Finite Group Databasebeta

← #24918696

Subgroups — #24918696

Order 10624.

362 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 115 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×#14581791C211yesmaximal, characteristic, complemented
2#19128237C211yesmaximal, characteristic
3#19128109C212yesmaximal, complemented
4#19128062C212yesmaximal, complemented
5#19128034C211yesmaximal, characteristic, complemented
6#19128020C211yesmaximal, characteristic
7#19128019C211yesmaximal, characteristic
8#19127984C212yesmaximal, complemented
9#19127980C212yesmaximal, complemented
10#19127755C211yesmaximal, characteristic, complemented
11#484×C83C211yesmaximal, characteristic, complemented
12C22×Dic166—21no—
13#14581791—21no—
14#14581791—21no—
15#14581791C2212yes—
16#14581791C2212yes—
17#14581787—21no—
18#14581783C2211yescharacteristic
19#14581781C2211yescharacteristic
20#14581776—21no—
21#14581766C2211yescharacteristic
22Q8×Dic83C2211yescharacteristic
23#14581754C2211yescharacteristic
24(C22⋊D83)⋊C4—21no—
25#14581750—21no—
26C2×C4⋊Dic83C2211yescharacteristic
27C2×Dic83⋊C4—21no—
28C2×C4×Dic83C2211yescharacteristic
29#14581725C2212yes—
30#14581724C2212yes—
31#14581723C2212yes—
32Dic83⋊C8C2212yes—
33#14581716C2212yes—
34#14581714C2212yes—
35#14581712C2212yes—
36C22⋊(C83⋊C8)C2211yescharacteristic
37#14581702C2212yes—
38C22⋊Q8×C83C2211yescharacteristic
39C2×C4⋊C4×C83C2211yescharacteristic
40#107×C83C2212yes—
41#103×C83C2212yes—
42C22⋊C8×C83C2211yescharacteristic
43C2×Dic166—22no—
44C2×Dic166—22no—
45C2×Dic166—44no—
46C2×D332C2311yescharacteristic
47C2×C22⋊D83—21no—
48C22×Dic83—21no—
49C22×Dic83D411yescharacteristic
50C22×Dic83D411yescharacteristic
51C2×C4×D83—21no—
52C2×C83⋊C8C2312yes—
53C4⋊Dic83—21no—
54C4⋊Dic83—21no—
55C4⋊Dic83—21no—
56C4⋊Dic83C2311yescharacteristic
57C4⋊Dic83C2312yes—
58Dic83⋊C4—21no—
59Dic83⋊C4—22no—
60Dic83⋊C4—22no—
61(C2×D83)⋊C4—21no—
62(C2×D83)⋊C4—21no—
63(C2×D83)⋊C4—21no—
64(C2×D83)⋊C4—21no—
65C4×Dic83—21no—
66C4×Dic83D411yescharacteristic
67C4×Dic83D411yescharacteristic
68C4×Dic83C2311yescharacteristic
69C4×Dic83C2311yescharacteristic
70C2×Q8×C83C2311yescharacteristic
71C22×C4×C83—21no—
72C22×C4×C83C2311yescharacteristic
73C2×C8×C83C2312yes—
74C4⋊C4×C83—21no—
75C4⋊C4×C83—21no—
76C4⋊C4×C83C2311yescharacteristic
77C4⋊C4×C83C2312yes—
78C22⋊C4×C83—21no—
79C23×C83C2×D411yescharacteristic
80C22×D83—21no—
81Q8×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83—21no—
84C2×C4×C83—21no—
85C2×C4×C83—22no—
86C2×C4×C83—42no—
87C2×C4×C83C2×D411yescharacteristic
88C2×C4×C83C2411yescharacteristic
89C22⋊D83—42no—
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83—21no—
93C2×Dic83—22no—
94C2×Dic83—42no—
95C2×Dic83C2×D411yescharacteristic
96C2×Dic83C2×D411yescharacteristic
97C2×Dic83C2×D411yescharacteristic
98C2×Dic83C2×D411yescharacteristic
99D332—22no—
100C4×D83—42no—
101Dic166—44no—
102Dic166—44no—
103C8×C83—22no—
104C83⋊C8—22no—
105C22×C83—21no—
106C22×C83—21no—
107C22×C83#13712yes—
108C22×C83C22×D411yescharacteristic
109C2×D83—41no—
110C2×D83—41no—
111C2×D83—41no—
112C4×C83—21no—
113C4×C83—41no—
114C4×C83—42no—
115C4×C83—41no—
116C4×C832−1+412yes—
117Dic83—21no—
118Dic83—21no—
119Dic83—21no—
120Dic83—42no—
121Dic83—41no—
122Dic83AGL(1,ℤ/8)12yes—
123C2×C83—22no—
124C2×C83#53611yescharacteristic
125C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
126C2×C83C2×#13711yescharacteristic
127D83—81no—
128#3239—831nomaximal, Sylow
129C83#323911yescharacteristic, complemented, Sylow
130C2×C4⋊Q8—831no—
131#529—831no—
132#501—832no—
133#489—832no—
134#484—831no—
135#484D8311yescharacteristic, complemented
136#481—831no—
137#481—831no—
138#475—832no—
139#474—832no—
140#432—831no—
141C22×Q8—1661no—
142C4⋊Q8—832no—
143C4⋊Q8—832no—
144C4⋊Q8—1661no—
145C4⋊Q8—1661no—
146#126—1661no—
147#125—831no—
148#124—831no—
149#123—1661no—
150C22⋊Q8C2×D8311yescharacteristic
151#121—831no—
152C4×Q8—831no—
153C4×D4—831no—
154C4×D4—1661no—
155#117—1661no—
156C2×C4⋊C4—831no—
157C2×C4⋊C4—1661no—
158C2×C4⋊C4C2×D8311yescharacteristic
159C2×C42—831no—
160#107—832no—
161#107C2×D8312yes—
162#106—832no—
163#106—832no—
164C4⋊C8—832no—
165#103—832no—
166#103C2×D8312yes—
167#102—832no—
168#102—832no—
169C22⋊C8—831no—
170C22⋊C8C2×D8311yescharacteristic
171#97—832no—
172C2×Q8—3324no—
173C2×Q8—1662no—
174C2×Q8—1662no—
175C2×Q8C22×D8311yescharacteristic
176C2×D4—831no—
177C2×D4—1661no—
178C22×C4—21no—
179C22×C4—831no—
180C22×C4—831no—
181C22×C4—1661no—
182C22×C4—1661no—
183C22×C4C22×D8311yescharacteristic
184C2×C8—832no—
185C2×C8C22×D8312yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—832no—
189C4⋊C4—831no—
190C4⋊C4—1662no—
191C4⋊C4—1662no—
192C4⋊C4—1661no—
193C4⋊C4—1661no—
194C4⋊C4—1661no—
195C4⋊C4—1661no—
196C4⋊C4C22×D8312yes—
197C4⋊C4C22×D8311yescharacteristic
198C22⋊C4—21no—
199C22⋊C4—1661no—
200C22⋊C4—1661no—
201C22⋊C4—1661no—
202C22⋊C4—1661no—
203C42—831no—
204C42—831no—
205C42—831no—
206C42—831no—
207C42—1661no—
208C23—1661no—
209C23D4×D8311yescharacteristic
210Q8—22no—
211Q8—3324no—
212Q8—3324no—
213D4—3322no—
214D4—1662no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—42no—
220C2×C4—3322no—
221C2×C4—3322no—
222C2×C4—831no—
223C2×C4—831no—
224C2×C4—831no—
225C2×C4—831no—
226C2×C4—1661no—
227C2×C4—1662no—
228C2×C4—1662no—
229C2×C4—1662no—
230C2×C4D4×D8311yescharacteristic
231C2×C4C23×D8311yescharacteristic
232C8—22no—
233C8—1662no—
234C22—21no—
235C22—21no—
236C22—3321no—
237C22—3321no—
238C22—3321no—
239C22#1458182412yes—
240C22C2×D4×D8311yescharacteristic
241C4—21no—
242C4—42no—
243C4—41no—
244C4—41no—
245C4—3322no—
246C4—3321no—
247C4—832no—
248C4—1661no—
249C4—1661no—
250C4—1661no—
251C4#1458183912yes—
252C2—22no—
253C2—6641no—
254C2#1912827311yescharacteristic
255C2AGL(1,ℤ/8)×D8311yescharacteristic
256C2C2×#1458182411yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#484×C83C2
socleC22×C83C22×D4
prime radical#14581766C22
hypercenter#484D83
hypocenterC83#3239
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. C2×#14581791C2semidirect
3. #19128109C2semidirect
4. #19128062C2semidirect
5. #19128034C2semidirect
8. #19127984C2semidirect
9. #19127980C2semidirect
10. #19127755C2semidirect
11. #484×C83C2semidirect
129. C83#3239semidirect
135. #484D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817912
#191282372
#191281092
#191280622
#191280342
#191280202
#191280192
#191279842
#191279802
#191277552
#484×C832
#323983