Finite Group Databasebeta

← #24915428

Subgroups — #24915428

Order 10624.

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

normal subgroups only

This group has 408 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#19127642C212yesmaximal, complemented
2#19127637C211yesmaximal, characteristic
3#19127634C212yesmaximal, complemented
4#19127596C212yesmaximal, complemented
5#19127596C212yesmaximal, complemented
6#19127589C212yesmaximal
7#19127573C212yesmaximal
8#19127573C211yesmaximal, characteristic
9#399×C83C211yesmaximal, characteristic
10C23×Dic83C2211yescharacteristic
11C2×C4⋊Dic83C2211yescharacteristic
12C2×Dic83⋊C4C2212yes—
13C2×Dic83⋊C4C2212yes—
14C2×Dic83⋊C4C2212yes—
15C2×Dic83⋊C4C2212yes—
16C2×Dic83⋊C4C2212yes—
17C2×C22⋊Dic83C2212yes—
18C2×C22⋊Dic83C2211yescharacteristic
19C2×C4×Dic83C2212yes—
20C83⋊#95C2212yes—
21C83⋊#95C2212yes—
22C83⋊#95C2212yes—
23C83⋊#95C2212yes—
24C83⋊#95C2212yes—
25C83⋊#95C2211yescharacteristic
26C2×C4⋊C4×C83C2211yescharacteristic
27C2×C4⋊C4×C83C2212yes—
28C2×C22⋊C4×C83C2212yes—
29C2×C22⋊C4×C83C2211yescharacteristic
30#95×C83C2211yescharacteristic
31C22×Dic83—42no—
32C22×Dic83—41no—
33C22×Dic83C2312yes—
34C22×Dic83C2312yes—
35C22×Dic83C2312yes—
36C22×Dic83C2312yes—
37C4⋊Dic83—41no—
38Dic83⋊C4—24no—
39Dic83⋊C4—24no—
40Dic83⋊C4—42no—
41Dic83⋊C4—42no—
42Dic83⋊C4—42no—
43C22⋊Dic83—24no—
44C22⋊Dic83Q814yes—
45C4×Dic83—42no—
46C24×C83C2311yescharacteristic
47C22×C4×C83C2312yes—
48C22×C4×C83C2311yescharacteristic
49C22×C4×C83C2312yes—
50C22×C4×C83C2311yescharacteristic
51C4⋊C4×C83—41no—
52C4⋊C4×C83—42no—
53C22⋊C4×C83—42no—
54C22⋊C4×C83—41no—
55C23×C83—22no—
56C23×C83—21no—
57C23×C83—22no—
58C23×C83—21no—
59C23×C83C2×Q812yes—
60C23×C83C2411yescharacteristic
61C2×C4×C83—22no—
62C2×C4×C83—22no—
63C2×C4×C83—22no—
64C2×C4×C83—21no—
65C2×C4×C83—22no—
66C2×C4×C83—21no—
67C2×C4×C83—22no—
68C2×C4×C83—22no—
69C2×C4×C83—22no—
70C2×C4×C83—22no—
71C2×Dic83—22no—
72C2×Dic83—22no—
73C2×Dic83—22no—
74C2×Dic83—22no—
75C2×Dic83—22no—
76C2×Dic83—22no—
77C2×Dic83—22no—
78C2×Dic83—22no—
79C2×Dic83—22no—
80C2×Dic83—22no—
81C2×Dic83—44no—
82C2×Dic83C2×Q814yes—
83C2×Dic83#4114yes—
84C22×C83—22no—
85C22×C83—24no—
86C22×C83—42no—
87C22×C83—41no—
88C22×C83—41no—
89C22×C83C22×Q811yescharacteristic
90C22×C83C2×#4112yes—
91C22×C83H(5,𝔽2)12yes—
92C22×C83H(5,𝔽2)11yescharacteristic
93C22×C832−1+411yescharacteristic
94C4×C83—42no—
95C4×C83—41no—
96C4×C83—42no—
97C4×C83—41no—
98Dic83—24no—
99Dic83—24no—
100Dic83—42no—
101Dic83—42no—
102C2×C83—42no—
103C2×C83#53912yes—
104C2×C83#54411yescharacteristic
105C2×C83#55412yes—
106C2×C83#55911yescharacteristic
107C2×C83#56011yescharacteristic
108#2807—831nomaximal, Sylow
109C83#280711yescharacteristic, complemented, Sylow
110#400—832no—
111#399—831no—
112#399D8311yescharacteristic
113#398—832no—
114#389—832no—
115#389—832no—
116#388—832no—
117#385—831no—
118#385—832no—
119C23×C4—831no—
120C2×C4⋊C4—832no—
121C2×C4⋊C4—832no—
122C2×C4⋊C4—832no—
123C2×C4⋊C4—831no—
124C2×C4⋊C4—832no—
125C2×C4⋊C4—832no—
126C2×C4⋊C4C2×D8311yescharacteristic
127C2×C4⋊C4C2×D8312yes—
128C2×C22⋊C4—831no—
129C2×C22⋊C4—832no—
130C2×C22⋊C4C2×D8312yes—
131C2×C22⋊C4C2×D8311yescharacteristic
132C2×C42—832no—
133#95—832no—
134#95—832no—
135#95—832no—
136#95—832no—
137#95—832no—
138#95—831no—
139#95C2×D8311yescharacteristic
140C24C22×D8311yescharacteristic
141C22×C4—3321no—
142C22×C4—3322no—
143C22×C4—832no—
144C22×C4—832no—
145C22×C4—832no—
146C22×C4—832no—
147C22×C4C22×D8312yes—
148C22×C4C22×D8311yescharacteristic
149C22×C4C22×D8312yes—
150C22×C4C22×D8311yescharacteristic
151C4⋊C4—41no—
152C4⋊C4—42no—
153C4⋊C4—3322no—
154C4⋊C4—3322no—
155C4⋊C4—3322no—
156C4⋊C4—3321no—
157C4⋊C4—1664no—
158C4⋊C4—1664no—
159C22⋊C4—42no—
160C22⋊C4—41no—
161C22⋊C4—834no—
162C22⋊C4—1664no—
163C42—3322no—
164C23—22no—
165C23—21no—
166C23—22no—
167C23—21no—
168C23Q8×D8312yes—
169C23C23×D8311yescharacteristic
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—3324no—
181C2×C4—834no—
182C2×C4—834no—
183C2×C4—1662no—
184C2×C4—1662no—
185C2×C4—1662no—
186C2×C4—1662no—
187C2×C4—1662no—
188C2×C4—1662no—
189C2×C4—1662no—
190C2×C4—1662no—
191C2×C4—1662no—
192C2×C4—1662no—
193C22—22no—
194C22—24no—
195C22—41no—
196C22—41no—
197C22—42no—
198C22C2×Q8×D8311yescharacteristic
199C22#41×D8312yes—
200C22#1458183711yescharacteristic
201C22#1458183712yes—
202C22#1458183911yescharacteristic
203C4—41no—
204C4—42no—
205C4—41no—
206C4—42no—
207C4—3322no—
208C4—3322no—
209C4—1664no—
210C4—1664no—
211C2—42no—
212C2#1912829412yes—
213C2#1912833011yescharacteristic
214C2#1912840512yes—
215C2#1912843511yescharacteristic
216C2#1912843811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#399×C83C2
prime radicalC24×C83C23
hypercenter#399D83
hypocenterC83#2807
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. #19127642C2semidirect
3. #19127634C2semidirect
4. #19127596C2semidirect
5. #19127596C2semidirect
109. C83#2807semidirect

Maximal subgroups

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

subgroupindex
#191276422
#191276372
#191276342
#191275962
#191275892
#191275732
#399×C832
#280783