Finite Group Databasebeta

← #24916934

Subgroups — #24916934

Order 10624.

484 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 135 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×#14581825C211yesmaximal, characteristic, complemented
2C2×#14581822C211yesmaximal, characteristic, complemented
3#19128247C211yesmaximal, characteristic
4C2×#14581757C211yesmaximal, characteristic, complemented
5#19127926C212yesmaximal, complemented
6#19127922C212yesmaximal, complemented
7#19127910C212yesmaximal, complemented
8#19127900C212yesmaximal, complemented
9#19127724C211yesmaximal, characteristic, complemented
10#19127700C211yesmaximal, characteristic, complemented
11#417×C83C211yesmaximal, characteristic, complemented
12C2×#13050590C2211yescharacteristic
13C22×Dic166C2211yescharacteristic
14#14581825—24no—
15#14581822—24no—
16C2×Dic332C2212yes—
17C2×#13050580C2212yes—
18C2×#13050579C2212yes—
19C2×#13050578C2212yes—
20C2×#13050574C2211yescharacteristic
21#14581790C2211yescharacteristic
22#14581788C2211yescharacteristic
23#14581781C2211yescharacteristic
24#14581779C2211yescharacteristic
25#14581764—21no—
26#14581764—21no—
27#14581757—21no—
28#14581757—21no—
29#14581757C2212yes—
30#14581757C2212yes—
31C4×C22⋊D83—21no—
32C4×C22⋊D83—21no—
33C2×Dic83⋊C4—21no—
34C2×C4×Dic83—21no—
35C2×C4×Dic83C2211yescharacteristic
36Dic83⋊C8C2212yes—
37Dic83⋊C8C2212yes—
38#14581715C2212yes—
39#14581712C2212yes—
40C2×#41×C83C2211yescharacteristic
41C2×M16×C83C2211yescharacteristic
42C2×C4⋊C4×C83C2211yescharacteristic
43#103×C83C2212yes—
44#102×C83C2212yes—
45#13050590—22no—
46#13050590—22no—
47C2×Dic166—21no—
48C2×Dic166—22no—
49C2×Dic166D414yescomplemented
50C2×Dic166C2311yescharacteristic
51C2×Dic166C2312yes—
52C2×D332C2311yescharacteristic
53C2×C22⋊D83—21no—
54C2×C22⋊D83—21no—
55C2×C22⋊D83—21no—
56C22×Dic83—21no—
57C22×Dic83D411yescharacteristic
58C22×Dic83D411yescharacteristic
59C2×C4×D83—21no—
60Dic332—24no—
61#13050580—24no—
62#13050579—24no—
63#13050578—24no—
64#13050574—22no—
65C2×C83⋊C8C2312yes—
66Dic83⋊C4—21no—
67Dic83⋊C4—21no—
68Dic83⋊C4—22no—
69Dic83⋊C4—22no—
70C22⋊Dic83—21no—
71C22⋊Dic83—21no—
72(C2×D83)⋊C4—21no—
73(C2×D83)⋊C4—21no—
74C4×Dic83—22no—
75C4×Dic83—22no—
76C4×Dic83D411yescharacteristic
77C4×Dic83D411yescharacteristic
78C4×Dic83C2311yescharacteristic
79C4×Dic83C2311yescharacteristic
80#41×C83—22no—
81#41×C83—22no—
82C2×Q8×C83C2311yescharacteristic
83C2×D4×C83—21no—
84C2×D4×C83C2311yescharacteristic
85C22×C4×C83—21no—
86C22×C4×C83—21no—
87C22×C4×C83C2311yescharacteristic
88M16×C83—22no—
89C2×C8×C83C2312yes—
90C4⋊C4×C83—21no—
91C4⋊C4×C83C2312yes—
92C23×C83—21no—
93C23×C83C2×D411yescharacteristic
94C22×D83—21no—
95Q8×C83—21no—
96Q8×C83—21no—
97D4×C83—21no—
98D4×C83—21no—
99D4×C83—42no—
100C2×C4×C83—22no—
101C2×C4×C83—21no—
102C2×C4×C83—42no—
103C2×C4×C83—42no—
104C2×C4×C83C2×D411yescharacteristic
105C2×C4×C83C2×D412yes—
106C2×C4×C83#4112yes—
107C2×C4×C83C2411yescharacteristic
108C22⋊D83—42no—
109C22⋊D83—42no—
110C22⋊D83—42no—
111C2×Dic83—24no—
112C2×Dic83—22no—
113C2×Dic83—22no—
114C2×Dic83—21no—
115C2×Dic83—41no—
116C2×Dic83C2×D411yescharacteristic
117C2×Dic83C2×D411yescharacteristic
118C2×Dic83C2×D411yescharacteristic
119C2×Dic83C2×D411yescharacteristic
120D332—21no—
121D332—21no—
122C4×D83—42no—
123Dic166—24no—
124Dic166—22no—
125Dic166—21no—
126Dic166—21no—
127Dic166C2×D414yes—
128C8×C83—22no—
129C83⋊C8—22no—
130C22×C83—21no—
131C22×C83—21no—
132C22×C83—41no—
133C22×C83—41no—
134C22×C83—41no—
135C22×C83#12112yes—
136C22×C83C22×D411yescharacteristic
137C2×D83—41no—
138C2×D83—41no—
139C2×D83—41no—
140C4×C83—41no—
141C4×C83—42no—
142C4×C83#12112yes—
143C4×C83C22×D411yescharacteristic
144C4×C83C2×#4111yescharacteristic
145Dic83—24no—
146Dic83—21no—
147Dic83—21no—
148Dic83—21no—
149Dic83—41no—
150Dic83#13712yescomplemented
151C2×C83—22no—
152C2×C83—81no—
153C2×C83C2×#12111yescharacteristic
154C2×C83C2×#13711yescharacteristic
155C2×C83#57511yescharacteristic
156D83—81no—
157#3037—831nomaximal, Sylow
158C83#303711yescharacteristic, complemented, Sylow
159C2×#137—831no—
160C2×#137—831no—
161#532—831no—
162C2×C4×Q8—831no—
163#464—832no—
164#464—832no—
165#462—832no—
166#460—832no—
167#423—831no—
168#417—831no—
169#417D8311yescharacteristic, complemented
170C2×#41—831no—
171C2×#41C2×D8311yescharacteristic
172C22×Q8—831no—
173#137—1664no—
174#137—1664no—
175C2×Q16—832no—
176C2×Q16—832no—
177C2×SD16—832no—
178C2×SD16—832no—
179C2×M16—831no—
180C2×M16C2×D8311yescharacteristic
181C4⋊Q8—831no—
182#127—831no—
183#124—831no—
184#124—831no—
185#121—1661no—
186#121—1661no—
187C4×Q8—832no—
188C4×Q8—832no—
189C4×Q8—1661no—
190C4×Q8—1661no—
191C4×D4—1661no—
192C4×D4—1661no—
193C2×C4⋊C4—1661no—
194C2×C4⋊C4C2×D8311yescharacteristic
195C2×C42—831no—
196C2×C42—1661no—
197C4⋊C8—832no—
198C4⋊C8—832no—
199#103—832no—
200#103C2×D8312yes—
201#102—832no—
202#102C2×D8312yes—
203#41—22no—
204#41—22no—
205#41—1662no—
206#41—1662no—
207C2×Q8—834no—
208C2×Q8—832no—
209C2×Q8—831no—
210C2×Q8—1661no—
211C2×Q8—1662no—
212C2×Q8C22×D8311yescharacteristic
213C2×D4—21no—
214C2×D4—831no—
215C2×D4—1661no—
216C2×D4—1661no—
217C2×D4—1661no—
218C2×D4C22×D8311yescharacteristic
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—831no—
222C22×C4—831no—
223C22×C4—1661no—
224C22×C4—1661no—
225C22×C4C22×D8311yescharacteristic
226Q16—1664no—
227Q16—1664no—
228SD16—1664no—
229SD16—1664no—
230M16—22no—
231M16—1662no—
232C2×C8—832no—
233C2×C8C22×D8312yes—
234C4⋊C4—21no—
235C4⋊C4—1662no—
236C4⋊C4—1662no—
237C4⋊C4—1661no—
238C4⋊C4—1661no—
239C4⋊C4C22×D8312yes—
240C22⋊C4—1661no—
241C22⋊C4—1661no—
242C22⋊C4—1661no—
243C22⋊C4—1661no—
244C42—831no—
245C42—831no—
246C42—831no—
247C42—831no—
248C42—1662no—
249C42—1662no—
250C23—21no—
251C23—1661no—
252C23D4×D8311yescharacteristic
253Q8—21no—
254Q8—21no—
255Q8—834no—
256Q8—1661no—
257Q8—1661no—
258Q8—1664no—
259Q8—1662no—
260D4—21no—
261D4—21no—
262D4—42no—
263D4—3322no—
264D4—3322no—
265D4—3322no—
266D4—1661no—
267D4—1661no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—3322no—
273C2×C4—3321no—
274C2×C4—831no—
275C2×C4—831no—
276C2×C4—831no—
277C2×C4—831no—
278C2×C4—1661no—
279C2×C4—1662no—
280C2×C4—1664no—
281C2×C4—1662no—
282C2×C4D4×D8311yescharacteristic
283C2×C4D4×D8312yes—
284C2×C4#1305059012yes—
285C2×C4C23×D8311yescharacteristic
286C8—22no—
287C8—1662no—
288C22—21no—
289C22—21no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—3321no—
294C22—3321no—
295C22—3321no—
296C22#1458176512yes—
297C22C2×D4×D8311yescharacteristic
298C4—41no—
299C4—42no—
300C4—3321no—
301C4—832no—
302C4—1664no—
303C4—1661no—
304C4—1661no—
305C4—1661no—
306C4#1458176512yes—
307C4C2×D4×D8311yescharacteristic
308C4C2×#1305059011yescharacteristic
309C2—22no—
310C2—81no—
311C2—6641no—
312C2C2×#1458176511yescharacteristic
313C2#137×D8311yescharacteristic
314C2#1912849711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#417×C83C2
socleC22×C83C22×D4
prime radical#19128247C2
hypercenter#417D83
hypocenterC83#3037
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#14581825C2semidirect
2. C2×#14581822C2semidirect
4. C2×#14581757C2semidirect
5. #19127926C2semidirect
6. #19127922C2semidirect
7. #19127910C2semidirect
8. #19127900C2semidirect
9. #19127724C2semidirect
10. #19127700C2semidirect
11. #417×C83C2semidirect
49. C2×Dic166D4semidirect
150. Dic83#137semidirect
158. C83#3037semidirect
169. #417D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818252
C2×#145818222
#191282472
C2×#145817572
#191279262
#191279222
#191279102
#191279002
#191277242
#191277002
#417×C832
#303783