Finite Group Databasebeta

← #49130555

Subgroups — #49130555

Order 36224.

700 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 149 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×#22583657C211yesmaximal, characteristic
2C22×#17985382C211yesmaximal, characteristic, complemented
3#31827039C211yesmaximal, characteristic, complemented
4#31826624C214yesmaximal, complemented
5#31826619C214yesmaximal, complemented
6#31826475C211yesmaximal, characteristic, complemented
7#31826461C211yesmaximal, characteristic, complemented
8#31826433C211yesmaximal, characteristic, complemented
9C2×#137×C283C211yesmaximal, characteristic, complemented
10#22583657—24no—
11#22583657—24no—
12C2×#17985392—21no—
13C2×#17985392C2211yescharacteristic
14C2×#17985391—22no—
15C2×#17985391C2212yes—
16C22×Dic566—21no—
17C22×Dic566C2211yescharacteristic
18C2×Q8×D283C2212yes—
19C2×#17985382—24no—
20C2×#17985382C2214yes—
21C22×C283⋊C8C2211yescharacteristic
22#22583608—21no—
23#22583599—21no—
24#22583593—22no—
25#22583586C2214yes—
26#22583569C2211yescharacteristic
27#22583535C2214yes—
28#22583534C2212yes—
29#22583531C2212yes—
30(C2×D283)⋊C8C2214yes—
31C2×#41×C283C2211yescharacteristic
32C22×Q8×C283C2211yescharacteristic
33#137×C283—24no—
34C2×Q16×C283C2212yes—
35C2×SD16×C283C2212yes—
36C2×M16×C283C2211yescharacteristic
37#17985392—28no—
38#17985392—22no—
39#17985392—22no—
40#17985391—216no—
41#17985391—24no—
42#17985391—24no—
43C2×Dic566—28no—
44C2×Dic566—22no—
45C2×Dic566—21no—
46C2×Dic566—22no—
47C2×Dic566D414yes—
48C2×Dic566D411yescharacteristic
49C2×Dic566C2312yes—
50Q8×D283—24no—
51Q8×D283—24no—
52C2×D1132D411yescharacteristic
53C2×C22⋊D283—22no—
54C2×C22⋊D283D412yes—
55C22×Dic283—22no—
56C22×Dic283—21no—
57C2×C4×D283—22no—
58C2×C4×D283C2312yes—
59#17985382—28no—
60C2×C283⋊C8—22no—
61C2×C283⋊C8C2312yes—
62C4⋊Dic283C2312yes—
63Dic283⋊C4—24no—
64C22⋊Dic283—21no—
65(C2×D283)⋊C4—24no—
66C4×Dic283—21no—
67#41×C283—22no—
68#41×C283D414yes—
69C2×Q8×C283—21no—
70C2×Q8×C283—22no—
71C2×Q8×C283—22no—
72C2×Q8×C283C2312yes—
73C2×Q8×C283C2311yescharacteristic
74C2×D4×C283—21no—
75C2×D4×C283C2311yescharacteristic
76C22×C4×C283—21no—
77C22×C4×C283—21no—
78C22×C4×C283C2311yescharacteristic
79Q16×C283—24no—
80SD16×C283—24no—
81M16×C283—22no—
82C2×C8×C283C2312yes—
83C22×D283C2×D412yes—
84Dic566—28no—
85Dic566—24no—
86Dic566—22no—
87Dic566—22no—
88Dic566C2×D414yes—
89D1132—22no—
90C22⋊D283—28no—
91C22⋊D283—24no—
92C2×Dic283—28no—
93C2×Dic283—24no—
94C2×Dic283—22no—
95C2×Dic283—22no—
96C2×Dic283C2×D412yes—
97C4×D283—28no—
98C4×D283—22no—
99C4×D283—22no—
100C283⋊C8—22no—
101C23×C283—21no—
102C23×C283C2×D411yescharacteristic
103Q8×C283—21no—
104Q8×C283—22no—
105Q8×C283—22no—
106Q8×C283—42no—
107Q8×C283C2×D412yes—
108D4×C283—24no—
109D4×C283—21no—
110D4×C283C2×D412yes—
111C2×C4×C283—21no—
112C2×C4×C283—24no—
113C2×C4×C283—22no—
114C2×C4×C283—42no—
115C2×C4×C283C2×D411yescharacteristic
116C2×C4×C283C2×D412yes—
117C2×C4×C283C2×D412yes—
118C2×C4×C283C2411yescharacteristic
119C8×C283—22no—
120C22×C283—22no—
121C22×C283—22no—
122C22×C283—42no—
123C22×C283#12012yes—
124C22×C283C22×D411yescharacteristic
125C2×D283—22no—
126C2×D283—24no—
127C4×C283—22no—
128C4×C283—42no—
129C4×C283#12012yes—
130C4×C283C22×D411yescharacteristic
131C4×C283C22×D411yescharacteristic
132Dic283—24no—
133Dic283—22no—
134Dic283—42no—
135C2×C283—22no—
136C2×C283—42no—
137C2×C283C2×#12011yescharacteristic
138C2×C283#57812yes—
139D283—42no—
140C283#301111yescharacteristic, complemented, Sylow
141#3011—2831nomaximal, Sylow
142C2×2−1+4—2831no—
143C2×#137D28311yescharacteristic, complemented
144C22×Q16—2831no—
145#536—2831no—
146#452—2834no—
147#451—2834no—
148#419—2831no—
149#416—2831no—
150#408—2831no—
1512−1+4—5664no—
1522−1+4—5664no—
153C2×#41—2831no—
154C2×#41—2832no—
155C2×#41—5661no—
156C2×#41—5662no—
157C2×#41C2×D28311yescharacteristic
158C22×Q8—2832no—
159C22×Q8—2831no—
160C22×Q8—5661no—
161C22×Q8C2×D28311yescharacteristic
162#137—24no—
163C2×Q16—2834no—
164C2×Q16—5664no—
165C2×Q16C2×D28312yes—
166C2×SD16C2×D28312yes—
167C2×M16C2×D28311yescharacteristic
168C22×C8—2831no—
169C4⋊Q8—5661no—
170#124—5661no—
171#123—5662no—
172C22⋊Q8—2834no—
173#117—2831no—
174#103—2832no—
175#103—2834no—
176#102—2832no—
177C22⋊C8—2834no—
178#41—22no—
179#41—5668no—
180#41—5664no—
181#41—56616no—
182#41—5664no—
183#41—5662no—
184#41—5662no—
185#41C22⋊D28314yes—
186C2×Q8—21no—
187C2×Q8—22no—
188C2×Q8—22no—
189C2×Q8—2831no—
190C2×Q8—2834no—
191C2×Q8—2832no—
192C2×Q8—5662no—
193C2×Q8—5668no—
194C2×Q8—5664no—
195C2×Q8—5664no—
196C2×Q8—5661no—
197C2×Q8—5662no—
198C2×Q8C22×D28312yes—
199C2×Q8C22×D28311yescharacteristic
200C2×D4—21no—
201C2×D4—2832no—
202C2×D4—2831no—
203C2×D4—5662no—
204C2×D4C22×D28311yescharacteristic
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—2832no—
208C22×C4—5662no—
209C22×C4—5662no—
210C22×C4—5661no—
211C22×C4C22×D28311yescharacteristic
212Q16—24no—
213Q16—5668no—
214SD16—24no—
215M16—22no—
216C2×C8—2832no—
217C2×C8—5662no—
218C2×C8C22×D28312yes—
219C4⋊C4—2832no—
220C4⋊C4—5664no—
221C22⋊C4—5664no—
222C22⋊C4—5661no—
223C42—5661no—
224C23—21no—
225C23—2832no—
226C23D4×D28311yescharacteristic
227Q8—21no—
228Q8—22no—
229Q8—22no—
230Q8—42no—
231Q8—2834no—
232Q8—5668no—
233Q8—5662no—
234Q8—5664no—
235Q8—5662no—
236Q8C2×C22⋊D28312yes—
237D4—24no—
238D4—21no—
239D4—5668no—
240D4—5664no—
241D4—5662no—
242D4C2×C22⋊D28312yes—
243C2×C4—21no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—2832no—
248C2×C4—5668no—
249C2×C4—5668no—
250C2×C4—5664no—
251C2×C4—5662no—
252C2×C4—5662no—
253C2×C4—5662no—
254C2×C4—5662no—
255C2×C4D4×D28311yescharacteristic
256C2×C4D4×D28312yes—
257C2×C4C2×C22⋊D28312yes—
258C2×C4C23×D28311yescharacteristic
259C8—22no—
260C8—5662no—
261C22—22no—
262C22—22no—
263C22—42no—
264C22—5662no—
265C22—5664no—
266C22#2258357812yes—
267C22C2×D4×D28311yescharacteristic
268C4—22no—
269C4—42no—
270C4—5662no—
271C4—5664no—
272C4—11322no—
273C4#2258357812yes—
274C4C2×D4×D28311yescharacteristic
275C4C22×C22⋊D28311yescharacteristic
276C2—22no—
277C2—42no—
278C2—11322no—
279C2C2×#2258357811yescharacteristic
280C2#3182728112yes—
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
FittingC2×#137×C283C2
socleC22×C283C22×D4
prime radicalC2×#22583657C2
hypercenterC2×#137D283
hypocenterC283#3011
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C22×#17985382C2semidirect
3. #31827039C2semidirect
4. #31826624C2semidirect
5. #31826619C2semidirect
6. #31826475C2semidirect
7. #31826461C2semidirect
8. #31826433C2semidirect
9. C2×#137×C283C2semidirect
140. C283#3011semidirect
143. C2×#137D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836572
C22×#179853822
#318270392
#318266242
#318266192
#318264752
#318264612
#318264332
C2×#137×C2832
#3011283