Finite Group Databasebeta

← #24918615

Subgroups — #24918615

Order 10624.

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

normal subgroups only

This group has 360 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#19128256C211yesmaximal, characteristic
2C4⋊Q8×D83C211yesmaximal, characteristic
3#19128104C211yesmaximal, characteristic, complemented
4#19128053C211yesmaximal, characteristic, complemented
5#19128052C211yesmaximal, characteristic, complemented
6#19128038C214yesmaximal
7#19127995C214yesmaximal, complemented
8#19127752C211yesmaximal, characteristic
9#500×C83C211yesmaximal, characteristic, complemented
10C2×Q8×D83—21no—
11#14581792C2211yescharacteristic
12#14581792C2211yescharacteristic
13#14581791—21no—
14#14581791C2212yes—
15#14581790—21no—
16#14581783—21no—
17#14581772C2212yes—
18#14581768—21no—
19Q8×Dic83—21no—
20#14581757C2212yes—
21#14581750—21no—
22C4⋊C4×D83—21no—
23C4⋊C4×D83C2212yes—
24C42×D83C2211yescharacteristic
25#14581725C2214yes—
26Dic83⋊C8C2212yes—
27#14581717C2214yes—
28#14581715C2214yes—
29(C2×D83)⋊C8C2212yes—
30C4×C83⋊C8C2211yescharacteristic
31C4⋊Q8×C83C2211yescharacteristic
32C4⋊Q8×C83C2211yescharacteristic
33#107×C83C2214yes—
34C4×C8×C83C2211yescharacteristic
35C2×Dic166—21no—
36C2×Dic166—22no—
37C2×Dic166C2312yes—
38Q8×D83—44no—
39C2×C4×D83—22no—
40C2×C4×D83D411yescharacteristic
41C2×C4×D83D411yescharacteristic
42C2×C4×D83C2311yescharacteristic
43C2×C83⋊C8C2312yes—
44C4⋊Dic83—21no—
45C4⋊Dic83—21no—
46C4⋊Dic83—21no—
47C4⋊Dic83—22no—
48C4⋊Dic83C2312yes—
49Dic83⋊C4—22no—
50Dic83⋊C4—22no—
51Dic83⋊C4—22no—
52(C2×D83)⋊C4—22no—
53C4×Dic83—22no—
54C4×Dic83D411yescharacteristic
55C4×Dic83D411yescharacteristic
56C4×Dic83C2311yescharacteristic
57C2×Q8×C83—21no—
58C2×Q8×C83—21no—
59C2×C8×C83C2312yes—
60C4⋊C4×C83—21no—
61C4⋊C4×C83—21no—
62C4⋊C4×C83C2312yes—
63C4⋊C4×C83C2312yes—
64C42×C83C2311yescharacteristic
65C22×D83C2×D411yescharacteristic
66Q8×C83—42no—
67Q8×C83—42no—
68C2×C4×C83—22no—
69C2×C4×C83—22no—
70C2×C4×C83C2×D411yescharacteristic
71C2×C4×C83C2×D411yescharacteristic
72C2×C4×C83C2411yescharacteristic
73C2×Dic83—22no—
74C2×Dic83—22no—
75C2×Dic83C2×D411yescharacteristic
76C2×Dic83C2×D411yescharacteristic
77C2×Dic83C2×D411yescharacteristic
78C4×D83—22no—
79C4×D83—22no—
80C4×D83—44no—
81C4×D83Q1614yes—
82Dic166—24no—
83Dic166—44no—
84Dic166—42no—
85C8×C83—22no—
86C83⋊C8—22no—
87C22×C83C22×D411yescharacteristic
88C2×D83—21no—
89C2×D83—21no—
90C2×D83C2×Q1612yes—
91C4×C83—42no—
92C4×C83—42no—
93C4×C83C2×Q1612yes—
94C4×C83AGL(1,ℤ/8)12yes—
95C4×C832−1+412yes—
96Dic83—21no—
97Dic83—21no—
98Dic83—42no—
99Dic83—42no—
100Dic83C2×Q1612yes—
101C2×C83#53611yescharacteristic
102C2×C83C22×Q1611yescharacteristic
103C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
104D83—22no—
105#3230—831nomaximal, Sylow
106C83#323011yescharacteristic, complemented, Sylow
107#533—831no—
108C2×C4⋊Q8—831no—
109#500—831no—
110#500D8311yescharacteristic, complemented
111#487—831no—
112#487—831no—
113#484—834no—
114#477—834no—
115#431—831no—
116C22×Q8—1661no—
117C4⋊Q8—832no—
118C4⋊Q8—831no—
119C4⋊Q8—831no—
120C4⋊Q8—1661no—
121C4⋊Q8—1661no—
122C4⋊Q8C2×D8311yescharacteristic
123C4⋊Q8C2×D8311yescharacteristic
124#125—1661no—
125C22⋊Q8—832no—
126C22⋊Q8—1661no—
127C4×Q8—832no—
128C4×Q8—1661no—
129#117—1661no—
130C2×C4⋊C4—832no—
131C2×C4⋊C4—1661no—
132C2×C42—831no—
133#107—834no—
134#107C2×D8314yes—
135C4⋊C8—832no—
136#103—834no—
137#103—834no—
138C22⋊C8—832no—
139C4×C8—831no—
140C4×C8C2×D8311yescharacteristic
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—3324no—
144C2×Q8—832no—
145C2×Q8—1662no—
146C2×Q8—1661no—
147C22×C4—831no—
148C22×C4—831no—
149C22×C4—831no—
150C22×C4—1662no—
151C2×C8—832no—
152C2×C8C22×D8312yes—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—832no—
156C4⋊C4—1662no—
157C4⋊C4—1662no—
158C4⋊C4—1662no—
159C4⋊C4—1662no—
160C4⋊C4—1661no—
161C4⋊C4—1661no—
162C4⋊C4—1661no—
163C4⋊C4C22×D8312yes—
164C4⋊C4C22×D8312yes—
165C22⋊C4—1662no—
166C42—831no—
167C42—831no—
168C42—831no—
169C42—1662no—
170C42C22×D8311yescharacteristic
171C23—831no—
172Q8—42no—
173Q8—42no—
174Q8—3324no—
175Q8—3322no—
176Q8—1664no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—3324no—
180C2×C4—831no—
181C2×C4—834no—
182C2×C4—831no—
183C2×C4—831no—
184C2×C4—1662no—
185C2×C4—1662no—
186C2×C4—1662no—
187C2×C4—1662no—
188C2×C4D4×D8311yescharacteristic
189C2×C4D4×D8311yescharacteristic
190C2×C4C23×D8311yescharacteristic
191C8—22no—
192C8—1662no—
193C22—832no—
194C22—1661no—
195C22—1661no—
196C22C2×D4×D8311yescharacteristic
197C4—42no—
198C4—42no—
199C4—3322no—
200C4—3322no—
201C4—832no—
202C4—1661no—
203C4—1661no—
204C4Q16×D8312yes—
205C4#1458181612yes—
206C4#1458184012yes—
207C2—1662no—
208C2#1912827611yescharacteristic
209C2C2×Q16×D8311yescharacteristic
210C2C2×#1458181611yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#500×C83C2
socleC22×C83C22×D4
prime radicalC22×D83C2×D4
hypercenter#500D83
hypocenterC83#3230
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
3. #19128104C2semidirect
4. #19128053C2semidirect
5. #19128052C2semidirect
7. #19127995C2semidirect
9. #500×C83C2semidirect
106. C83#3230semidirect
110. #500D83semidirect

Maximal subgroups

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

subgroupindex
#191282562
C4⋊Q8×D832
#191281042
#191280532
#191280522
#191280382
#191279952
#191277522
#500×C832
#323083