Finite Group Databasebeta

← #24913614

Subgroups — #24913614

Order 10624.

498 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 137 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×#14581775C211yesmaximal, characteristic
2#19127639C211yesmaximal, characteristic, complemented
3#19127636C211yesmaximal, characteristic, complemented
4#19127629C212yesmaximal, complemented
5#19127620C211yesmaximal, characteristic, complemented
6#19127616C212yesmaximal
7#19127612C211yesmaximal, characteristic, complemented
8#19127596C212yesmaximal, complemented
9#19127591C212yesmaximal, complemented
10C4×C22⋊Dic83C211yesmaximal, characteristic, complemented
11C2×C22⋊Q8×C83C211yesmaximal, characteristic, complemented
12C22×C22⋊D83C2211yescharacteristic
13#14581775—24no—
14C2×Dic83⋊C4C2211yescharacteristic
15C2×Dic83⋊C4C2212yes—
16C2×Dic83⋊C4C2211yescharacteristic
17C2×Dic83⋊C4C2212yes—
18C2×C22⋊Dic83C2212yes—
19C2×C22⋊Dic83C2211yescharacteristic
20C2×(C2×D83)⋊C4C2211yescharacteristic
21C2×(C2×D83)⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2211yescharacteristic
23C2×(C2×D83)⋊C4C2212yes—
24C2×C4×Dic83C2212yes—
25C83⋊#95C2212yes—
26C83⋊#95C2212yes—
27C83⋊#95C2212yes—
28C83⋊#95C2212yes—
29C83⋊#95C2211yescharacteristic
30C83⋊#95C2211yescharacteristic
31C22×Q8×C83C2211yescharacteristic
32C23×C4×C83C2211yescharacteristic
33C22⋊Q8×C83—24no—
34C2×C4⋊C4×C83C2212yes—
35C2×C4⋊C4×C83C2211yescharacteristic
36C2×C22⋊C4×C83C2212yes—
37C23×D83C2311yescharacteristic
38C2×C22⋊D83—22no—
39C2×C22⋊D83—42no—
40C22×Dic83C2311yescharacteristic
41C22×Dic83C2312yes—
42C22×Dic83C2311yescharacteristic
43C22×Dic83C2311yescharacteristic
44C22×Dic83C2312yes—
45Dic83⋊C4—22no—
46Dic83⋊C4—22no—
47Dic83⋊C4—24no—
48Dic83⋊C4—42no—
49C22⋊Dic83—22no—
50C22⋊Dic83—24no—
51(C2×D83)⋊C4—22no—
52(C2×D83)⋊C4—22no—
53(C2×D83)⋊C4—24no—
54(C2×D83)⋊C4—42no—
55C4×Dic83—42no—
56C24×C83C2311yescharacteristic
57C2×Q8×C83—22no—
58C2×Q8×C83—42no—
59C22×C4×C83—42no—
60C22×C4×C83D414yes—
61C22×C4×C83C2312yes—
62C22×C4×C83C2312yes—
63C22×C4×C83C2311yescharacteristic
64C22×C4×C83C2311yescharacteristic
65C4⋊C4×C83—22no—
66C4⋊C4×C83—24no—
67C22⋊C4×C83—24no—
68C23×C83—22no—
69C23×C83—22no—
70C23×C83—21no—
71C23×C83—21no—
72C23×C83C2×D412yes—
73C23×C83C2411yescharacteristic
74C22×D83—21no—
75C22×D83—22no—
76C22×D83—22no—
77C22×D83—21no—
78C22×D83—21no—
79Q8×C83—44no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83—22no—
84C2×C4×C83—22no—
85C2×C4×C83—22no—
86C2×C4×C83—22no—
87C2×C4×C83—22no—
88C2×C4×C83—44no—
89C2×C4×C83C2×D412yes—
90C2×C4×C83C2×D412yes—
91C22⋊D83—44no—
92C2×Dic83—22no—
93C2×Dic83—22no—
94C2×Dic83—21no—
95C2×Dic83—21no—
96C2×Dic83—21no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83—22no—
100C2×Dic83—22no—
101C2×Dic83—21no—
102C2×Dic83—21no—
103C2×Dic83—22no—
104C2×Dic83#4114yes—
105C22×C83—22no—
106C22×C83—22no—
107C22×C83—22no—
108C22×C83—42no—
109C22×C83—42no—
110C22×C83C22×D411yescharacteristic
111C22×C83C2×#4112yes—
112C22×C83H(5,𝔽2)12yes—
113C22×C832−1+411yescharacteristic
114C22×C832−1+411yescharacteristic
115C2×D83—41no—
116C2×D83—41no—
117C2×D83—41no—
118C2×D83—42no—
119C2×D83—42no—
120C4×C83—22no—
121C4×C83—22no—
122C4×C83—42no—
123C4×C83—42no—
124Dic83—24no—
125Dic83—41no—
126Dic83—41no—
127Dic83—41no—
128Dic83—42no—
129C2×C83—42no—
130C2×C83#53211yescharacteristic
131C2×C83#53411yescharacteristic
132C2×C83#53611yescharacteristic
133C2×C83#53912yes—
134C2×C83#54212yes—
135D83—81no—
136#2663—831nomaximal, Sylow
137C83#266311yescharacteristic, complemented, Sylow
138C2×#123—831no—
139C2×C22⋊Q8D8311yescharacteristic, complemented
140#400—831no—
141#399—831no—
142#397—832no—
143#395—831no—
144#394—832no—
145#393—831no—
146#389—832no—
147#388—832no—
148C4×C22⋊C4—831no—
149C22×Q8C2×D8311yescharacteristic
150C22×D4—831no—
151C23×C4C2×D8311yescharacteristic
152#123—1664no—
153C22⋊Q8—24no—
154C2×C4⋊C4—832no—
155C2×C4⋊C4—831no—
156C2×C4⋊C4—831no—
157C2×C4⋊C4—832no—
158C2×C4⋊C4C2×D8312yes—
159C2×C4⋊C4C2×D8311yescharacteristic
160C2×C22⋊C4—832no—
161C2×C22⋊C4—831no—
162C2×C22⋊C4—832no—
163C2×C22⋊C4—831no—
164C2×C22⋊C4—831no—
165C2×C22⋊C4—832no—
166C2×C22⋊C4C2×D8312yes—
167C2×C42—832no—
168#95—831no—
169#95—831no—
170#95—832no—
171#95—832no—
172#95—832no—
173#95—832no—
174C24—831no—
175C24C22×D8311yescharacteristic
176C2×Q8—22no—
177C2×Q8—42no—
178C2×D4—3322no—
179C2×D4—1662no—
180C22×C4—42no—
181C22×C4—832no—
182C22×C4—832no—
183C22×C4—831no—
184C22×C4—831no—
185C22×C4—831no—
186C22×C4C22⋊D8314yes—
187C22×C4C22×D8312yes—
188C22×C4C22×D8312yes—
189C22×C4C22×D8311yescharacteristic
190C22×C4C22×D8311yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—24no—
193C4⋊C4—3322no—
194C4⋊C4—1664no—
195C4⋊C4—1662no—
196C4⋊C4—1662no—
197C22⋊C4—24no—
198C22⋊C4—3322no—
199C22⋊C4—1664no—
200C22⋊C4—1664no—
201C22⋊C4—1662no—
202C22⋊C4—1662no—
203C22⋊C4—1662no—
204C42—3322no—
205C23—22no—
206C23—22no—
207C23—21no—
208C23—21no—
209C23—1661no—
210C23—1662no—
211C23—1662no—
212C23—1661no—
213C23—1661no—
214C23C2×C22⋊D8312yes—
215C23C23×D8311yescharacteristic
216Q8—44no—
217D4—3324no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—44no—
227C2×C4—834no—
228C2×C4—1661no—
229C2×C4—1661no—
230C2×C4—1661no—
231C2×C4—1662no—
232C2×C4—1662no—
233C2×C4—1662no—
234C2×C4—1662no—
235C2×C4—1662no—
236C2×C4—1661no—
237C2×C4—1661no—
238C2×C4—1662no—
239C2×C4—1662no—
240C2×C4C2×C22⋊D8312yes—
241C2×C4C2×C22⋊D8312yes—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—42no—
246C22—42no—
247C22—3322no—
248C22—3321no—
249C22—3321no—
250C22—3321no—
251C22—3322no—
252C22C22×C22⋊D8311yescharacteristic
253C22#41×D8312yes—
254C22#1458183712yes—
255C22#1458184011yescharacteristic
256C22#1458184011yescharacteristic
257C4—22no—
258C4—22no—
259C4—42no—
260C4—42no—
261C4—3322no—
262C4—3321no—
263C4—3321no—
264C4—3321no—
265C4—1664no—
266C2—42no—
267C2—6641no—
268C2#1912824711yescharacteristic
269C2#1912826011yescharacteristic
270C2#1912827511yescharacteristic
271C2#1912829712yes—
272C2#1912832112yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×C22⋊Q8×C83C2
prime radicalC22×C22⋊D83C22
hypercenterC2×C22⋊Q8D83
hypocenterC83#2663
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #19127639C2semidirect
3. #19127636C2semidirect
4. #19127629C2semidirect
5. #19127620C2semidirect
7. #19127612C2semidirect
8. #19127596C2semidirect
9. #19127591C2semidirect
10. C4×C22⋊Dic83C2semidirect
11. C2×C22⋊Q8×C83C2semidirect
137. C83#2663semidirect
139. C2×C22⋊Q8D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817752
#191276392
#191276362
#191276292
#191276202
#191276162
#191276122
#191275962
#191275912
C4×C22⋊Dic832
C2×C22⋊Q8×C832
#266383