Finite Group Databasebeta

← #24906642

Subgroups — #24906642

Order 10624.

308 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 65 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#19128047C211yesmaximal, characteristic, complemented
2#19127981C211yesmaximal, characteristic, complemented
3#19127906C211yesmaximal, characteristic, complemented
4#19127445C211yesmaximal, characteristic, complemented
5D4⋊(C83⋊C8)C211yesmaximal, characteristic, complemented
6D4⋊(C83⋊C8)C211yesmaximal, characteristic, complemented
7D42×C83C211yesmaximal, characteristic
8C2×#13050578—21no—
9#14581792C2211yescharacteristic, complemented
10#14581725—21no—
11C4⋊(C83⋊C8)C2211yescharacteristic, complemented
12C4⋊(C83⋊C8)C2211yescharacteristic, complemented
13#14581715—21no—
14#14581713—21no—
15#14581713—22no—
16C4×C83⋊C8C2211yescharacteristic, complemented
17C22×D4×C83—21no—
18C22×D4×C83—21no—
19#127×C83C2211yescharacteristic
20#121×C83—21no—
21#121×C83—21no—
22#120×C83—22no—
23C4×D4×C83C2211yescharacteristic
24C4×D4×C83C2211yescharacteristic
25C2×Dic166—21no—
26#13050578—42no—
27C2×C83⋊C8—21no—
28C2×C83⋊C8—21no—
29C2×C83⋊C8—21no—
30C4⋊Dic83—21no—
31C4⋊Dic83—21no—
32C24×C83—22no—
33C2×D4×C83—22no—
34C2×D4×C83—22no—
35C2×D4×C83—21no—
36C2×D4×C83—21no—
37C2×D4×C83—42no—
38C2×D4×C83—42no—
39C2×D4×C83—42no—
40C2×D4×C83—42no—
41C2×D4×C83D411yescharacteristic
42C2×D4×C83D411yescharacteristic
43C2×D4×C83D412yes—
44C22×C4×C83—21no—
45C22×C4×C83—21no—
46C4⋊C4×C83D411yescharacteristic
47C4⋊C4×C83D411yescharacteristic
48C22⋊C4×C83—22no—
49C22⋊C4×C83—21no—
50C22⋊C4×C83—21no—
51C42×C83C2311yescharacteristic
52C23×C83—21no—
53C23×C83—21no—
54C23×C83—22no—
55C23×C83—44no—
56C23×C83—44no—
57C23×C83—82no—
58D4×C83—21no—
59D4×C83—21no—
60D4×C83—42no—
61D4×C83—44no—
62D4×C83—44no—
63D4×C83—41no—
64D4×C83—41no—
65D4×C83—41no—
66D4×C83—41no—
67D4×C83D812yes—
68D4×C83SD1612yescomplemented
69C2×C4×C83—21no—
70C2×C4×C83—21no—
71C2×C4×C83—41no—
72C2×C4×C83—41no—
73C2×C4×C83—41no—
74C2×C4×C83—41no—
75C2×C4×C83C2×D411yescharacteristic
76C2×C4×C83C2×D411yescharacteristic
77C2×C4×C83C2×D411yescharacteristic
78C2×Dic83—41no—
79Dic166—41no—
80Dic166—41no—
81C83⋊C8—41no—
82C83⋊C8—41no—
83C83⋊C8—41no—
84C22×C83—22no—
85C22×C83—22no—
86C22×C83—41no—
87C22×C83—41no—
88C22×C83—41no—
89C22×C83—41no—
90C22×C83—42no—
91C22×C83—42no—
92C22×C83—42no—
93C22×C83—88no—
94C22×C83#12011yescharacteristic
95C4×C83—21no—
96C4×C83—41no—
97C4×C83—41no—
98C4×C83C2×D811yescharacteristic
99C4×C83C2×SD1611yescharacteristic
100C4×C83AGL(1,ℤ/8)11yescharacteristic
101C4×C83AGL(1,ℤ/8)11yescharacteristic
102Dic83—81no—
103C2×C83—42no—
104C2×C83—42no—
105C2×C83—82no—
106C2×C83#44711yescharacteristic
107C2×C83#45011yescharacteristic
108C2×C83#45511yescharacteristic
109#1639—831nomaximal, Sylow
110C83#163911yescharacteristic, complemented, Sylow
111D42D8311yescharacteristic
112#486—831no—
113#474—831no—
114#461—831no—
115#331—831no—
116D4⋊C8—831no—
117D4⋊C8—831no—
118C22×D4—21no—
119C22×D4—21no—
120C2×SD16—1661no—
121C4⋊Q8—831no—
122#127C2×D8311yescharacteristic
123#121—21no—
124#121—21no—
125#120—22no—
126C4×D4C2×D8311yescharacteristic
127C4×D4C2×D8311yescharacteristic
128#107—1661no—
129C4⋊C8—831no—
130C4⋊C8—831no—
131#103—1661no—
132#102—1661no—
133#102—1662no—
134C4×C8—831no—
135C24—22no—
136C2×Q8—1661no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—42no—
144C2×D4—42no—
145C2×D4C22⋊D8311yescharacteristic
146C2×D4C22⋊D8311yescharacteristic
147C2×D4C22⋊D8312yes—
148C22×C4—21no—
149C22×C4—21no—
150SD16—3322no—
151C2×C8—1661no—
152C2×C8—1661no—
153C2×C8—1661no—
154C4⋊C4—1661no—
155C4⋊C4—1661no—
156C4⋊C4C22⋊D8311yescharacteristic
157C4⋊C4C22⋊D8311yescharacteristic
158C22⋊C4—22no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C42C22×D8311yescharacteristic
162C23—21no—
163C23—21no—
164C23—22no—
165C23—44no—
166C23—44no—
167C23—82no—
168Q8—3321no—
169Q8—3321no—
170D4—21no—
171D4—21no—
172D4—42no—
173D4—44no—
174D4—44no—
175D4—41no—
176D4—41no—
177D4—41no—
178D4—41no—
179D4#1305057512yes—
180D4#1305057812yescomplemented
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—3321no—
188C2×C4C2×C22⋊D8311yescharacteristic
189C2×C4C2×C22⋊D8311yescharacteristic
190C2×C4C2×C22⋊D8311yescharacteristic
191C8—3321no—
192C8—3321no—
193C8—3321no—
194C22—22no—
195C22—22no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22—41no—
200C22—42no—
201C22—42no—
202C22—42no—
203C22—88no—
204C22#1458176111yescharacteristic
205C4—21no—
206C4—41no—
207C4—41no—
208C4—6641no—
209C4C2×#1305057511yescharacteristic
210C4C2×#1305057811yescharacteristic
211C4#1458181811yescharacteristic
212C4#1458181811yescharacteristic
213C2—42no—
214C2—42no—
215C2—82no—
216C2#1912783511yescharacteristic
217C2#1912785211yescharacteristic
218C2#1912787911yescharacteristic
subgroupG/H
centerC22#14581761
commutator, cosocleC42×C83C23
FrattiniC42C22×D83
Fitting, prime radicalD42×C83C2
socleC22×C83#120
hypercenterD42D83
hypocenterC83#1639
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #19128047C2semidirect
2. #19127981C2semidirect
3. #19127906C2semidirect
4. #19127445C2semidirect
5. D4⋊(C83⋊C8)C2semidirect
6. D4⋊(C83⋊C8)C2semidirect
9. #14581792C22semidirect
11. C4⋊(C83⋊C8)C22semidirect
12. C4⋊(C83⋊C8)C22semidirect
16. C4×C83⋊C8C22semidirect
68. D4×C83SD16semidirect
110. C83#1639semidirect
180. D4#13050578semidirect

Maximal subgroups

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

subgroupindex
#191280472
#191279812
#191279062
#191274452
D4⋊(C83⋊C8)2
D42×C832
#163983