Finite Group Databasebeta

← #35433168

Subgroups — #35433168

Order 21248.

1226 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127707C212yesmaximal
2C2×#19127603C211yesmaximal, characteristic
3#24909870C211yesmaximal, characteristic, complemented
4#24908878C214yesmaximal, complemented
5#24908868C214yesmaximal, complemented
6#24907183C212yesmaximal, complemented
7C83×#1728C211yesmaximal, characteristic, complemented
8C23×D332C2211yescharacteristic
9C2×#14581751C2212yes—
10C22×C4⋊Dic83C2211yescharacteristic
11C22×(C2×D83)⋊C4—22no—
12#19127707—28no—
13C2×#14581714C2218yes—
14#19127603—22no—
15#19127603C2212yes—
16#19127603C2212yes—
17#19127571C2212yescomplemented
18#19127567C2212yescomplemented
19#19127561—22no—
20C4×C4⋊Dic83—22no—
21#19127464C2218yescomplemented
22C22×M16×C83C2212yes—
23C22×C42×C83C2211yescharacteristic
24#336×C83C2212yescomplemented
25#336×C83C2212yescomplemented
26C24×D83—21no—
27C22×D332—21no—
28C22×D332—24no—
29C22×D332—24no—
30C22×D332—22no—
31C22×D332D414yes—
32C22×D332C2312yes—
33C23×Dic83—21no—
34#14581751—24no—
35#14581751—24no—
36C2×C4⋊Dic83—22no—
37C2×C4⋊Dic83—22no—
38C2×C4⋊Dic83—21no—
39C2×C4⋊Dic83—24no—
40C2×C4⋊Dic83C2314yes—
41C2×C4⋊Dic83C2312yes—
42C2×C22⋊Dic83—22no—
43C2×(C2×D83)⋊C4—24no—
44C2×(C2×D83)⋊C4—24no—
45C2×(C2×D83)⋊C4—48no—
46C2×C4×Dic83—22no—
47#14581714—216no—
48C83⋊#95—24no—
49C23×C4×C83D412yes—
50C23×C4×C83C2311yescharacteristic
51C2×M16×C83—24no—
52C2×M16×C83—24no—
53C2×M16×C83—24no—
54C22×C8×C83C2314yes—
55C2×C42×C83—24no—
56C2×C42×C83D412yes—
57C2×C42×C83C2311yescharacteristic
58C2×C42×C83C2311yescharacteristic
59C23×D83—24no—
60C23×D83—22no—
61C23×D83—44no—
62C23×D83—44no—
63C23×D83—42no—
64C23×D83—42no—
65C2×D332—216no—
66C2×D332—24no—
67C2×D332—24no—
68C2×D332—24no—
69C2×D332—22no—
70C2×D332—42no—
71C2×D332—42no—
72C2×D332—48no—
73C2×D332C2×D414yes—
74C22×Dic83—22no—
75C22×Dic83—24no—
76C22×Dic83—44no—
77C22×Dic83—42no—
78C4⋊Dic83—24no—
79C4⋊Dic83—24no—
80C4⋊Dic83—24no—
81C4⋊Dic83—44no—
82C4⋊Dic83—42no—
83C4⋊Dic83—44no—
84C22⋊Dic83—44no—
85(C2×D83)⋊C4—48no—
86(C2×D83)⋊C4—48no—
87C4×Dic83—44no—
88C24×C83C2×D411yescharacteristic
89C22×C4×C83—24no—
90C22×C4×C83—24no—
91C22×C4×C83—42no—
92C22×C4×C83C2×D412yes—
93C22×C4×C83C2×D412yes—
94C22×C4×C83C2×D412yes—
95C22×C4×C83C2×D411yescharacteristic
96C22×C4×C83#4114yes—
97C22×C4×C83#4114yes—
98C22×C4×C83#4112yes—
99C22×C4×C83C2411yescharacteristic
100M16×C83—48no—
101C2×C8×C83—24no—
102C2×C8×C83—24no—
103C2×C8×C83—24no—
104C42×C83—24no—
105C42×C83—22no—
106C42×C83—22no—
107C23×C83—22no—
108C23×C83—22no—
109C23×C83#12112yes—
110C23×C83#12312yes—
111C23×C83#12412yes—
112C23×C83C22×D411yescharacteristic
113C22×D83—24no—
114C22×D83—416no—
115C22×D83—42no—
116C22×D83—42no—
117C22×D83—44no—
118C22×D83—44no—
119C22×D83—48no—
120C22×D83—88no—
121C22×D83—84no—
122C22×D83—84no—
123C2×C4×C83—22no—
124C2×C4×C83—22no—
125C2×C4×C83—22no—
126C2×C4×C83—22no—
127C2×C4×C83—22no—
128C2×C4×C83—48no—
129C2×C4×C83#12112yes—
130C2×C4×C83#12318yes—
131C2×C4×C83#12312yes—
132C2×C4×C83#12412yes—
133C2×C4×C83#12618yes—
134C2×C4×C83AGL(1,ℤ/8)14yes—
135C2×C4×C83C22×D411yescharacteristic
136C2×C4×C83C2×#4112yes—
137C2×C4×C83C2×#4112yes—
138C2×C4×C83C2×#4111yescharacteristic
139C2×Dic83—44no—
140C2×Dic83—42no—
141C2×Dic83—44no—
142C2×Dic83—44no—
143C2×Dic83—44no—
144C2×Dic83—84no—
145D332—28no—
146D332—44no—
147D332—48no—
148C8×C83—44no—
149C22×C83—24no—
150C22×C83—22no—
151C22×C83—22no—
152C22×C83—24no—
153C22×C83#39714yes—
154C22×C83C2×#12111yescharacteristic
155C22×C83C2×#12311yescharacteristic
156C22×C83C2×#12411yescharacteristic
157C22×C83C2×AGL(1,ℤ/8)12yes—
158C22×C83#57512yes—
159C2×D83—44no—
160C2×D83—48no—
161C2×D83—816no—
162C2×D83—84no—
163C2×D83—82no—
164C2×D83—82no—
165C4×C83—24no—
166C4×C83—42no—
167C4×C83#39714yes—
168C4×C83C2×#12312yes—
169C4×C83C2×#12612yes—
170Dic83—82no—
171Dic83—84no—
172#23628—831nomaximal, Sylow
173C2×C83—24no—
174C2×C83C2×#39711yescharacteristic
175C2×C83#303612yes—
176C2×C83#309412yes—
177C2×C83#313811yescharacteristic
178C2×C83#314011yescharacteristic
179D83—84no—
180C2×#418—832no—
181C2×#390—831no—
182#2285—831no—
183#2073—834no—
184#2071—834no—
185#1736—832no—
186#1728D8311yescharacteristic, complemented
187C83#2362811yescharacteristic, complemented, Sylow
188C23×D4—831no—
189C22×M16C2×D8312yes—
190C2×#117—832no—
191C22×C4⋊C4—831no—
192C22×C22⋊C4—1662no—
193C22×C42C2×D8311yescharacteristic
194#418—1668no—
195C2×#102—838no—
196#390—832no—
197#390—832no—
198#390—1662no—
199#384—832no—
200#382—832no—
201#380—1662no—
202C4×C4⋊C4—1662no—
203#340—838no—
204#336C2×D8312yescomplemented
205#336C2×D8312yescomplemented
206C25—1661no—
207C22×D4—834no—
208C22×D4—832no—
209C22×D4—1661no—
210C22×D4—1664no—
211C22×D4—1664no—
212C22×D4—1662no—
213C23×C4—1661no—
214C23×C4D33212yes—
215C23×C4C22×D8311yescharacteristic
216C2×M16—24no—
217C2×M16—24no—
218C2×M16—24no—
219C22×C8C22×D8314yes—
220#117—1664no—
221#117—1664no—
222C2×C4⋊C4—832no—
223C2×C4⋊C4—834no—
224C2×C4⋊C4—1661no—
225C2×C4⋊C4—1662no—
226C2×C4⋊C4—1662no—
227C2×C4⋊C4—1664no—
228C2×C22⋊C4—3328no—
229C2×C22⋊C4—1662no—
230C2×C22⋊C4—1664no—
231C2×C22⋊C4—1664no—
232C2×C42—24no—
233C2×C42—1662no—
234C2×C42D33212yes—
235C2×C42C22×D8311yescharacteristic
236C2×C42C22×D8311yescharacteristic
237#102—16616no—
238#95—1664no—
239C24—3324no—
240C24—3324no—
241C24—3322no—
242C24—3322no—
243C24—1664no—
244C24—1662no—
245C24C2×D33211yescharacteristic
246C2×D4—3322no—
247C2×D4—3322no—
248C2×D4—3328no—
249C2×D4—834no—
250C2×D4—1664no—
251C2×D4—16616no—
252C2×D4—1664no—
253C2×D4—1664no—
254C2×D4—1662no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—42no—
258C22×C4—3324no—
259C22×C4—3322no—
260C22×C4—1664no—
261C22×C4—1662no—
262C22×C4D4×D8312yes—
263C22×C4C2×D33212yes—
264C22×C4C2×D33212yes—
265C22×C4C2×D33211yescharacteristic
266C22×C4#1305058914yes—
267C22×C4#1305059014yes—
268C22×C4#1305059112yes—
269C22×C4C23×D8311yescharacteristic
270M16—48no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—24no—
274C4⋊C4—3324no—
275C4⋊C4—3322no—
276C4⋊C4—3324no—
277C4⋊C4—1664no—
278C4⋊C4—1664no—
279C4⋊C4—1664no—
280C22⋊C4—3328no—
281C22⋊C4—3324no—
282C22⋊C4—3328no—
283C42—24no—
284C42—22no—
285C42—22no—
286C42—3324no—
287C23—22no—
288C23—22no—
289C23—33216no—
290C23—3328no—
291C23—3322no—
292C23—3322no—
293C23—3324no—
294C23—3324no—
295C23—6644no—
296C23—6644no—
297C23—6648no—
298C23—1664no—
299C23#1458176712yes—
300C23#1458177612yes—
301C23#1458178012yes—
302C23C22×D33211yescharacteristic
303D4—3328no—
304D4—3324no—
305D4—1668no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—48no—
312C2×C4—3324no—
313C2×C4—3322no—
314C2×C4—3324no—
315C2×C4—3324no—
316C2×C4—3324no—
317C2×C4—6644no—
318C2×C4#1458176712yes—
319C2×C4#1458177418yes—
320C2×C4#1458177612yes—
321C2×C4#1458178012yes—
322C2×C4#1458178718yes—
323C2×C4#1458182014yes—
324C2×C4C2×D4×D8311yescharacteristic
325C2×C4C2×#1305058912yes—
326C2×C4C2×#1305059012yes—
327C2×C4C2×#1305059111yescharacteristic
328C8—44no—
329C22—22no—
330C22—22no—
331C22—24no—
332C22—24no—
333C22—3324no—
334C22—3328no—
335C22—66416no—
336C22—6642no—
337C22—6642no—
338C22—6644no—
339C22#1912762814yes—
340C22C2×#1458176711yescharacteristic
341C22C2×#1458177611yescharacteristic
342C22C2×#1458178011yescharacteristic
343C22C2×#1458182012yes—
344C22#1912850112yes—
345C4—24no—
346C4—42no—
347C4—6642no—
348C4—6644no—
349C4#1912762814yes—
350C4C2×#1458177412yes—
351C4C2×#1458178712yes—
352C2—24no—
353C2—6644no—
354C2C2×#1912762811yescharacteristic
355C2#2491692912yes—
356C2#2491740412yes—
357C2#2491783811yescharacteristic
358C2#2491785011yescharacteristic
subgroupG/H
centerC23C22×D332
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#1728C2
socleC23×C83C22×D4
prime radicalC23×D332C22
hypercenter#1728D83
hypocenterC83#23628
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1728

Complements

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

NH ≅ G/Ntype
3. #24909870C2semidirect
4. #24908878C2semidirect
5. #24908868C2semidirect
6. #24907183C2semidirect
7. C83×#1728C2semidirect
17. #19127571C22semidirect
18. #19127567C22semidirect
21. #19127464C22semidirect
24. #336×C83C22semidirect
25. #336×C83C22semidirect
186. #1728D83semidirect
187. C83#23628semidirect
204. #336C2×D83semidirect
205. #336C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191277072
C2×#191276032
#249098702
#249088782
#249088682
#249071832
C83×#17282
#2362883