4×C283—22no—
69C22×C4×C283D412yes—
70C22×C4×C283C2311yescharacteristic
71C22×D283—21no—
72Dic566—22no—
73D1132—22no—
74C22⋊D283—41no—
75C22⋊D283—41no—
76C2×Dic283—22no—
77C2×Dic283—21no—
78C4×D283—42no—
79C283⋊C8—24no—
80C23×C283—22no—
81C23×C283C2×D411yescharacteristic
82Q8×C283—28no—
83Q8×C283—22no—
84Q8×C283—24no—
85Q8×C283—24no—
86Q8×C283C2×D414yes—
87D4×C283—28no—
88D4×C283—24no—
89D4×C283—24no—
90D4×C283—22no—
91D4×C283C2×D414yes—
92C2×C4×C283—216no—
93C2×C4×C283—22no—
94C2×C4×C283—22no—
95C2×C4×C283—22no—
96C2×C4×C283—24no—
97C2×C4×C283—24no—
98C2×C4×C283C2×D414yes—
99C2×C4×C283C2×D411yescharacteristic
100C2×C4×C283C2×D412yes—
101C2×C4×C283C2×D412yes—
102C2×C4×C283C2411yescharacteristic
103C22×C283—24no—
104C22×C283—41no—
105C22×C283—44no—
106C22×C283#12011yescharacteristic
107C22×C283#12011yescharacteristic
108C22×C283C22×D411yescharacteristic
109C2×D283—42no—
110C2×D283—41no—
111C4×C283—24no—
112C4×C283—24no—
113C4×C283#12012yes—
114C4×C283C22×D412yes—
115Dic283—42no—
116Dic283—41no—
117C2×C283—21no—
118C2×C283—21no—
119C2×C283—21no—
120C2×C283—44no—
121C2×C283C2×#12011yescharacteristic
122D283—81no—
123C283#302011yescharacteristic, complemented, Sylow
124#3020—2831nomaximal, Sylow
125C2×2−1+4D28311yescharacteristic, complemented
126#575—2832no—
127#536—2831no—
128#456—2834no—
129#454—2834no—
130#421—2832no—
131C2×#101—2831no—
1322−1+4—24no—
1332−1+4—22no—
1342−1+4C2×D28314yes—
135C2×#41—22no—
136C2×#41—22no—
137C2×#41—2831no—
138C2×#41C2×D28312yes—
139C22×Q8—22no—
140C22×Q8C2×D28311yescharacteristic
141#137—2834no—
142#137—5662no—
143AGL(1,ℤ/8)—2834no—
144AGL(1,ℤ/8)—5662no—
145#135—5664no—
146C2×M16—2832no—
147C4⋊Q8—2832no—
148#124—2832no—
149#123—5662no—
150C22⋊Q8—5662no—
151#117—2831no—
152#104—2838no—
153#101—2834no—
154#41—28no—
155#41—28no—
156#41—28no—
157#41—28no—
158#41—22no—
159#41—22no—
160#41—5662no—
161#41—5662no—
162#41C22⋊D28314yes—
163#41C22×D28314yes—
164C2×Q8—22no—
165C2×Q8—28no—
166C2×Q8—28no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—2831no—
170C2×Q8C22⋊D28312yes—
171C2×Q8C22⋊D28312yes—
172C2×Q8C22×D28312yes—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—2831no—
176C2×D4—5661no—
177C2×D4C22⋊D28312yes—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—5661no—
181C22×C4C22⋊D28312yes—
182C22×C4C22×D28311yescharacteristic
183Q16—5664no—
184SD16—5664no—
185SD16—5664no—
186D8—5664no—
187M16—2834no—
188M16—5662no—
189C2×C8—5662no—
190C4⋊C4—5664no—
191C4⋊C4—5661no—
192C22⋊C4—5661no—
193C22⋊C4—5664no—
194C42—2832no—
195C23—22no—
196C23—5661no—
197C23C2×C22⋊D28311yescharacteristic
198Q8—28no—
199Q8—22no—
200Q8—24no—
201Q8—24no—
202Q8—5662no—
203Q8C2×C22⋊D28314yes—
204D4—28no—
205D4—24no—
206D4—24no—
207D4—22no—
208D4—5662no—
209D4—11321no—
210D4—11321no—
211D4C2×C22⋊D28314yes—
212C2×C4—216no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—5662no—
219C2×C4—5661no—
220C2×C4—11322no—
221C2×C4C2×C22⋊D28314yes—
222C2×C4C2×C22⋊D28311yescharacteristic
223C2×C4C2×C22⋊D28312yes—
224C2×C4C2×C22⋊D28312yes—
225C2×C4C23×D28311yescharacteristic
226C8—5664no—
227C22—24no—
228C22—41no—
229C22—44no—
230C22—11322no—
231C22—11321no—
232C22#2258357911yescharacteristic
233C22#2258357911yescharacteristic
234C22C22×C22⋊D28311yescharacteristic
235C4—24no—
236C4—24no—
237C4—11322no—
238C4—11321no—
239C4#2258357912yes—
240C4C22×C22⋊D28312yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—44no—
245C2—22641no—
246C2C2×#2258357911yescharacteristic
subgroupG/H
centerC2C2×#22583579
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
FittingC2×2−1+4×C283C2
socleC2×C283C2×#120
hypercenterC2×2−1+4D283
hypocenterC283#3020
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #31827263C2semidirect
2. #31827039C2semidirect
3. #31826648C2semidirect
4. #31826638C2semidirect
5. #31826482C2semidirect
6. C2×#22583529C2semidirect
7. C2×2−1+4×C283C2semidirect
10. #22583640C22semidirect
12. #22583635C22semidirect
15. #22583608C22semidirect
16. #22583599C22semidirect
20. #22583537C22semidirect
21. #22583529C22semidirect
123. C283#3020semidirect
125. C2×2−1+4D283semidirect

Maximal subgroups

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

subgroupindex
#318272632
#318270392
#318266482
#318266382
#318264822
C2×#225835292
C2×2−1+4×C2832
#3020283