Finite Group Databasebeta

← #24917085

Subgroups — #24917085

Order 10624.

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

normal subgroups only

This group has 548 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×#14581817C211yesmaximal, characteristic, complemented
2C2×M16×D83C211yesmaximal, characteristic, complemented
3#19128208C211yesmaximal, characteristic, complemented
4C2×#14581767C211yesmaximal, characteristic, complemented
5#19127953C212yesmaximal, complemented
6#19127950C212yesmaximal, complemented
7#19127936C212yesmaximal, complemented
8#19127933C212yesmaximal, complemented
9#19127741C211yesmaximal, characteristic, complemented
10#19127707C211yesmaximal, characteristic, complemented
11#417×C83C211yesmaximal, characteristic, complemented
12C22×D332—21no—
13C22×D332C2211yescharacteristic
14C22×C4×D83C2211yescharacteristic
15#14581817—24no—
16C2×#13050577C2212yescomplemented
17C2×#13050575C2212yescomplemented
18C2×#13050574C2211yescharacteristic
19C2×#13050573C2212yescomplemented
20M16×D83—24no—
21C2×C8×D83C2212yescomplemented
22#14581773—21no—
23#14581768C2211yescharacteristic, complemented
24#14581768C2211yescharacteristic, complemented
25#14581767—21no—
26#14581767—21no—
27#14581767C2212yes—
28#14581767C2212yes—
29#14581763C2211yescharacteristic, complemented
30#14581763C2211yescharacteristic, complemented
31#14581760—21no—
32C4×C22⋊D83—21no—
33C4×C22⋊D83—21no—
34#14581751C2211yescharacteristic
35C2×(C2×D83)⋊C4—21no—
36#14581725C2212yescomplemented
37#14581723C2212yescomplemented
38#14581714C2212yescomplemented
39#14581714C2212yescomplemented
40C2×#41×C83C2211yescharacteristic
41C2×M16×C83C2211yescharacteristic
42C2×C4⋊C4×C83C2211yescharacteristic
43#103×C83C2212yescomplemented
44#102×C83C2212yescomplemented
45C23×D83—21no—
46C23×D83D411yescharacteristic
47C2×D332—22no—
48C2×D332—22no—
49C2×D332—22no—
50C2×D332—22no—
51C2×D332—21no—
52C2×D332—44no—
53C2×D332C2312yes—
54C2×C22⋊D83—21no—
55C2×C22⋊D83—21no—
56C22×Dic83D411yescharacteristic
57C2×C4×D83—22no—
58C2×C4×D83—22no—
59C2×C4×D83D411yescharacteristic
60C2×C4×D83D411yescharacteristic
61C2×C4×D83C2311yescharacteristic
62C2×C4×D83C2311yescharacteristic
63#13050577—24no—
64#13050575—24no—
65#13050574D414yescomplemented
66#13050573—24no—
67C2×C83⋊C8C2312yes—
68C8×D83—24no—
69C4⋊Dic83C2311yescharacteristic
70C4⋊Dic83C2311yescharacteristic
71Dic83⋊C4—21no—
72Dic83⋊C4—21no—
73C22⋊Dic83—21no—
74C22⋊Dic83—21no—
75C22⋊Dic83—21no—
76(C2×D83)⋊C4—21no—
77(C2×D83)⋊C4—21no—
78(C2×D83)⋊C4—22no—
79(C2×D83)⋊C4—22no—
80C4×Dic83—21no—
81#41×C83—22no—
82#41×C83—22no—
83C2×Q8×C83C2311yescharacteristic
84C2×D4×C83—21no—
85C2×D4×C83C2311yescharacteristic
86C22×C4×C83—21no—
87C22×C4×C83—21no—
88C22×C4×C83C2311yescharacteristic
89M16×C83—22no—
90C2×C8×C83C2312yes—
91C4⋊C4×C83—21no—
92C4⋊C4×C83C2312yes—
93C23×C83—21no—
94C23×C83C2×D411yescharacteristic
95C22×D83—22no—
96C22×D83—22no—
97C22×D83—22no—
98C22×D83—41no—
99C22×D83—42no—
100C22×D83—42no—
101C22×D83—42no—
102C22×D83C2×D411yescharacteristic
103C22×D83C2×D411yescharacteristic
104Q8×C83—21no—
105Q8×C83—21no—
106D4×C83—21no—
107D4×C83—21no—
108D4×C83—42no—
109C2×C4×C83—22no—
110C2×C4×C83—21no—
111C2×C4×C83—42no—
112C2×C4×C83—42no—
113C2×C4×C83C2×D411yescharacteristic
114C2×C4×C83C2×D412yes—
115C2×C4×C83#4112yes—
116C2×C4×C83C2411yescharacteristic
117C22⋊D83—42no—
118C22⋊D83—42no—
119C2×Dic83—22no—
120C2×Dic83—21no—
121C2×Dic83—21no—
122C2×Dic83C2×D411yescharacteristic
123C2×Dic83C2×D411yescharacteristic
124D332—22no—
125D332—22no—
126D332—44no—
127D332—44no—
128D332—42no—
129C4×D83—22no—
130C4×D83—22no—
131C4×D83—21no—
132C4×D83—21no—
133C4×D83—21no—
134C4×D83—21no—
135C8×C83—22no—
136C83⋊C8C2×D414yescomplemented
137C22×C83—21no—
138C22×C83—21no—
139C22×C83—41no—
140C22×C83—41no—
141C22×C83—41no—
142C22×C83#12112yes—
143C22×C83C22×D411yescharacteristic
144C2×D83—21no—
145C2×D83—21no—
146C2×D83—21no—
147C2×D83—21no—
148C2×D83—21no—
149C2×D83—44no—
150C2×D83—42no—
151C2×D83—42no—
152C2×D83—42no—
153C2×D83—84no—
154C2×D83AGL(1,ℤ/8)12yescomplemented
155C4×C83—41no—
156C4×C83—42no—
157C4×C83#12112yes—
158C4×C83C22×D411yescharacteristic
159C4×C83C2×#4111yescharacteristic
160Dic83—21no—
161Dic83—21no—
162Dic83—41no—
163Dic83—41no—
164C2×C83—22no—
165C2×C83—81no—
166C2×C83C2×#12111yescharacteristic
167C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
168C2×C83#57511yescharacteristic
169D83—22no—
170D83—41no—
171D83—82no—
172#3055—831nomaximal, Sylow
173C83#305511yescharacteristic, complemented, Sylow
174C2×AGL(1,ℤ/8)—831no—
175C22×M16—831no—
176#525—831no—
177C2×#121—831no—
178#469—832no—
179#468—832no—
180#466—832no—
181#465—832no—
182#428—831no—
183#418—831no—
184#417D8311yescharacteristic, complemented
185C2×#41C2×D8311yescharacteristic
186C22×D4—831no—
187C22×D4—1661no—
188C23×C4—831no—
189AGL(1,ℤ/8)—1664no—
190C2×SD16—832no—
191C2×D8—832no—
192C2×M16—831no—
193C2×M16—832no—
194C2×M16—1664no—
195C2×M16C2×D8311yescharacteristic
196C22×C8—832no—
197#123—1661no—
198C22⋊Q8—831no—
199C22⋊Q8—831no—
200#121—831no—
201#121—831no—
202#121—832no—
203#121—832no—
204#121—1661no—
205#121—1661no—
206#120—1661no—
207C4×D4—1661no—
208C4×D4—1661no—
209#117—831no—
210C2×C4⋊C4C2×D8311yescharacteristic
211C2×C22⋊C4—1661no—
212#107—832no—
213#106—832no—
214#103C2×D8312yescomplemented
215#102—832no—
216#102—832no—
217#102C2×D8312yescomplemented
218C24—831no—
219C24—1661no—
220#41—22no—
221#41—22no—
222C2×Q8C22×D8311yescharacteristic
223C2×D4—21no—
224C2×D4—3324no—
225C2×D4—832no—
226C2×D4—1662no—
227C2×D4—1662no—
228C2×D4—1661no—
229C2×D4—1661no—
230C2×D4—1661no—
231C2×D4—1662no—
232C2×D4—1662no—
233C2×D4C22×D8311yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—831no—
237C22×C4—831no—
238C22×C4—831no—
239C22×C4—831no—
240C22×C4—831no—
241C22×C4—1662no—
242C22×C4—1662no—
243C22×C4C22×D8311yescharacteristic
244SD16—1664no—
245D8—1664no—
246M16—22no—
247M16—834no—
248M16—1664no—
249C2×C8—832no—
250C2×C8—1664no—
251C2×C8C22×D8312yes—
252C4⋊C4—21no—
253C4⋊C4—831no—
254C4⋊C4—831no—
255C4⋊C4—1661no—
256C4⋊C4—1661no—
257C4⋊C4C22×D8312yes—
258C22⋊C4—1662no—
259C22⋊C4—1662no—
260C22⋊C4—1661no—
261C22⋊C4—1661no—
262C22⋊C4—1661no—
263C22⋊C4—1661no—
264C22⋊C4—1661no—
265C42—1661no—
266C23—21no—
267C23—3321no—
268C23—3322no—
269C23—3322no—
270C23—3322no—
271C23—831no—
272C23—831no—
273C23—1662no—
274C23—1662no—
275C23—1662no—
276C23D4×D8311yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—21no—
280D4—21no—
281D4—42no—
282D4—3324no—
283D4—3324no—
284D4—3322no—
285D4—3322no—
286D4—3322no—
287D4—1662no—
288D4—1662no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—831no—
294C2×C4—831no—
295C2×C4—1661no—
296C2×C4—1661no—
297C2×C4—1662no—
298C2×C4—1662no—
299C2×C4—1662no—
300C2×C4—1661no—
301C2×C4—1661no—
302C2×C4—1661no—
303C2×C4—1661no—
304C2×C4D4×D8311yescharacteristic
305C2×C4D4×D8312yes—
306C2×C4#1305059012yes—
307C2×C4C23×D8311yescharacteristic
308C8—22no—
309C8—834no—
310C22—21no—
311C22—21no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—3324no—
316C22—3322no—
317C22—3322no—
318C22—3322no—
319C22—832no—
320C22—6644no—
321C22—1661no—
322C22—1661no—
323C22—1661no—
324C22—1661no—
325C22—1661no—
326C22#1458176512yes—
327C22C2×D4×D8311yescharacteristic
328C4—41no—
329C4—42no—
330C4—3321no—
331C4—3321no—
332C4—1661no—
333C4—1661no—
334C4#1458176512yes—
335C4C2×D4×D8311yescharacteristic
336C4C2×#1305059011yescharacteristic
337C2—22no—
338C2—81no—
339C2—3321no—
340C2—6642no—
341C2—1662no—
342C2C2×#1458176511yescharacteristic
343C2AGL(1,ℤ/8)×D8311yescharacteristic
344C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#417×C83C2
socleC22×C83C22×D4
hypercenter#417D83
hypocenterC83#3055
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×#14581817C2semidirect
2. C2×M16×D83C2semidirect
3. #19128208C2semidirect
4. C2×#14581767C2semidirect
5. #19127953C2semidirect
6. #19127950C2semidirect
7. #19127936C2semidirect
8. #19127933C2semidirect
9. #19127741C2semidirect
10. #19127707C2semidirect
11. #417×C83C2semidirect
16. C2×#13050577C22semidirect
17. C2×#13050575C22semidirect
19. C2×#13050573C22semidirect
21. C2×C8×D83C22semidirect
23. #14581768C22semidirect
24. #14581768C22semidirect
29. #14581763C22semidirect
30. #14581763C22semidirect
36. #14581725C22semidirect
37. #14581723C22semidirect
38. #14581714C22semidirect
39. #14581714C22semidirect
43. #103×C83C22semidirect
44. #102×C83C22semidirect
65. #13050574D4semidirect
136. C83⋊C8C2×D4semidirect
154. C2×D83AGL(1,ℤ/8)semidirect
173. C83#3055semidirect
184. #417D83semidirect
214. #103C2×D83semidirect
217. #102C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818172
C2×M16×D832
#191282082
C2×#145817672
#191279532
#191279502
#191279362
#191279332
#191277412
#191277072
#417×C832
#305583