Finite Group Databasebeta

← #49130541

Subgroups — #49130541

Order 36224.

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

normal subgroups only

This group has 752 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
2C2×#22583640C211yesmaximal, characteristic, complemented
3C22×#17985381C211yesmaximal, characteristic, complemented
4#31827041C211yesmaximal, characteristic, complemented
5#31826630C214yesmaximal, complemented
6#31826617C214yesmaximal, complemented
7#31826469C211yesmaximal, characteristic, complemented
8#31826431C211yesmaximal, characteristic, complemented
9#416×C283C211yesmaximal, characteristic, complemented
10#22583656—24no—
11#22583656—24no—
12C2×#17985393C2212yes—
13C2×#17985392—21no—
14C2×#17985392C2211yescharacteristic
15C22×Dic566C2211yescharacteristic, complemented
16C22×D1132—21no—
17C22×D1132C2211yescharacteristic
18C2×D4×D283—22no—
19C2×D4×D283C2212yes—
20#22583640—24no—
21C2×#17985382C2212yescomplemented
22C2×#17985381—24no—
23C2×#17985381C2214yescomplemented
24C2×#17985380C2212yescomplemented
25C2×#17985376C2211yescharacteristic, complemented
26#22583610—21no—
27#22583598—21no—
28#22583594—22no—
29#22583590C2214yescomplemented
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—
39#17985393—24no—
40#17985393—24no—
41#17985392—28no—
42#17985392—22no—
43#17985392—22no—
44C2×Dic566—21no—
45C2×Dic566—22no—
46C2×Dic566—22no—
47C2×Dic566D411yescharacteristic
48C2×Dic566C2312yes—
49C2×D1132—28no—
50C2×D1132—22no—
51C2×D1132—21no—
52C2×D1132—22no—
53C2×D1132D411yescharacteristic
54C2×D1132D414yes—
55C2×D1132C2312yes—
56C2×C22⋊D283—22no—
57C2×C22⋊D283D412yes—
58D4×D283—216no—
59D4×D283—24no—
60D4×D283—24no—
61C22×Dic283—21no—
62C2×C4×D283—22no—
63C2×C4×D283C2312yes—
64#17985382—24no—
65#17985381—28no—
66#17985380—24no—
67#17985376—22no—
68C2×C283⋊C8C2312yes—
69C4⋊Dic283—24no—
70(C2×D283)⋊C4—24no—
71#41×C283—22no—
72#41×C283D414yes—
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—84no—
87C22×D283—82no—
88C22×D283C2×D412yes—
89Dic566—22no—
90Dic566—22no—
91Dic566—22no—
92Dic566—42no—
93D1132—28no—
94D1132—22no—
95D1132—24no—
96D1132—22no—
97D1132C2×D414yes—
98C22⋊D283—28no—
99C22⋊D283—24no—
100C2×Dic283—22no—
101C2×Dic283—42no—
102C2×Dic283C2×D412yes—
103C4×D283—28no—
104C4×D283—22no—
105C4×D283—22no—
106C283⋊C8—22no—
107C23×C283—21no—
108C23×C283C2×D411yescharacteristic
109Q8×C283—21no—
110Q8×C283C2×D412yes—
111D4×C283—24no—
112D4×C283—21no—
113D4×C283C2×D412yes—
114C2×C4×C283—21no—
115C2×C4×C283—24no—
116C2×C4×C283—22no—
117C2×C4×C283C2×D411yescharacteristic
118C2×C4×C283C2×D412yes—
119C2×C4×C283C2×D412yes—
120C2×C4×C283C2411yescharacteristic
121C8×C283—22no—
122C22×C283—22no—
123C22×C283—22no—
124C22×C283—42no—
125C22×C283#12012yes—
126C22×C283C22×D411yescharacteristic
127C2×D283—22no—
128C2×D283—24no—
129C2×D283—24no—
130C2×D283—44no—
131C2×D283—84no—
132C2×D283—88no—
133C4×C283—22no—
134C4×C283—42no—
135C4×C283#12012yes—
136C4×C283C22×D411yescharacteristic
137C4×C283C22×D411yescharacteristic
138Dic283—22no—
139Dic283—42no—
140C2×C283—22no—
141C2×C283—42no—
142C2×C283C2×#12011yescharacteristic
143C2×C283#57712yes—
144D283—42no—
145D283—44no—
146C283#300911yescharacteristic, complemented, Sylow
147#3009—2831nomaximal, Sylow
148C2×H(5,𝔽2)—2831no—
149C2×#137—2831no—
150C22×SD16—2831no—
151#536—2831no—
152#452—2834no—
153#450—2834no—
154#418—2831no—
155#416D28311yescharacteristic, complemented
156#408—2831no—
157H(5,𝔽2)—5664no—
158H(5,𝔽2)—5664no—
159C2×#41—2831no—
160C2×#41—2832no—
161C2×#41—5661no—
162C2×#41C2×D28311yescharacteristic
163C22×Q8—2831no—
164C22×D4—2832no—
165C22×D4—2831no—
166C22×D4—5661no—
167C22×D4—5662no—
168#137—5664no—
169C2×Q16—2832no—
170C2×SD16—2832no—
171C2×SD16—2834no—
172C2×SD16—5664no—
173C2×M16—2831no—
174C22×C8C2×D28311yescharacteristic, complemented
175C4⋊Q8—5661no—
176#124—5661no—
177#123—5662no—
178C22⋊Q8—2834no—
179#117C2×D28311yescharacteristic, complemented
180#103C2×D28312yescomplemented
181#102—2834no—
182#102C2×D28312yescomplemented
183C22⋊C8—2834no—
184C24—5662no—
185C24—5661no—
186#41—22no—
187#41—5668no—
188#41—5664no—
189#41—5664no—
190#41—5662no—
191#41—5662no—
192#41D113214yes—
193C2×Q8—2831no—
194C2×Q8—2832no—
195C2×Q8—5661no—
196C2×Q8—5662no—
197C2×Q8—5662no—
198C2×Q8C22×D28311yescharacteristic
199C2×D4—21no—
200C2×D4—2832no—
201C2×D4—2834no—
202C2×D4—2831no—
203C2×D4—2832no—
204C2×D4—5662no—
205C2×D4—5668no—
206C2×D4—5662no—
207C2×D4—5664no—
208C2×D4—56616no—
209C2×D4—5664no—
210C2×D4—5661no—
211C2×D4—5662no—
212C2×D4C22×D28311yescharacteristic
213C22×C4—21no—
214C22×C4—2832no—
215C22×C4—5662no—
216C22×C4—5661no—
217C22×C4C22×D28311yescharacteristic
218Q16—5664no—
219SD16—5668no—
220SD16—5664no—
221M16—5662no—
222C2×C8—22no—
223C2×C8—2832no—
224C2×C8C22×D28312yes—
225C4⋊C4—5664no—
226C4⋊C4C22×D28312yes—
227C22⋊C4—21no—
228C22⋊C4—5664no—
229C42—21no—
230C23—21no—
231C23—2832no—
232C23—5668no—
233C23—5662no—
234C23—5664no—
235C23—22642no—
236C23—22644no—
237C23D4×D28311yescharacteristic
238Q8—21no—
239Q8—5662no—
240Q8—5662no—
241Q8—5662no—
242Q8—11322no—
243Q8C2×D113212yes—
244D4—24no—
245D4—21no—
246D4—2834no—
247D4—5668no—
248D4—5668no—
249D4—5664no—
250D4—5664no—
251D4—5662no—
252D4—5662no—
253D4C2×D113212yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—2832no—
258C2×C4—5668no—
259C2×C4—5662no—
260C2×C4—5662no—
261C2×C4—5662no—
262C2×C4—11322no—
263C2×C4D4×D28311yescharacteristic
264C2×C4D4×D28312yes—
265C2×C4C2×D113212yes—
266C2×C4C23×D28311yescharacteristic
267C8—22no—
268C8—5662no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—5664no—
273C22—5662no—
274C22—5664no—
275C22—11324no—
276C22—22644no—
277C22—22648no—
278C22#2258358012yes—
279C22C2×D4×D28311yescharacteristic
280C4—22no—
281C4—42no—
282C4—5662no—
283C4—11322no—
284C4#2258358012yes—
285C4C2×D4×D28311yescharacteristic
286C4C22×D113211yescharacteristic
287C2—22no—
288C2—42no—
289C2—11322no—
290C2—11324no—
291C2C2×#2258358011yescharacteristic
292C2#3182727212yes—
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#416×C283C2
socleC22×C283C22×D4
prime radicalC2×#22583656C2
hypercenter#416D283
hypocenterC283#3009
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
2. C2×#22583640C2semidirect
3. C22×#17985381C2semidirect
4. #31827041C2semidirect
5. #31826630C2semidirect
6. #31826617C2semidirect
7. #31826469C2semidirect
8. #31826431C2semidirect
9. #416×C283C2semidirect
15. C22×Dic566C22semidirect
21. C2×#17985382C22semidirect
23. C2×#17985381C22semidirect
24. C2×#17985380C22semidirect
25. C2×#17985376C22semidirect
29. #22583590C22semidirect
30. #22583530C22semidirect
31. (C2×D283)⋊C8C22semidirect
33. C22×C8×C283C22semidirect
34. #117×C283C22semidirect
35. #103×C283C22semidirect
36. #102×C283C22semidirect
146. C283#3009semidirect
155. #416D283semidirect
174. C22×C8C2×D283semidirect
179. #117C2×D283semidirect
180. #103C2×D283semidirect
182. #102C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836562
C2×#225836402
C22×#179853812
#318270412
#318266302
#318266172
#318264692
#318264312
#416×C2832
#3009283