Finite Group Databasebeta

← #49129738

Subgroups — #49129738

Order 36224.

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

normal subgroups only

This group has 756 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×#22583656C211yesmaximal, characteristic
2#31827231C212yesmaximal, complemented
3#31826449C211yesmaximal, characteristic, complemented
4#31826448C214yesmaximal, complemented
5C2×#22583526C212yesmaximal, complemented
6#100×D283C214yesmaximal, complemented
7#413×C283C211yesmaximal, characteristic, complemented
8#22583656—24no—
9#22583656—24no—
10C2×#17985393C2212yes—
11C2×#17985392C412yes—
12C2×#17985392C2211yescharacteristic
13C22×D1132C412yes—
14C22×D1132C2211yescharacteristic
15C2×D4×D283C414yes—
16C2×D4×D283C2212yes—
17#22583618—24no—
18#22583616—24no—
19C2×#17985376C2212yescomplemented
20M16×D283C2218yescomplemented
21#22583529C2212yescomplemented
22#22583527C2212yescomplemented
23#22583526C2218yescomplemented
24C2×#41×C283C2211yescharacteristic
25C2×M16×C283C2212yescomplemented
26#101×C283C2212yescomplemented
27#100×C283C2212yescomplemented
28C23×D283C2×C412yes—
29C23×D283C2×C414yes—
30#17985393—28no—
31#17985392—24no—
32#17985392—24no—
33#17985392D414yes—
34#17985392D414yes—
35C2×Dic566C2×C411yescharacteristic
36C2×D1132—24no—
37C2×D1132—24no—
38C2×D1132—22no—
39C2×D1132—22no—
40C2×D1132C2×C414yes—
41C2×D1132C2×C412yes—
42C2×D1132C2×C411yescharacteristic
43C2×D1132C2312yes—
44C2×C22⋊D283C2×C414yes—
45C2×C22⋊D283C2×C412yes—
46D4×D283—216no—
47D4×D283—28no—
48C2×C4×D283C2×C414yes—
49C2×C4×D283C2312yes—
50#17985376—22no—
51#17985376C2314yes—
52#17985375—28no—
53C2×C283⋊C8—22no—
54C8×D283—28no—
55#41×C283—22no—
56#41×C283—22no—
57C2×Q8×C283C2311yescharacteristic
58C2×D4×C283C2×C412yes—
59C2×D4×C283C2311yescharacteristic
60C22×C4×C283C2×C412yes—
61C22×C4×C283C2311yescharacteristic
62M16×C283—22no—
63M16×C283C2314yes—
64C2×C8×C283—22no—
65C22×D283—24no—
66C22×D283—28no—
67C22×D283—84no—
68C22×D283—82no—
69C22×D283C22×C414yes—
70C22×D283C22×C412yes—
71Dic566C22⋊C412yes—
72Dic566C22⋊C412yes—
73D1132—28no—
74D1132—22no—
75D1132—22no—
76D1132—24no—
77D1132C22⋊C412yes—
78D1132C22⋊C412yes—
79C22⋊D283—28no—
80C22⋊D283C22⋊C418yes—
81C2×Dic283C22×C412yes—
82C4×D283—28no—
83C4×D283C2×D418yes—
84C283⋊C8—24no—
85C23×C283C22×C412yes—
86C23×C283C22×C411yescharacteristic
87Q8×C283—22no—
88D4×C283—24no—
89D4×C283—22no—
90C2×C4×C283—22no—
91C2×C4×C283—22no—
92C2×C4×C283C22×C412yes—
93C2×C4×C283C22×C411yescharacteristic
94C2×C4×C283C2×D412yes—
95C2×C4×C283C2×D412yes—
96C2×C4×C283C2411yescharacteristic
97C8×C283—24no—
98C22×C283—22no—
99C22×C283—41no—
100C22×C283—42no—
101C22×C283C2×C22⋊C412yes—
102C22×C283C23×C411yescharacteristic
103C2×D283—24no—
104C2×D283—42no—
105C2×D283—44no—
106C2×D283—88no—
107C2×D283—84no—
108C2×D283C2×C22⋊C414yes—
109C4×C283—22no—
110C4×C283C2×C22⋊C411yescharacteristic
111C4×C283C2×C22⋊C411yescharacteristic
112C4×C283C22×D412yes—
113Dic283C2×C22⋊C414yes—
114C2×C283—21no—
115C2×C283—22no—
116C2×C283—42no—
117C2×C283C22×C22⋊C411yescharacteristic
118D283—24no—
119D283—44no—
120C283#288811yescharacteristic, complemented, Sylow
121#2888—2831nomaximal, Sylow
122C2×H(5,𝔽2)—2831no—
123#568—2832no—
124#413—2831no—
125#413—2834no—
126#413D28311yescharacteristic, complemented
127C2×#100—2832no—
128C2×#100—2834no—
129H(5,𝔽2)—5664no—
130H(5,𝔽2)—5664no—
131C2×#41—2832no—
132C2×#41—2832no—
133C2×#41—2831no—
134C2×#41C2×D28311yescharacteristic
135C22×D4—2834no—
136C22×D4—2832no—
137C22×D4—2832no—
138C22×D4—2831no—
139#131—5664no—
140#131—5664no—
141C2×M16—2832no—
142C2×M16—2838no—
143C2×M16C2×D28312yescomplemented
144#101—2832no—
145#101C2×D28312yescomplemented
146#100—2832no—
147#100—2838no—
148#100C2×D28312yescomplemented
149C24—2834no—
150C24—2832no—
151#41—22no—
152#41—22no—
153#41—2834no—
154#41—2834no—
155#41—5664no—
156#41—5664no—
157#41—5668no—
158C2×Q8—2831no—
159C2×Q8C22×D28311yescharacteristic
160C2×D4—2832no—
161C2×D4—2834no—
162C2×D4—2831no—
163C2×D4—2834no—
164C2×D4—2832no—
165C2×D4—2832no—
166C2×D4—5664no—
167C2×D4—5664no—
168C2×D4—5668no—
169C2×D4—56616no—
170C2×D4—5662no—
171C2×D4—5662no—
172C2×D4C4×D28312yes—
173C2×D4C22×D28311yescharacteristic
174C22×C4—2832no—
175C22×C4—2834no—
176C22×C4C4×D28312yes—
177C22×C4C22×D28311yescharacteristic
178M16—22no—
179M16—2834no—
180M16—5668no—
181M16—5662no—
182M16C22×D28314yes—
183C2×C8—22no—
184C2×C8—5668no—
185C2×C8—5662no—
186C23—2832no—
187C23—2834no—
188C23—5668no—
189C23—5664no—
190C23—22642no—
191C23—22644no—
192C23C2×C4×D28312yes—
193C23C2×C4×D28311yescharacteristic
194Q8—22no—
195Q8—2832no—
196Q8—2832no—
197D4—24no—
198D4—22no—
199D4—2838no—
200D4—2832no—
201D4—2832no—
202D4—5668no—
203D4—5668no—
204D4—5662no—
205D4—5662no—
206D4—5664no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—2832no—
210C2×C4—2838no—
211C2×C4—5668no—
212C2×C4C2×C4×D28312yes—
213C2×C4C2×C4×D28311yescharacteristic
214C2×C4D4×D28312yes—
215C2×C4D4×D28312yes—
216C2×C4C23×D28311yescharacteristic
217C8—24no—
218C8—5664no—
219C22—22no—
220C22—41no—
221C22—42no—
222C22—2834no—
223C22—5664no—
224C22—11322no—
225C22—11324no—
226C22—22644no—
227C22—22648no—
228C22C22⋊C4×D28312yes—
229C22C22×C4×D28311yescharacteristic
230C4—22no—
231C4—2834no—
232C4C22⋊C4×D28311yescharacteristic
233C4C22⋊C4×D28311yescharacteristic
234C4C2×D4×D28312yes—
235C2—21no—
236C2—22no—
237C2—42no—
238C2—5664no—
239C2—11324no—
240C2C2×C22⋊C4×D28311yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D283
commutatorC22×C283C23×C4
FrattiniC2×C4C23×D283
Fitting#413×C283C2
socleC2×C283C22×C22⋊C4
prime radicalC2×#22583656C2
cosocleC2×C4×C283C24
hypercenter#413D283
hypocenterC283#2888
LCSG ▷ C22×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #31827231C2semidirect
3. #31826449C2semidirect
4. #31826448C2semidirect
5. C2×#22583526C2semidirect
6. #100×D283C2semidirect
7. #413×C283C2semidirect
19. C2×#17985376C22semidirect
20. M16×D283C22semidirect
21. #22583529C22semidirect
22. #22583527C22semidirect
23. #22583526C22semidirect
25. C2×M16×C283C22semidirect
26. #101×C283C22semidirect
27. #100×C283C22semidirect
120. C283#2888semidirect
126. #413D283semidirect
143. C2×M16C2×D283semidirect
145. #101C2×D283semidirect
148. #100C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836562
#318272312
#318264492
#318264482
C2×#225835262
#100×D2832
#413×C2832
#2888283