Finite Group Databasebeta

← #24918693

Subgroups — #24918693

Order 10624.

362 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 125 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#19128235C211yesmaximal, characteristic
2#19128107C212yesmaximal, complemented
3#19128063C212yesmaximal, complemented
4#19128036C212yesmaximal, complemented
5#19128019C212yesmaximal
6#19127983C212yesmaximal, complemented
7#19127978C212yesmaximal, complemented
8#19127758C211yesmaximal, characteristic, complemented
9C2×C4⋊Q8×C83C211yesmaximal, characteristic, complemented
10#14581786—21no—
11#14581784C2211yescharacteristic
12#14581780C2211yescharacteristic
13#14581775—21no—
14#14581771C2211yescharacteristic
15#14581766C2211yescharacteristic
16C4×Dic166C2211yescharacteristic
17C4×D332C2211yescharacteristic
18C4×C22⋊D83—21no—
19#14581749—21no—
20#14581725C2214yes—
21#14581723C2214yes—
22C4⋊(C83⋊C8)C2212yes—
23#14581715C2214yes—
24#14581712C2214yes—
25C22⋊(C83⋊C8)C2212yes—
26#14581703C2212yes—
27C22×Q8×C83—21no—
28C4⋊Q8×C83—21no—
29C4⋊Q8×C83—21no—
30C4⋊Q8×C83C2212yes—
31C4⋊Q8×C83C2212yes—
32C2×C4⋊C4×C83—21no—
33C2×C4⋊C4×C83C2212yes—
34C2×C42×C83C2211yescharacteristic
35C2×Dic166C2311yescharacteristic
36C2×D332C2311yescharacteristic
37C2×C22⋊D83—21no—
38C2×C4×D83—21no—
39C2×C83⋊C8C2314yes—
40C4⋊Dic83C2311yescharacteristic
41C4⋊Dic83C2311yescharacteristic
42Dic83⋊C4—21no—
43Dic83⋊C4—21no—
44Dic83⋊C4—21no—
45Dic83⋊C4—21no—
46C22⋊Dic83—21no—
47(C2×D83)⋊C4—21no—
48(C2×D83)⋊C4—21no—
49(C2×D83)⋊C4—21no—
50(C2×D83)⋊C4—21no—
51C4×Dic83—21no—
52C2×Q8×C83—22no—
53C2×Q8×C83—22no—
54C2×Q8×C83—44no—
55C22×C4×C83—22no—
56C22×C4×C83D411yescharacteristic
57C22×C4×C83D411yescharacteristic
58C22×C4×C83C2311yescharacteristic
59C4⋊C4×C83—22no—
60C4⋊C4×C83—22no—
61C4⋊C4×C83—22no—
62C4⋊C4×C83C2314yes—
63C42×C83D411yescharacteristic
64C42×C83D411yescharacteristic
65C42×C83C2311yescharacteristic
66C42×C83C2311yescharacteristic
67C23×C83C2×D411yescharacteristic
68C22×D83—21no—
69Q8×C83—44no—
70Q8×C83—44no—
71C2×C4×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83—22no—
74C2×C4×C83—24no—
75C2×C4×C83—44no—
76C2×C4×C83C2×D411yescharacteristic
77C2×C4×C83C2×D411yescharacteristic
78C2×C4×C83C2×D411yescharacteristic
79C2×C4×C83C2×D411yescharacteristic
80C2×C4×C83C2×D411yescharacteristic
81C2×C4×C83C2411yescharacteristic
82C22⋊D83—42no—
83C2×Dic83—21no—
84C2×Dic83—21no—
85C2×Dic83—21no—
86D332—22no—
87C4×D83—42no—
88Dic166—22no—
89C83⋊C8—24no—
90C22×C83—21no—
91C22×C83—21no—
92C22×C83#13712yes—
93C22×C83C22×D411yescharacteristic
94C2×D83—41no—
95C2×D83—41no—
96C2×D83—41no—
97C4×C83—21no—
98C4×C83—21no—
99C4×C83—21no—
100C4×C83—21no—
101C4×C83—44no—
102C4×C83AGL(1,ℤ/8)12yes—
103C4×C832−1+412yes—
104Dic83—41no—
105Dic83—41no—
106Dic83—41no—
107C2×C83—22no—
108C2×C83#53611yescharacteristic
109C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
110C2×C83C2×#13711yescharacteristic
111D83—81no—
112#3239—831nomaximal, Sylow
113C83#323911yescharacteristic, complemented, Sylow
114C2×C4⋊Q8D8311yescharacteristic, complemented
115#529—831no—
116#501—832no—
117#489—832no—
118#484—832no—
119#481—832no—
120#475—832no—
121#474—832no—
122#432—831no—
123C22×Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8C2×D8312yes—
127C4⋊Q8C2×D8312yes—
128#126—1661no—
129#125—831no—
130#124—831no—
131#123—1661no—
132C22⋊Q8—831no—
133#121—831no—
134C4×Q8—831no—
135C4×D4—831no—
136C4×D4—1661no—
137#117—1661no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4C2×D8312yes—
140C2×C42C2×D8311yescharacteristic
141#107—834no—
142#106—834no—
143C4⋊C8—832no—
144#103—834no—
145#102—834no—
146C22⋊C8—832no—
147#97—832no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—44no—
151C2×Q8—831no—
152C2×D4—831no—
153C2×D4—1661no—
154C22×C4—22no—
155C22×C4—1661no—
156C22×C4C22⋊D8311yescharacteristic
157C22×C4C22⋊D8311yescharacteristic
158C22×C4C22×D8311yescharacteristic
159C2×C8—834no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—831no—
164C4⋊C4—831no—
165C4⋊C4—1661no—
166C4⋊C4—1661no—
167C4⋊C4—1661no—
168C4⋊C4—1661no—
169C4⋊C4C22×D8314yes—
170C22⋊C4—1661no—
171C22⋊C4—1661no—
172C22⋊C4—1661no—
173C22⋊C4—1661no—
174C22⋊C4—1661no—
175C42—1661no—
176C42C22⋊D8311yescharacteristic
177C42C22⋊D8311yescharacteristic
178C42C22×D8311yescharacteristic
179C42C22×D8311yescharacteristic
180C23—1661no—
181C23C2×C22⋊D8311yescharacteristic
182Q8—44no—
183Q8—44no—
184Q8—1662no—
185D4—3322no—
186D4—1662no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—44no—
192C2×C4—3322no—
193C2×C4—1661no—
194C2×C4—1661no—
195C2×C4—1661no—
196C2×C4C2×C22⋊D8311yescharacteristic
197C2×C4C2×C22⋊D8311yescharacteristic
198C2×C4C2×C22⋊D8311yescharacteristic
199C2×C4C2×C22⋊D8311yescharacteristic
200C2×C4C2×C22⋊D8311yescharacteristic
201C2×C4C23×D8311yescharacteristic
202C8—1664no—
203C22—21no—
204C22—21no—
205C22—3321no—
206C22—3321no—
207C22—3321no—
208C22#1458182312yes—
209C22C22×C22⋊D8311yescharacteristic
210C4—21no—
211C4—21no—
212C4—21no—
213C4—21no—
214C4—44no—
215C4—3321no—
216C4—3321no—
217C4—3321no—
218C4#1458181812yes—
219C4#1458184012yes—
220C2—22no—
221C2—6641no—
222C2#1912827511yescharacteristic
223C2C2×#1458181811yescharacteristic
224C2C2×#1458182311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×C4⋊Q8×C83C2
socleC22×C83C22×D4
prime radical#14581766C22
hypercenterC2×C4⋊Q8D83
hypocenterC83#3239
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #19128107C2semidirect
3. #19128063C2semidirect
4. #19128036C2semidirect
6. #19127983C2semidirect
7. #19127978C2semidirect
8. #19127758C2semidirect
9. C2×C4⋊Q8×C83C2semidirect
113. C83#3239semidirect
114. C2×C4⋊Q8D83semidirect

Maximal subgroups

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

subgroupindex
#191282352
#191281072
#191280632
#191280362
#191280192
#191279832
#191279782
#191277582
C2×C4⋊Q8×C832
#323983