Finite Group Databasebeta

← #49130739

Subgroups — #49130739

Order 36224.

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

normal subgroups only

This group has 584 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×#22583635C211yesmaximal, characteristic, complemented
2C2×#22583606C211yesmaximal, characteristic, complemented
3#41×Dic283C211yesmaximal, characteristic, complemented
4#31826663C214yesmaximal, complemented
5#31826658C214yesmaximal, complemented
6#31826488C211yesmaximal, characteristic, complemented
7#31826471C211yesmaximal, characteristic, complemented
8#31826470C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C283C211yesmaximal, characteristic, complemented
10C22×D1132C2211yescharacteristic
11C22×C22⋊D283—22no—
12#22583635—24no—
13C2×#17985379C2212yescomplemented
14C2×#17985377C2212yescomplemented
15C2×#17985376C2211yescharacteristic
16#22583606—21no—
17#22583606—21no—
18#22583606C2214yes—
19Q8×Dic283C2212yescomplemented
20D4×Dic283—22no—
21D4×Dic283C2212yescomplemented
22#22583569—21no—
23#22583569C2211yescharacteristic
24C2×C4×Dic283—21no—
25C2×C4×Dic283C2211yescharacteristic
26Dic283⋊C8C2214yescomplemented
27#22583532C2214yescomplemented
28#22583531C2214yescomplemented
29C2×#41×C283C2211yescharacteristic
30C22×D4×C283C2211yescharacteristic
31AGL(1,ℤ/8)×C283—24no—
32C2×SD16×C283C2212yescomplemented
33C2×D8×C283C2212yescomplemented
34C2×M16×C283C2211yescharacteristic
35C23×D283—21no—
36C2×D1132—22no—
37C2×D1132—21no—
38C2×D1132—22no—
39C2×D1132C2312yes—
40C2×C22⋊D283—24no—
41C2×C22⋊D283—24no—
42C2×C22⋊D283—48no—
43C22×Dic283—22no—
44C22×Dic283D412yes—
45#17985379—24no—
46#17985377—24no—
47#17985376—22no—
48C2×C283⋊C8C2312yes—
49C4⋊Dic283—22no—
50C4⋊Dic283C2312yes—
51C22⋊Dic283—22no—
52C22⋊Dic283—21no—
53C4×Dic283—22no—
54C4×Dic283—21no—
55C4×Dic283D411yescharacteristic
56C4×Dic283D411yescharacteristic
57C4×Dic283C2312yes—
58C24×C283—21no—
59#41×C283—22no—
60#41×C283D414yescomplemented
61C2×Q8×C283C2311yescharacteristic
62C2×D4×C283—21no—
63C2×D4×C283—22no—
64C2×D4×C283—21no—
65C2×D4×C283—22no—
66C2×D4×C283C2312yes—
67C2×D4×C283C2311yescharacteristic
68C22×C4×C283—21no—
69C22×C4×C283C2311yescharacteristic
70SD16×C283—24no—
71D8×C283—24no—
72M16×C283—22no—
73C2×C8×C283C2312yes—
74C22×D283—22no—
75C22×D283—44no—
76C22×D283—42no—
77D1132—22no—
78D1132—22no—
79D1132—42no—
80C22⋊D283—48no—
81C22⋊D283—48no—
82C2×Dic283—24no—
83C2×Dic283—22no—
84C2×Dic283—44no—
85C2×Dic283C2×D412yes—
86C2×Dic283C2×D412yes—
87C283⋊C8—22no—
88C23×C283—21no—
89C23×C283—22no—
90C23×C283—44no—
91C23×C283—42no—
92C23×C283C2×D411yescharacteristic
93Q8×C283—21no—
94Q8×C283C2×D412yescomplemented
95D4×C283—24no—
96D4×C283—22no—
97D4×C283—21no—
98D4×C283—22no—
99D4×C283—42no—
100D4×C283C2×D412yescomplemented
101C2×C4×C283—21no—
102C2×C4×C283—24no—
103C2×C4×C283C2×D411yescharacteristic
104C2×C4×C283C2×D412yes—
105C2×C4×C283#4112yes—
106C2×C4×C283C2411yescharacteristic
107C8×C283—22no—
108C22×C283—22no—
109C22×C283—22no—
110C22×C283—42no—
111C22×C283—42no—
112C22×C283—44no—
113C22×C283—84no—
114C22×C283#12112yes—
115C22×C283C22×D411yescharacteristic
116C2×D283—42no—
117C2×D283—44no—
118C2×D283—84no—
119C4×C283—22no—
120C4×C283#12112yes—
121C4×C283C22×D411yescharacteristic
122C4×C283C2×#4111yescharacteristic
123Dic283—22no—
124Dic283—42no—
125Dic283AGL(1,ℤ/8)14yescomplemented
126C2×C283—22no—
127C2×C283—42no—
128C2×C283—82no—
129C2×C283C2×#12111yescharacteristic
130C2×C283C2×AGL(1,ℤ/8)12yes—
131D283—82no—
132C283#303811yescharacteristic, complemented, Sylow
133#3038—2831nomaximal, Sylow
134C2×AGL(1,ℤ/8)—2831no—
135C2×AGL(1,ℤ/8)D28311yescharacteristic, complemented
136C2×#127—2831no—
137C4×#41—2831no—
138#460—2834no—
139#459—2834no—
140#423—2831no—
141#418—2831no—
142#418—2831no—
143C2×#41C2×D28311yescharacteristic
144C22×D4—2831no—
145C22×D4—5662no—
146C22×D4C2×D28311yescharacteristic
147AGL(1,ℤ/8)—24no—
148AGL(1,ℤ/8)—5664no—
149C2×SD16—2832no—
150C2×SD16C2×D28312yescomplemented
151C2×D8—2832no—
152C2×D8C2×D28312yescomplemented
153C2×M16—2831no—
154C2×M16C2×D28311yescharacteristic
155#127—2834no—
156#127—5661no—
157#127—5661no—
158C4×Q8—2832no—
159C4×D4—2832no—
160C4×D4—5662no—
161#117—2831no—
162#117—5661no—
163C2×C42—2831no—
164C2×C42—5661no—
165C4⋊C8—2834no—
166#102—2834no—
167#102—2834no—
168C24—21no—
169C24—5661no—
170#41—22no—
171#41C22⋊D28314yescomplemented
172C2×Q8C22×D28311yescharacteristic
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—2832no—
178C2×D4—5664no—
179C2×D4—5664no—
180C2×D4—5661no—
181C2×D4—5662no—
182C2×D4—5662no—
183C2×D4—11328no—
184C2×D4C22×D28312yes—
185C2×D4C22×D28311yescharacteristic
186C22×C4—21no—
187C22×C4—2832no—
188C22×C4—5662no—
189C22×C4C22×D28311yescharacteristic
190SD16—24no—
191SD16—5664no—
192D8—24no—
193D8—5664no—
194M16—22no—
195M16—5662no—
196C2×C8—2832no—
197C2×C8C22×D28312yes—
198C4⋊C4—2832no—
199C4⋊C4—5662no—
200C22⋊C4—5662no—
201C22⋊C4—5661no—
202C42—2831no—
203C42—2831no—
204C42—2832no—
205C42—5662no—
206C42—5661no—
207C23—21no—
208C23—22no—
209C23—44no—
210C23—42no—
211C23—5662no—
212C23—11324no—
213C23—11322no—
214C23D4×D28311yescharacteristic
215Q8—21no—
216Q8C2×C22⋊D28312yescomplemented
217D4—24no—
218D4—22no—
219D4—21no—
220D4—22no—
221D4—42no—
222D4—5662no—
223D4—5662no—
224D4—11328no—
225D4—11328no—
226D4—11322no—
227D4C2×C22⋊D28312yescomplemented
228C2×C4—21no—
229C2×C4—24no—
230C2×C4—2832no—
231C2×C4—2832no—
232C2×C4—5664no—
233C2×C4—5662no—
234C2×C4—11324no—
235C2×C4D4×D28311yescharacteristic
236C2×C4C2×C22⋊D28312yes—
237C2×C4#1798539112yes—
238C2×C4C23×D28311yescharacteristic
239C8—22no—
240C8—5662no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—42no—
245C22—44no—
246C22—84no—
247C22—11322no—
248C22—11324no—
249C22—22644no—
250C22#2258358212yes—
251C22C2×D4×D28311yescharacteristic
252C4—22no—
253C4—2834no—
254C4—5662no—
255C4—11322no—
256C4#2258358212yes—
257C4C22×C22⋊D28311yescharacteristic
258C4C2×#1798539111yescharacteristic
259C2—22no—
260C2—42no—
261C2—82no—
262C2—22642no—
263C2C2×#2258358211yescharacteristic
264C2AGL(1,ℤ/8)×D28312yes—
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
FittingC2×AGL(1,ℤ/8)×C283C2
socleC22×C283C22×D4
hypercenterC2×AGL(1,ℤ/8)D283
hypocenterC283#3038
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#22583635C2semidirect
2. C2×#22583606C2semidirect
3. #41×Dic283C2semidirect
4. #31826663C2semidirect
5. #31826658C2semidirect
6. #31826488C2semidirect
7. #31826471C2semidirect
8. #31826470C2semidirect
9. C2×AGL(1,ℤ/8)×C283C2semidirect
13. C2×#17985379C22semidirect
14. C2×#17985377C22semidirect
19. Q8×Dic283C22semidirect
21. D4×Dic283C22semidirect
26. Dic283⋊C8C22semidirect
27. #22583532C22semidirect
28. #22583531C22semidirect
32. C2×SD16×C283C22semidirect
33. C2×D8×C283C22semidirect
60. #41×C283D4semidirect
94. Q8×C283C2×D4semidirect
100. D4×C283C2×D4semidirect
125. Dic283AGL(1,ℤ/8)semidirect
132. C283#3038semidirect
135. C2×AGL(1,ℤ/8)D283semidirect
150. C2×SD16C2×D283semidirect
152. C2×D8C2×D283semidirect
171. #41C22⋊D283semidirect
216. Q8C2×C22⋊D283semidirect
227. D4C2×C22⋊D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836352
C2×#225836062
#41×Dic2832
#318266632
#318266582
#318264882
#318264712
#318264702
C2×AGL(1,ℤ/8)×C2832
#3038283