Finite Group Databasebeta

← #49130624

Subgroups — #49130624

Order 36224.

698 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 145 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×#22583658C211yesmaximal, characteristic, complemented
2#31827263C212yesmaximal, complemented
3#31826647C214yesmaximal, complemented
4#31826637C214yesmaximal, complemented
5#31826483C212yesmaximal, complemented
6C2×#22583529C211yesmaximal, characteristic, complemented
7#536×C283C211yesmaximal, characteristic, complemented
8#22583658—24no—
9#22583658—22no—
10#22583658C2214yes—
11C2×#17985393—22no—
12C2×#17985392—22no—
13C2×#17985392C2212yes—
14C2×Q8×D283—22no—
15#22583641C2214yescomplemented
16#22583640—22no—
17#22583636C2214yescomplemented
18#22583635—22no—
19#22583630—24no—
20C2×#17985376C2212yes—
21#22583538C2218yescomplemented
22#22583529C2214yescomplemented
23C2×#41×C283C2211yescharacteristic
24C22×Q8×C283C2211yescharacteristic
25C4⋊Q8×C283C2212yescomplemented
26#124×C283C2212yescomplemented
27#123×C283—22no—
28C22⋊Q8×C283—22no—
29#117×C283C2211yescharacteristic
30#17985393—28no—
31#17985393—28no—
32#17985392—28no—
33#17985392—28no—
34#17985392—22no—
35#17985392—22no—
36#17985392D414yes—
37#17985392C2314yes—
38C2×Dic566—22no—
39C2×Dic566D412yes—
40Q8×D283—28no—
41Q8×D283—28no—
42C2×D1132—22no—
43C2×D1132D412yes—
44C2×C22⋊D283—22no—
45C2×C4×D283—24no—
46C2×C4×D283—22no—
47#17985382—24no—
48#17985380—24no—
49#17985379—24no—
50#17985377—24no—
51#17985376—22no—
52#17985376C2314yes—
53C2×C283⋊C8—22no—
54#41×C283—22no—
55#41×C283—22no—
56C2×Q8×C283—22no—
57C2×Q8×C283—22no—
58C2×Q8×C283D412yes—
59C2×Q8×C283C2311yescharacteristic
60C2×Q8×C283C2312yes—
61C2×D4×C283—21no—
62C2×D4×C283C2311yescharacteristic
63C22×C4×C283—21no—
64C22×C4×C283D412yes—
65C22×C4×C283C2311yescharacteristic
66C4⋊C4×C283—21no—
67C4⋊C4×C283—24no—
68C22⋊C4×C283—21no—
69C22⋊C4×C283—24no—
70C42×C283C2312yes—
71C22×D283—22no—
72Dic566—28no—
73Dic566—22no—
74Dic566C2×D414yes—
75D1132—28no—
76D1132—22no—
77D1132C2×D414yes—
78C22⋊D283—24no—
79C22⋊D283—24no—
80C2×Dic283—22no—
81C4×D283—216no—
82C4×D283—24no—
83C4×D283—24no—
84C283⋊C8—24no—
85C23×C283—21no—
86C23×C283C2×D411yescharacteristic
87Q8×C283—24no—
88Q8×C283—24no—
89Q8×C283—22no—
90D4×C283—22no—
91D4×C283—41no—
92D4×C283—41no—
93C2×C4×C283—22no—
94C2×C4×C283—22no—
95C2×C4×C283—22no—
96C2×C4×C283—21no—
97C2×C4×C283—42no—
98C2×C4×C283C2×D412yes—
99C2×C4×C283C2×D412yes—
100C2×C4×C283C2×D414yes—
101C2×C4×C283C2×D411yescharacteristic
102C2×C4×C283C2411yescharacteristic
103C22×C283—41no—
104C22×C283—42no—
105C22×C283—41no—
106C22×C283#12011yescharacteristic
107C22×C283#12011yescharacteristic
108C22×C283C22×D411yescharacteristic
109C2×D283—24no—
110C2×D283—44no—
111C4×C283—24no—
112C4×C283—42no—
113C4×C283—41no—
114C4×C283#12012yes—
115C4×C283C22×D412yes—
116Dic283—24no—
117C2×C283—21no—
118C2×C283—21no—
119C2×C283—21no—
120C2×C283—81no—
121C2×C283C2×#12011yescharacteristic
122D283—44no—
123C283#302011yescharacteristic, complemented, Sylow
124#3020—2831nomaximal, Sylow
125C2×2−1+4—2831no—
126#575—2832no—
127#536D28311yescharacteristic, complemented
128#456—2834no—
129#454—2834no—
130#421—2832no—
131C2×#101—2831no—
1322−1+4—2834no—
1332−1+4—5664no—
1342−1+4—5662no—
135C2×#41—2832no—
136C2×#41—5662no—
137C2×#41—5662no—
138C2×#41C2×D28311yescharacteristic
139C22×Q8—5662no—
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⋊Q8C2×D28312yescomplemented
148#124C2×D28312yescomplemented
149#123—22no—
150C22⋊Q8—22no—
151#117C2×D28311yescharacteristic
152#104—2838no—
153#101—2834no—
154#41—22no—
155#41—22no—
156#41—2834no—
157#41—2834no—
158#41—5668no—
159#41—5668no—
160#41—5668no—
161#41—5662no—
162#41—5662no—
163#41—5668no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—2832no—
167C2×Q8—5662no—
168C2×Q8—5668no—
169C2×Q8—5668no—
170C2×Q8C22⋊D28312yes—
171C2×Q8C22×D28311yescharacteristic
172C2×Q8C22×D28312yes—
173C2×D4—21no—
174C2×D4—2832no—
175C2×D4—5662no—
176C2×D4—5662no—
177C2×D4C22×D28311yescharacteristic
178C22×C4—21no—
179C22×C4—5664no—
180C22×C4—5662no—
181C22×C4C22⋊D28312yes—
182C22×C4C22×D28311yescharacteristic
183Q16—5664no—
184SD16—5664no—
185SD16—5664no—
186D8—5664no—
187M16—2834no—
188M16—5662no—
189C2×C8—5662no—
190C4⋊C4—21no—
191C4⋊C4—24no—
192C22⋊C4—21no—
193C22⋊C4—24no—
194C42C22×D28312yes—
195C23—21no—
196C23—5662no—
197C23C2×C22⋊D28311yescharacteristic
198Q8—24no—
199Q8—24no—
200Q8—22no—
201Q8—2834no—
202Q8—5668no—
203Q8—5662no—
204D4—22no—
205D4—41no—
206D4—41no—
207D4—2834no—
208D4—5668no—
209D4—5664no—
210D4—5664no—
211D4—5662no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—42no—
217C2×C4—56616no—
218C2×C4—5662no—
219C2×C4—5664no—
220C2×C4—5664no—
221C2×C4D4×D28312yes—
222C2×C4D4×D28312yes—
223C2×C4C2×C22⋊D28314yes—
224C2×C4C2×C22⋊D28311yescharacteristic
225C2×C4C23×D28311yescharacteristic
226C8—5664no—
227C22—41no—
228C22—42no—
229C22—41no—
230C22—5664no—
231C22—11324no—
232C22#2258357811yescharacteristic
233C22#2258357811yescharacteristic
234C22C22×C22⋊D28311yescharacteristic
235C4—24no—
236C4—42no—
237C4—41no—
238C4—5664no—
239C4#2258357812yes—
240C4C2×D4×D28312yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—81no—
245C2—11324no—
246C2C2×#2258357811yescharacteristic
subgroupG/H
centerC2C2×#22583578
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#536×C283C2
socleC2×C283C2×#120
hypercenter#536D283
hypocenterC283#3020
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#22583658C2semidirect
2. #31827263C2semidirect
3. #31826647C2semidirect
4. #31826637C2semidirect
5. #31826483C2semidirect
6. C2×#22583529C2semidirect
7. #536×C283C2semidirect
15. #22583641C22semidirect
17. #22583636C22semidirect
21. #22583538C22semidirect
22. #22583529C22semidirect
25. C4⋊Q8×C283C22semidirect
26. #124×C283C22semidirect
123. C283#3020semidirect
127. #536D283semidirect
147. C4⋊Q8C2×D283semidirect
148. #124C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836582
#318272632
#318266472
#318266372
#318264832
C2×#225835292
#536×C2832
#3020283