Finite Group Databasebeta

← #24919890

Subgroups — #24919890

Order 10624.

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

normal subgroups only

This group has 340 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
1#19128422C212yesmaximal
2#19128105C211yesmaximal, characteristic, complemented
3#19128040C212yesmaximal, complemented
4#19128039C212yesmaximal, complemented
5#19127984C212yesmaximal, complemented
6#19127939C211yesmaximal, characteristic, complemented
7Q16×Dic83C211yesmaximal, characteristic, complemented
8C8×C22⋊D83C211yesmaximal, characteristic
9#19127739C211yesmaximal, characteristic, complemented
10#19127738C211yesmaximal, characteristic, complemented
11#467×C83C211yesmaximal, characteristic, complemented
12C2×C8×D83C2211yescharacteristic
13#14581791C2212yes—
14#14581787—22no—
15#14581785—22no—
16#14581778—22no—
17#14581768C2212yes—
18Q8×Dic83C2212yes—
19#14581757—22no—
20C4×C22⋊D83C2211yescharacteristic
21#14581751C2212yes—
22#14581750C2212yes—
23#14581726C2211yescharacteristic
24#14581726C2211yescharacteristic
25#14581725C2212yes—
26#14581724C2212yes—
27#14581723C2212yes—
28Dic83⋊C8C2211yescharacteristic
29#14581716C2212yes—
30#14581716C2212yes—
31C22⋊(C83⋊C8)C2211yescharacteristic
32(C2×D83)⋊C8C2211yescharacteristic
33C8×Dic83C2211yescharacteristic
34C2×Q16×C83C2211yescharacteristic
35C22×C8×C83C2211yescharacteristic
36C22⋊Q8×C83C2212yes—
37#107×C83C2211yescharacteristic
38#103×C83C2212yes—
39C2×Dic166—22no—
40C2×C22⋊D83D411yescharacteristic
41C2×C4×D83C2311yescharacteristic
42C2×C83⋊C8C2311yescharacteristic
43C2×C83⋊C8C2311yescharacteristic
44C8×D83—22no—
45C4⋊Dic83—22no—
46C4⋊Dic83C2312yes—
47C4⋊Dic83C2312yes—
48Dic83⋊C4—22no—
49Dic83⋊C4—22no—
50Dic83⋊C4D411yescharacteristic
51C22⋊Dic83—22no—
52C22⋊Dic83D411yescharacteristic
53(C2×D83)⋊C4—22no—
54(C2×D83)⋊C4—22no—
55(C2×D83)⋊C4D411yescharacteristic
56C4×Dic83—22no—
57C4×Dic83—22no—
58C4×Dic83—22no—
59C4×Dic83C2311yescharacteristic
60C2×Q8×C83C2312yes—
61C22×C4×C83C2311yescharacteristic
62Q16×C83—22no—
63C2×C8×C83—22no—
64C2×C8×C83C2311yescharacteristic
65C2×C8×C83C2311yescharacteristic
66C4⋊C4×C83—22no—
67C4⋊C4×C83C2312yes—
68C22⋊C4×C83—22no—
69C23×C83C2×D411yescharacteristic
70C22×D83C2×D411yescharacteristic
71Q8×C83—22no—
72Q8×C83—22no—
73C2×C4×C83—21no—
74C2×C4×C83—21no—
75C2×C4×C83—22no—
76C2×C4×C83—22no—
77C2×C4×C83C2×D411yescharacteristic
78C2×C4×C83C2411yescharacteristic
79C22⋊D83—22no—
80C2×Dic83—22no—
81C2×Dic83—22no—
82C2×Dic83C2×D411yescharacteristic
83C2×Dic83C2×D411yescharacteristic
84C2×Dic83C2×D411yescharacteristic
85C4×D83—21no—
86C4×D83—21no—
87Dic166—44no—
88C8×C83—21no—
89C8×C83#4112yes—
90C83⋊C8—21no—
91C83⋊C8—21no—
92C22×C83—21no—
93C22×C83—21no—
94C22×C83—21no—
95C22×C83C22×D411yescharacteristic
96C2×D83—21no—
97C2×D83—21no—
98C2×D83—21no—
99C4×C83—21no—
100C4×C83—42no—
101C4×C83—42no—
102C4×C83C2×#4111yescharacteristic
103C4×C832−1+411yescharacteristic
104Dic83—21no—
105Dic83—21no—
106Dic83—42no—
107Dic83—42no—
108Dic83#13512yes—
109C2×C83—41no—
110C2×C83#54711yescharacteristic
111C2×C83C2×#13511yescharacteristic
112C2×C83#57811yescharacteristic
113D83—41no—
114#3342—831nomaximal, Sylow
115C83#334211yescharacteristic, complemented, Sylow
116#556—832no—
117#500—831no—
118#485—832no—
119#485—832no—
120#475—832no—
121#467—831no—
122#467D8311yescharacteristic, complemented
123C4×Q16—831no—
124C8×D4—831no—
125#427—831no—
126#427—831no—
127C2×Q16C2×D8311yescharacteristic
128C22×C8—831no—
129C22×C8C2×D8311yescharacteristic
130C4⋊Q8—832no—
131#126—1662no—
132#126—1662no—
133#124—1662no—
134C22⋊Q8—832no—
135C22⋊Q8C2×D8312yes—
136C4×Q8—832no—
137C4×Q8—1662no—
138C4×D4—831no—
139#117—832no—
140#117—832no—
141#107—831no—
142#107—831no—
143#107—832no—
144#107C2×D8311yescharacteristic
145#106—832no—
146#106—832no—
147C4⋊C8—831no—
148#103—832no—
149#103—832no—
150#103C2×D8312yes—
151C22⋊C8—831no—
152C22⋊C8—831no—
153C4×C8—831no—
154C2×Q8—1662no—
155C2×Q8C22×D8312yes—
156C2×D4—831no—
157C22×C4—831no—
158C22×C4C22×D8311yescharacteristic
159Q16—22no—
160C2×C8—22no—
161C2×C8—831no—
162C2×C8—831no—
163C2×C8—1662no—
164C2×C8C22×D8311yescharacteristic
165C2×C8C22×D8311yescharacteristic
166C4⋊C4—22no—
167C4⋊C4—831no—
168C4⋊C4—832no—
169C4⋊C4—832no—
170C4⋊C4—1662no—
171C4⋊C4—1662no—
172C4⋊C4—1662no—
173C4⋊C4C22×D8312yes—
174C22⋊C4—22no—
175C22⋊C4—831no—
176C22⋊C4—831no—
177C22⋊C4—1662no—
178C22⋊C4—1662no—
179C22⋊C4—1662no—
180C42—831no—
181C42—1662no—
182C42—1662no—
183C42—1662no—
184C23—831no—
185C23D4×D8311yescharacteristic
186Q8—22no—
187Q8—22no—
188Q8—3324no—
189D4—1662no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—831no—
195C2×C4—831no—
196C2×C4—831no—
197C2×C4—1662no—
198C2×C4—1662no—
199C2×C4—1661no—
200C2×C4—1661no—
201C2×C4D4×D8311yescharacteristic
202C2×C4C23×D8311yescharacteristic
203C8—21no—
204C8—1661no—
205C8—1661no—
206C8#1305058912yes—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—1661no—
211C22—1661no—
212C22—1661no—
213C22C2×D4×D8311yescharacteristic
214C4—21no—
215C4—42no—
216C4—42no—
217C4—3322no—
218C4—3322no—
219C4—832no—
220C4—1661no—
221C4—1661no—
222C4C2×#1305058911yescharacteristic
223C4#1458183911yescharacteristic
224C2—41no—
225C2—3321no—
226C2#1912835111yescharacteristic
227C2#135×D8311yescharacteristic
228C2#1912851411yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#467×C83C2
socleC22×C83C22×D4
prime radicalC2×C22⋊D83D4
hypercenter#467D83
hypocenterC83#3342
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
2. #19128105C2semidirect
3. #19128040C2semidirect
4. #19128039C2semidirect
5. #19127984C2semidirect
6. #19127939C2semidirect
7. Q16×Dic83C2semidirect
9. #19127739C2semidirect
10. #19127738C2semidirect
11. #467×C83C2semidirect
115. C83#3342semidirect
122. #467D83semidirect

Maximal subgroups

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

subgroupindex
#191284222
#191281052
#191280402
#191280392
#191279842
#191279392
Q16×Dic832
C8×C22⋊D832
#191277392
#191277382
#467×C832
#334283