Finite Group Databasebeta

← #24911775

Subgroups — #24911775

Order 10624.

584 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 151 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×#14581754C211yesmaximal, characteristic
2C2×C4⋊C4×D83C211yesmaximal, characteristic, complemented
3#19127640C211yesmaximal, characteristic, complemented
4#19127638C211yesmaximal, characteristic, complemented
5#19127636C211yesmaximal, characteristic, complemented
6#19127598C212yesmaximal, complemented
7#19127579C212yesmaximal
8#19127575C212yesmaximal, complemented
9#19127560C212yesmaximal, complemented
10#19127560C211yesmaximal, characteristic, complemented
11#400×C83C211yesmaximal, characteristic, complemented
12C22×D332C2211yescharacteristic
13C22×C4×D83C2211yescharacteristic
14C22×C4×D83C2211yescharacteristic
15C22×C4×D83C2212yes—
16#14581754—24no—
17C2×C4⋊Dic83C2211yescharacteristic
18C2×C4⋊Dic83C2212yes—
19C2×Dic83⋊C4C2212yes—
20C2×Dic83⋊C4C2212yes—
21C2×Dic83⋊C4C2212yes—
22C4⋊C4×D83—24no—
23C2×(C2×D83)⋊C4C2212yes—
24C2×(C2×D83)⋊C4C2211yescharacteristic
25C2×(C2×D83)⋊C4C2211yescharacteristic
26C2×(C2×D83)⋊C4C2212yes—
27C2×(C2×D83)⋊C4C2212yes—
28C2×C4×Dic83C2211yescharacteristic
29C83⋊#95C2212yes—
30C83⋊#95C2211yescharacteristic
31C83⋊#95C2212yes—
32C2×C4⋊C4×C83C2211yescharacteristic
33C2×C4⋊C4×C83C2211yescharacteristic
34#95×C83C2212yes—
35#95×C83C2211yescharacteristic
36#95×C83C2212yes—
37C23×D83C2311yescharacteristic
38C23×D83C2311yescharacteristic
39C2×D332—22no—
40C2×D332—22no—
41C2×D332Q814yes—
42C22×Dic83C2312yes—
43C22×Dic83C2312yes—
44C22×Dic83C2311yescharacteristic
45C22×Dic83C2311yescharacteristic
46C2×C4×D83—24no—
47C2×C4×D83—24no—
48C2×C4×D83—22no—
49C2×C4×D83—24no—
50C2×C4×D83—42no—
51C2×C4×D83—42no—
52C2×C4×D83D414yes—
53C4⋊Dic83—22no—
54C4⋊Dic83—42no—
55Dic83⋊C4—24no—
56Dic83⋊C4—24no—
57Dic83⋊C4—42no—
58(C2×D83)⋊C4—22no—
59(C2×D83)⋊C4—24no—
60(C2×D83)⋊C4—22no—
61(C2×D83)⋊C4—42no—
62(C2×D83)⋊C4—42no—
63C4×Dic83—22no—
64C22×C4×C83C2311yescharacteristic
65C22×C4×C83C2312yes—
66C22×C4×C83C2312yes—
67C22×C4×C83C2311yescharacteristic
68C22×C4×C83C2311yescharacteristic
69C4⋊C4×C83—22no—
70C4⋊C4×C83—22no—
71C23×C83C2411yescharacteristic
72C22×D83—22no—
73C22×D83—22no—
74C22×D83—21no—
75C22×D83—22no—
76C22×D83—21no—
77C22×D83—22no—
78C22×D83C2×D412yes—
79C22×D83C2×Q812yes—
80C22×D83C2×Q812yes—
81C22×D83#4112yes—
82C2×C4×C83—21no—
83C2×C4×C83—22no—
84C2×C4×C83—21no—
85C2×C4×C83—21no—
86C2×C4×C83—22no—
87C2×C4×C83—22no—
88C2×C4×C83—22no—
89C2×C4×C83—22no—
90C2×C4×C83—22no—
91C2×C4×C83—22no—
92C2×C4×C83—22no—
93C2×C4×C83C2×D412yes—
94C2×C4×C83C2×Q812yes—
95C2×Dic83—21no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83—22no—
100C2×Dic83—22no—
101C2×Dic83—22no—
102C2×Dic83—22no—
103C2×Dic83C2×D412yes—
104C2×Dic83#4114yes—
105D332—28no—
106C4×D83—28no—
107C4×D83—48no—
108C4×D83—44no—
109C22×C83C22×D411yescharacteristic
110C22×C83C22×Q811yescharacteristic
111C22×C83C2×#4111yescharacteristic
112C22×C83C2×#4112yes—
113C22×C83H(5,𝔽2)12yes—
114C2×D83—22no—
115C2×D83—22no—
116C2×D83—24no—
117C2×D83—24no—
118C2×D83—22no—
119C2×D83—24no—
120C2×D83—41no—
121C2×D83—41no—
122C2×D83—42no—
123C2×D83C22⋊Q814yes—
124C4×C83—22no—
125C4×C83—22no—
126C4×C83—41no—
127C4×C83—42no—
128C4×C83—42no—
129Dic83—42no—
130Dic83—41no—
131Dic83—42no—
132Dic83#12314yes—
133C2×C83C2×C22⋊Q811yescharacteristic
134C2×C83C2×#12311yescharacteristic
135C2×C83#53511yescharacteristic
136C2×C83#55312yes—
137C2×C83#55412yes—
138D83—24no—
139D83—42no—
140#2487—831nomaximal, Sylow
141C83#248711yescharacteristic, complemented, Sylow
142C2×C4×D4—831no—
143C22×C4⋊C4—831no—
144#400—831no—
145#400D8311yescharacteristic, complemented
146#399—831no—
147#399—831no—
148#389—832no—
149#386—832no—
150#385—832no—
151#380—832no—
152#380—831no—
153C22×D4—831no—
154C23×C4—832no—
155C23×C4—831no—
156C23×C4—831no—
157C4×D4—1664no—
158C2×C4⋊C4—832no—
159C2×C4⋊C4—832no—
160C2×C4⋊C4—832no—
161C2×C4⋊C4—831no—
162C2×C4⋊C4—832no—
163C2×C4⋊C4—1664no—
164C2×C4⋊C4C2×D8311yescharacteristic
165C2×C4⋊C4C2×D8311yescharacteristic
166C2×C22⋊C4—832no—
167C2×C22⋊C4—832no—
168C2×C22⋊C4—831no—
169C2×C22⋊C4—831no—
170C2×C22⋊C4—832no—
171C2×C42—831no—
172#95—832no—
173#95—831no—
174#95—832no—
175#95C2×D8312yes—
176#95C2×D8311yescharacteristic
177#95C2×D8312yes—
178C24—831no—
179C24—831no—
180C2×D4—834no—
181C2×D4—1662no—
182C2×D4—1662no—
183C22×C4—3322no—
184C22×C4—3322no—
185C22×C4—831no—
186C22×C4—834no—
187C22×C4—832no—
188C22×C4—832no—
189C22×C4—831no—
190C22×C4—1662no—
191C22×C4—1664no—
192C22×C4—1664no—
193C22×C4—1664no—
194C22×C4C22×D8311yescharacteristic
195C22×C4C22×D8312yes—
196C22×C4C22×D8312yes—
197C22×C4C22×D8311yescharacteristic
198C22×C4C22×D8311yescharacteristic
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—3322no—
202C4⋊C4—3322no—
203C4⋊C4—1664no—
204C4⋊C4—1664no—
205C4⋊C4—1662no—
206C22⋊C4—3322no—
207C22⋊C4—3322no—
208C22⋊C4—1662no—
209C22⋊C4—1664no—
210C22⋊C4—1662no—
211C42—1662no—
212C23—832no—
213C23—832no—
214C23—832no—
215C23—832no—
216C23—1661no—
217C23—1662no—
218C23—1661no—
219C23—1662no—
220C23—1662no—
221C23—1662no—
222C23C23×D8311yescharacteristic
223D4—1668no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—3328no—
236C2×C4—3324no—
237C2×C4—832no—
238C2×C4—834no—
239C2×C4—1668no—
240C2×C4—1661no—
241C2×C4—1662no—
242C2×C4—1662no—
243C2×C4—1662no—
244C2×C4—1662no—
245C2×C4—1662no—
246C2×C4—1662no—
247C2×C4—1662no—
248C2×C4D4×D8312yes—
249C2×C4Q8×D8312yes—
250C22—3322no—
251C22—3321no—
252C22—3321no—
253C22—834no—
254C22—1664no—
255C22—1662no—
256C22—1664no—
257C22—1664no—
258C22—1662no—
259C22—1662no—
260C22C2×D4×D8311yescharacteristic
261C22C2×Q8×D8311yescharacteristic
262C22#41×D8311yescharacteristic
263C22#41×D8312yes—
264C22#1458183812yes—
265C4—22no—
266C4—22no—
267C4—42no—
268C4—41no—
269C4—42no—
270C4—3322no—
271C4—3322no—
272C4—3321no—
273C4—834no—
274C2—3322no—
275C2—1664no—
276C2C22⋊Q8×D8311yescharacteristic
277C2#123×D8311yescharacteristic
278C2#1912826911yescharacteristic
279C2#1912840212yes—
280C2#1912840712yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#400×C83C2
prime radicalC22×D332C22
hypercenter#400D83
hypocenterC83#2487
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D83C2semidirect
3. #19127640C2semidirect
4. #19127638C2semidirect
5. #19127636C2semidirect
6. #19127598C2semidirect
8. #19127575C2semidirect
9. #19127560C2semidirect
10. #19127560C2semidirect
11. #400×C83C2semidirect
141. C83#2487semidirect
145. #400D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817542
C2×C4⋊C4×D832
#191276402
#191276382
#191276362
#191275982
#191275792
#191275752
#191275602
#400×C832
#248783