Finite Group Databasebeta

← #24917152

Subgroups — #24917152

Order 10624.

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

normal subgroups only

This group has 472 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×#14581823C211yesmaximal, characteristic, complemented
2C2×#14581798C211yesmaximal, characteristic, complemented
3#19128282C211yesmaximal, characteristic
4#19128275C211yesmaximal, characteristic, complemented
5#19127958C212yesmaximal, complemented
6#19127950C212yesmaximal, complemented
7#19127941C212yesmaximal, complemented
8#19127933C212yesmaximal, complemented
9#19127741C211yesmaximal, characteristic, complemented
10#19127698C211yesmaximal, characteristic, complemented
11C2×#103×C83C211yesmaximal, characteristic, complemented
12C2×#13050590C2211yescharacteristic
13C2×#13050590C2211yescharacteristic
14#14581823—24no—
15C2×#13050580C2212yes—
16C2×#13050577C2212yes—
17#14581798—24no—
18C2×#13050574C2211yescharacteristic
19C2×#13050573C2212yes—
20C2×C8×D83C2212yes—
21#14581790—21no—
22#14581781—21no—
23#14581775—22no—
24#14581772—21no—
25#14581772C2212yes—
26#14581768C2212yes—
27#14581768C2212yes—
28#14581767—21no—
29#14581767C2212yes—
30#14581766—22no—
31#14581751C2211yescharacteristic
32#14581725C2212yes—
33#14581723C2212yes—
34#14581717C2211yescharacteristic
35#14581717C2211yescharacteristic
36#14581714C2211yescharacteristic
37#14581714C2211yescharacteristic
38C22×Q8×C83C2211yescharacteristic
39C22×C8×C83C2211yescharacteristic
40C2×C4⋊C4×C83C2211yescharacteristic
41#103×C83C2212yes—
42#103×C83C2212yes—
43#13050590—22no—
44#13050590—22no—
45#13050590—22no—
46#13050590—22no—
47C2×Dic166D411yescharacteristic
48C2×Dic166C2311yescharacteristic
49C2×D332—22no—
50C2×D332D411yescharacteristic
51C2×D332C2311yescharacteristic
52C2×C22⋊D83—21no—
53C2×C22⋊D83D412yes—
54C2×C4×D83—21no—
55C2×C4×D83C2312yes—
56#13050580—24no—
57#13050577—24no—
58#13050574D414yescomplemented
59#13050573—24no—
60C2×C83⋊C8C2312yes—
61C8×D83—24no—
62C4⋊Dic83—22no—
63C4⋊Dic83C2311yescharacteristic
64C4⋊Dic83C2311yescharacteristic
65Dic83⋊C4—22no—
66Dic83⋊C4—22no—
67C22⋊Dic83—21no—
68(C2×D83)⋊C4—22no—
69(C2×D83)⋊C4—22no—
70(C2×D83)⋊C4—22no—
71(C2×D83)⋊C4—22no—
72C4×Dic83—21no—
73C2×Q8×C83—21no—
74C2×Q8×C83—22no—
75C2×Q8×C83—22no—
76C2×Q8×C83C2312yes—
77C22×C4×C83—21no—
78C22×C4×C83—21no—
79C22×C4×C83C2311yescharacteristic
80C2×C8×C83—22no—
81C2×C8×C83C2311yescharacteristic
82C2×C8×C83C2311yescharacteristic
83C4⋊C4×C83—21no—
84C4⋊C4×C83C2312yes—
85C23×C83C2×D411yescharacteristic
86C22×D83—21no—
87C22×D83C2×D412yes—
88Q8×C83—22no—
89Q8×C83—22no—
90Q8×C83—42no—
91C2×C4×C83—22no—
92C2×C4×C83—22no—
93C2×C4×C83—42no—
94C2×C4×C83—81no—
95C2×C4×C83C2×D411yescharacteristic
96C2×C4×C83C2×D412yes—
97C2×C4×C83#4112yes—
98C2×C4×C83C2411yescharacteristic
99C22⋊D83—24no—
100C22⋊D83—42no—
101C2×Dic83—21no—
102C2×Dic83—21no—
103C2×Dic83—21no—
104C2×Dic83C2×D412yes—
105D332—22no—
106D332—21no—
107D332—21no—
108D332—82no—
109C4×D83—22no—
110C4×D83—22no—
111C4×D83—42no—
112Dic166—22no—
113Dic166—21no—
114Dic166—21no—
115C8×C83—21no—
116C8×C83—21no—
117C83⋊C8C2×D414yes—
118C22×C83—21no—
119C22×C83—21no—
120C22×C83#12112yes—
121C22×C83C22×D411yescharacteristic
122C2×D83—22no—
123C2×D83—22no—
124C2×D83—22no—
125C2×D83—41no—
126C2×D83—41no—
127C2×D83—41no—
128C4×C83—42no—
129C4×C83—42no—
130C4×C83#12112yes—
131C4×C83C22×D411yescharacteristic
132C4×C83C2×#4111yescharacteristic
133Dic83—22no—
134Dic83—41no—
135Dic83—41no—
136Dic83—41no—
137C2×C83—22no—
138C2×C83C2×#12111yescharacteristic
139C2×C83#57711yescharacteristic
140C2×C83#57811yescharacteristic
141D83—42no—
142D83—81no—
143#3063—831nomaximal, Sylow
144C83#306311yescharacteristic, complemented, Sylow
145C2×#137—831no—
146C2×#131—831no—
147#537—831no—
148#536—831no—
149#470—832no—
150#468—832no—
151#467—832no—
152#465—832no—
153#428—831no—
154#416—831no—
155C2×#103D8311yescharacteristic, complemented
156C2×#41—831no—
157C2×#41—831no—
158C22×Q8C2×D8311yescharacteristic
159#137—1664no—
160C2×Q16—832no—
161C2×SD16—832no—
162#131—1664no—
163C2×M16—831no—
164C2×M16—832no—
165C22×C8—832no—
166C22×C8C2×D8311yescharacteristic
167C4⋊Q8—1661no—
168#124—1661no—
169#123—1662no—
170C22⋊Q8—832no—
171C22⋊Q8—832no—
172C22⋊Q8—832no—
173C22⋊Q8—1661no—
174#121—832no—
175#121—1661no—
176#121—1662no—
177#117—831no—
178C2×C4⋊C4C2×D8311yescharacteristic
179#107—832no—
180#106—832no—
181#103—831no—
182#103—831no—
183#103C2×D8312yes—
184#103C2×D8312yes—
185#102—831no—
186#102—831no—
187#41—1662no—
188#41—1662no—
189#41—1662no—
190#41—1662no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—831no—
195C2×Q8—831no—
196C2×Q8C22×D8312yes—
197C2×D4—832no—
198C2×D4—831no—
199C2×D4—831no—
200C2×D4—1662no—
201C2×D4—1661no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—832no—
205C22×C4—1661no—
206C22×C4C22×D8311yescharacteristic
207Q16—1664no—
208SD16—1664no—
209M16—834no—
210M16—1664no—
211C2×C8—22no—
212C2×C8—832no—
213C2×C8—1664no—
214C2×C8C22×D8311yescharacteristic
215C2×C8C22×D8311yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—831no—
218C4⋊C4—831no—
219C4⋊C4—1662no—
220C4⋊C4—1662no—
221C4⋊C4—1662no—
222C4⋊C4C22×D8312yes—
223C22⋊C4—1662no—
224C22⋊C4—1662no—
225C22⋊C4—1662no—
226C22⋊C4—1662no—
227C22⋊C4—1661no—
228C42—1661no—
229C23—832no—
230C23—1661no—
231C23D4×D8311yescharacteristic
232Q8—22no—
233Q8—22no—
234Q8—42no—
235Q8—1661no—
236Q8—1661no—
237Q8—1662no—
238D4—3322no—
239D4—6642no—
240D4—1664no—
241D4—1662no—
242D4—1661no—
243D4—1661no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—81no—
248C2×C4—3322no—
249C2×C4—832no—
250C2×C4—1661no—
251C2×C4—1661no—
252C2×C4—1661no—
253C2×C4—1662no—
254C2×C4—1662no—
255C2×C4D4×D8311yescharacteristic
256C2×C4D4×D8312yes—
257C2×C4#1305059012yes—
258C2×C4C23×D8311yescharacteristic
259C8—21no—
260C8—21no—
261C8—834no—
262C22—21no—
263C22—21no—
264C22—3321no—
265C22—3321no—
266C22—3321no—
267C22—1662no—
268C22—1662no—
269C22—1662no—
270C22#1458176512yes—
271C22C2×D4×D8311yescharacteristic
272C4—42no—
273C4—42no—
274C4—3321no—
275C4—3321no—
276C4—3321no—
277C4—1662no—
278C4#1458176512yes—
279C4C2×D4×D8311yescharacteristic
280C4C2×#1305059011yescharacteristic
281C2—22no—
282C2—3322no—
283C2—6641no—
284C2C2×#1458176511yescharacteristic
285C2#1912850711yescharacteristic
286C2#1912851411yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×#103×C83C2
socleC22×C83C22×D4
prime radical#19128282C2
hypercenterC2×#103D83
hypocenterC83#3063
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. C2×#14581823C2semidirect
2. C2×#14581798C2semidirect
4. #19128275C2semidirect
5. #19127958C2semidirect
6. #19127950C2semidirect
7. #19127941C2semidirect
8. #19127933C2semidirect
9. #19127741C2semidirect
10. #19127698C2semidirect
11. C2×#103×C83C2semidirect
58. #13050574D4semidirect
144. C83#3063semidirect
155. C2×#103D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818232
C2×#145817982
#191282822
#191282752
#191279582
#191279502
#191279412
#191279332
#191277412
#191276982
C2×#103×C832
#306383