Finite Group Databasebeta

← #49131737

Subgroups — #49131737

Order 36224.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×#22583643C211yesmaximal, characteristic, complemented
2C2×#22583638C211yesmaximal, characteristic, complemented
3C2×#22583632C211yesmaximal, characteristic, complemented
4C22×#17985381C211yesmaximal, characteristic, complemented
5#31827041C211yesmaximal, characteristic, complemented
6#31827034C211yesmaximal, characteristic, complemented
7#31826859C212yesmaximal, complemented
8#31826854C212yesmaximal, complemented
9#31826849C212yesmaximal, complemented
10#31826838C212yesmaximal, complemented
11#405×C283C211yesmaximal, characteristic, complemented
12C2×#17985392C2211yescharacteristic
13C2×#17985392C2211yescharacteristic
14C22×Dic566C2211yescharacteristic
15C22×D1132C2211yescharacteristic
16#22583643—24no—
17#22583638—24no—
18#22583632—24no—
19C2×Dic1132C2211yescharacteristic, complemented
20C2×Dic1132C2212yescomplemented
21C2×#17985381—24no—
22C2×#17985381C2212yes—
23C2×#17985381C2212yes—
24C2×#17985381C2212yes—
25C2×#17985381C2211yescharacteristic, complemented
26C2×#17985381C2211yescharacteristic, complemented
27C2×#17985381C2212yescomplemented
28C2×D2264C2212yes—
29C2×D2264C2211yescharacteristic, complemented
30#22583610C2212yescomplemented
31#22583607C2212yes—
32#22583598C2212yes—
33#22583598C2212yescomplemented
34#22583594—22no—
35#22583590—22no—
36#22583585—22no—
37#22583580—22no—
38C2×M16×C283C2211yescharacteristic
39C22×C8×C283C2211yescharacteristic
40#117×C283C2211yescharacteristic
41#97×C283C2212yescomplemented
42C4×C8×C283C2212yescomplemented
43C23×D283—21no—
44#17985392—22no—
45#17985392—22no—
46#17985392—22no—
47#17985392—22no—
48C2×Dic566—21no—
49C2×Dic566—22no—
50C2×Dic566—22no—
51C2×Dic566C2311yescharacteristic
52C2×Dic566C2312yes—
53C2×Dic566C2311yescharacteristic
54C2×D1132—22no—
55C2×D1132—22no—
56C2×D1132—21no—
57C2×D1132—24no—
58C2×D1132C2312yes—
59C2×D1132C2311yescharacteristic
60C2×D1132C2311yescharacteristic
61C2×C22⋊D283—21no—
62C2×C22⋊D283—21no—
63C22×Dic283—21no—
64C2×C4×D283—21no—
65C2×C4×D283—21no—
66Dic1132—22no—
67Dic1132—24no—
68#17985381—24no—
69#17985381—24no—
70#17985381—24no—
71#17985381—22no—
72#17985381—22no—
73#17985381—24no—
74D2264—24no—
75D2264—22no—
76C4⋊Dic283—24no—
77(C2×D283)⋊C4—24no—
78(C2×D283)⋊C4—24no—
79C22×C4×C283C2311yescharacteristic
80M16×C283D414yescomplemented
81C2×C8×C283D414yescomplemented
82C2×C8×C283C2312yes—
83C2×C8×C283C2311yescharacteristic
84C2×C8×C283C2311yescharacteristic
85C4⋊C4×C283D412yes—
86C22⋊C4×C283D412yes—
87C42×C283C2312yes—
88C22×D283—21no—
89C22×D283—22no—
90C22×D283—21no—
91C22×D283—42no—
92C22×D283—82no—
93Dic566—22no—
94Dic566—21no—
95Dic566—22no—
96Dic566—21no—
97Dic566—21no—
98Dic566—21no—
99Dic566—42no—
100D1132—21no—
101D1132—22no—
102D1132—21no—
103D1132—21no—
104D1132—22no—
105D1132—21no—
106D1132—42no—
107D1132—84no—
108C22⋊D283—42no—
109C22⋊D283—42no—
110C2×Dic283—21no—
111C2×Dic283—22no—
112C2×Dic283—21no—
113C2×Dic283—42no—
114C4×D283—42no—
115C4×D283—42no—
116C23×C283C2×D411yescharacteristic
117C2×C4×C283C2×D414yes—
118C2×C4×C283C2×D411yescharacteristic
119C2×C4×C283C2×D412yes—
120C2×C4×C283C2×D412yes—
121C2×C4×C283C2411yescharacteristic
122C8×C283C2×D412yes—
123C8×C283C2×D412yes—
124C8×C283C2×D414yes—
125C22×C283—22no—
126C22×C283#12712yes—
127C22×C283C22×D411yescharacteristic
128C2×D283—41no—
129C2×D283—41no—
130C2×D283—42no—
131C2×D283—42no—
132C2×D283—44no—
133C2×D283—42no—
134C2×D283—84no—
135C4×C283—24no—
136C4×C283#12712yes—
137C4×C283C22×D411yescharacteristic
138C4×C283C22×D411yescharacteristic
139Dic283—41no—
140Dic283—41no—
141Dic283—42no—
142C2×C283—22no—
143C2×C283C2×#12711yescharacteristic
144C2×C283#57712yes—
145D283—81no—
146D283—81no—
147D283—82no—
148C283#315411yescharacteristic, complemented, Sylow
149#3154—2831nomaximal, Sylow
150C2×#137—2831no—
151C2×AGL(1,ℤ/8)—2831no—
152C2×#135—2831no—
153C22×SD16—2831no—
154#536—2831no—
155#535—2831no—
156#497—2832no—
157#496—2832no—
158#495—2832no—
159#492—2832no—
160#405D28311yescharacteristic, complemented
161C2×#41—2831no—
162C2×#41—2831no—
163C22×Q8—2831no—
164C22×D4—2831no—
165#137—5664no—
166AGL(1,ℤ/8)—5664no—
167#135—5664no—
168C2×Q16—2831no—
169C2×Q16—2832no—
170C2×SD16—2832no—
171C2×SD16—2832no—
172C2×SD16—2831no—
173C2×SD16—2831no—
174C2×SD16—2832no—
175C2×SD16—2832no—
176C2×SD16—5664no—
177C2×D8—2831no—
178C2×D8—2832no—
179C2×M16C2×D28311yescharacteristic
180C22×C8C2×D28311yescharacteristic
181C4⋊Q8—2832no—
182#127—2832no—
183#124—2832no—
184#124—2832no—
185#123—5662no—
186C22⋊Q8—5662no—
187#121—5662no—
188#120—5662no—
189#117C2×D28311yescharacteristic
190#97C2×D28312yescomplemented
191C4×C8C2×D28312yescomplemented
192C24—5661no—
193#41—5662no—
194#41—5662no—
195#41—5662no—
196#41—5662no—
197C2×Q8—2831no—
198C2×Q8—2832no—
199C2×Q8—2831no—
200C2×Q8—5661no—
201C2×Q8—5662no—
202C2×Q8—5662no—
203C2×D4—2831no—
204C2×D4—2832no—
205C2×D4—2831no—
206C2×D4—5661no—
207C2×D4—5661no—
208C2×D4—5662no—
209C2×D4—5662no—
210C2×D4—5661no—
211C2×D4—5664no—
212C22×C4—5661no—
213C22×C4—5661no—
214C22×C4—5661no—
215C22×C4C22×D28311yescharacteristic
216Q16—5662no—
217Q16—5664no—
218SD16—5664no—
219SD16—5664no—
220SD16—5662no—
221SD16—5662no—
222SD16—5664no—
223SD16—5664no—
224D8—5662no—
225D8—5664no—
226M16D113214yescomplemented
227C2×C8D113214yescomplemented
228C2×C8C22×D28311yescharacteristic
229C2×C8C22×D28312yes—
230C2×C8C22×D28311yescharacteristic
231C4⋊C4—5664no—
232C4⋊C4D113212yes—
233C22⋊C4—5664no—
234C22⋊C4—5664no—
235C22⋊C4D113212yes—
236C42C22×D28312yes—
237C23—5661no—
238C23—5662no—
239C23—5661no—
240C23—11322no—
241C23—22642no—
242C23C2×D113211yescharacteristic
243Q8—5662no—
244Q8—5661no—
245Q8—5662no—
246Q8—5661no—
247Q8—5661no—
248Q8—5661no—
249Q8—11322no—
250D4—5661no—
251D4—5662no—
252D4—5661no—
253D4—5661no—
254D4—5662no—
255D4—5661no—
256D4—11322no—
257D4—11322no—
258D4—11322no—
259D4—22644no—
260C2×C4—5661no—
261C2×C4—5662no—
262C2×C4—5661no—
263C2×C4—11322no—
264C2×C4—11322no—
265C2×C4—11322no—
266C2×C4C2×D113214yes—
267C2×C4C2×D113211yescharacteristic
268C2×C4C2×D113212yes—
269C2×C4C2×D113212yes—
270C2×C4C23×D28311yescharacteristic
271C8C2×D113212yes—
272C8C2×D113212yes—
273C8C2×D113214yes—
274C22—22no—
275C22—11321no—
276C22—11321no—
277C22—11322no—
278C22—11322no—
279C22—11324no—
280C22—11322no—
281C22—22644no—
282C22#2258360712yes—
283C22C22×D113211yescharacteristic
284C4—24no—
285C4—11321no—
286C4—11321no—
287C4—11322no—
288C4#2258360712yes—
289C4C22×D113211yescharacteristic
290C4C22×D113211yescharacteristic
291C2—22no—
292C2—22642no—
293C2—22641no—
294C2—22641no—
295C2C2×#2258360711yescharacteristic
296C2#3182727712yes—
subgroupG/H
centerC22C22×D1132
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#405×C283C2
socleC22×C283C22×D4
hypercenter#405D283
hypocenterC283#3154
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ #117 ◁ #405

Complements

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

NH ≅ G/Ntype
1. C2×#22583643C2semidirect
2. C2×#22583638C2semidirect
3. C2×#22583632C2semidirect
4. C22×#17985381C2semidirect
5. #31827041C2semidirect
6. #31827034C2semidirect
7. #31826859C2semidirect
8. #31826854C2semidirect
9. #31826849C2semidirect
10. #31826838C2semidirect
11. #405×C283C2semidirect
19. C2×Dic1132C22semidirect
20. C2×Dic1132C22semidirect
25. C2×#17985381C22semidirect
26. C2×#17985381C22semidirect
27. C2×#17985381C22semidirect
29. C2×D2264C22semidirect
30. #22583610C22semidirect
33. #22583598C22semidirect
41. #97×C283C22semidirect
42. C4×C8×C283C22semidirect
80. M16×C283D4semidirect
81. C2×C8×C283D4semidirect
148. C283#3154semidirect
160. #405D283semidirect
190. #97C2×D283semidirect
191. C4×C8C2×D283semidirect
226. M16D1132semidirect
227. C2×C8D1132semidirect

Maximal subgroups

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

subgroupindex
C2×#225836432
C2×#225836382
C2×#225836322
C22×#179853812
#318270412
#318270342
#318268592
#318268542
#318268492
#318268382
#405×C2832
#3154283