Finite Group Databasebeta

← #49131658

Subgroups — #49131658

Order 36224.

392 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 117 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#31827014C211yesmaximal, characteristic
2#125×D283C211yesmaximal, characteristic, complemented
3#31826832C212yesmaximal, complemented
4#31826828C212yesmaximal, complemented
5#31826822C211yesmaximal, characteristic, complemented
6#31826820C211yesmaximal, characteristic, complemented
7#31826819C211yesmaximal, characteristic, complemented
8#31826465C212yesmaximal, complemented
9#31826463C212yesmaximal, complemented
10C4×#17985375C211yesmaximal, characteristic, complemented
11#488×C283C211yesmaximal, characteristic, complemented
12C2×#17985393C2211yescharacteristic
13C2×#17985391C2211yescharacteristic
14C2×#17985375C2212yes—
15#22583608C2211yescharacteristic
16#22583606C2211yescharacteristic
17#22583602C2211yescharacteristic
18#22583601C2212yes—
19#22583600—22no—
20#22583598C2211yescharacteristic
21#22583583—22no—
22(C22⋊D283)⋊C4—22no—
23C4⋊C4×D283—22no—
24C4⋊C4×D283C2212yes—
25C42×D283C2211yescharacteristic
26#22583535C2212yes—
27#22583534C2212yes—
28#22583533C2212yes—
29#22583532C2212yes—
30#22583531C2212yes—
31#22583530C2212yes—
32#22583520C2212yes—
33C4×C283⋊C8C2211yescharacteristic
34#125×C283C2211yescharacteristic
35#124×C283C2211yescharacteristic
36#103×C283C2212yes—
37#102×C283C2212yes—
38C4×C8×C283C2211yescharacteristic
39#17985393—24no—
40#17985391—24no—
41C2×Dic566C2311yescharacteristic
42C2×D1132—21no—
43C2×D1132C2311yescharacteristic
44C2×C22⋊D283—21no—
45C2×C22⋊D283—22no—
46C22×Dic283—21no—
47C2×C4×D283—22no—
48C2×C4×D283—21no—
49C2×C4×D283D412yes—
50C2×C4×D283C2311yescharacteristic
51#17985375—24no—
52C2×C283⋊C8C2312yes—
53C4⋊Dic283—22no—
54C4⋊Dic283C2312yes—
55Dic283⋊C4—22no—
56Dic283⋊C4—22no—
57Dic283⋊C4—22no—
58Dic283⋊C4—22no—
59(C2×D283)⋊C4—22no—
60C4×Dic283D412yes—
61C4×Dic283C2311yescharacteristic
62C2×Q8×C283C2311yescharacteristic
63C2×D4×C283C2311yescharacteristic
64C2×C8×C283C2312yes—
65C4⋊C4×C283—22no—
66C4⋊C4×C283C2312yes—
67C22⋊C4×C283—22no—
68C42×C283C2311yescharacteristic
69C22×D283—21no—
70C22×D283C2×D411yescharacteristic
71Dic566—22no—
72D1132—22no—
73D1132—42no—
74C22⋊D283—44no—
75C22⋊D283—42no—
76C2×Dic283—21no—
77C2×Dic283—22no—
78C2×Dic283—42no—
79C2×Dic283C2×D412yes—
80C2×Dic283C2×D411yescharacteristic
81C4×D283—42no—
82C4×D283—44no—
83C4×D283—42no—
84C4×D283#4114yes—
85C283⋊C8—22no—
86C23×C283—21no—
87Q8×C283—22no—
88D4×C283—22no—
89C2×C4×C283—21no—
90C2×C4×C283—22no—
91C2×C4×C283C2×D412yes—
92C2×C4×C283C2411yescharacteristic
93C8×C283—22no—
94C22×C283—41no—
95C22×C283—42no—
96C22×C283C22×D411yescharacteristic
97C2×D283—22no—
98C2×D283—41no—
99C2×D283—42no—
100C2×D283#12412yes—
101C4×C283—22no—
102C4×C283—41no—
103C4×C283—42no—
104C4×C283C2×#4112yes—
105Dic283—22no—
106Dic283—41no—
107Dic283—42no—
108Dic283#12412yes—
109C2×C283—81no—
110C2×C283C2×#12411yescharacteristic
111C2×C283#57512yes—
112D283—22no—
113D283—81no—
114C283#314211yescharacteristic, complemented, Sylow
115#3142—2831nomaximal, Sylow
116#532—2831no—
117C2×#125—2831no—
118#491—2832no—
119#490—2832no—
120#488—2831no—
121#488—2831no—
122#488—2831no—
123#488D28311yescharacteristic, complemented
124#417—2832no—
125#417—2832no—
126C4×M16—2831no—
127C2×#41—2831no—
128C2×#41—2831no—
129C2×M16—2832no—
130C4⋊Q8—2831no—
131#127—2831no—
132#125—2832no—
133#125—2831no—
134#125—5662no—
135#125C2×D28311yescharacteristic
136#124—2831no—
137#124C2×D28311yescharacteristic
138#121—5662no—
139C4×D4—5662no—
140C2×C4⋊C4—2832no—
141C2×C4⋊C4—5662no—
142C2×C42—2831no—
143#103—2832no—
144#103—2832no—
145#103—2832no—
146#103C2×D28312yes—
147#102—2832no—
148#102—2832no—
149#102—2832no—
150#102C2×D28312yes—
151#97—2832no—
152C4×C8—2831no—
153C4×C8C2×D28311yescharacteristic
154#41—5664no—
155#41—5664no—
156C2×Q8—2831no—
157C2×Q8C22×D28311yescharacteristic
158C2×D4—2831no—
159C2×D4—5661no—
160C2×D4—5662no—
161C2×D4—5661no—
162C2×D4C22×D28311yescharacteristic
163C22×C4—2831no—
164C22×C4—2832no—
165C22×C4—5661no—
166C22×C4—5662no—
167C22×C4—5661no—
168M16—5664no—
169C2×C8—2832no—
170C2×C8C22×D28312yes—
171C4⋊C4—22no—
172C4⋊C4—2832no—
173C4⋊C4—5662no—
174C4⋊C4—5662no—
175C4⋊C4—5662no—
176C4⋊C4—5662no—
177C4⋊C4—5662no—
178C4⋊C4C22×D28312yes—
179C22⋊C4—22no—
180C22⋊C4—5662no—
181C42—2832no—
182C42—2831no—
183C42C22×D28311yescharacteristic
184C23—21no—
185C23—2831no—
186C23—5661no—
187Q8—22no—
188Q8—5662no—
189D4—22no—
190D4—5662no—
191D4—11322no—
192D4—11324no—
193D4—11322no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—2831no—
197C2×C4—2834no—
198C2×C4—2832no—
199C2×C4—5661no—
200C2×C4—5662no—
201C2×C4—11322no—
202C2×C4—11322no—
203C2×C4—11324no—
204C2×C4—11322no—
205C2×C4D4×D28312yes—
206C2×C4C23×D28311yescharacteristic
207C8—22no—
208C8—5662no—
209C22—41no—
210C22—42no—
211C22—2832no—
212C22—5662no—
213C22—11321no—
214C22—11322no—
215C22C2×D4×D28311yescharacteristic
216C4—22no—
217C4—41no—
218C4—42no—
219C4—2832no—
220C4—5662no—
221C4—11321no—
222C4—11322no—
223C4#41×D28312yes—
224C2—81no—
225C2—5662no—
226C2—22641no—
227C2#124×D28311yescharacteristic
228C2#3182726212yes—
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#488×C283C2
socleC22×C283C22×D4
prime radical#31827014C2
hypercenter#488D283
hypocenterC283#3142
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
2. #125×D283C2semidirect
3. #31826832C2semidirect
4. #31826828C2semidirect
5. #31826822C2semidirect
6. #31826820C2semidirect
7. #31826819C2semidirect
8. #31826465C2semidirect
9. #31826463C2semidirect
10. C4×#17985375C2semidirect
11. #488×C283C2semidirect
114. C283#3142semidirect
123. #488D283semidirect

Maximal subgroups

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

subgroupindex
#318270142
#125×D2832
#318268322
#318268282
#318268222
#318268202
#318268192
#318264652
#318264632
C4×#179853752
#488×C2832
#3142283