Finite Group Databasebeta

← #24913244

Subgroups — #24913244

Order 10624.

526 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14581777C211yesmaximal, characteristic
2C2×C4⋊C4×D83C211yesmaximal, characteristic, complemented
3#19127642C212yesmaximal, complemented
4#19127636C212yesmaximal, complemented
5#19127631C211yesmaximal, characteristic, complemented
6#19127628C211yesmaximal, characteristic, complemented
7#19127627C211yesmaximal, characteristic, complemented
8#19127579C212yesmaximal
9#19127572C212yesmaximal, complemented
10#19127563C211yesmaximal, characteristic, complemented
11#398×C83C211yesmaximal, characteristic, complemented
12C22×D332C2211yescharacteristic
13C22×C4×D83C2211yescharacteristic
14C22×C4×D83C2212yes—
15#14581777—24no—
16C2×C4⋊Dic83C2211yescharacteristic
17C2×Dic83⋊C4C2212yes—
18C2×Dic83⋊C4C2211yescharacteristic
19C2×Dic83⋊C4C2212yes—
20C4⋊C4×D83—24no—
21C2×(C2×D83)⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2211yescharacteristic
23C2×(C2×D83)⋊C4C2211yescharacteristic
24C2×(C2×D83)⋊C4C2211yescharacteristic
25C2×(C2×D83)⋊C4C2212yes—
26C2×(C2×D83)⋊C4C2212yes—
27C83⋊#95C2211yescharacteristic
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2212yes—
32C2×C4⋊C4×C83C2211yescharacteristic
33C2×C4⋊C4×C83C2211yescharacteristic
34C2×C4⋊C4×C83C2212yes—
35#95×C83C2211yescharacteristic
36#95×C83C2212yes—
37C23×D83C2311yescharacteristic
38C23×D83C2311yescharacteristic
39C2×D332—22no—
40C2×D332—24no—
41C22×Dic83C2312yes—
42C22×Dic83C2311yescharacteristic
43C22×Dic83C2312yes—
44C22×Dic83C2311yescharacteristic
45C2×C4×D83—24no—
46C2×C4×D83—24no—
47C2×C4×D83—42no—
48C2×C4×D83—42no—
49C2×C4×D83D414yes—
50C4⋊Dic83—22no—
51Dic83⋊C4—22no—
52Dic83⋊C4—24no—
53Dic83⋊C4—42no—
54(C2×D83)⋊C4—22no—
55(C2×D83)⋊C4—24no—
56(C2×D83)⋊C4—22no—
57(C2×D83)⋊C4—22no—
58(C2×D83)⋊C4—42no—
59(C2×D83)⋊C4—42no—
60C22×C4×C83C2312yes—
61C22×C4×C83C2311yescharacteristic
62C22×C4×C83C2312yes—
63C22×C4×C83C2311yescharacteristic
64C22×C4×C83C2311yescharacteristic
65C4⋊C4×C83—22no—
66C4⋊C4×C83—22no—
67C4⋊C4×C83—42no—
68C23×C83C2411yescharacteristic
69C22×D83—21no—
70C22×D83—21no—
71C22×D83—22no—
72C22×D83—21no—
73C22×D83—22no—
74C22×D83—21no—
75C22×D83—21no—
76C22×D83—22no—
77C22×D83C2×D412yes—
78C22×D83#4114yes—
79C2×C4×C83—22no—
80C2×C4×C83—21no—
81C2×C4×C83—22no—
82C2×C4×C83—21no—
83C2×C4×C83—22no—
84C2×C4×C83—22no—
85C2×C4×C83—21no—
86C2×C4×C83—22no—
87C2×C4×C83—22no—
88C2×C4×C83—22no—
89C2×C4×C83—21no—
90C2×C4×C83—22no—
91C2×C4×C83C2×D412yes—
92C2×Dic83—22no—
93C2×Dic83—21no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83—22no—
100C2×Dic83—22no—
101C2×Dic83C2×D412yes—
102D332—44no—
103C4×D83—28no—
104C4×D83—48no—
105C22×C83C22×D411yescharacteristic
106C22×C83C2×#4112yes—
107C22×C83H(5,𝔽2)11yescharacteristic
108C22×C83H(5,𝔽2)11yescharacteristic
109C22×C832−1+412yes—
110C2×D83—22no—
111C2×D83—22no—
112C2×D83—24no—
113C2×D83—24no—
114C2×D83—41no—
115C2×D83—41no—
116C2×D83—41no—
117C2×D83—42no—
118C2×D83—42no—
119C2×D83#12314yes—
120C4×C83—22no—
121C4×C83—41no—
122C4×C83—42no—
123C4×C83—41no—
124C4×C83—42no—
125Dic83—22no—
126Dic83—42no—
127Dic83—41no—
128Dic83—42no—
129C2×C83C2×#12311yescharacteristic
130C2×C83#53511yescharacteristic
131C2×C83#53611yescharacteristic
132C2×C83#53912yes—
133C2×C83#53912yes—
134D83—24no—
135D83—81no—
136#2623—831nomaximal, Sylow
137C83#262311yescharacteristic, complemented, Sylow
138C2×#123—831no—
139C22×C4⋊C4—831no—
140#400—832no—
141#399—832no—
142#398—831no—
143#398D8311yescharacteristic, complemented
144#397—831no—
145#397—831no—
146#386—832no—
147#385—832no—
148#381—831no—
149C22×D4—831no—
150C23×C4—832no—
151C23×C4—831no—
152#123—1664no—
153C2×C4⋊C4—832no—
154C2×C4⋊C4—831no—
155C2×C4⋊C4—832no—
156C2×C4⋊C4—831no—
157C2×C4⋊C4—1664no—
158C2×C4⋊C4C2×D8311yescharacteristic
159C2×C4⋊C4C2×D8311yescharacteristic
160C2×C4⋊C4C2×D8312yes—
161C2×C22⋊C4—832no—
162C2×C22⋊C4—832no—
163C2×C22⋊C4—831no—
164C2×C22⋊C4—831no—
165C2×C22⋊C4—831no—
166C2×C22⋊C4—832no—
167#95—831no—
168#95—832no—
169#95—832no—
170#95—832no—
171#95—832no—
172#95C2×D8311yescharacteristic
173#95C2×D8312yes—
174C24—831no—
175C24—831no—
176C2×D4—1662no—
177C2×D4—1664no—
178C22×C4—3322no—
179C22×C4—3322no—
180C22×C4—831no—
181C22×C4—834no—
182C22×C4—831no—
183C22×C4—832no—
184C22×C4—832no—
185C22×C4—1664no—
186C22×C4—1664no—
187C22×C4C22×D8312yes—
188C22×C4C22×D8311yescharacteristic
189C22×C4C22×D8312yes—
190C22×C4C22×D8311yescharacteristic
191C22×C4C22×D8311yescharacteristic
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—42no—
195C4⋊C4—3322no—
196C4⋊C4—1664no—
197C4⋊C4—1662no—
198C4⋊C4—1662no—
199C22⋊C4—3322no—
200C22⋊C4—3322no—
201C22⋊C4—1662no—
202C22⋊C4—1662no—
203C22⋊C4—1664no—
204C22⋊C4—1662no—
205C23—832no—
206C23—834no—
207C23—1661no—
208C23—1662no—
209C23—1661no—
210C23—1661no—
211C23—1662no—
212C23—1661no—
213C23—1661no—
214C23—1662no—
215C23C23×D8311yescharacteristic
216D4—3324no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—3328no—
230C2×C4—832no—
231C2×C4—1668no—
232C2×C4—1661no—
233C2×C4—1662no—
234C2×C4—1662no—
235C2×C4—1662no—
236C2×C4—1662no—
237C2×C4—1662no—
238C2×C4—1662no—
239C2×C4—1662no—
240C2×C4—1662no—
241C2×C4D4×D8312yes—
242C22—3321no—
243C22—3321no—
244C22—3321no—
245C22—3322no—
246C22—3322no—
247C22—834no—
248C22—1662no—
249C22—1662no—
250C22—1664no—
251C22—1664no—
252C22C2×D4×D8311yescharacteristic
253C22#41×D8312yes—
254C22#1458183711yescharacteristic
255C22#1458183811yescharacteristic
256C22#1458184012yes—
257C4—22no—
258C4—41no—
259C4—42no—
260C4—41no—
261C4—42no—
262C4—3322no—
263C4—3321no—
264C4—3322no—
265C4—1662no—
266C2—6641no—
267C2—1664no—
268C2#123×D8311yescharacteristic
269C2#1912826711yescharacteristic
270C2#1912827611yescharacteristic
271C2#1912829712yes—
272C2#1912829912yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#398×C83C2
prime radicalC22×D332C22
hypercenter#398D83
hypocenterC83#2623
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D83C2semidirect
3. #19127642C2semidirect
4. #19127636C2semidirect
5. #19127631C2semidirect
6. #19127628C2semidirect
7. #19127627C2semidirect
9. #19127572C2semidirect
10. #19127563C2semidirect
11. #398×C83C2semidirect
137. C83#2623semidirect
143. #398D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817772
C2×C4⋊C4×D832
#191276422
#191276362
#191276312
#191276282
#191276272
#191275792
#191275722
#191275632
#398×C832
#262383