Finite Group Databasebeta

← #24918177

Subgroups — #24918177

Order 10624.

402 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 129 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#19128104C211yesmaximal, characteristic, complemented
3#19128073C211yesmaximal, characteristic, complemented
4#19128053C211yesmaximal, characteristic, complemented
5#19128047C211yesmaximal, characteristic, complemented
6#19127945C214yesmaximal, complemented
7#19127801C214yesmaximal, complemented
8C4×#13050574C211yesmaximal, characteristic, complemented
9#532×C83C211yesmaximal, characteristic
10C2×#13050578C2214yes—
11C2×#13050574C2212yes—
12#14581792C2211yescharacteristic
13#14581792C2211yescharacteristic
14#14581784—21no—
15#14581771—21no—
16#14581771C2212yes—
17C4×Dic166—21no—
18C4×Dic166C2212yes—
19#14581751—21no—
20#14581725C2214yes—
21#14581715C2214yes—
22#14581713C2214yes—
23#14581703C2212yes—
24C4×C83⋊C8C2211yescharacteristic
25C4×C83⋊C8C2211yescharacteristic
26C2×#41×C83—21no—
27C4⋊Q8×C83C2211yescharacteristic
28#127×C83C2211yescharacteristic
29#124×C83—21no—
30#121×C83—21no—
31#121×C83C2212yes—
32C4×D4×C83—21no—
33C4×D4×C83C2212yes—
34C2×C42×C83C2211yescharacteristic
35C2×Dic166—21no—
36C2×Dic166C2312yes—
37#13050578—28no—
38#13050574—24no—
39C2×C83⋊C8C2312yes—
40C2×C83⋊C8C2312yes—
41C4⋊Dic83—21no—
42C4⋊Dic83—21no—
43C4⋊Dic83—21no—
44C4⋊Dic83C2312yes—
45Dic83⋊C4—22no—
46Dic83⋊C4—22no—
47C22⋊Dic83—22no—
48C4×Dic83—22no—
49#41×C83—44no—
50C2×Q8×C83—21no—
51C2×D4×C83—22no—
52C2×D4×C83—21no—
53C2×D4×C83—21no—
54C2×D4×C83C2312yes—
55C22×C4×C83—22no—
56C22×C4×C83D411yescharacteristic
57C22×C4×C83D411yescharacteristic
58C22×C4×C83C2311yescharacteristic
59C4⋊C4×C83—21no—
60C4⋊C4×C83C2312yes—
61C22⋊C4×C83—22no—
62C22⋊C4×C83—22no—
63C42×C83D411yescharacteristic
64C42×C83D411yescharacteristic
65C42×C83C2311yescharacteristic
66C42×C83C2311yescharacteristic
67C23×C83—22no—
68C23×C83C2×D411yescharacteristic
69Q8×C83—42no—
70D4×C83—24no—
71D4×C83—44no—
72D4×C83—42no—
73D4×C83—42no—
74C2×C4×C83—22no—
75C2×C4×C83—22no—
76C2×C4×C83—22no—
77C2×C4×C83—22no—
78C2×C4×C83—44no—
79C2×C4×C83C2×D411yescharacteristic
80C2×C4×C83C2×D411yescharacteristic
81C2×C4×C83C2×D411yescharacteristic
82C2×C4×C83C2×D411yescharacteristic
83C2×C4×C83C2×D411yescharacteristic
84C2×C4×C83C2411yescharacteristic
85C2×Dic83—22no—
86C2×Dic83—22no—
87Dic166—24no—
88Dic166—42no—
89C83⋊C8—22no—
90C83⋊C8#4114yescomplemented
91C22×C83—21no—
92C22×C83—21no—
93C22×C83—21no—
94C22×C83—42no—
95C22×C83—42no—
96C22×C83—42no—
97C22×C83C22×D411yescharacteristic
98C4×C83—21no—
99C4×C83—21no—
100C4×C83—21no—
101C4×C83—42no—
102C4×C83AGL(1,ℤ/8)12yes—
103C4×C83#13712yes—
104C4×C83C2×#4112yes—
105Dic83—42no—
106Dic83—42no—
107C2×C83—41no—
108C2×C83—82no—
109C2×C83#53211yescharacteristic
110C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
111C2×C83C2×#13711yescharacteristic
112#3182—831nomaximal, Sylow
113C83#318211yescharacteristic, complemented, Sylow
114#533—831no—
115#532D8311yescharacteristic
116#500—831no—
117#492—831no—
118#487—831no—
119#486—831no—
120#468—834no—
121#440—834no—
122C4×M16—831no—
123C2×#41—21no—
124C2×SD16—834no—
125C2×M16—832no—
126C4⋊Q8—831no—
127C4⋊Q8—831no—
128C4⋊Q8C2×D8311yescharacteristic
129#127C2×D8311yescharacteristic
130#125—1661no—
131#124—21no—
132C22⋊Q8—832no—
133C22⋊Q8—1661no—
134#121—21no—
135#121C2×D8312yes—
136C4×Q8—832no—
137C4×Q8—1661no—
138C4×D4—21no—
139C4×D4C2×D8312yes—
140#117—1661no—
141C2×C42C2×D8311yescharacteristic
142#107—834no—
143#103—834no—
144#102—834no—
145#97—832no—
146C4×C8—831no—
147C4×C8—831no—
148#41—44no—
149C2×Q8—21no—
150C2×Q8—832no—
151C2×Q8—1661no—
152C2×D4—22no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4C22×D8312yes—
156C22×C4—22no—
157C22×C4C22⋊D8311yescharacteristic
158C22×C4C22⋊D8311yescharacteristic
159C22×C4C22×D8311yescharacteristic
160SD16—1668no—
161M16—1664no—
162C2×C8—832no—
163C2×C8—832no—
164C4⋊C4—21no—
165C4⋊C4—832no—
166C4⋊C4—1661no—
167C4⋊C4—1661no—
168C4⋊C4—1662no—
169C4⋊C4—1662no—
170C4⋊C4—1661no—
171C4⋊C4C22×D8312yes—
172C22⋊C4—22no—
173C22⋊C4—22no—
174C22⋊C4—1662no—
175C42—1662no—
176C42C22⋊D8311yescharacteristic
177C42C22⋊D8311yescharacteristic
178C42C22×D8311yescharacteristic
179C42C22×D8311yescharacteristic
180C23—22no—
181C23C2×C22⋊D8311yescharacteristic
182Q8—42no—
183Q8—3322no—
184Q8—1664no—
185D4—24no—
186D4—44no—
187D4—42no—
188D4—42no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
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—834no—
203C8—1662no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22C22×C22⋊D8311yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—42no—
215C4—3322no—
216C4—3322no—
217C4#1458181812yes—
218C4#1458182212yes—
219C4#41×D8312yes—
220C2—41no—
221C2—82no—
222C2#1912824711yescharacteristic
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#3182
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. #19128104C2semidirect
3. #19128073C2semidirect
4. #19128053C2semidirect
5. #19128047C2semidirect
6. #19127945C2semidirect
7. #19127801C2semidirect
8. C4×#13050574C2semidirect
90. C83⋊C8#41semidirect
113. C83#3182semidirect

Maximal subgroups

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

subgroupindex
#191282582
#191281042
#191280732
#191280532
#191280472
#191279452
#191278012
C4×#130505742
#532×C832
#318283