Finite Group Databasebeta

← #49131739

Subgroups — #49131739

Order 36224.

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

normal subgroups only

This group has 564 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×#22583641C211yesmaximal, characteristic, complemented
2C2×#22583635C211yesmaximal, characteristic, complemented
3C2×#22583630C211yesmaximal, characteristic, complemented
4C22×#17985379C211yesmaximal, characteristic, complemented
5#31827034C211yesmaximal, characteristic, complemented
6#31826857C212yesmaximal, complemented
7#31826855C212yesmaximal, complemented
8#31826847C212yesmaximal, complemented
9#31826839C212yesmaximal, complemented
10#31826421C211yesmaximal, characteristic, complemented
11#536×C283C211yesmaximal, characteristic, complemented
12C2×#17985392C2211yescharacteristic
13C22×D1132C2211yescharacteristic
14#22583641—24no—
15#22583635—24no—
16#22583630—24no—
17C2×#17985382C2211yescharacteristic, complemented
18C2×#17985382C2212yescomplemented
19C2×#17985380C2211yescharacteristic, complemented
20C2×#17985379—24no—
21C2×#17985379C2212yes—
22C2×#17985379C2212yes—
23C2×#17985379C2212yes—
24C2×#17985379C2211yescharacteristic, complemented
25C2×#17985379C2212yescomplemented
26C2×#17985377C2212yes—
27C2×#17985377C2211yescharacteristic, complemented
28C2×#17985376C2211yescharacteristic
29C22×C283⋊C8C2211yescharacteristic
30#22583607C2212yes—
31#22583598C2212yes—
32#22583585—22no—
33#22583580—22no—
34#22583521C2212yescomplemented
35C4×C283⋊C8C2212yescomplemented
36C2×#41×C283C2211yescharacteristic
37C22×Q8×C283C2211yescharacteristic
38C4⋊Q8×C283C2212yescomplemented
39#124×C283C2212yescomplemented
40#123×C283—22no—
41C22⋊Q8×C283—22no—
42#117×C283C2211yescharacteristic
43C23×D283—21no—
44#17985392—22no—
45#17985392—22no—
46C2×Dic566C2311yescharacteristic
47C2×D1132—22no—
48C2×D1132—22no—
49C2×D1132—21no—
50C2×D1132—24no—
51C2×D1132C2312yes—
52C2×D1132C2311yescharacteristic
53C2×C22⋊D283—21no—
54C2×C4×D283—21no—
55#17985382—22no—
56#17985382—24no—
57#17985380—22no—
58#17985379—24no—
59#17985379—24no—
60#17985379—24no—
61#17985379—22no—
62#17985379—24no—
63#17985377—24no—
64#17985377—22no—
65#17985376D414yescomplemented
66C2×C283⋊C8D414yescomplemented
67C2×C283⋊C8C2311yescharacteristic
68C2×C283⋊C8C2311yescharacteristic
69C2×C283⋊C8C2312yes—
70(C2×D283)⋊C4—24no—
71#41×C283—22no—
72#41×C283—22no—
73C2×Q8×C283—21no—
74C2×Q8×C283—22no—
75C2×Q8×C283—22no—
76C2×Q8×C283C2312yes—
77C2×Q8×C283C2311yescharacteristic
78C2×D4×C283—21no—
79C2×D4×C283C2311yescharacteristic
80C22×C4×C283—21no—
81C22×C4×C283—21no—
82C22×C4×C283C2311yescharacteristic
83C4⋊C4×C283—24no—
84C4⋊C4×C283D412yes—
85C22⋊C4×C283—24no—
86C22⋊C4×C283D412yes—
87C42×C283C2312yes—
88C22×D283—22no—
89C22×D283—21no—
90C22×D283—42no—
91C22×D283—82no—
92Dic566—21no—
93Dic566—21no—
94D1132—22no—
95D1132—21no—
96D1132—22no—
97D1132—21no—
98D1132—42no—
99D1132—84no—
100C22⋊D283—42no—
101C2×Dic283—21no—
102C4×D283—42no—
103C283⋊C8C2×D412yes—
104C283⋊C8C2×D412yes—
105C283⋊C8C2×D414yes—
106C23×C283—21no—
107C23×C283C2×D411yescharacteristic
108Q8×C283—22no—
109Q8×C283—22no—
110Q8×C283—21no—
111Q8×C283—21no—
112Q8×C283—42no—
113D4×C283—21no—
114D4×C283—21no—
115D4×C283—42no—
116C2×C4×C283—22no—
117C2×C4×C283—21no—
118C2×C4×C283—42no—
119C2×C4×C283—42no—
120C2×C4×C283C2×D412yes—
121C2×C4×C283C2×D412yes—
122C2×C4×C283C2×D414yes—
123C2×C4×C283C2×D411yescharacteristic
124C2×C4×C283C2411yescharacteristic
125C22×C283—22no—
126C22×C283—41no—
127C22×C283—42no—
128C22×C283#12712yes—
129C22×C283C22×D411yescharacteristic
130C2×D283—41no—
131C2×D283—42no—
132C2×D283—42no—
133C2×D283—44no—
134C2×D283—84no—
135C4×C283—24no—
136C4×C283—41no—
137C4×C283—42no—
138C4×C283#12712yes—
139C4×C283C22×D411yescharacteristic
140C4×C283C22×D411yescharacteristic
141Dic283—41no—
142C2×C283—22no—
143C2×C283—81no—
144C2×C283C2×#12711yescharacteristic
145C2×C283#57712yes—
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#536D28311yescharacteristic, complemented
155#535—2831no—
156#497—2832no—
157#496—2832no—
158#495—2832no—
159#492—2832no—
160#405—2831no—
161C2×#41—2831no—
162C2×#41C2×D28311yescharacteristic
163C22×Q8C2×D28311yescharacteristic
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—2832no—
174C2×SD16—2832no—
175C2×SD16—2831no—
176C2×SD16—5664no—
177C2×D8—2831no—
178C2×D8—2832no—
179C2×M16—2831no—
180C22×C8—2831no—
181C4⋊Q8C2×D28312yescomplemented
182#127—2832no—
183#124—2832no—
184#124C2×D28312yescomplemented
185#123—22no—
186C22⋊Q8—22no—
187#121—5662no—
188#120—5662no—
189#117C2×D28311yescharacteristic
190#97—2832no—
191C4×C8—2832no—
192C24—5661no—
193#41—22no—
194#41—22no—
195#41—5662no—
196#41—5662no—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—2831no—
201C2×Q8C22×D28312yes—
202C2×Q8C22×D28311yescharacteristic
203C2×D4—21no—
204C2×D4—2832no—
205C2×D4—2831no—
206C2×D4—5661no—
207C2×D4—5662no—
208C2×D4—5662no—
209C2×D4—5661no—
210C2×D4—5664no—
211C2×D4C22×D28311yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—5661no—
215C22×C4C22×D28311yescharacteristic
216Q16—5662no—
217Q16—5664no—
218SD16—5662no—
219SD16—5664no—
220SD16—5664no—
221SD16—5662no—
222SD16—5664no—
223SD16—5664no—
224D8—5662no—
225D8—5664no—
226M16—2834no—
227C2×C8—2834no—
228C2×C8—2831no—
229C2×C8—2832no—
230C2×C8—2831no—
231C4⋊C4—24no—
232C4⋊C4C22⋊D28312yes—
233C22⋊C4—24no—
234C22⋊C4—5664no—
235C22⋊C4C22⋊D28312yes—
236C42C22×D28312yes—
237C23—21no—
238C23—5662no—
239C23—5661no—
240C23—11322no—
241C23—22642no—
242C23C2×C22⋊D28311yescharacteristic
243Q8—22no—
244Q8—21no—
245Q8—22no—
246Q8—21no—
247Q8—42no—
248Q8—5661no—
249Q8—5661no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—5662no—
254D4—5661no—
255D4—5662no—
256D4—5661no—
257D4—11322no—
258D4—11322no—
259D4—22644no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—5661no—
265C2×C4—11322no—
266C2×C4D4×D28312yes—
267C2×C4D4×D28312yes—
268C2×C4C2×C22⋊D28314yes—
269C2×C4C2×C22⋊D28311yescharacteristic
270C2×C4C23×D28311yescharacteristic
271C8—2832no—
272C8—2832no—
273C8—2834no—
274C22—22no—
275C22—41no—
276C22—42no—
277C22—11321no—
278C22—11322no—
279C22—11322no—
280C22—11324no—
281C22—22644no—
282C22#2258360612yes—
283C22C22×C22⋊D28311yescharacteristic
284C4—24no—
285C4—41no—
286C4—42no—
287C4—11321no—
288C4#2258360612yes—
289C4C2×D4×D28311yescharacteristic
290C4C2×D4×D28311yescharacteristic
291C2—22no—
292C2—81no—
293C2—22642no—
294C2—22641no—
295C2C2×#2258360611yescharacteristic
296C2#3182727312yes—
subgroupG/H
centerC22C22×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#536×C283C2
socleC22×C283C22×D4
hypercenter#536D283
hypocenterC283#3154
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#22583641C2semidirect
2. C2×#22583635C2semidirect
3. C2×#22583630C2semidirect
4. C22×#17985379C2semidirect
5. #31827034C2semidirect
6. #31826857C2semidirect
7. #31826855C2semidirect
8. #31826847C2semidirect
9. #31826839C2semidirect
10. #31826421C2semidirect
11. #536×C283C2semidirect
17. C2×#17985382C22semidirect
18. C2×#17985382C22semidirect
19. C2×#17985380C22semidirect
24. C2×#17985379C22semidirect
25. C2×#17985379C22semidirect
27. C2×#17985377C22semidirect
34. #22583521C22semidirect
35. C4×C283⋊C8C22semidirect
38. C4⋊Q8×C283C22semidirect
39. #124×C283C22semidirect
65. #17985376D4semidirect
66. C2×C283⋊C8D4semidirect
148. C283#3154semidirect
154. #536D283semidirect
181. C4⋊Q8C2×D283semidirect
184. #124C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836412
C2×#225836352
C2×#225836302
C22×#179853792
#318270342
#318268572
#318268552
#318268472
#318268392
#318264212
#536×C2832
#3154283