Finite Group Databasebeta

← #24916722

Subgroups — #24916722

Order 10624.

754 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#13050591C211yesmaximal, characteristic
2C2×#14581824C212yesmaximal, complemented
3C22⋊Q8×D83C211yesmaximal, characteristic, complemented
4#19127866C212yesmaximal, complemented
5#19127862C212yesmaximal, complemented
6#19127857C211yesmaximal, characteristic, complemented
7#19127854C211yesmaximal, characteristic, complemented
8#19127851C211yesmaximal, characteristic, complemented
9#19127701C212yesmaximal, complemented
10#19127663C211yesmaximal, characteristic, complemented
11#451×C83C211yesmaximal, characteristic, complemented
12C2×#13050591—28no—
13C2×#13050591—21no—
14C2×#13050591—21no—
15C2×#13050591C2212yes—
16C2×#13050591C2212yes—
17C2×Q8×D83C2211yescharacteristic, complemented
18C22×D332—21no—
19C22×D332C2211yescharacteristic
20C22×C4×D83—21no—
21C22×C4×D83C2211yescharacteristic
22#14581824—28no—
23C2×#13050580C2212yescomplemented
24C2×#13050579C2212yescomplemented
25C2×#13050577C2212yescomplemented
26C2×#13050573C2212yescomplemented
27#14581772C2211yescharacteristic, complemented
28#14581771C2211yescharacteristic
29#14581770—21no—
30#14581769—21no—
31#14581768C2211yescharacteristic, complemented
32C4⋊C4×D83—21no—
33C4⋊C4×D83C2211yescharacteristic, complemented
34C22⋊C4×D83—21no—
35#14581716C2212yescomplemented
36#14581714C2212yescomplemented
37#14581712C2212yescomplemented
38C22⋊(C83⋊C8)C2211yescharacteristic
39(C2×D83)⋊C8C2212yescomplemented
40C22×Q8×C83C2211yescharacteristic
41C2×Q16×C83C2212yescomplemented
42C22⋊Q8×C83C2211yescharacteristic
43#103×C83C2212yescomplemented
44C22⋊C8×C83C2211yescharacteristic
45C23×D83—21no—
46C23×D83D411yescharacteristic
47#13050591—28no—
48#13050591—28no—
49#13050591—24no—
50#13050591—24no—
51#13050591—24no—
52#13050591D418yes—
53C2×Dic166—21no—
54C2×Dic166C2311yescharacteristic
55Q8×D83—24no—
56C2×D332—28no—
57C2×D332—22no—
58C2×D332—22no—
59C2×D332—21no—
60C2×D332—24no—
61C2×D332C2312yes—
62C22×Dic83D411yescharacteristic
63C2×C4×D83—28no—
64C2×C4×D83—22no—
65C2×C4×D83—22no—
66C2×C4×D83—24no—
67C2×C4×D83—21no—
68C2×C4×D83—21no—
69C2×C4×D83D411yescharacteristic
70C2×C4×D83D411yescharacteristic
71C2×C4×D83C2311yescharacteristic
72C2×C4×D83C2311yescharacteristic
73#13050580—24no—
74#13050579—24no—
75#13050577—24no—
76#13050573—24no—
77C2×C83⋊C8C2312yes—
78C4⋊Dic83—21no—
79C4⋊Dic83C2311yescharacteristic
80Dic83⋊C4—21no—
81Dic83⋊C4—21no—
82Dic83⋊C4—21no—
83C22⋊Dic83—21no—
84(C2×D83)⋊C4—21no—
85(C2×D83)⋊C4—21no—
86C2×Q8×C83—24no—
87C2×Q8×C83—21no—
88C2×Q8×C83C2311yescharacteristic
89C2×Q8×C83C2312yes—
90C22×C4×C83—21no—
91C22×C4×C83C2311yescharacteristic
92Q16×C83—24no—
93C2×C8×C83C2312yes—
94C4⋊C4×C83—21no—
95C4⋊C4×C83C2311yescharacteristic
96C22⋊C4×C83—21no—
97C23×C83C2×D411yescharacteristic
98C22×D83—24no—
99C22×D83—22no—
100C22×D83—22no—
101C22×D83—22no—
102C22×D83—41no—
103C22×D83—42no—
104C22×D83—42no—
105C22×D83C2×D411yescharacteristic
106C22×D83C2×D411yescharacteristic
107Q8×C83—24no—
108Q8×C83—22no—
109Q8×C83—22no—
110Q8×C83C2×D414yes—
111C2×C4×C83—24no—
112C2×C4×C83—22no—
113C2×C4×C83—22no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83C2×D411yescharacteristic
117C2×C4×C83C2411yescharacteristic
118C2×Dic83—22no—
119C2×Dic83—21no—
120C2×Dic83—21no—
121C2×Dic83C2×D411yescharacteristic
122C2×Dic83C2×D411yescharacteristic
123D332—28no—
124D332—24no—
125D332—28no—
126D332—22no—
127D332C2×D414yes—
128C4×D83—28no—
129C4×D83—22no—
130C4×D83—22no—
131C4×D83—28no—
132C4×D83—22no—
133C4×D83—42no—
134C4×D83—42no—
135C4×D83C2×D414yes—
136Dic166—22no—
137Dic166—42no—
138C8×C83—22no—
139C83⋊C8—22no—
140C22×C83—21no—
141C22×C83—21no—
142C22×C83#13712yescomplemented
143C22×C83C22×D411yescharacteristic
144C2×D83—21no—
145C2×D83—21no—
146C2×D83—21no—
147C2×D83—24no—
148C2×D83—21no—
149C2×D83—21no—
150C2×D83—44no—
151C2×D83—48no—
152C2×D83—42no—
153C2×D83—42no—
154C2×D83#12012yes—
155C4×C83—24no—
156C4×C83—21no—
157C4×C83—41no—
158C4×C83—41no—
159C4×C83C22×D412yes—
160Dic83—21no—
161Dic83—41no—
162Dic83—41no—
163Dic83#12012yes—
164C2×C83—22no—
165C2×C83C2×#12011yescharacteristic
166C2×C83C2×#13711yescharacteristic
167C2×C83#57511yescharacteristic
168D83—22no—
169D83—41no—
170D83—44no—
171#3007—831nomaximal, Sylow
172C83#300711yescharacteristic, complemented, Sylow
173C22×#41—831no—
174C2×#137—832no—
175C2×C22⋊Q8—831no—
176#452—832no—
177#452—832no—
178#451—831no—
179#451D8311yescharacteristic, complemented
180#450—831no—
181#450—831no—
182#417—832no—
183#407—831no—
184C2×#41—832no—
185C2×#41—832no—
186C2×#41—1661no—
187C2×#41—1661no—
188C2×#41—1668no—
189C22×Q8—831no—
190C22×Q8C2×D8311yescharacteristic
191C22×D4—831no—
192C22×D4—1661no—
193C23×C4—831no—
194C23×C4—1661no—
195#137—1668no—
196C2×Q16—832no—
197C2×Q16C2×D8312yescomplemented
198C2×SD16—832no—
199C2×SD16—832no—
200C2×M16—832no—
201C22⋊Q8—831no—
202C22⋊Q8—831no—
203C22⋊Q8—831no—
204C22⋊Q8—1661no—
205C22⋊Q8—1661no—
206C22⋊Q8C2×D8311yescharacteristic
207C2×C4⋊C4—831no—
208C2×C4⋊C4—1661no—
209C2×C22⋊C4—1661no—
210#103—832no—
211#103C2×D8312yescomplemented
212#102—832no—
213#102—832no—
214C22⋊C8—831no—
215C22⋊C8—832no—
216C22⋊C8C2×D8311yescharacteristic
217C24—831no—
218C24—1661no—
219#41—838no—
220#41—1664no—
221#41—1664no—
222#41—1668no—
223#41—1668no—
224#41—1664no—
225C2×Q8—24no—
226C2×Q8—21no—
227C2×Q8—831no—
228C2×Q8—1661no—
229C2×Q8—1664no—
230C2×Q8C22×D8311yescharacteristic
231C2×Q8C22×D8312yes—
232C2×D4—832no—
233C2×D4—1664no—
234C2×D4—1662no—
235C2×D4—1662no—
236C2×D4—1668no—
237C2×D4—1661no—
238C22×C4—21no—
239C22×C4—831no—
240C22×C4—831no—
241C22×C4—831no—
242C22×C4—831no—
243C22×C4—831no—
244C22×C4—1661no—
245C22×C4—1661no—
246C22×C4—1664no—
247C22×C4—1662no—
248C22×C4—1662no—
249C22×C4—1668no—
250C22×C4C22×D8311yescharacteristic
251Q16—24no—
252Q16—1664no—
253SD16—1664no—
254SD16—1664no—
255M16—1664no—
256C2×C8—832no—
257C2×C8C22×D8312yes—
258C4⋊C4—21no—
259C4⋊C4—831no—
260C4⋊C4—1661no—
261C4⋊C4—1661no—
262C4⋊C4—1661no—
263C4⋊C4—1661no—
264C4⋊C4C22×D8311yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—1661no—
267C22⋊C4—1661no—
268C22⋊C4—1661no—
269C23—3322no—
270C23—3321no—
271C23—3322no—
272C23—831no—
273C23—831no—
274C23—1662no—
275C23—1664no—
276C23—1662no—
277C23—1662no—
278C23D4×D8311yescharacteristic
279Q8—24no—
280Q8—22no—
281Q8—22no—
282Q8—3322no—
283Q8—1662no—
284Q8D4×D8314yes—
285D4—834no—
286D4—1668no—
287D4—1668no—
288D4—1664no—
289D4—1662no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—3322no—
296C2×C4—3322no—
297C2×C4—831no—
298C2×C4—834no—
299C2×C4—831no—
300C2×C4—1662no—
301C2×C4—1668no—
302C2×C4—1662no—
303C2×C4—1668no—
304C2×C4—1661no—
305C2×C4—1661no—
306C2×C4—1662no—
307C2×C4—1662no—
308C2×C4D4×D8311yescharacteristic
309C2×C4C23×D8311yescharacteristic
310C8—22no—
311C8—1662no—
312C22—21no—
313C22—21no—
314C22—3328no—
315C22—3324no—
316C22—3322no—
317C22—3322no—
318C22—832no—
319C22—1664no—
320C22—1661no—
321C22—1661no—
322C22—1661no—
323C22—1661no—
324C22—1661no—
325C22#1458182412yescomplemented
326C22C2×D4×D8311yescharacteristic
327C4—24no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—3321no—
332C4—3321no—
333C4—832no—
334C4—1661no—
335C4C2×D4×D8312yes—
336C2—22no—
337C2—3324no—
338C2—3321no—
339C2—1662no—
340C2#120×D8311yescharacteristic
341C2C2×#1458182411yescharacteristic
342C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#451×C83C2
socleC22×C83C22×D4
prime radicalC22×#13050591C2
hypercenter#451D83
hypocenterC83#3007
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×#14581824C2semidirect
3. C22⋊Q8×D83C2semidirect
4. #19127866C2semidirect
5. #19127862C2semidirect
6. #19127857C2semidirect
7. #19127854C2semidirect
8. #19127851C2semidirect
9. #19127701C2semidirect
10. #19127663C2semidirect
11. #451×C83C2semidirect
17. C2×Q8×D83C22semidirect
23. C2×#13050580C22semidirect
24. C2×#13050579C22semidirect
25. C2×#13050577C22semidirect
26. C2×#13050573C22semidirect
27. #14581772C22semidirect
31. #14581768C22semidirect
33. C4⋊C4×D83C22semidirect
35. #14581716C22semidirect
36. #14581714C22semidirect
37. #14581712C22semidirect
39. (C2×D83)⋊C8C22semidirect
41. C2×Q16×C83C22semidirect
43. #103×C83C22semidirect
142. C22×C83#137semidirect
172. C83#3007semidirect
179. #451D83semidirect
197. C2×Q16C2×D83semidirect
211. #103C2×D83semidirect
325. C22#14581824semidirect

Maximal subgroups

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

subgroupindex
C22×#130505912
C2×#145818242
C22⋊Q8×D832
#191278662
#191278622
#191278572
#191278542
#191278512
#191277012
#191276632
#451×C832
#300783