Finite Group Databasebeta

← #24918850

Subgroups — #24918850

Order 10624.

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

normal subgroups only

This group has 386 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#19128258C211yesmaximal, characteristic, complemented
2#19127993C212yesmaximal, complemented
3#19127981C212yesmaximal, complemented
4#19127945C214yesmaximal, complemented
5#19127906C212yesmaximal, complemented
6#19127898C212yesmaximal, complemented
7#19127725C211yesmaximal, characteristic, complemented
8#532×C83C211yesmaximal, characteristic
9C2×#13050578C2214yes—
10C2×#13050574C2212yes—
11#14581792C2211yescharacteristic
12#14581792C2211yescharacteristic
13#14581784—21no—
14#14581771—21no—
15#14581771C2212yes—
16C4×Dic166—21no—
17C4×Dic166C2212yes—
18#14581751—21no—
19#14581725C2214yes—
20C4⋊(C83⋊C8)C2212yes—
21C4⋊(C83⋊C8)C2212yes—
22#14581715C2214yes—
23#14581713C2214yes—
24C2×#41×C83—21no—
25C4⋊Q8×C83C2211yescharacteristic
26#127×C83C2211yescharacteristic
27#124×C83—21no—
28#121×C83—21no—
29#121×C83C2212yes—
30C4×D4×C83—21no—
31C4×D4×C83C2212yes—
32C2×C42×C83C2211yescharacteristic
33C2×Dic166—21no—
34C2×Dic166C2312yes—
35#13050578—44no—
36#13050574—42no—
37C2×C83⋊C8C2314yes—
38C4⋊Dic83—21no—
39C4⋊Dic83—21no—
40C4⋊Dic83—21no—
41C4⋊Dic83C2312yes—
42Dic83⋊C4—22no—
43Dic83⋊C4—22no—
44C22⋊Dic83—22no—
45C4×Dic83—22no—
46#41×C83—44no—
47C2×Q8×C83—21no—
48C2×D4×C83—22no—
49C2×D4×C83—21no—
50C2×D4×C83—21no—
51C2×D4×C83C2312yes—
52C22×C4×C83—22no—
53C22×C4×C83D411yescharacteristic
54C22×C4×C83D411yescharacteristic
55C22×C4×C83C2311yescharacteristic
56C4⋊C4×C83—21no—
57C4⋊C4×C83C2312yes—
58C22⋊C4×C83—22no—
59C22⋊C4×C83—22no—
60C42×C83D411yescharacteristic
61C42×C83D411yescharacteristic
62C42×C83C2311yescharacteristic
63C42×C83C2311yescharacteristic
64C23×C83—22no—
65C23×C83C2×D411yescharacteristic
66Q8×C83—42no—
67D4×C83—22no—
68D4×C83—22no—
69D4×C83—42no—
70D4×C83—44no—
71D4×C83—42no—
72C2×C4×C83—22no—
73C2×C4×C83—22no—
74C2×C4×C83—21no—
75C2×C4×C83—21no—
76C2×C4×C83—22no—
77C2×C4×C83—44no—
78C2×C4×C83C2×D411yescharacteristic
79C2×C4×C83C2×D411yescharacteristic
80C2×C4×C83C2×D411yescharacteristic
81C2×C4×C83C2×D411yescharacteristic
82C2×C4×C83C2×D411yescharacteristic
83C2×C4×C83C2411yescharacteristic
84C2×Dic83—22no—
85C2×Dic83—22no—
86Dic166—22no—
87Dic166—22no—
88Dic166—42no—
89C83⋊C8—24no—
90C22×C83—21no—
91C22×C83—21no—
92C22×C83—21no—
93C22×C83—42no—
94C22×C83—42no—
95C22×C83—42no—
96C22×C83C22×D411yescharacteristic
97C4×C83—21no—
98C4×C83—21no—
99C4×C83—21no—
100C4×C83—42no—
101C4×C83AGL(1,ℤ/8)12yes—
102C4×C83#13712yes—
103C4×C83H(5,𝔽2)11yescharacteristic
104C4×C832−1+411yescharacteristic
105Dic83—42no—
106Dic83—42no—
107C2×C83—41no—
108C2×C83—82no—
109C2×C83#53711yescharacteristic
110C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
111C2×C83C2×#13711yescharacteristic
112#3254—831nomaximal, Sylow
113C83#325411yescharacteristic, complemented, Sylow
114#533—831no—
115#532D8311yescharacteristic
116#477—832no—
117#474—832no—
118#468—834no—
119#461—832no—
120#460—832no—
121#423—831no—
122C2×#41—21no—
123C2×SD16—834no—
124C2×M16—832no—
125C4⋊Q8—831no—
126C4⋊Q8—831no—
127C4⋊Q8C2×D8311yescharacteristic
128#127C2×D8311yescharacteristic
129#125—1661no—
130#124—21no—
131C22⋊Q8—832no—
132C22⋊Q8—1661no—
133#121—21no—
134#121C2×D8312yes—
135C4×Q8—832no—
136C4×Q8—1661no—
137C4×D4—21no—
138C4×D4C2×D8312yes—
139#117—1661no—
140C2×C42C2×D8311yescharacteristic
141#107—834no—
142C4⋊C8—832no—
143C4⋊C8—832no—
144#103—834no—
145#102—834no—
146#41—44no—
147C2×Q8—21no—
148C2×Q8—832no—
149C2×Q8—1661no—
150C2×D4—22no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4C22×D8312yes—
154C22×C4—22no—
155C22×C4C22⋊D8311yescharacteristic
156C22×C4C22⋊D8311yescharacteristic
157C22×C4C22×D8311yescharacteristic
158SD16—3324no—
159M16—3322no—
160C2×C8—834no—
161C4⋊C4—21no—
162C4⋊C4—832no—
163C4⋊C4—1661no—
164C4⋊C4—1661no—
165C4⋊C4—1662no—
166C4⋊C4—1662no—
167C4⋊C4—1661no—
168C4⋊C4C22×D8312yes—
169C22⋊C4—22no—
170C22⋊C4—22no—
171C22⋊C4—1662no—
172C42—1662no—
173C42C22⋊D8311yescharacteristic
174C42C22⋊D8311yescharacteristic
175C42C22×D8311yescharacteristic
176C42C22×D8311yescharacteristic
177C23—22no—
178C23C2×C22⋊D8311yescharacteristic
179Q8—42no—
180Q8—3322no—
181Q8—1662no—
182Q8—1662no—
183D4—22no—
184D4—22no—
185D4—42no—
186D4—44no—
187D4—42no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—44no—
194C2×C4—1662no—
195C2×C4—1662no—
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—21no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22C22×C22⋊D8311yescharacteristic
210C4—21no—
211C4—21no—
212C4—21no—
213C4—42no—
214C4—3322no—
215C4—3322no—
216C4#1458181812yes—
217C4#1458182212yes—
218C4#1458183811yescharacteristic
219C4#1458183911yescharacteristic
220C2—41no—
221C2—82no—
222C2#1912827911yescharacteristic
223C2C2×#1458181811yescharacteristic
224C2C2×#1458182211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting, prime radical#532×C83C2
socleC22×C83C22×D4
hypercenter#532D83
hypocenterC83#3254
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #19128258C2semidirect
2. #19127993C2semidirect
3. #19127981C2semidirect
4. #19127945C2semidirect
5. #19127906C2semidirect
6. #19127898C2semidirect
7. #19127725C2semidirect
113. C83#3254semidirect

Maximal subgroups

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

subgroupindex
#191282582
#191279932
#191279812
#191279452
#191279062
#191278982
#191277252
#532×C832
#325483