Finite Group Databasebeta

← #49130532

Subgroups — #49130532

Order 36224.

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

normal subgroups only

This group has 784 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×#22583656C211yesmaximal, characteristic, complemented
2C2×#22583635C211yesmaximal, characteristic, complemented
3C22×D2264C211yesmaximal, characteristic, complemented
4#31827034C211yesmaximal, characteristic, complemented
5#31826613C214yesmaximal, complemented
6#31826600C214yesmaximal, complemented
7#31826469C211yesmaximal, characteristic, complemented
8#31826431C211yesmaximal, characteristic, complemented
9#416×C283C211yesmaximal, characteristic, complemented
10#22583656—24no—
11#22583656—24no—
12C2×#17985393C2212yescomplemented
13C2×#17985392—21no—
14C2×#17985392C2211yescharacteristic
15C22×D1132—21no—
16C22×D1132C2211yescharacteristic
17C22×D1132C2211yescharacteristic, complemented
18C2×D4×D283—22no—
19C2×D4×D283C2212yescomplemented
20#22583635—24no—
21C2×#17985379C2212yescomplemented
22C2×D2264—24no—
23C2×D2264C2214yescomplemented
24C2×#17985377C2212yescomplemented
25C2×#17985376C2211yescharacteristic, complemented
26#22583607—21no—
27#22583598—21no—
28#22583585C2214yescomplemented
29#22583580—22no—
30#22583530C2214yescomplemented
31(C2×D283)⋊C8C2214yescomplemented
32C2×#41×C283C2211yescharacteristic
33C22×C8×C283C2211yescharacteristic, complemented
34#117×C283C2211yescharacteristic, complemented
35#103×C283C2212yescomplemented
36#102×C283C2212yescomplemented
37C23×D283—22no—
38C23×D283—21no—
39C23×D283—21no—
40#17985393—24no—
41#17985393—24no—
42#17985392—28no—
43#17985392—22no—
44#17985392—22no—
45C2×Dic566D411yescharacteristic
46C2×D1132—28no—
47C2×D1132—22no—
48C2×D1132—24no—
49C2×D1132—21no—
50C2×D1132—22no—
51C2×D1132—22no—
52C2×D1132—22no—
53C2×D1132—21no—
54C2×D1132D411yescharacteristic
55C2×D1132D414yescomplemented
56C2×D1132C2312yes—
57C2×D1132C2312yes—
58C2×C22⋊D283—22no—
59C2×C22⋊D283D412yes—
60D4×D283—216no—
61D4×D283—24no—
62D4×D283—24no—
63C2×C4×D283—22no—
64C2×C4×D283C2312yes—
65#17985379—24no—
66D2264—28no—
67#17985377—24no—
68#17985376—22no—
69C2×C283⋊C8C2312yes—
70(C2×D283)⋊C4—24no—
71#41×C283—22no—
72#41×C283D414yescomplemented
73C2×Q8×C283C2311yescharacteristic
74C2×D4×C283—21no—
75C2×D4×C283C2311yescharacteristic
76C22×C4×C283—21no—
77C22×C4×C283C2311yescharacteristic
78C2×C8×C283—22no—
79C2×C8×C283C2312yes—
80C4⋊C4×C283C2312yes—
81C22⋊C4×C283—21no—
82C42×C283—21no—
83C22×D283—22no—
84C22×D283—28no—
85C22×D283—24no—
86C22×D283—22no—
87C22×D283—42no—
88C22×D283—84no—
89C22×D283—82no—
90C22×D283—82no—
91C22×D283C2×D412yes—
92Dic566—22no—
93D1132—28no—
94D1132—22no—
95D1132—24no—
96D1132—22no—
97D1132—22no—
98D1132—22no—
99D1132—42no—
100D1132—84no—
101D1132C2×D414yes—
102C22⋊D283—28no—
103C22⋊D283—24no—
104C2×Dic283C2×D412yes—
105C4×D283—28no—
106C4×D283—22no—
107C4×D283—22no—
108C283⋊C8—22no—
109C23×C283—21no—
110C23×C283C2×D411yescharacteristic
111Q8×C283—21no—
112Q8×C283C2×D412yes—
113D4×C283—24no—
114D4×C283—21no—
115D4×C283C2×D412yes—
116C2×C4×C283—21no—
117C2×C4×C283—24no—
118C2×C4×C283—22no—
119C2×C4×C283C2×D411yescharacteristic
120C2×C4×C283C2×D412yes—
121C2×C4×C283C2×D412yes—
122C2×C4×C283C2411yescharacteristic
123C8×C283—22no—
124C22×C283—22no—
125C22×C283—22no—
126C22×C283—42no—
127C22×C283#12012yes—
128C22×C283C22×D411yescharacteristic
129C2×D283—22no—
130C2×D283—24no—
131C2×D283—24no—
132C2×D283—44no—
133C2×D283—42no—
134C2×D283—44no—
135C2×D283—84no—
136C2×D283—88no—
137C2×D283—84no—
138C4×C283—22no—
139C4×C283—42no—
140C4×C283#12012yes—
141C4×C283C22×D411yescharacteristic
142C4×C283C22×D411yescharacteristic
143Dic283—22no—
144C2×C283—22no—
145C2×C283—42no—
146C2×C283C2×#12011yescharacteristic
147C2×C283#57612yes—
148D283—42no—
149D283—44no—
150D283—82no—
151C283#300811yescharacteristic, complemented, Sylow
152#3008—2831nomaximal, Sylow
153C2×H(5,𝔽2)—2831no—
154C2×AGL(1,ℤ/8)—2831no—
155C22×D8—2831no—
156#535—2831no—
157#449—2834no—
158#447—2834no—
159#418—2831no—
160#416D28311yescharacteristic, complemented
161#408—2831no—
162H(5,𝔽2)—5664no—
163H(5,𝔽2)—5664no—
164C2×#41—2831no—
165C2×#41—2832no—
166C2×#41—5661no—
167C2×#41C2×D28311yescharacteristic
168C22×D4—2832no—
169C22×D4—2831no—
170C22×D4—2831no—
171C22×D4—5661no—
172C22×D4—5662no—
173AGL(1,ℤ/8)—5664no—
174C2×SD16—2832no—
175C2×D8—2832no—
176C2×D8—2834no—
177C2×D8—5664no—
178C2×M16—2831no—
179C22×C8C2×D28311yescharacteristic, complemented
180#127—5661no—
181#124—5661no—
182#121—2834no—
183#120—5662no—
184#117C2×D28311yescharacteristic, complemented
185#103C2×D28312yescomplemented
186#102—2834no—
187#102C2×D28312yescomplemented
188C22⋊C8—2834no—
189C24—5662no—
190C24—5661no—
191C24—5661no—
192#41—22no—
193#41—5668no—
194#41—5664no—
195#41—5664no—
196#41—5662no—
197#41—5662no—
198#41D113214yescomplemented
199C2×Q8—2831no—
200C2×Q8C22×D28311yescharacteristic
201C2×D4—21no—
202C2×D4—2832no—
203C2×D4—2834no—
204C2×D4—2831no—
205C2×D4—2832no—
206C2×D4—2832no—
207C2×D4—5662no—
208C2×D4—5668no—
209C2×D4—5664no—
210C2×D4—5662no—
211C2×D4—5664no—
212C2×D4—56616no—
213C2×D4—5664no—
214C2×D4—5661no—
215C2×D4—5662no—
216C2×D4—5661no—
217C2×D4—5662no—
218C2×D4—5662no—
219C2×D4C22×D28311yescharacteristic
220C22×C4—21no—
221C22×C4—2832no—
222C22×C4—5662no—
223C22×C4C22×D28311yescharacteristic
224SD16—5664no—
225D8—5668no—
226D8—5664no—
227M16—5662no—
228C2×C8—22no—
229C2×C8—2832no—
230C2×C8C22×D28312yes—
231C4⋊C4C22×D28312yes—
232C22⋊C4—21no—
233C22⋊C4—5664no—
234C42—21no—
235C23—21no—
236C23—2832no—
237C23—5668no—
238C23—5662no—
239C23—5664no—
240C23—5662no—
241C23—11322no—
242C23—22642no—
243C23—22642no—
244C23—22644no—
245C23D4×D28311yescharacteristic
246Q8—21no—
247Q8—5662no—
248Q8C2×D113212yes—
249D4—24no—
250D4—21no—
251D4—2834no—
252D4—5668no—
253D4—5668no—
254D4—5664no—
255D4—5664no—
256D4—5662no—
257D4—5662no—
258D4—5662no—
259D4—5662no—
260D4—11322no—
261D4—22644no—
262D4C2×D113212yes—
263C2×C4—21no—
264C2×C4—24no—
265C2×C4—22no—
266C2×C4—2832no—
267C2×C4—5668no—
268C2×C4—5662no—
269C2×C4—5662no—
270C2×C4D4×D28311yescharacteristic
271C2×C4D4×D28312yes—
272C2×C4C2×D113212yes—
273C2×C4C23×D28311yescharacteristic
274C8—22no—
275C8—5662no—
276C22—22no—
277C22—22no—
278C22—42no—
279C22—5664no—
280C22—5662no—
281C22—5664no—
282C22—11324no—
283C22—11322no—
284C22—11324no—
285C22—22644no—
286C22—22644no—
287C22—22648no—
288C22#2258358012yes—
289C22C2×D4×D28311yescharacteristic
290C4—22no—
291C4—42no—
292C4—5662no—
293C4#2258358012yes—
294C4C2×D4×D28311yescharacteristic
295C4C22×D113211yescharacteristic
296C2—22no—
297C2—42no—
298C2—11322no—
299C2—11324no—
300C2—22642no—
301C2C2×#2258358011yescharacteristic
302C2#3182726712yes—
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#416×C283C2
socleC22×C283C22×D4
hypercenter#416D283
hypocenterC283#3008
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#22583656C2semidirect
2. C2×#22583635C2semidirect
3. C22×D2264C2semidirect
4. #31827034C2semidirect
5. #31826613C2semidirect
6. #31826600C2semidirect
7. #31826469C2semidirect
8. #31826431C2semidirect
9. #416×C283C2semidirect
12. C2×#17985393C22semidirect
17. C22×D1132C22semidirect
19. C2×D4×D283C22semidirect
21. C2×#17985379C22semidirect
23. C2×D2264C22semidirect
24. C2×#17985377C22semidirect
25. C2×#17985376C22semidirect
28. #22583585C22semidirect
30. #22583530C22semidirect
31. (C2×D283)⋊C8C22semidirect
33. C22×C8×C283C22semidirect
34. #117×C283C22semidirect
35. #103×C283C22semidirect
36. #102×C283C22semidirect
55. C2×D1132D4semidirect
72. #41×C283D4semidirect
151. C283#3008semidirect
160. #416D283semidirect
179. C22×C8C2×D283semidirect
184. #117C2×D283semidirect
185. #103C2×D283semidirect
187. #102C2×D283semidirect
198. #41D1132semidirect

Maximal subgroups

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

subgroupindex
C2×#225836562
C2×#225836352
C22×D22642
#318270342
#318266132
#318266002
#318264692
#318264312
#416×C2832
#3008283