Finite Group Databasebeta

← #31824140

Subgroups — #31824140

Order 17984.

248 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 97 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
1#22582874C211yesmaximal, characteristic, complemented
2#22582874C211yesmaximal, characteristic, complemented
3#22582868C211yesmaximal, characteristic, complemented
4#22582864C211yesmaximal, characteristic, complemented
5#22582864C211yesmaximal, characteristic, complemented
6#22582863C211yesmaximal, characteristic, complemented
7#22582855C211yesmaximal, characteristic, complemented
8#22582849C211yesmaximal, characteristic
9#22582841C211yesmaximal, characteristic, complemented
10C4×C22⋊D281C211yesmaximal, characteristic, complemented
11#22582837C211yesmaximal, characteristic, complemented
12#22582835C211yesmaximal, characteristic, complemented
13C2×#17985115C211yesmaximal, characteristic, complemented
14C22⋊C4×D281C211yesmaximal, characteristic, complemented
15C22⋊Q8×C281C211yesmaximal, characteristic, complemented
16C23×D281C2211yescharacteristic
17C2×D1124C2211yescharacteristic
18C2×D1124C2211yescharacteristic
19C2×C22⋊D281C2211yescharacteristic
20C22×Dic281C2211yescharacteristic
21C2×C4×D281C2211yescharacteristic
22C2×C4×D281C2211yescharacteristic
23C2×C4×D281C2211yescharacteristic, complemented
24C4⋊Dic281C2211yescharacteristic, complemented
25C4⋊Dic281C2211yescharacteristic, complemented
26#17985117C2211yescharacteristic, complemented
27#17985117C2211yescharacteristic, complemented
28#17985117C2211yescharacteristic, complemented
29C22⋊Dic281C2211yescharacteristic
30#17985115C2211yescharacteristic
31#17985115C2211yescharacteristic
32#17985115C2211yescharacteristic
33#17985115C2211yescharacteristic
34#17985115C2211yescharacteristic
35#17985115C2211yescharacteristic
36#17985115C2211yescharacteristic
37#17985115C2211yescharacteristic
38#17985115C2211yescharacteristic, complemented
39#17985115C2211yescharacteristic, complemented
40#17985115C2211yescharacteristic, complemented
41C4×Dic281C2211yescharacteristic, complemented
42C4×Dic281C2211yescharacteristic, complemented
43C4×Dic281C2211yescharacteristic, complemented
44C2×Q8×C281C2211yescharacteristic, complemented
45C22×C4×C281C2211yescharacteristic
46C4⋊C4×C281C2211yescharacteristic, complemented
47C4⋊C4×C281C2211yescharacteristic, complemented
48C4⋊C4×C281C2211yescharacteristic, complemented
49C22⋊C4×C281C2211yescharacteristic
50C22⋊C4×C281C2211yescharacteristic
51C22×D281—22no—
52C22×D281—22no—
53C22×D281—41no—
54C22×D281C2311yescharacteristic
55C22×D281C2311yescharacteristic
56C22×D281C2311yescharacteristic
57D1124—22no—
58D1124—41no—
59C22⋊D281—22no—
60C2×Dic281—22no—
61C2×Dic281C2311yescharacteristic
62C2×Dic281C2311yescharacteristic
63C2×Dic281C2311yescharacteristic
64C2×Dic281C2311yescharacteristic
65C2×Dic281C2311yescharacteristic
66C4×D281—22no—
67C4×D281—41no—
68C4×D281—41no—
69C23×C281C2311yescharacteristic
70Q8×C281—41no—
71C2×C4×C281—41no—
72C2×C4×C281C2311yescharacteristic
73C2×C4×C281C2311yescharacteristic
74C2×C4×C281C2311yescharacteristic
75C2×C4×C281C2311yescharacteristic
76C2×C4×C281C2311yescharacteristic
77C2×C4×C281C2311yescharacteristic
78C22×C281—21no—
79C22×C281—21no—
80C22×C281#4112yes—
81C22×C281C2411yescharacteristic
82C2×D281—21no—
83C2×D281—21no—
84C2×D281—21no—
85C2×D281—21no—
86C2×D281—21no—
87C2×D281—21no—
88C2×D281—21no—
89C2×D281—21no—
90C2×D281—44no—
91C2×D281#4112yes—
92C4×C281—21no—
93C4×C281—21no—
94C4×C281—21no—
95C4×C281—21no—
96C4×C281—21no—
97C4×C281—21no—
98Dic281—21no—
99Dic281—21no—
100Dic281—21no—
101Dic281—21no—
102Dic281—21no—
103C2×C281—22no—
104C2×C281C2×#4111yescharacteristic
105C2×C281C2×#4111yescharacteristic
106C2×C281H(5,𝔽2)11yescharacteristic
107D281—22no—
108D281—41no—
109D281—41no—
110C281#55111yescharacteristic, complemented, Sylow
111#551—2811nomaximal, Sylow
112#126—2811no—
113#126—2811no—
114#124—2811no—
115#123—2811no—
116#123—2811no—
117#123—2811no—
118C22⋊Q8—2811no—
119C22⋊Q8D28111yescharacteristic, complemented
120#120—2811no—
121C4×D4—2811no—
122C4×D4—2811no—
123#117—2811no—
124#117—2811no—
125C2×C22⋊C4—2811no—
126C2×C22⋊C4—2811no—
127C24—2811no—
128C2×Q8C2×D28111yescharacteristic, complemented
129C2×D4—2811no—
130C2×D4—2811no—
131C2×D4—2811no—
132C22×C4—2811no—
133C22×C4—2811no—
134C22×C4—2811no—
135C22×C4—2811no—
136C22×C4C2×D28111yescharacteristic
137C4⋊C4—2811no—
138C4⋊C4—2811no—
139C4⋊C4—2811no—
140C4⋊C4—2811no—
141C4⋊C4—2811no—
142C4⋊C4C2×D28111yescharacteristic, complemented
143C4⋊C4C2×D28111yescharacteristic, complemented
144C4⋊C4C2×D28111yescharacteristic, complemented
145C22⋊C4—2811no—
146C22⋊C4—2811no—
147C22⋊C4—2811no—
148C22⋊C4—2811no—
149C22⋊C4—2811no—
150C22⋊C4—2811no—
151C22⋊C4—2811no—
152C22⋊C4—2811no—
153C22⋊C4—2811no—
154C22⋊C4—2811no—
155C22⋊C4—2811no—
156C22⋊C4—2811no—
157C22⋊C4C2×D28111yescharacteristic
158C22⋊C4C2×D28111yescharacteristic
159C42—2811no—
160C42—2811no—
161C42—2811no—
162C23—2811no—
163C23—2811no—
164C23—2811no—
165C23—5622no—
166C23—5622no—
167C23—11241no—
168C23C22×D28111yescharacteristic
169Q8—41no—
170D4—5622no—
171D4—5622no—
172D4—11241no—
173C2×C4—41no—
174C2×C4—2811no—
175C2×C4—2811no—
176C2×C4—2811no—
177C2×C4—2811no—
178C2×C4—2811no—
179C2×C4—5622no—
180C2×C4—5622no—
181C2×C4—11241no—
182C2×C4—11241no—
183C2×C4C22×D28111yescharacteristic
184C2×C4C22×D28111yescharacteristic
185C2×C4C22×D28111yescharacteristic
186C2×C4C22×D28111yescharacteristic
187C2×C4C22×D28111yescharacteristic
188C2×C4C22×D28111yescharacteristic
189C22—21no—
190C22—21no—
191C22—2812no—
192C22—5621no—
193C22—5621no—
194C22—5621no—
195C22—5621no—
196C22—5621no—
197C22—5621no—
198C22—5621no—
199C22—5621no—
200C22—11244no—
201C22#1798513912yes—
202C22C23×D28111yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—5621no—
210C4—5621no—
211C4—5621no—
212C4—5621no—
213C4—5621no—
214C2—22no—
215C2—5622no—
216C2—11241no—
217C2—11241no—
218C2#41×D28111yescharacteristic
219C2C2×#1798513911yescharacteristic
220C2#2258292411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D281
commutator, socle, cosocleC22×C281C24
FittingC22⋊Q8×C281C2
prime radical#22582849C2
hypercenterC22⋊Q8D281
hypocenterC281#551
LCSG ▷ C22×C281 ▷ C281
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #22582874C2semidirect
2. #22582874C2semidirect
3. #22582868C2semidirect
4. #22582864C2semidirect
5. #22582864C2semidirect
6. #22582863C2semidirect
7. #22582855C2semidirect
9. #22582841C2semidirect
10. C4×C22⋊D281C2semidirect
11. #22582837C2semidirect
12. #22582835C2semidirect
13. C2×#17985115C2semidirect
14. C22⋊C4×D281C2semidirect
15. C22⋊Q8×C281C2semidirect
23. C2×C4×D281C22semidirect
24. C4⋊Dic281C22semidirect
25. C4⋊Dic281C22semidirect
26. #17985117C22semidirect
27. #17985117C22semidirect
28. #17985117C22semidirect
38. #17985115C22semidirect
39. #17985115C22semidirect
40. #17985115C22semidirect
41. C4×Dic281C22semidirect
42. C4×Dic281C22semidirect
43. C4×Dic281C22semidirect
44. C2×Q8×C281C22semidirect
46. C4⋊C4×C281C22semidirect
47. C4⋊C4×C281C22semidirect
48. C4⋊C4×C281C22semidirect
110. C281#551semidirect
119. C22⋊Q8D281semidirect
128. C2×Q8C2×D281semidirect
142. C4⋊C4C2×D281semidirect
143. C4⋊C4C2×D281semidirect
144. C4⋊C4C2×D281semidirect

Maximal subgroups

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

subgroupindex
#225828742
#225828682
#225828642
#225828632
#225828552
#225828492
#225828412
C4×C22⋊D2812
#225828372
#225828352
C2×#179851152
C22⋊C4×D2812
C22⋊Q8×C2812
#551281