Finite Group Databasebeta

← #24916947

Subgroups — #24916947

Order 10624.

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

normal subgroups only

This group has 480 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×#14581824C212yesmaximal
2C4⋊Q8×D83C211yesmaximal, characteristic, complemented
3C4×#13050591C211yesmaximal, characteristic
4#19127911C211yesmaximal, characteristic, complemented
5#19127909C212yesmaximal, complemented
6#19127907C211yesmaximal, characteristic, complemented
7#19127905C212yesmaximal, complemented
8#19127903C211yesmaximal, characteristic, complemented
9#19127723C211yesmaximal, characteristic, complemented
10#19127708C212yesmaximal, complemented
11#462×C83C211yesmaximal, characteristic, complemented
12C2×#13050591C2211yescharacteristic
13C2×Q8×D83C2212yes—
14#14581824—28no—
15C2×#13050580C2212yes—
16C2×#13050579C2212yes—
17C2×#13050577C2212yes—
18C2×#13050573C2212yes—
19#14581792C2211yescharacteristic
20#14581791—21no—
21#14581791—21no—
22#14581790C2212yes—
23Q8×Dic83C2211yescharacteristic
24C4×D332—21no—
25C4×D332C2211yescharacteristic
26#14581754—21no—
27#14581754C2211yescharacteristic
28#14581750—21no—
29#14581750C2211yescharacteristic
30C4⋊C4×D83—21no—
31C4⋊C4×D83—21no—
32C42×D83—21no—
33C42×D83C2211yescharacteristic
34C4⋊(C83⋊C8)C2211yescharacteristic
35Dic83⋊C8C2212yes—
36#14581717C2212yes—
37#14581716C2212yes—
38#14581715C2212yes—
39C2×Q16×C83C2212yes—
40C4⋊Q8×C83C2211yescharacteristic
41C4×Q8×C83C2211yescharacteristic
42C4⋊C8×C83C2211yescharacteristic
43#103×C83C2212yes—
44#13050591—22no—
45#13050591D414yes—
46C2×Dic166—22no—
47C2×Dic166C2312yes—
48Q8×D83—24no—
49Q8×D83—24no—
50C2×D332—21no—
51C2×D332C2311yescharacteristic
52C2×C4×D83—21no—
53C2×C4×D83—22no—
54C2×C4×D83—21no—
55C2×C4×D83—21no—
56C2×C4×D83D411yescharacteristic
57C2×C4×D83D411yescharacteristic
58C2×C4×D83C2311yescharacteristic
59#13050580—24no—
60#13050579—24no—
61#13050577—24no—
62#13050573—24no—
63C2×C83⋊C8C2312yes—
64C4⋊Dic83—21no—
65C4⋊Dic83—21no—
66C4⋊Dic83—21no—
67C4⋊Dic83C2311yescharacteristic
68Dic83⋊C4—22no—
69Dic83⋊C4—22no—
70(C2×D83)⋊C4—21no—
71(C2×D83)⋊C4—21no—
72(C2×D83)⋊C4—21no—
73C4×Dic83—21no—
74C4×Dic83—21no—
75C4×Dic83D411yescharacteristic
76C4×Dic83D411yescharacteristic
77C4×Dic83C2311yescharacteristic
78C2×Q8×C83C2312yes—
79C2×Q8×C83C2311yescharacteristic
80Q16×C83—24no—
81C2×C8×C83C2312yes—
82C4⋊C4×C83—21no—
83C4⋊C4×C83—21no—
84C4⋊C4×C83—21no—
85C4⋊C4×C83C2311yescharacteristic
86C42×C83—21no—
87C42×C83C2311yescharacteristic
88C22×D83—21no—
89C22×D83C2×D411yescharacteristic
90Q8×C83—21no—
91Q8×C83—22no—
92Q8×C83—22no—
93Q8×C83C2×D412yes—
94C2×C4×C83—21no—
95C2×C4×C83—21no—
96C2×C4×C83—22no—
97C2×C4×C83C2×D411yescharacteristic
98C2×C4×C83C2×D411yescharacteristic
99C2×C4×C83C2411yescharacteristic
100C2×Dic83—21no—
101C2×Dic83—22no—
102C2×Dic83C2×D411yescharacteristic
103C2×Dic83C2×D411yescharacteristic
104C2×Dic83C2×D411yescharacteristic
105D332—24no—
106D332—21no—
107D332C2×D412yes—
108C4×D83—22no—
109C4×D83—24no—
110C4×D83—22no—
111C4×D83—44no—
112C4×D83—42no—
113C4×D83—42no—
114C4×D83C2×D412yes—
115C4×D83#4112yes—
116Dic166—22no—
117Dic166—22no—
118Dic166—44no—
119C8×C83—22no—
120C83⋊C8—22no—
121C22×C83C22×D411yescharacteristic
122C2×D83—21no—
123C2×D83—21no—
124C2×D83—22no—
125C2×D83—41no—
126C2×D83—41no—
127C2×D83#12112yes—
128C4×C83—22no—
129C4×C83—21no—
130C4×C83—42no—
131C4×C83—41no—
132C4×C83#13712yescomplemented
133C4×C83C22×D411yescharacteristic
134C4×C83C2×#4111yescharacteristic
135Dic83—21no—
136Dic83—42no—
137Dic83—41no—
138Dic83#12112yes—
139Dic83#13712yescomplemented
140C2×C83C2×#12111yescharacteristic
141C2×C83C2×#13711yescharacteristic
142C2×C83C2×#13711yescharacteristic
143D83—22no—
144D83—42no—
145#3039—831nomaximal, Sylow
146C83#303911yescharacteristic, complemented, Sylow
147C2×#137—832no—
148C2×C4⋊Q8—831no—
149C4×#41—831no—
150#462—832no—
151#462—831no—
152#462D8311yescharacteristic, complemented
153#461—831no—
154#461—831no—
155#461—832no—
156#423—831no—
157#419—832no—
158C2×#41—831no—
159C22×Q8—832no—
160#137—1668no—
161C2×Q16—832no—
162C2×Q16C2×D8312yes—
163C2×SD16—832no—
164C2×SD16—832no—
165C2×M16—832no—
166C4⋊Q8—832no—
167C4⋊Q8—831no—
168C4⋊Q8—1661no—
169C4⋊Q8—1661no—
170C4⋊Q8C2×D8311yescharacteristic
171C4×Q8—831no—
172C4×Q8C2×D8311yescharacteristic
173C4×D4—831no—
174C4×D4—831no—
175C4×D4—1661no—
176C4×D4—1661no—
177#117—831no—
178#117—1661no—
179C2×C4⋊C4—1661no—
180C2×C4⋊C4—1661no—
181C2×C42—831no—
182C2×C42—1661no—
183C4⋊C8—832no—
184C4⋊C8—831no—
185C4⋊C8C2×D8311yescharacteristic
186#103—832no—
187#103—832no—
188#103—832no—
189#103C2×D8312yes—
190#41—834no—
191#41—1662no—
192C2×Q8—832no—
193C2×Q8—1662no—
194C2×Q8—1664no—
195C2×Q8—1664no—
196C2×Q8C22×D8312yes—
197C2×Q8C22×D8311yescharacteristic
198C2×D4—831no—
199C2×D4—1661no—
200C22×C4—831no—
201C22×C4—831no—
202C22×C4—831no—
203C22×C4—1661no—
204C22×C4—1662no—
205C22×C4—1661no—
206C22×C4—1661no—
207Q16—24no—
208Q16—1664no—
209SD16—1664no—
210SD16—1664no—
211M16—1664no—
212C2×C8—832no—
213C2×C8C22×D8312yes—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—831no—
218C4⋊C4—1662no—
219C4⋊C4—1662no—
220C4⋊C4—1661no—
221C4⋊C4—1661no—
222C4⋊C4—1661no—
223C4⋊C4C22×D8311yescharacteristic
224C22⋊C4—1661no—
225C22⋊C4—1661no—
226C22⋊C4—1661no—
227C42—21no—
228C42—831no—
229C42—831no—
230C42—831no—
231C42—1661no—
232C42—1661no—
233C42C22×D8311yescharacteristic
234C23—831no—
235C23—1661no—
236Q8—21no—
237Q8—22no—
238Q8—22no—
239Q8—3324no—
240Q8—1662no—
241Q8—1662no—
242Q8D4×D8312yes—
243D4—832no—
244D4—1664no—
245D4—1661no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—3324no—
250C2×C4—3322no—
251C2×C4—3322no—
252C2×C4—831no—
253C2×C4—831no—
254C2×C4—832no—
255C2×C4—832no—
256C2×C4—831no—
257C2×C4—1662no—
258C2×C4—1664no—
259C2×C4—1661no—
260C2×C4—1662no—
261C2×C4—1662no—
262C2×C4D4×D8311yescharacteristic
263C2×C4D4×D8311yescharacteristic
264C2×C4C23×D8311yescharacteristic
265C8—22no—
266C8—1662no—
267C22—3321no—
268C22—3321no—
269C22—832no—
270C22—1662no—
271C22—1661no—
272C22—1661no—
273C22C2×D4×D8311yescharacteristic
274C4—22no—
275C4—21no—
276C4—42no—
277C4—41no—
278C4—3322no—
279C4—3321no—
280C4—832no—
281C4—832no—
282C4—1661no—
283C4#1458182412yescomplemented
284C4C2×D4×D8311yescharacteristic
285C4#41×D8311yescharacteristic
286C2—3322no—
287C2—1662no—
288C2#121×D8311yescharacteristic
289C2#137×D8311yescharacteristic
290C2C2×#1458182411yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#462×C83C2
socleC22×C83C22×D4
prime radicalC2×#13050591C22
hypercenter#462D83
hypocenterC83#3039
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D83C2semidirect
4. #19127911C2semidirect
5. #19127909C2semidirect
6. #19127907C2semidirect
7. #19127905C2semidirect
8. #19127903C2semidirect
9. #19127723C2semidirect
10. #19127708C2semidirect
11. #462×C83C2semidirect
132. C4×C83#137semidirect
139. Dic83#137semidirect
146. C83#3039semidirect
152. #462D83semidirect
283. C4#14581824semidirect

Maximal subgroups

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

subgroupindex
C2×#145818242
C4⋊Q8×D832
C4×#130505912
#191279112
#191279092
#191279072
#191279052
#191279032
#191277232
#191277082
#462×C832
#303983