Finite Group Databasebeta

← #49130531

Subgroups — #49130531

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×#22583638C211yesmaximal, characteristic, complemented
3C22×D2264C211yesmaximal, characteristic, complemented
4#31827029C211yesmaximal, characteristic, complemented
5#31826612C214yesmaximal, complemented
6#31826601C214yesmaximal, complemented
7#31826471C211yesmaximal, characteristic, complemented
8#31826462C211yesmaximal, characteristic, complemented
9#408×C283C211yesmaximal, characteristic, complemented
10#22583656—24no—
11#22583656—24no—
12C2×#17985393—21no—
13C2×#17985392C2211yescharacteristic
14C2×#17985392C2212yescomplemented
15C22×D1132C2211yescharacteristic
16C22×D1132C2211yescharacteristic, complemented
17C22×D1132C2212yescomplemented
18C2×D4×D283—22no—
19C2×D4×D283—21no—
20#22583638—24no—
21C2×#17985381C2212yescomplemented
22C2×D2264—24no—
23C2×D2264C2214yescomplemented
24C2×D2264C2212yescomplemented
25#22583606—21no—
26#22583599—21no—
27#22583584C2214yescomplemented
28#22583578—22no—
29#22583569C2211yescharacteristic, complemented
30#22583535C2212yescomplemented
31#22583532C2212yescomplemented
32#22583532C2214yescomplemented
33C2×#41×C283C2211yescharacteristic
34C2×M16×C283C2211yescharacteristic, complemented
35C22×C8×C283C2211yescharacteristic, complemented
36C22⋊C8×C283C2214yescomplemented
37C23×D283—21no—
38C23×D283—22no—
39C23×D283—21no—
40#17985393—28no—
41#17985392—24no—
42#17985392—24no—
43#17985392—22no—
44#17985392D414yescomplemented
45C2×Dic566C2311yescharacteristic
46C2×D1132—24no—
47C2×D1132—24no—
48C2×D1132—21no—
49C2×D1132—22no—
50C2×D1132—22no—
51C2×D1132—22no—
52C2×D1132—22no—
53C2×D1132—21no—
54C2×D1132D414yescomplemented
55C2×D1132C2312yes—
56C2×D1132C2312yes—
57C2×D1132C2311yescharacteristic
58C2×C22⋊D283—21no—
59C2×C22⋊D283—22no—
60C2×C22⋊D283—24no—
61D4×D283—216no—
62D4×D283—28no—
63C2×C4×D283—21no—
64C2×C4×D283—22no—
65#17985381—24no—
66D2264—28no—
67D2264—24no—
68C4⋊Dic283C2312yes—
69C22⋊Dic283—21no—
70(C2×D283)⋊C4—24no—
71C4×Dic283—21no—
72#41×C283—22no—
73#41×C283—22no—
74C2×Q8×C283D411yescharacteristic
75C2×D4×C283D412yes—
76C2×D4×C283D411yescharacteristic
77C22×C4×C283C2312yes—
78C22×C4×C283C2311yescharacteristic
79M16×C283—22no—
80C2×C8×C283—22no—
81C2×C8×C283C2312yes—
82C2×C8×C283C2312yes—
83C22×D283—21no—
84C22×D283—22no—
85C22×D283—28no—
86C22×D283—24no—
87C22×D283—22no—
88C22×D283—42no—
89C22×D283—82no—
90C22×D283—84no—
91C22×D283—82no—
92Dic566—21no—
93Dic566C2×D412yes—
94D1132—28no—
95D1132—24no—
96D1132—22no—
97D1132—21no—
98D1132—22no—
99D1132—22no—
100D1132—42no—
101D1132C2×D412yes—
102D1132C2×D414yes—
103C22⋊D283—28no—
104C22⋊D283—24no—
105C22⋊D283—84no—
106C2×Dic283—21no—
107C2×Dic283—22no—
108C4×D283—28no—
109C4×D283—24no—
110C23×C283C2×D412yes—
111C23×C283C2×D411yescharacteristic
112Q8×C283—22no—
113D4×C283—24no—
114D4×C283—22no—
115C2×C4×C283—22no—
116C2×C4×C283—22no—
117C2×C4×C283C2×D412yes—
118C2×C4×C283C2×D412yes—
119C2×C4×C283C2×D412yes—
120C2×C4×C283C2×D411yescharacteristic
121C2×C4×C283C2411yescharacteristic
122C8×C283—22no—
123C8×C283—22no—
124C22×C283—22no—
125C22×C283—22no—
126C22×C283—24no—
127C22×C283#12012yes—
128C22×C283C22×D411yescharacteristic
129C2×D283—24no—
130C2×D283—22no—
131C2×D283—44no—
132C2×D283—42no—
133C2×D283—42no—
134C2×D283—44no—
135C2×D283—84no—
136C2×D283—88no—
137C2×D283—84no—
138C4×C283—22no—
139C4×C283#12012yes—
140C4×C283C22×D411yescharacteristic
141C4×C283C22×D411yescharacteristic
142Dic283—22no—
143Dic283—42no—
144C2×C283—22no—
145C2×C283—42no—
146C2×C283C2×#12011yescharacteristic
147C2×C283#57612yes—
148D283—44no—
149D283—42no—
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#416—2831no—
161#408D28311yescharacteristic, complemented
162H(5,𝔽2)—5664no—
163H(5,𝔽2)—5664no—
164C2×#41—2831no—
165C2×#41—2832no—
166C2×#41—5661no—
167C2×#41C2×D28311yescharacteristic
168C22×D4—2831no—
169C22×D4—2831no—
170C22×D4—2832no—
171C22×D4—5662no—
172C22×D4—5661no—
173AGL(1,ℤ/8)—5664no—
174C2×SD16—2832no—
175C2×D8—2834no—
176C2×D8—2832no—
177C2×D8—5664no—
178C2×M16C2×D28311yescharacteristic, complemented
179C22×C8C2×D28311yescharacteristic, complemented
180#127—5661no—
181#124—5661no—
182#121—2834no—
183#120—5662no—
184#117—2831no—
185#103—2832no—
186#102—2832no—
187#102—2834no—
188C22⋊C8C2×D28314yescomplemented
189C24—5662no—
190C24—5661no—
191C24—5661no—
192#41—22no—
193#41—22no—
194#41—2834no—
195#41—5664no—
196#41—5664no—
197#41—5668no—
198#41—5662no—
199C2×Q8—2831no—
200C2×Q8D113211yescharacteristic
201C2×D4—2834no—
202C2×D4—2832no—
203C2×D4—2831no—
204C2×D4—2832no—
205C2×D4—5664no—
206C2×D4—5664no—
207C2×D4—56616no—
208C2×D4—5668no—
209C2×D4—5662no—
210C2×D4—5661no—
211C2×D4—5664no—
212C2×D4—5662no—
213C2×D4—5662no—
214C2×D4—5662no—
215C2×D4—5662no—
216C2×D4—5661no—
217C2×D4—5661no—
218C2×D4D113212yes—
219C2×D4D113211yescharacteristic
220C22×C4—5662no—
221C22×C4—5661no—
222C22×C4C22×D28312yes—
223C22×C4C22×D28311yescharacteristic
224SD16—5664no—
225D8—5668no—
226D8—5664no—
227M16—22no—
228C2×C8—22no—
229C2×C8C22×D28312yes—
230C2×C8C22×D28312yes—
231C4⋊C4—2832no—
232C22⋊C4—5661no—
233C22⋊C4—5664no—
234C42—5661no—
235C23—5668no—
236C23—5662no—
237C23—5661no—
238C23—5664no—
239C23—5662no—
240C23—11322no—
241C23—22642no—
242C23—22642no—
243C23—22644no—
244C23C2×D113212yes—
245C23C2×D113211yescharacteristic
246Q8—22no—
247Q8—2832no—
248Q8—5661no—
249D4—24no—
250D4—22no—
251D4—2834no—
252D4—2832no—
253D4—5668no—
254D4—5668no—
255D4—5664no—
256D4—5664no—
257D4—5661no—
258D4—5662no—
259D4—5662no—
260D4—5662no—
261D4—11322no—
262D4—22644no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—5668no—
266C2×C4—5661no—
267C2×C4—5664no—
268C2×C4—5662no—
269C2×C4D4×D28312yes—
270C2×C4D4×D28312yes—
271C2×C4C2×D113212yes—
272C2×C4C2×D113211yescharacteristic
273C2×C4C23×D28311yescharacteristic
274C8—22no—
275C8—22no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—5664no—
280C22—5662no—
281C22—11324no—
282C22—11322no—
283C22—11322no—
284C22—11324no—
285C22—22644no—
286C22—22644no—
287C22—22648no—
288C22#2258358012yes—
289C22C22×D113211yescharacteristic
290C4—22no—
291C4—5662no—
292C4—11322no—
293C4#2258358012yes—
294C4C2×D4×D28311yescharacteristic
295C4C2×D4×D28311yescharacteristic
296C2—22no—
297C2—42no—
298C2—11324no—
299C2—11322no—
300C2—22642no—
301C2C2×#2258358011yescharacteristic
302C2#3182727012yes—
subgroupG/H
centerC22C22×D1132
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#408×C283C2
socleC22×C283C22×D4
hypercenter#408D283
hypocenterC283#3008
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. C2×#22583656C2semidirect
2. C2×#22583638C2semidirect
3. C22×D2264C2semidirect
4. #31827029C2semidirect
5. #31826612C2semidirect
6. #31826601C2semidirect
7. #31826471C2semidirect
8. #31826462C2semidirect
9. #408×C283C2semidirect
14. C2×#17985392C22semidirect
16. C22×D1132C22semidirect
17. C22×D1132C22semidirect
21. C2×#17985381C22semidirect
23. C2×D2264C22semidirect
24. C2×D2264C22semidirect
27. #22583584C22semidirect
29. #22583569C22semidirect
30. #22583535C22semidirect
31. #22583532C22semidirect
32. #22583532C22semidirect
34. C2×M16×C283C22semidirect
35. C22×C8×C283C22semidirect
36. C22⋊C8×C283C22semidirect
44. #17985392D4semidirect
54. C2×D1132D4semidirect
151. C283#3008semidirect
161. #408D283semidirect
178. C2×M16C2×D283semidirect
179. C22×C8C2×D283semidirect
188. C22⋊C8C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836562
C2×#225836382
C22×D22642
#318270292
#318266122
#318266012
#318264712
#318264622
#408×C2832
#3008283