Finite Group Databasebeta

← #24920353

Subgroups — #24920353

Order 10624.

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

normal subgroups only

This group has 324 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#19128430C211yesmaximal, characteristic
2#19128426C211yesmaximal, characteristic
3#19128104C211yesmaximal, characteristic
4#19128101C212yesmaximal
5#19128011C212yesmaximal
6#19128006C212yesmaximal
7#19127987C211yesmaximal, characteristic
8#19127825C211yesmaximal, characteristic
9C4×#13050580C211yesmaximal, characteristic
10#19127793C212yesmaximal
11#475×C83C211yesmaximal, characteristic
12C2×#13050580C2211yescharacteristic
13#14581792—21no—
14#14581792C2211yescharacteristic
15#14581790—21no—
16#14581783—22no—
17#14581783C2212yes—
18#14581783C2212yes—
19#14581782—22no—
20Q8×Dic83—21no—
21Q8×Dic83C2212yes—
22C4×Dic166—21no—
23C4×Dic166C2211yescharacteristic
24C4×Dic166C2211yescharacteristic
25#14581757C2212yes—
26#14581725C2211yescharacteristic
27#14581725C2212yes—
28#14581725C2212yes—
29#14581724C2212yes—
30Dic83⋊C8C2212yes—
31#14581717C2212yes—
32#14581716C2211yescharacteristic
33#14581715C2211yescharacteristic
34C4×C83⋊C8C2211yescharacteristic
35C8×Dic83C2212yes—
36C4⋊Q8×C83C2211yescharacteristic
37C4×Q8×C83C2211yescharacteristic
38#106×C83C2212yes—
39C4⋊C8×C83C2211yescharacteristic
40#103×C83C2212yes—
41C2×Dic166—21no—
42C2×Dic166D411yescharacteristic
43C2×Dic166C2311yescharacteristic
44#13050580Q814yescomplemented
45C2×C83⋊C8C2311yescharacteristic
46C2×C83⋊C8C2311yescharacteristic
47C4⋊Dic83—21no—
48C4⋊Dic83—22no—
49C4⋊Dic83—22no—
50C4⋊Dic83—22no—
51C4⋊Dic83D411yescharacteristic
52C4⋊Dic83C2311yescharacteristic
53C4⋊Dic83C2312yes—
54Dic83⋊C4—21no—
55Dic83⋊C4—22no—
56Dic83⋊C4—22no—
57Dic83⋊C4—22no—
58Dic83⋊C4D412yes—
59C4×Dic83—21no—
60C4×Dic83—22no—
61C4×Dic83—22no—
62C4×Dic83C2312yes—
63C2×Q8×C83—21no—
64C2×Q8×C83C2311yescharacteristic
65C2×C8×C83C2312yes—
66C4⋊C4×C83—21no—
67C4⋊C4×C83—21no—
68C4⋊C4×C83C2311yescharacteristic
69C4⋊C4×C83C2312yes—
70C42×C83—21no—
71C42×C83C2311yescharacteristic
72Q8×C83—21no—
73Q8×C83—42no—
74Q8×C83C2×Q812yes—
75C2×C4×C83—21no—
76C2×C4×C83—21no—
77C2×C4×C83—22no—
78C2×C4×C83C2×D411yescharacteristic
79C2×C4×C83C2×D411yescharacteristic
80C2×C4×C83C2411yescharacteristic
81C2×Dic83—21no—
82C2×Dic83—21no—
83C2×Dic83—22no—
84C2×Dic83C2×D412yes—
85C2×Dic83C2×D412yes—
86Dic166—22no—
87Dic166—21no—
88Dic166—42no—
89Dic166C2×Q812yes—
90C8×C83—22no—
91C83⋊C8—21no—
92C83⋊C8C2×Q812yes—
93C22×C83C22×D411yescharacteristic
94C4×C83—22no—
95C4×C83—21no—
96C4×C83—42no—
97C4×C83—41no—
98C4×C83#13512yes—
99C4×C83C22×Q811yescharacteristic
100C4×C832−1+411yescharacteristic
101Dic83—22no—
102Dic83—22no—
103Dic83—22no—
104Dic83—42no—
105Dic83—41no—
106C2×C83D4×Q811yescharacteristic
107C2×C83C2×#13511yescharacteristic
108C2×C83#57811yescharacteristic
109#3383—831nomaximal, Sylow
110C83#338311yescharacteristic, complemented, Sylow
111#557—831no—
112#557—831no—
113#500—831no—
114#499—832no—
115#479—832no—
116#479—832no—
117#475—831no—
118#475D8311yescharacteristic
119C8×Q8—831no—
120C4×Q16—831no—
121C4×Q16—832no—
122C2×Q16—831no—
123C4⋊Q8—831no—
124C4⋊Q8—1661no—
125C4⋊Q8—1661no—
126C4⋊Q8C2×D8311yescharacteristic
127#125—832no—
128#125—832no—
129#125—1662no—
130#125—1662no—
131C4×Q8—831no—
132C4×Q8—832no—
133C4×Q8—832no—
134C4×Q8—831no—
135C4×Q8—1661no—
136C4×Q8—1661no—
137C4×Q8C2×D8311yescharacteristic
138#107—831no—
139#107—832no—
140#107—832no—
141#106—832no—
142#106C2×D8312yes—
143C4⋊C8—832no—
144C4⋊C8C2×D8311yescharacteristic
145#103—831no—
146#103—832no—
147#103—831no—
148#103C2×D8312yes—
149C4×C8—832no—
150C4×C8—831no—
151C2×Q8—21no—
152C2×Q8—831no—
153C2×Q8—831no—
154C2×Q8—1661no—
155C2×Q8C22×D8311yescharacteristic
156Q16—834no—
157C2×C8—831no—
158C2×C8—831no—
159C2×C8C22×D8312yes—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—831no—
163C4⋊C4—832no—
164C4⋊C4—831no—
165C4⋊C4—832no—
166C4⋊C4—1662no—
167C4⋊C4—1662no—
168C4⋊C4—1662no—
169C4⋊C4—1662no—
170C4⋊C4—1662no—
171C4⋊C4—1662no—
172C4⋊C4—1661no—
173C4⋊C4—1661no—
174C4⋊C4C22×D8311yescharacteristic
175C4⋊C4C22×D8312yes—
176C42—21no—
177C42—832no—
178C42—1662no—
179C42—1662no—
180C42—1661no—
181C42C22×D8311yescharacteristic
182Q8—21no—
183Q8—42no—
184Q8—3322no—
185Q8—832no—
186Q8—1662no—
187Q8—1661no—
188Q8Q8×D8312yes—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—832no—
193C2×C4—832no—
194C2×C4—1661no—
195C2×C4—1662no—
196C2×C4—1661no—
197C2×C4D4×D8311yescharacteristic
198C2×C4D4×D8311yescharacteristic
199C2×C4C23×D8311yescharacteristic
200C8—22no—
201C8—832no—
202C8—1661no—
203C22C2×D4×D8311yescharacteristic
204C4—22no—
205C4—21no—
206C4—42no—
207C4—41no—
208C4—3322no—
209C4—3321no—
210C4—1662no—
211C4—1662no—
212C4—1662no—
213C4#1458181312yes—
214C4C2×Q8×D8311yescharacteristic
215C4#1458183911yescharacteristic
216C2#1912836411yescharacteristic
217C2C2×#1458181311yescharacteristic
218C2#1912851711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#475×C83C2
socle, prime radicalC22×C83C22×D4
hypercenter#475D83
hypocenterC83#3383
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
44. #13050580Q8semidirect
110. C83#3383semidirect

Maximal subgroups

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

subgroupindex
#191284302
#191284262
#191281042
#191281012
#191280112
#191280062
#191279872
#191278252
C4×#130505802
#191277932
#475×C832
#338383