Finite Group Databasebeta

← #24918283

Subgroups — #24918283

Order 10624.

410 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C22⋊Q8×D83C212yesmaximal
2#19128033C212yesmaximal, complemented
3#19128030C212yesmaximal, complemented
4#19127959C211yesmaximal, characteristic, complemented
5#19127957C212yesmaximal, complemented
6#19127858C212yesmaximal, complemented
7#19127855C212yesmaximal, complemented
8#19127664C211yesmaximal, characteristic
9#470×C83C211yesmaximal, characteristic, complemented
10C2×Q8×D83C2212yes—
11C22×C4×D83C2211yescharacteristic
12C2×Dic332C2211yescharacteristic, complemented
13C2×#13050580C2212yescomplemented
14C2×#13050574C2211yescharacteristic
15#14581772C2212yes—
16#14581771C2212yes—
17#14581770—22no—
18#14581769—22no—
19#14581768C2212yes—
20C4⋊C4×D83—22no—
21C4⋊C4×D83C2212yes—
22C22⋊C4×D83—22no—
23#14581724C2211yescharacteristic, complemented
24#14581723C2212yescomplemented
25#14581717C2212yescomplemented
26#14581716C2212yescomplemented
27#14581715C2212yescomplemented
28(C2×D83)⋊C8C2212yes—
29(C2×D83)⋊C8C2212yes—
30C2×Q16×C83C2211yescharacteristic, complemented
31C2×M16×C83C2211yescharacteristic
32C22⋊Q8×C83C2212yes—
33#106×C83C2211yescharacteristic, complemented
34#103×C83C2212yescomplemented
35C23×D83D411yescharacteristic
36C2×Dic166—22no—
37C2×Dic166C2312yes—
38Q8×D83—42no—
39Q8×D83—42no—
40C22×Dic83D411yescharacteristic
41C2×C4×D83—22no—
42C2×C4×D83—22no—
43C2×C4×D83—41no—
44C2×C4×D83—41no—
45C2×C4×D83D412yes—
46C2×C4×D83C2312yes—
47Dic332—41no—
48#13050580—42no—
49#13050574—41no—
50C2×C83⋊C8C2312yes—
51C4⋊Dic83—22no—
52C4⋊Dic83C2312yes—
53Dic83⋊C4—22no—
54Dic83⋊C4—22no—
55Dic83⋊C4—22no—
56C22⋊Dic83—22no—
57(C2×D83)⋊C4—22no—
58(C2×D83)⋊C4—22no—
59C2×Q8×C83C2312yes—
60C22×C4×C83C2311yescharacteristic
61Q16×C83—41no—
62M16×C83—41no—
63C2×C8×C83C2311yescharacteristic
64C2×C8×C83C2311yescharacteristic
65C4⋊C4×C83—22no—
66C4⋊C4×C83C2312yes—
67C22⋊C4×C83—22no—
68C23×C83C2×D411yescharacteristic
69C22×D83—24no—
70C22×D83—41no—
71C22×D83C2×D412yes—
72Q8×C83—22no—
73Q8×C83—22no—
74C2×C4×C83—21no—
75C2×C4×C83—21no—
76C2×C4×C83—22no—
77C2×C4×C83—22no—
78C2×C4×C83C2×D411yescharacteristic
79C2×C4×C83C2411yescharacteristic
80C2×Dic83—22no—
81C2×Dic83—22no—
82C2×Dic83—41no—
83C2×Dic83C2×D412yes—
84C4×D83—22no—
85C4×D83—22no—
86C4×D83—42no—
87C4×D83—44no—
88C4×D83—44no—
89Dic166—22no—
90Dic166—22no—
91Dic166—44no—
92C8×C83—21no—
93C8×C83—21no—
94C83⋊C8—22no—
95C22×C83—21no—
96C22×C83—22no—
97C22×C83C22×D411yescharacteristic
98C2×D83—22no—
99C2×D83—22no—
100C2×D83—44no—
101C2×D83#13714yes—
102C4×C83—21no—
103C4×C83—42no—
104C4×C83—42no—
105C4×C83H(5,𝔽2)11yescharacteristic
106C4×C83H(5,𝔽2)11yescharacteristic
107Dic83—22no—
108Dic83—42no—
109Dic83—42no—
110C2×C83—41no—
111C2×C83#53411yescharacteristic
112C2×C83C2×#13712yes—
113D83—24no—
114#3195—831nomaximal, Sylow
115C83#319511yescharacteristic, complemented, Sylow
116C2×C22⋊Q8—832no—
117#483—832no—
118#483—832no—
119#470—831no—
120#470—832no—
121#470D8311yescharacteristic, complemented
122#451—832no—
123#451—832no—
124#407—831no—
125C22×Q8—832no—
126C23×C4—831no—
127C2×Q16—832no—
128C2×Q16—831no—
129C2×Q16C2×D8311yescharacteristic, complemented
130C2×M16—831no—
131C2×M16C2×D8311yescharacteristic
132C22⋊Q8—832no—
133C22⋊Q8—832no—
134C22⋊Q8—832no—
135C22⋊Q8—1662no—
136C22⋊Q8—1662no—
137C22⋊Q8C2×D8312yes—
138C2×C4⋊C4—832no—
139C2×C4⋊C4—1662no—
140C2×C22⋊C4—1662no—
141#106—831no—
142#106—832no—
143#106C2×D8311yescharacteristic, complemented
144#103—832no—
145#103—832no—
146#103—832no—
147#103C2×D8312yescomplemented
148C22⋊C8—832no—
149C22⋊C8—832no—
150C24—831no—
151C2×Q8—3322no—
152C2×Q8—3322no—
153C2×Q8—832no—
154C2×Q8—1662no—
155C2×Q8C22×D8312yes—
156C22×C4—3321no—
157C22×C4—3321no—
158C22×C4—831no—
159C22×C4—832no—
160C22×C4—832no—
161C22×C4—1662no—
162C22×C4—1662no—
163C22×C4C22×D8311yescharacteristic
164Q16—41no—
165Q16—3321no—
166Q16—3322no—
167M16—41no—
168M16—3321no—
169C2×C8—832no—
170C2×C8C22×D8311yescharacteristic
171C2×C8C22×D8311yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—832no—
174C4⋊C4—1662no—
175C4⋊C4—1662no—
176C4⋊C4—1662no—
177C4⋊C4—1662no—
178C4⋊C4C22×D8312yes—
179C22⋊C4—22no—
180C22⋊C4—1662no—
181C22⋊C4—1662no—
182C22⋊C4—1662no—
183C23—3321no—
184C23—832no—
185C23—1664no—
186C23D4×D8311yescharacteristic
187Q8—22no—
188Q8—22no—
189Q8—3324no—
190Q8—1662no—
191Q8—1662no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—3322no—
197C2×C4—3324no—
198C2×C4—3324no—
199C2×C4—3321no—
200C2×C4—832no—
201C2×C4—1662no—
202C2×C4—1662no—
203C2×C4—1662no—
204C2×C4—1662no—
205C2×C4D4×D8311yescharacteristic
206C2×C4C23×D8311yescharacteristic
207C8—21no—
208C8—21no—
209C8—1662no—
210C22—22no—
211C22—21no—
212C22—3324no—
213C22—834no—
214C22—1662no—
215C22—1662no—
216C22C2×D4×D8311yescharacteristic
217C4—21no—
218C4—42no—
219C4—42no—
220C4—3322no—
221C4—3322no—
222C4—1662no—
223C4#1458183711yescharacteristic
224C4#1458183811yescharacteristic
225C2—41no—
226C2—1664no—
227C2#1912825911yescharacteristic
228C2#137×D8312yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#470×C83C2
socleC22×C83C22×D4
prime radicalC23×D83D4
hypercenter#470D83
hypocenterC83#3195
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
2. #19128033C2semidirect
3. #19128030C2semidirect
4. #19127959C2semidirect
5. #19127957C2semidirect
6. #19127858C2semidirect
7. #19127855C2semidirect
9. #470×C83C2semidirect
12. C2×Dic332C22semidirect
13. C2×#13050580C22semidirect
23. #14581724C22semidirect
24. #14581723C22semidirect
25. #14581717C22semidirect
26. #14581716C22semidirect
27. #14581715C22semidirect
30. C2×Q16×C83C22semidirect
33. #106×C83C22semidirect
34. #103×C83C22semidirect
115. C83#3195semidirect
121. #470D83semidirect
129. C2×Q16C2×D83semidirect
143. #106C2×D83semidirect
147. #103C2×D83semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D832
#191280332
#191280302
#191279592
#191279572
#191278582
#191278552
#191276642
#470×C832
#319583