Finite Group Databasebeta

← #49131651

Subgroups — #49131651

Order 36224.

424 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 125 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#31827013C211yesmaximal, characteristic
2C4⋊Q8×D283C211yesmaximal, characteristic, complemented
3#31826823C214yesmaximal, complemented
4#31826817C211yesmaximal, characteristic, complemented
5#31826816C211yesmaximal, characteristic, complemented
6#31826811C211yesmaximal, characteristic, complemented
7#31826463C214yesmaximal, complemented
8C4×#17985375C211yesmaximal, characteristic, complemented
9#486×C283C211yesmaximal, characteristic, complemented
10C2×#17985391C2212yes—
11C2×Q8×D283—21no—
12C2×#17985375C2212yes—
13#22583610C2211yescharacteristic
14#22583610C2211yescharacteristic
15#22583609—21no—
16#22583609C2212yes—
17#22583608—21no—
18#22583597C2212yes—
19#22583581—21no—
20#22583581—21no—
21D4×Dic283—21no—
22D4×Dic283—21no—
23C4⋊C4×D283—21no—
24C4⋊C4×D283C2212yes—
25C42×D283C2211yescharacteristic
26#22583535C2214yes—
27#22583533C2214yes—
28#22583531C2214yes—
29#22583520C2212yes—
30C4×C283⋊C8C2211yescharacteristic
31C4⋊Q8×C283C2211yescharacteristic
32#127×C283C2211yescharacteristic
33#102×C283C2214yes—
34C4×C8×C283C2211yescharacteristic
35#17985391—28no—
36C2×Dic566—22no—
37C2×Dic566—21no—
38C2×Dic566C2312yes—
39Q8×D283—44no—
40C2×C22⋊D283—22no—
41C22×Dic283—22no—
42C2×C4×D283—22no—
43C2×C4×D283D411yescharacteristic
44C2×C4×D283D411yescharacteristic
45C2×C4×D283C2311yescharacteristic
46#17985375—24no—
47C2×C283⋊C8C2312yes—
48C4⋊Dic283—21no—
49C4⋊Dic283—21no—
50C4⋊Dic283—21no—
51C4⋊Dic283C2312yes—
52Dic283⋊C4—22no—
53Dic283⋊C4—22no—
54C22⋊Dic283—22no—
55C22⋊Dic283—22no—
56C4×Dic283D411yescharacteristic
57C4×Dic283D411yescharacteristic
58C4×Dic283C2311yescharacteristic
59C2×Q8×C283—21no—
60C2×D4×C283—21no—
61C2×D4×C283—21no—
62C2×D4×C283C2312yes—
63C2×C8×C283C2312yes—
64C4⋊C4×C283—21no—
65C4⋊C4×C283C2312yes—
66C42×C283C2311yescharacteristic
67C22×D283C2×D411yescharacteristic
68Dic566—24no—
69Dic566—44no—
70Dic566—42no—
71C22⋊D283—44no—
72C2×Dic283—22no—
73C2×Dic283—22no—
74C2×Dic283—44no—
75C2×Dic283C2×D411yescharacteristic
76C2×Dic283C2×D411yescharacteristic
77C2×Dic283C2×D411yescharacteristic
78C4×D283—22no—
79C4×D283—22no—
80C4×D283—44no—
81C4×D283#4114yes—
82C283⋊C8—22no—
83C23×C283—22no—
84Q8×C283—42no—
85D4×C283—24no—
86D4×C283—42no—
87D4×C283—42no—
88C2×C4×C283—22no—
89C2×C4×C283C2×D411yescharacteristic
90C2×C4×C283C2×D411yescharacteristic
91C2×C4×C283C2411yescharacteristic
92C8×C283—22no—
93C22×C283—42no—
94C22×C283—42no—
95C22×C283—42no—
96C22×C283C22×D411yescharacteristic
97C2×D283—21no—
98C2×D283—21no—
99C2×D283#12412yes—
100C4×C283—42no—
101C4×C283AGL(1,ℤ/8)12yes—
102C4×C283#13712yes—
103C4×C283C2×#4112yes—
104Dic283—21no—
105Dic283—21no—
106Dic283—42no—
107Dic283—42no—
108Dic283#12412yes—
109C2×C283—82no—
110C2×C283C2×#12411yescharacteristic
111C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
112C2×C283C2×#13711yescharacteristic
113D283—22no—
114C283#314111yescharacteristic, complemented, Sylow
115#3141—2831nomaximal, Sylow
116#532—2831no—
117C2×C4⋊Q8—2831no—
118#489—2834no—
119#487—2831no—
120#487—2831no—
121#486—2831no—
122#486D28311yescharacteristic, complemented
123#417—2834no—
124C4×M16—2831no—
125C2×#41—2832no—
126C22×Q8—5661no—
127C2×M16—2832no—
128C4⋊Q8—2832no—
129C4⋊Q8—2831no—
130C4⋊Q8—2831no—
131C4⋊Q8—5661no—
132C4⋊Q8—5661no—
133C4⋊Q8C2×D28311yescharacteristic
134#127C2×D28311yescharacteristic
135#124—2832no—
136#121—5661no—
137#121—5661no—
138C4×D4—5661no—
139C4×D4—5661no—
140C2×C4⋊C4—2832no—
141C2×C4⋊C4—5661no—
142C2×C42—2831no—
143#103—2834no—
144#103—2834no—
145#102—2834no—
146#102C2×D28314yes—
147#97—2832no—
148C4×C8—2831no—
149C4×C8C2×D28311yescharacteristic
150#41—5668no—
151C2×Q8—21no—
152C2×Q8—2832no—
153C2×Q8—5662no—
154C2×Q8—5661no—
155C2×Q8—11324no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—5662no—
159C2×D4C22×D28312yes—
160C22×C4—2831no—
161C22×C4—2831no—
162C22×C4—2831no—
163C22×C4—5662no—
164C22×C4—5662no—
165M16—5664no—
166C2×C8—2832no—
167C2×C8C22×D28312yes—
168C4⋊C4—21no—
169C4⋊C4—2832no—
170C4⋊C4—5662no—
171C4⋊C4—5662no—
172C4⋊C4—5661no—
173C4⋊C4—5661no—
174C4⋊C4—5661no—
175C4⋊C4C22×D28312yes—
176C22⋊C4—5662no—
177C22⋊C4—5662no—
178C42—2831no—
179C42—2831no—
180C42—2831no—
181C42C22×D28311yescharacteristic
182C23—22no—
183C23—2831no—
184Q8—42no—
185Q8—5664no—
186Q8—11324no—
187Q8—11322no—
188D4—24no—
189D4—42no—
190D4—42no—
191D4—11324no—
192C2×C4—22no—
193C2×C4—2831no—
194C2×C4—2834no—
195C2×C4—2831no—
196C2×C4—2831no—
197C2×C4—5662no—
198C2×C4—5662no—
199C2×C4—5662no—
200C2×C4—5662no—
201C2×C4—11324no—
202C2×C4—11324no—
203C2×C4D4×D28311yescharacteristic
204C2×C4D4×D28311yescharacteristic
205C2×C4C23×D28311yescharacteristic
206C8—22no—
207C8—5662no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—2832no—
212C22—5661no—
213C22—5661no—
214C22C2×D4×D28311yescharacteristic
215C4—42no—
216C4—2832no—
217C4—5661no—
218C4—5661no—
219C4—11322no—
220C4—11322no—
221C4#2258363412yes—
222C4#2258363912yes—
223C4#41×D28312yes—
224C2—82no—
225C2—5662no—
226C2#124×D28311yescharacteristic
227C2C2×#2258363411yescharacteristic
228C2C2×#2258363911yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#486×C283C2
socleC22×C283C22×D4
prime radical#31827013C2
hypercenter#486D283
hypocenterC283#3141
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D283C2semidirect
3. #31826823C2semidirect
4. #31826817C2semidirect
5. #31826816C2semidirect
6. #31826811C2semidirect
7. #31826463C2semidirect
8. C4×#17985375C2semidirect
9. #486×C283C2semidirect
114. C283#3141semidirect
122. #486D283semidirect

Maximal subgroups

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

subgroupindex
#318270132
C4⋊Q8×D2832
#318268232
#318268172
#318268162
#318268112
#318264632
C4×#179853752
#486×C2832
#3141283