Finite Group Databasebeta

← #24916990

Subgroups — #24916990

Order 10624.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14581823C211yesmaximal, characteristic
2#19128275C211yesmaximal, characteristic
3#19128179C211yesmaximal, characteristic, complemented
4#19127925C214yesmaximal, complemented
5#19127909C214yesmaximal, complemented
6#19127727C211yesmaximal, characteristic, complemented
7#19127710C211yesmaximal, characteristic, complemented
8#19127698C211yesmaximal, characteristic
9C22×Q16×C83C211yesmaximal, characteristic, complemented
10C2×#13050590C2211yescharacteristic
11#14581823—24no—
12C2×#13050580C2212yes—
13C2×#13050577C2212yes—
14C2×#13050574C2211yescharacteristic
15#14581790C2212yes—
16#14581781C2212yes—
17#14581775—22no—
18#14581768—22no—
19Q8×Dic83—22no—
20Q8×Dic83C2214yes—
21#14581751—22no—
22#14581751C2211yescharacteristic
23#14581751C2211yescharacteristic
24Dic83⋊C8C2214yes—
25#14581717C2212yes—
26#14581716C2214yes—
27#14581714C2212yes—
28C22×Q8×C83C2211yescharacteristic
29C22×Q8×C83C2211yescharacteristic
30C2×Q16×C83—24no—
31C2×Q16×C83C2214yes—
32C22×C8×C83C2211yescharacteristic
33#13050590—22no—
34#13050590—22no—
35C2×Dic166C2311yescharacteristic
36C2×D332C2311yescharacteristic
37C2×C22⋊D83—21no—
38C2×C4×D83—21no—
39#13050580—24no—
40#13050577—24no—
41#13050574—22no—
42C2×C83⋊C8C2312yes—
43C4⋊Dic83—24no—
44C4⋊Dic83D412yes—
45C4⋊Dic83C2312yes—
46Dic83⋊C4—24no—
47C22⋊Dic83—21no—
48C22⋊Dic83D412yes—
49(C2×D83)⋊C4—24no—
50C4×Dic83—21no—
51C4×Dic83—24no—
52C4×Dic83C2312yes—
53C2×Q8×C83—21no—
54C2×Q8×C83—21no—
55C2×Q8×C83—22no—
56C2×Q8×C83—22no—
57C2×Q8×C83—22no—
58C2×Q8×C83D414yes—
59C2×Q8×C83C2312yes—
60C2×Q8×C83C2312yes—
61C22×C4×C83—21no—
62C22×C4×C83—21no—
63C22×C4×C83C2311yescharacteristic
64Q16×C83—28no—
65C2×C8×C83—22no—
66C2×C8×C83C2312yes—
67C23×C83C2×D411yescharacteristic
68C22×D83—21no—
69Q8×C83—24no—
70Q8×C83—22no—
71Q8×C83—22no—
72Q8×C83—22no—
73Q8×C83—42no—
74Q8×C83C2×D414yes—
75C2×C4×C83—24no—
76C2×C4×C83—22no—
77C2×C4×C83—22no—
78C2×C4×C83—42no—
79C2×C4×C83C2×D411yescharacteristic
80C2×C4×C83C2×D412yes—
81C2×C4×C83#4112yes—
82C2×C4×C83C2411yescharacteristic
83C22⋊D83—42no—
84C2×Dic83—22no—
85C2×Dic83—21no—
86C2×Dic83C2×D414yes—
87D332—21no—
88D332—21no—
89C4×D83—42no—
90Dic166—21no—
91Dic166—21no—
92C8×C83—22no—
93C83⋊C8—22no—
94C22×C83—22no—
95C22×C83#12112yes—
96C22×C83C22×D411yescharacteristic
97C2×D83—41no—
98C2×D83—42no—
99C4×C83—24no—
100C4×C83—42no—
101C4×C83#12112yes—
102C4×C83C22×D411yescharacteristic
103C4×C83C2×#4111yescharacteristic
104Dic83—24no—
105Dic83—41no—
106Dic83—42no—
107C2×C83—22no—
108C2×C83C2×#12111yescharacteristic
109C2×C83#57812yes—
110D83—81no—
111#3044—831nomaximal, Sylow
112C83#304411yescharacteristic, complemented, Sylow
113C2×#137—831no—
114C22×Q16D8311yescharacteristic, complemented
115#536—831no—
116#519—831no—
117#464—834no—
118#462—834no—
119#424—831no—
120#419—831no—
121#416—831no—
122C2×#41—831no—
123C22×Q8C2×D8311yescharacteristic
124C22×Q8C2×D8311yescharacteristic
125#137—1664no—
126C2×Q16—24no—
127C2×Q16—832no—
128C2×Q16C2×D8314yes—
129C2×SD16—832no—
130C2×M16—831no—
131C22×C8C2×D8311yescharacteristic
132C4⋊Q8—832no—
133#124—832no—
134#123—1662no—
135C22⋊Q8—1662no—
136C4×Q8—834no—
137C4×Q8—1662no—
138#117—831no—
139#117—831no—
140#117—1662no—
141C4⋊C8—834no—
142#103—834no—
143#103—832no—
144#102—832no—
145#41—1662no—
146#41—1662no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—22no—
152C2×Q8—831no—
153C2×Q8C22⋊D8314yes—
154C2×Q8C22×D8312yes—
155C2×Q8C22×D8312yes—
156C2×D4—831no—
157C2×D4—1661no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—1661no—
161C22×C4C22×D8311yescharacteristic
162Q16—28no—
163Q16—1664no—
164SD16—1664no—
165M16—1662no—
166C2×C8—22no—
167C2×C8—832no—
168C2×C8C22×D8312yes—
169C4⋊C4—832no—
170C4⋊C4—832no—
171C4⋊C4—1664no—
172C4⋊C4—1664no—
173C22⋊C4—832no—
174C22⋊C4—1664no—
175C22⋊C4—1661no—
176C42—832no—
177C42—1664no—
178C42—1661no—
179C23—1661no—
180C23D4×D8311yescharacteristic
181Q8—24no—
182Q8—22no—
183Q8—22no—
184Q8—22no—
185Q8—42no—
186Q8—1661no—
187Q8—1661no—
188Q8C2×C22⋊D8314yes—
189D4—3322no—
190D4—1661no—
191D4—1661no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—42no—
196C2×C4—3322no—
197C2×C4—834no—
198C2×C4—1662no—
199C2×C4—1661no—
200C2×C4D4×D8311yescharacteristic
201C2×C4C2×C22⋊D8312yes—
202C2×C4#1305058912yes—
203C2×C4C23×D8311yescharacteristic
204C8—22no—
205C8—1662no—
206C22—22no—
207C22—3321no—
208C22—3322no—
209C22#1458176412yes—
210C22C2×D4×D8311yescharacteristic
211C4—24no—
212C4—42no—
213C4—3321no—
214C4—3322no—
215C4—1664no—
216C4#1458176412yes—
217C4C22×C22⋊D8311yescharacteristic
218C4C2×#1305058911yescharacteristic
219C2—22no—
220C2—6641no—
221C2C2×#1458176411yescharacteristic
222C2#1912851412yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC22×Q16×C83C2
socleC22×C83C22×D4
prime radicalC2×#13050590C22
hypercenterC22×Q16D83
hypocenterC83#3044
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
3. #19128179C2semidirect
4. #19127925C2semidirect
5. #19127909C2semidirect
6. #19127727C2semidirect
7. #19127710C2semidirect
9. C22×Q16×C83C2semidirect
112. C83#3044semidirect
114. C22×Q16D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818232
#191282752
#191281792
#191279252
#191279092
#191277272
#191277102
#191276982
C22×Q16×C832
#304483