Finite Group Databasebeta

← #24916994

Subgroups — #24916994

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×#14581825C211yesmaximal, characteristic
2C22×Dic332C211yesmaximal, characteristic, complemented
3#19128277C211yesmaximal, characteristic
4#19128178C211yesmaximal, characteristic, complemented
5#19127924C214yesmaximal, complemented
6#19127912C214yesmaximal, complemented
7#19127711C211yesmaximal, characteristic, complemented
8#19127698C211yesmaximal, characteristic
9#424×C83C211yesmaximal, characteristic, complemented
10C2×#13050590C2211yescharacteristic
11C22×Dic166C2211yescharacteristic
12C22×Dic166C2211yescharacteristic
13#14581825—24no—
14C2×Dic332—24no—
15C2×Dic332C2212yes—
16C2×Dic332C2214yes—
17C2×#13050579C2212yes—
18#14581792C2212yes—
19#14581780C2212yes—
20#14581776—22no—
21#14581772—22no—
22C4×Dic166C2214yes—
23#14581757—22no—
24#14581751C2211yescharacteristic
25#14581748—22no—
26#14581717C2212yes—
27#14581717C2214yes—
28#14581714C2212yes—
29C2×M16×C83C2211yescharacteristic
30C22×C8×C83C2211yescharacteristic
31#117×C83C2211yescharacteristic
32C4⋊C8×C83C2214yes—
33#13050590—22no—
34#13050590—22no—
35C2×Dic166—21no—
36C2×Dic166—22no—
37C2×Dic166—22no—
38C2×Dic166—22no—
39C2×Dic166—21no—
40C2×Dic166D414yes—
41C2×Dic166C2312yes—
42C2×Dic166C2311yescharacteristic
43C2×Dic166C2312yes—
44C2×D332C2311yescharacteristic
45C2×C22⋊D83—21no—
46C22×Dic83—21no—
47C22×Dic83—21no—
48C2×C4×D83—21no—
49Dic332—28no—
50Dic332—24no—
51#13050579—24no—
52C4⋊Dic83—24no—
53C4⋊Dic83C2312yes—
54Dic83⋊C4—24no—
55C22⋊Dic83—21no—
56(C2×D83)⋊C4—24no—
57C4×Dic83—24no—
58C4×Dic83—21no—
59C22×C4×C83C2311yescharacteristic
60M16×C83—22no—
61C2×C8×C83—22no—
62C2×C8×C83C2312yes—
63C2×C8×C83C2312yes—
64C4⋊C4×C83D412yes—
65C22⋊C4×C83D412yes—
66C42×C83C2312yes—
67C23×C83C2×D411yescharacteristic
68C22×D83—21no—
69C2×C4×C83C2×D412yes—
70C2×C4×C83C2×D414yes—
71C2×C4×C83C2×D411yescharacteristic
72C2×C4×C83#4112yes—
73C2×C4×C83C2411yescharacteristic
74C22⋊D83—42no—
75C2×Dic83—24no—
76C2×Dic83—22no—
77C2×Dic83—22no—
78C2×Dic83—22no—
79C2×Dic83—21no—
80C2×Dic83—42no—
81D332—21no—
82D332—21no—
83C4×D83—42no—
84Dic166—24no—
85Dic166—22no—
86Dic166—22no—
87Dic166—21no—
88Dic166—22no—
89Dic166—21no—
90Dic166—42no—
91Dic166C2×D414yes—
92C8×C83—22no—
93C8×C83—22no—
94C22×C83—22no—
95C22×C83#12112yes—
96C22×C83C22×D411yescharacteristic
97C2×D83—41no—
98C2×D83—42no—
99C4×C83—24no—
100C4×C83#12112yes—
101C4×C83C22×D411yescharacteristic
102C4×C83C2×#4111yescharacteristic
103Dic83—24no—
104Dic83—42no—
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×Q16—831no—
115#536—831no—
116#519—831no—
117#464—834no—
118#462—834no—
119#424D8311yescharacteristic, complemented
120#419—831no—
121#416—831no—
122C2×#41—831no—
123C22×Q8—831no—
124C22×Q8—831no—
125#137—1664no—
126C2×Q16—834no—
127C2×Q16—832no—
128C2×Q16—1664no—
129C2×SD16—832no—
130C2×M16C2×D8311yescharacteristic
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—1662no—
140#117C2×D8311yescharacteristic
141C4⋊C8C2×D8314yes—
142#103—832no—
143#103—834no—
144#102—832no—
145#41—1662no—
146#41—1662no—
147C2×Q8—834no—
148C2×Q8—832no—
149C2×Q8—832no—
150C2×Q8—831no—
151C2×Q8—1662no—
152C2×Q8—1662no—
153C2×Q8—1661no—
154C2×Q8—1661no—
155C2×Q8—1662no—
156C2×D4—831no—
157C2×D4—1661no—
158C22×C4—1661no—
159C22×C4—1661no—
160C22×C4—1661no—
161C22×C4C22×D8311yescharacteristic
162Q16—1668no—
163Q16—1664no—
164SD16—1664no—
165M16—22no—
166C2×C8—22no—
167C2×C8C22×D8312yes—
168C2×C8C22×D8312yes—
169C4⋊C4—832no—
170C4⋊C4—1664no—
171C4⋊C4—1664no—
172C4⋊C4D33212yes—
173C22⋊C4—1661no—
174C22⋊C4—1664no—
175C22⋊C4D33212yes—
176C42—1661no—
177C42—1664no—
178C42C22×D8312yes—
179C23—1661no—
180C23C2×D33211yescharacteristic
181Q8—3322no—
182Q8—834no—
183Q8—1662no—
184Q8—1661no—
185Q8—1662no—
186Q8—1661no—
187Q8—1664no—
188Q8—1662no—
189D4—3322no—
190D4—1661no—
191D4—1661no—
192C2×C4—3322no—
193C2×C4—3322no—
194C2×C4—1662no—
195C2×C4—1662no—
196C2×C4—1661no—
197C2×C4—1664no—
198C2×C4—1662no—
199C2×C4D4×D8312yes—
200C2×C4C2×D33214yes—
201C2×C4C2×D33211yescharacteristic
202C2×C4#1305059112yes—
203C2×C4C23×D8311yescharacteristic
204C8—22no—
205C8—22no—
206C22—22no—
207C22—3321no—
208C22—3322no—
209C22#1458176712yes—
210C22C22×D33211yescharacteristic
211C4—24no—
212C4—3321no—
213C4—3322no—
214C4—3322no—
215C4—1664no—
216C4#1458176712yes—
217C4C2×D4×D8311yescharacteristic
218C4C2×#1305059111yescharacteristic
219C2—22no—
220C2—6641no—
221C2C2×#1458176711yescharacteristic
222C2#1912851812yes—
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#424×C83C2
socleC22×C83C22×D4
prime radicalC2×#13050590C22
hypercenter#424D83
hypocenterC83#3044
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
2. C22×Dic332C2semidirect
4. #19128178C2semidirect
5. #19127924C2semidirect
6. #19127912C2semidirect
7. #19127711C2semidirect
9. #424×C83C2semidirect
112. C83#3044semidirect
119. #424D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818252
C22×Dic3322
#191282772
#191281782
#191279242
#191279122
#191277112
#191276982
#424×C832
#304483