Finite Group Databasebeta

← #24914494

Subgroups — #24914494

Order 10624.

456 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C2×#14581791C212yesmaximal
2#19127634C212yesmaximal
3#19127622C212yesmaximal
4#19127621C212yesmaximal
5#19127607C211yesmaximal, characteristic
6#19127599C211yesmaximal, characteristic
7#19127587C212yesmaximal
8#19127584C212yesmaximal
9#389×C83C211yesmaximal, characteristic
10C22×Dic166C2212yes—
11C22×Dic166C2211yescharacteristic
12#14581791—28no—
13C2×C4⋊Dic83C2212yes—
14C2×C4⋊Dic83C2211yescharacteristic
15C2×C4⋊Dic83C2211yescharacteristic
16C2×C4⋊Dic83C2211yescharacteristic
17C2×Dic83⋊C4C2212yes—
18C2×Dic83⋊C4C2212yes—
19C2×Dic83⋊C4C2212yes—
20C2×C4×Dic83C2212yes—
21C83⋊#95C2212yes—
22C83⋊#95C2212yes—
23C83⋊#95C2212yes—
24C83⋊#95C2212yes—
25C83⋊#95C2212yes—
26C83⋊#95C2212yes—
27C2×C4⋊C4×C83C2212yes—
28C2×C4⋊C4×C83C2211yescharacteristic
29C2×C4⋊C4×C83C2211yescharacteristic
30C2×C42×C83C2211yescharacteristic
31#95×C83C2212yes—
32C2×Dic166—24no—
33C2×Dic166—41no—
34C2×Dic166—41no—
35C2×Dic166—42no—
36C2×Dic166—42no—
37C2×Dic166D414yes—
38C22×Dic83C2312yes—
39C22×Dic83C2312yes—
40C22×Dic83C2312yes—
41C22×Dic83C2312yes—
42C4⋊Dic83—24no—
43C4⋊Dic83—41no—
44C4⋊Dic83—41no—
45C4⋊Dic83Q814yescomplemented
46Dic83⋊C4—24no—
47Dic83⋊C4—24no—
48Dic83⋊C4—42no—
49C4×Dic83—24no—
50C22×C4×C83C2312yes—
51C22×C4×C83C2311yescharacteristic
52C22×C4×C83C2311yescharacteristic
53C22×C4×C83C2312yes—
54C22×C4×C83C2311yescharacteristic
55C4⋊C4×C83—24no—
56C4⋊C4×C83—41no—
57C4⋊C4×C83Q814yescomplemented
58C42×C83—41no—
59C23×C83C2411yescharacteristic
60C2×C4×C83—22no—
61C2×C4×C83—21no—
62C2×C4×C83—21no—
63C2×C4×C83—21no—
64C2×C4×C83—21no—
65C2×C4×C83—21no—
66C2×C4×C83—22no—
67C2×C4×C83—21no—
68C2×C4×C83—22no—
69C2×C4×C83—22no—
70C2×C4×C83—22no—
71C2×C4×C83C2×D412yes—
72C2×C4×C83C2×Q812yes—
73C2×C4×C83C2×Q812yes—
74C2×C4×C83C2×Q814yes—
75C2×Dic83—22no—
76C2×Dic83—22no—
77C2×Dic83—22no—
78C2×Dic83—22no—
79C2×Dic83—22no—
80C2×Dic83—22no—
81C2×Dic83—22no—
82C2×Dic83—22no—
83C2×Dic83—22no—
84C2×Dic83—22no—
85C2×Dic83C2×D414yes—
86C2×Dic83C2×Q814yes—
87Dic166—44no—
88Dic166—48no—
89C22×C83C22×D411yescharacteristic
90C22×C83C22×Q811yescharacteristic
91C22×C83C22×Q811yescharacteristic
92C22×C83H(5,𝔽2)11yescharacteristic
93C22×C83H(5,𝔽2)11yescharacteristic
94C22×C832−1+411yescharacteristic
95C22×C832−1+411yescharacteristic
96C4×C83—22no—
97C4×C83—22no—
98C4×C83—22no—
99C4×C83—24no—
100C4×C83—42no—
101Dic83—24no—
102Dic83—24no—
103Dic83—42no—
104Dic83—42no—
105C2×C83#53411yescharacteristic
106C2×C83#54411yescharacteristic
107C2×C83#54411yescharacteristic
108C2×C83D4×Q811yescharacteristic
109C2×C83D4×Q811yescharacteristic
110C2×C83Q8211yescharacteristic
111C2×C83#56311yescharacteristic
112#2734—831nomaximal, Sylow
113C83#273411yescharacteristic, complemented, Sylow
114C2×C4⋊Q8—832no—
115#398—832no—
116#395—832no—
117#395—832no—
118#391—831no—
119#389—831no—
120#389D8311yescharacteristic
121#387—832no—
122#387—832no—
123C22×Q8—831no—
124C22×Q8—832no—
125C4⋊Q8—1668no—
126C2×C4⋊C4—831no—
127C2×C4⋊C4—831no—
128C2×C4⋊C4—832no—
129C2×C4⋊C4—831no—
130C2×C4⋊C4—832no—
131C2×C4⋊C4—832no—
132C2×C4⋊C4—832no—
133C2×C4⋊C4C2×D8312yes—
134C2×C4⋊C4C2×D8311yescharacteristic
135C2×C4⋊C4C2×D8311yescharacteristic
136C2×C42—832no—
137C2×C42C2×D8311yescharacteristic
138#95—832no—
139#95—832no—
140#95—832no—
141#95—832no—
142#95—832no—
143#95—832no—
144#95C2×D8312yes—
145C2×Q8—3321no—
146C2×Q8—3321no—
147C2×Q8—3322no—
148C2×Q8—3322no—
149C2×Q8—834no—
150C2×Q8—1664no—
151C22×C4—832no—
152C22×C4—832no—
153C22×C4—832no—
154C22×C4—832no—
155C22×C4C22×D8312yes—
156C22×C4C22×D8311yescharacteristic
157C22×C4C22×D8311yescharacteristic
158C22×C4C22×D8311yescharacteristic
159C22×C4C22×D8312yes—
160C4⋊C4—24no—
161C4⋊C4—41no—
162C4⋊C4—3322no—
163C4⋊C4—3321no—
164C4⋊C4—3321no—
165C4⋊C4—834no—
166C4⋊C4—1664no—
167C4⋊C4—1664no—
168C4⋊C4—1664no—
169C4⋊C4Dic16614yescomplemented
170C42—41no—
171C42—1664no—
172C23C23×D8311yescharacteristic
173Q8—3324no—
174Q8—3328no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—834no—
187C2×C4—834no—
188C2×C4—1662no—
189C2×C4—1662no—
190C2×C4—1662no—
191C2×C4—1662no—
192C2×C4—1662no—
193C2×C4—1662no—
194C2×C4—1662no—
195C2×C4—1662no—
196C2×C4—1662no—
197C2×C4—1662no—
198C2×C4D4×D8312yes—
199C2×C4Q8×D8312yes—
200C2×C4C2×Dic16612yes—
201C2×C4C2×Dic16614yes—
202C22C2×D4×D8311yescharacteristic
203C22C2×Q8×D8311yescharacteristic
204C22C22×Dic16611yescharacteristic
205C22#1458183711yescharacteristic
206C22#1458183811yescharacteristic
207C22#1458183911yescharacteristic
208C22#1458184011yescharacteristic
209C4—22no—
210C4—22no—
211C4—24no—
212C4—22no—
213C4—42no—
214C4—3322no—
215C4—3322no—
216C4—1664no—
217C4—1664no—
218C2#1912825911yescharacteristic
219C2#1912833011yescharacteristic
220C2#1912833211yescharacteristic
221C2#1912836611yescharacteristic
222C2D4×Dic16611yescharacteristic
223C2Q8×Dic16611yescharacteristic
224C2#1912845511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, prime radical, cosocleC23×C83C24
Fitting#389×C83C2
hypercenter#389D83
hypocenterC83#2734
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
45. C4⋊Dic83Q8semidirect
57. C4⋊C4×C83Q8semidirect
113. C83#2734semidirect
169. C4⋊C4Dic166semidirect

Maximal subgroups

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

subgroupindex
C2×#145817912
#191276342
#191276222
#191276212
#191276072
#191275992
#191275872
#191275842
#389×C832
#273483