Finite Group Databasebeta

← #49131644

Subgroups — #49131644

Order 36224.

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

normal subgroups only

This group has 400 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#31827021C211yesmaximal, characteristic, complemented
2#124×D283C211yesmaximal, characteristic
3#31826826C212yesmaximal, complemented
4#31826823C212yesmaximal, complemented
5#31826822C211yesmaximal, characteristic, complemented
6#31826820C211yesmaximal, characteristic, complemented
7#31826819C211yesmaximal, characteristic, complemented
8#31826472C212yesmaximal, complemented
9#31826468C212yesmaximal
10C4×#17985375C211yesmaximal, characteristic, complemented
11#488×C283C211yesmaximal, characteristic, complemented
12C2×Q8×D283C2211yescharacteristic
13C2×D4×D283C2211yescharacteristic
14C2×#17985375C2212yes—
15#22583609C2212yescomplemented
16#22583602C2211yescharacteristic, complemented
17#22583599C2211yescharacteristic
18#22583598C2211yescharacteristic
19#22583597C2211yescharacteristic
20#22583596—22no—
21#22583588—22no—
22#22583575—22no—
23#22583568C2212yes—
24C22⋊C4×D283—22no—
25C42×D283C2211yescharacteristic
26#22583535C2212yescomplemented
27#22583534C2212yescomplemented
28#22583533C2212yescomplemented
29#22583532C2212yes—
30#22583531C2212yes—
31#22583530C2212yes—
32#22583520C2212yescomplemented
33C4×C283⋊C8C2211yescharacteristic, complemented
34#125×C283C2211yescharacteristic, complemented
35#124×C283C2211yescharacteristic
36#103×C283C2212yescomplemented
37#102×C283C2212yes—
38C4×C8×C283C2211yescharacteristic, complemented
39C23×D283—21no—
40C2×Dic566—22no—
41C2×Dic566—21no—
42C2×Dic566C2311yescharacteristic
43Q8×D283—24no—
44C2×D1132C2311yescharacteristic
45C2×C22⋊D283—21no—
46D4×D283—24no—
47C2×C4×D283—21no—
48C2×C4×D283D412yes—
49C2×C4×D283C2311yescharacteristic
50#17985375—24no—
51C2×C283⋊C8C2312yes—
52C4⋊Dic283C2312yes—
53Dic283⋊C4—22no—
54Dic283⋊C4—22no—
55C22⋊Dic283—22no—
56(C2×D283)⋊C4—22no—
57(C2×D283)⋊C4—22no—
58(C2×D283)⋊C4—22no—
59C4×Dic283—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×D283—44no—
71C22×D283—42no—
72C22×D283C2×D411yescharacteristic
73Dic566—22no—
74Dic566—44no—
75Dic566—42no—
76D1132—22no—
77C22⋊D283—42no—
78C2×Dic283—22no—
79C2×Dic283—21no—
80C2×Dic283C2×D412yes—
81C2×Dic283C2×D411yescharacteristic
82C4×D283—42no—
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—84no—
101C2×D283#12412yes—
102C4×C283—22no—
103C4×C283—41no—
104C4×C283—42no—
105C4×C283C2×#4112yes—
106Dic283—22no—
107Dic283—41no—
108Dic283—42no—
109Dic283#12412yes—
110C2×C283—81no—
111C2×C283C2×#12411yescharacteristic
112C2×C283#57512yes—
113D283—22no—
114D283—81no—
115C283#314011yescharacteristic, complemented, Sylow
116#3140—2831nomaximal, Sylow
117#533—2831no—
118C2×#124—2831no—
119#489—2832no—
120#489—2832no—
121#488—2831no—
122#488—2831no—
123#488—2831no—
124#488D28311yescharacteristic, complemented
125#419—2832no—
126#418—2832no—
127C4×M16—2831no—
128C22×Q8—2831no—
129C22×D4—2831no—
130C2×M16—2832no—
131C4⋊Q8—2832no—
132#125—2831no—
133#125C2×D28311yescharacteristic, complemented
134#124—2831no—
135#124—2831no—
136#124—2831no—
137#124—5662no—
138#124C2×D28311yescharacteristic
139C22⋊Q8—5662no—
140C4×Q8—5662no—
141#117—2832no—
142C2×C22⋊C4—5662no—
143C2×C42—2831no—
144#103—2832no—
145#103—2832no—
146#103—2832no—
147#103C2×D28312yescomplemented
148#102—2832no—
149#102—2832no—
150#102—2832no—
151#102C2×D28312yes—
152#97—2832no—
153C4×C8—2831no—
154C4×C8C2×D28311yescharacteristic, complemented
155C24—5661no—
156C2×Q8—2831no—
157C2×Q8—5662no—
158C2×Q8—5661no—
159C2×Q8—5664no—
160C2×Q8C22×D28311yescharacteristic
161C2×D4—2831no—
162C2×D4—5664no—
163C2×D4—5661no—
164C2×D4C22×D28311yescharacteristic
165C22×C4—2831no—
166C22×C4—2832no—
167C22×C4—5661no—
168M16—5664no—
169C2×C8—2832no—
170C2×C8C22×D28312yes—
171C4⋊C4—22no—
172C4⋊C4—2832no—
173C4⋊C4—5662no—
174C4⋊C4—5662no—
175C4⋊C4C22×D28312yes—
176C22⋊C4—22no—
177C22⋊C4—5662no—
178C22⋊C4—5662no—
179C22⋊C4—5662no—
180C22⋊C4—5662no—
181C42—2832no—
182C42—2831no—
183C42—5662no—
184C42C22×D28311yescharacteristic
185C23—21no—
186C23—2831no—
187C23—5661no—
188C23—11322no—
189C23—11324no—
190Q8—22no—
191Q8—5662no—
192Q8—11322no—
193Q8—11324no—
194D4—22no—
195D4—5662no—
196D4—11322no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—2831no—
200C2×C4—2834no—
201C2×C4—2832no—
202C2×C4—5662no—
203C2×C4—5661no—
204C2×C4—11322no—
205C2×C4—11322no—
206C2×C4D4×D28312yes—
207C2×C4C23×D28311yescharacteristic
208C8—22no—
209C8—5662no—
210C22—41no—
211C22—42no—
212C22—2832no—
213C22—5662no—
214C22—11321no—
215C22—11322no—
216C22—22644no—
217C22C2×D4×D28311yescharacteristic
218C4—22no—
219C4—41no—
220C4—42no—
221C4—2832no—
222C4—5662no—
223C4—11321no—
224C4—11322no—
225C4#41×D28312yes—
226C2—81no—
227C2—5662no—
228C2—22641no—
229C2#124×D28311yescharacteristic
230C2#3182726212yes—
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#488×C283C2
socleC22×C283C22×D4
prime radicalC2×D4×D283C22
hypercenter#488D283
hypocenterC283#3140
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
1. #31827021C2semidirect
3. #31826826C2semidirect
4. #31826823C2semidirect
5. #31826822C2semidirect
6. #31826820C2semidirect
7. #31826819C2semidirect
8. #31826472C2semidirect
10. C4×#17985375C2semidirect
11. #488×C283C2semidirect
15. #22583609C22semidirect
16. #22583602C22semidirect
26. #22583535C22semidirect
27. #22583534C22semidirect
28. #22583533C22semidirect
32. #22583520C22semidirect
33. C4×C283⋊C8C22semidirect
34. #125×C283C22semidirect
36. #103×C283C22semidirect
38. C4×C8×C283C22semidirect
115. C283#3140semidirect
124. #488D283semidirect
133. #125C2×D283semidirect
147. #103C2×D283semidirect
154. C4×C8C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318270212
#124×D2832
#318268262
#318268232
#318268222
#318268202
#318268192
#318264722
#318264682
C4×#179853752
#488×C2832
#3140283