Finite Group Databasebeta

← #24918625

Subgroups — #24918625

Order 10624.

326 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 117 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
2#19128256C211yesmaximal, characteristic
3#19128104C211yesmaximal, characteristic, complemented
4#19128053C211yesmaximal, characteristic, complemented
5#19128052C211yesmaximal, characteristic, complemented
6#19128045C214yesmaximal
7#19128011C214yesmaximal, complemented
8#19127752C211yesmaximal, characteristic
9#498×C83C211yesmaximal, characteristic, complemented
10#14581792C2211yescharacteristic
11#14581792C2211yescharacteristic
12#14581783—21no—
13#14581783C2212yes—
14#14581772—21no—
15#14581772C2212yes—
16#14581768—21no—
17#14581768—21no—
18Q8×Dic83—21no—
19Q8×Dic83—21no—
20#14581757—21no—
21#14581757C2212yes—
22#14581750—21no—
23#14581750C2212yes—
24C42×D83C2211yescharacteristic
25#14581725C2214yes—
26Dic83⋊C8C2212yes—
27#14581717C2214yes—
28#14581715C2214yes—
29(C2×D83)⋊C8C2212yes—
30C4×C83⋊C8C2211yescharacteristic
31C4⋊Q8×C83C2211yescharacteristic
32C4⋊Q8×C83C2211yescharacteristic
33#106×C83C2214yes—
34C4×C8×C83C2211yescharacteristic
35C2×Dic166—21no—
36C2×Dic166C2312yes—
37C2×C4×D83D411yescharacteristic
38C2×C4×D83D411yescharacteristic
39C2×C4×D83C2311yescharacteristic
40C2×C83⋊C8C2312yes—
41C4⋊Dic83—21no—
42C4⋊Dic83—21no—
43C4⋊Dic83—21no—
44C4⋊Dic83—22no—
45C4⋊Dic83—22no—
46C4⋊Dic83C2312yes—
47Dic83⋊C4—22no—
48Dic83⋊C4—22no—
49(C2×D83)⋊C4—22no—
50(C2×D83)⋊C4—22no—
51C4×Dic83—22no—
52C4×Dic83—22no—
53C4×Dic83D411yescharacteristic
54C4×Dic83D411yescharacteristic
55C4×Dic83C2311yescharacteristic
56C2×Q8×C83—21no—
57C2×Q8×C83—21no—
58C2×C8×C83C2312yes—
59C4⋊C4×C83—21no—
60C4⋊C4×C83—21no—
61C4⋊C4×C83C2312yes—
62C4⋊C4×C83C2312yes—
63C42×C83C2311yescharacteristic
64C22×D83C2×D411yescharacteristic
65Q8×C83—42no—
66Q8×C83—42no—
67C2×C4×C83—22no—
68C2×C4×C83—22no—
69C2×C4×C83C2×D411yescharacteristic
70C2×C4×C83C2×D411yescharacteristic
71C2×C4×C83C2411yescharacteristic
72C2×Dic83—22no—
73C2×Dic83—22no—
74C2×Dic83C2×D411yescharacteristic
75C2×Dic83C2×D411yescharacteristic
76C2×Dic83C2×D411yescharacteristic
77C4×D83—22no—
78C4×D83—22no—
79C4×D83—22no—
80Dic166—24no—
81Dic166—42no—
82C8×C83—22no—
83C83⋊C8—22no—
84C22×C83C22×D411yescharacteristic
85C2×D83—21no—
86C2×D83—21no—
87C2×D83—21no—
88C4×C83—42no—
89C4×C83—42no—
90C4×C83#13512yes—
91C4×C83#13712yes—
92C4×C832−1+412yes—
93Dic83—21no—
94Dic83—21no—
95Dic83—21no—
96Dic83—42no—
97Dic83—42no—
98C2×C83#53611yescharacteristic
99C2×C83C2×#13511yescharacteristic
100C2×C83C2×#13711yescharacteristic
101D83—41no—
102#3231—831nomaximal, Sylow
103C83#323111yescharacteristic, complemented, Sylow
104#533—831no—
105#533—831no—
106#500—831no—
107#498D8311yescharacteristic, complemented
108#487—831no—
109#487—831no—
110#485—834no—
111#479—834no—
112#431—831no—
113C4⋊Q8—831no—
114C4⋊Q8—831no—
115C4⋊Q8C2×D8311yescharacteristic
116C4⋊Q8C2×D8311yescharacteristic
117#125—832no—
118#125—1661no—
119C22⋊Q8—832no—
120C22⋊Q8—1661no—
121C22⋊Q8—1661no—
122C22⋊Q8—1661no—
123C4×Q8—832no—
124C4×Q8—1661no—
125C4×Q8—1661no—
126C4×Q8—1661no—
127#117—832no—
128#117—1661no—
129C2×C42—831no—
130#107—834no—
131#106C2×D8314yes—
132C4⋊C8—832no—
133#103—834no—
134#103—834no—
135C22⋊C8—832no—
136C4×C8—831no—
137C4×C8C2×D8311yescharacteristic
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—832no—
141C2×Q8—1661no—
142C22×C4—831no—
143C22×C4—831no—
144C22×C4—831no—
145C2×C8—832no—
146C2×C8C22×D8312yes—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—832no—
150C4⋊C4—1662no—
151C4⋊C4—1662no—
152C4⋊C4—1662no—
153C4⋊C4—1662no—
154C4⋊C4—1661no—
155C4⋊C4—1661no—
156C4⋊C4—1661no—
157C4⋊C4C22×D8312yes—
158C4⋊C4C22×D8312yes—
159C22⋊C4—1662no—
160C22⋊C4—1662no—
161C42—831no—
162C42—831no—
163C42—831no—
164C42—1662no—
165C42—1662no—
166C42C22×D8311yescharacteristic
167C23—831no—
168Q8—42no—
169Q8—42no—
170Q8—3322no—
171Q8—1664no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—831no—
175C2×C4—831no—
176C2×C4—831no—
177C2×C4—1662no—
178C2×C4—1662no—
179C2×C4—1662no—
180C2×C4—1662no—
181C2×C4—1662no—
182C2×C4D4×D8311yescharacteristic
183C2×C4D4×D8311yescharacteristic
184C2×C4C23×D8311yescharacteristic
185C8—22no—
186C8—1662no—
187C22—1661no—
188C22—1661no—
189C22—1661no—
190C22C2×D4×D8311yescharacteristic
191C4—42no—
192C4—42no—
193C4—3322no—
194C4—3322no—
195C4—1661no—
196C4—1661no—
197C4—1661no—
198C4#1458181312yes—
199C4#1458182112yes—
200C4#1458184012yes—
201C2—3321no—
202C2#1912827611yescharacteristic
203C2C2×#1458181311yescharacteristic
204C2C2×#1458182111yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#498×C83C2
socleC22×C83C22×D4
prime radicalC22×D83C2×D4
hypercenter#498D83
hypocenterC83#3231
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
3. #19128104C2semidirect
4. #19128053C2semidirect
5. #19128052C2semidirect
7. #19128011C2semidirect
9. #498×C83C2semidirect
103. C83#3231semidirect
107. #498D83semidirect

Maximal subgroups

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

subgroupindex
#191282562
#191281042
#191280532
#191280522
#191280452
#191280112
#191277522
#498×C832
#323183