Finite Group Databasebeta

← #35434215

Subgroups — #35434215

Order 21248.

1314 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127744C212yesmaximal, complemented
2C2×#19127705C212yesmaximal
3C2×#19127603C211yesmaximal, characteristic
4#24908997C212yesmaximal, complemented
5#24908992C212yesmaximal, complemented
6#24908987C214yesmaximal, complemented
7#24907157C211yesmaximal, characteristic, complemented
8C83×#2285C211yesmaximal, characteristic, complemented
9C23×D332C2211yescharacteristic
10C22×#13050574C2212yes—
11C22×C4⋊Dic83C2211yescharacteristic
12C22×(C2×D83)⋊C4—22no—
13#19127744—28no—
14C2×#14581725C2214yescomplemented
15C2×#14581723C2214yescomplemented
16#19127705—28no—
17C2×#14581712C2214yes—
18C2×#14581712C2214yes—
19#19127603—22no—
20#19127603C2212yes—
21#19127603C2212yes—
22#19127456C2212yescomplemented
23#19127456C2212yescomplemented
24C2×#117×C83C2212yes—
25C22×C42×C83C2211yescharacteristic
26#384×C83C2211yescharacteristic, complemented
27#384×C83C2211yescharacteristic, complemented
28#382×C83C2212yescomplemented
29#380×C83—22no—
30C4×C4⋊C4×C83—22no—
31C24×D83—21no—
32C22×D332—24no—
33C22×D332—21no—
34C22×D332—24no—
35C22×D332—22no—
36C22×D332Q814yes—
37C22×D332C2312yes—
38C23×Dic83—21no—
39C2×#13050574—24no—
40C2×#13050574—24no—
41C2×#13050574D418yes—
42C22×C83⋊C8C2314yes—
43C2×C4⋊Dic83—24no—
44C2×C4⋊Dic83—21no—
45C2×C4⋊Dic83C2312yes—
46C2×(C2×D83)⋊C4—24no—
47C2×(C2×D83)⋊C4—24no—
48C2×(C2×D83)⋊C4—48no—
49#14581725—28no—
50#14581723—28no—
51#14581712—28no—
52#14581712—28no—
53C23×C4×C83D412yes—
54C23×C4×C83C2311yescharacteristic
55#117×C83—24no—
56#117×C83—24no—
57C2×C4⋊C4×C83—22no—
58C2×C4⋊C4×C83—22no—
59C2×C4⋊C4×C83C2312yes—
60C2×C4⋊C4×C83C2312yes—
61C2×C22⋊C4×C83—22no—
62C2×C42×C83—24no—
63C2×C42×C83—22no—
64C2×C42×C83D412yes—
65C2×C42×C83C2311yescharacteristic
66C2×C42×C83C2311yescharacteristic
67#95×C83—24no—
68C23×D83—24no—
69C23×D83—22no—
70C23×D83—44no—
71C23×D83—44no—
72C23×D83—42no—
73C23×D83—42no—
74C2×D332—216no—
75C2×D332—24no—
76C2×D332—24no—
77C2×D332—24no—
78C2×D332—22no—
79C2×D332—42no—
80C2×D332—48no—
81C2×D332—42no—
82C2×D332C2×Q814yes—
83C22×Dic83—22no—
84C22×Dic83—44no—
85C22×Dic83—42no—
86#13050574—216no—
87C2×C83⋊C8—24no—
88C2×C83⋊C8—24no—
89C2×C83⋊C8C2×D418yes—
90C4⋊Dic83—24no—
91C4⋊Dic83—42no—
92(C2×D83)⋊C4—48no—
93(C2×D83)⋊C4—48no—
94C24×C83C2×D411yescharacteristic
95C22×C4×C83—24no—
96C22×C4×C83—24no—
97C22×C4×C83—22no—
98C22×C4×C83—22no—
99C22×C4×C83—42no—
100C22×C4×C83C2×D414yes—
101C22×C4×C83C2×D412yes—
102C22×C4×C83C2×D412yes—
103C22×C4×C83C2×D411yescharacteristic
104C22×C4×C83C2×Q812yes—
105C22×C4×C83#4114yes—
106C22×C4×C83#4112yes—
107C22×C4×C83C2411yescharacteristic
108C4⋊C4×C83—22no—
109C4⋊C4×C83—22no—
110C4⋊C4×C83—22no—
111C4⋊C4×C83—22no—
112C4⋊C4×C83—44no—
113C4⋊C4×C83—44no—
114C22⋊C4×C83—44no—
115C42×C83—24no—
116C42×C83—22no—
117C42×C83—22no—
118C42×C83—44no—
119C23×C83—22no—
120C23×C83—22no—
121C23×C83C22⋊Q812yes—
122C23×C83#12312yes—
123C23×C83#12712yes—
124C23×C83C22×D411yescharacteristic
125C22×D83—24no—
126C22×D83—416no—
127C22×D83—42no—
128C22×D83—42no—
129C22×D83—44no—
130C22×D83—44no—
131C22×D83—48no—
132C22×D83—88no—
133C22×D83—84no—
134C22×D83—84no—
135C2×C4×C83—22no—
136C2×C4×C83—22no—
137C2×C4×C83—22no—
138C2×C4×C83—22no—
139C2×C4×C83—22no—
140C2×C4×C83—48no—
141C2×C4×C83—42no—
142C2×C4×C83—42no—
143C2×C4×C83—44no—
144C2×C4×C83—44no—
145C2×C4×C83C22⋊Q812yes—
146C2×C4×C83C22⋊Q818yes—
147C2×C4×C83#12312yes—
148C2×C4×C83#12318yes—
149C2×C4×C83#12712yes—
150C2×C4×C83AGL(1,ℤ/8)14yes—
151C2×C4×C83C22×D412yes—
152C2×C4×C83C22×Q811yescharacteristic
153C2×C4×C83C2×#4112yes—
154C2×C4×C83C2×#4111yescharacteristic
155C2×Dic83—42no—
156C2×Dic83—44no—
157C2×Dic83—84no—
158D332—28no—
159D332—44no—
160D332—48no—
161C83⋊C8—28no—
162C22×C83—24no—
163C22×C83—24no—
164C22×C83—22no—
165C22×C83—22no—
166C22×C83#39914yes—
167C22×C83C2×C22⋊Q811yescharacteristic
168C22×C83C2×#12311yescharacteristic
169C22×C83C2×#12711yescharacteristic
170C22×C83C2×AGL(1,ℤ/8)12yes—
171C22×C83#57512yes—
172C2×D83—48no—
173C2×D83—44no—
174C2×D83—816no—
175C2×D83—84no—
176C2×D83—82no—
177C2×D83—82no—
178C4×C83—24no—
179C4×C83—42no—
180C4×C83—82no—
181C4×C83—82no—
182C4×C83#39914yes—
183C4×C83C2×C22⋊Q812yes—
184C4×C83C2×#12312yes—
185Dic83—82no—
186#23731—831nomaximal, Sylow
187C2×C83—24no—
188C2×C83C2×#39911yescharacteristic
189C2×C83#307712yes—
190C2×C83#309412yes—
191C2×C83#315111yescharacteristic
192C2×C83#315311yescharacteristic
193D83—84no—
194C2×#428—832no—
195C2×#418—832no—
196C2×#390—831no—
197#2285D8311yescharacteristic, complemented
198#2094—832no—
199#2093—832no—
200#2092—834no—
201#1728—831no—
202C83#2373111yescharacteristic, complemented, Sylow
203C23×D4—831no—
204C22×M16—832no—
205C2×#117C2×D8312yes—
206C22×C4⋊C4—831no—
207C22×C22⋊C4—1662no—
208C22×C42C2×D8311yescharacteristic
209#428—1668no—
210C2×#107—834no—
211C2×#106—834no—
212#418—1668no—
213C2×#102—834no—
214C2×#102—834no—
215#390—832no—
216#390—832no—
217#390—1662no—
218#384C2×D8311yescharacteristic, complemented
219#384C2×D8311yescharacteristic, complemented
220#382C2×D8312yescomplemented
221#380—22no—
222C4×C4⋊C4—22no—
223#336—832no—
224#336—832no—
225C25—1661no—
226C22×D4—834no—
227C22×D4—832no—
228C22×D4—1664no—
229C22×D4—1661no—
230C22×D4—1664no—
231C22×D4—1662no—
232C23×C4—1661no—
233C23×C4C22⋊D8312yes—
234C23×C4C22×D8311yescharacteristic
235C2×M16—838no—
236C2×M16—1664no—
237C2×M16—1664no—
238C22×C8—834no—
239#117—24no—
240#117—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—832no—
244C2×C4⋊C4—1661no—
245C2×C4⋊C4—1664no—
246C2×C4⋊C4C22×D8312yes—
247C2×C4⋊C4C22×D8312yes—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—3328no—
250C2×C22⋊C4—1664no—
251C2×C22⋊C4—1664no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42C22⋊D8312yes—
255C2×C42C22×D8311yescharacteristic
256C2×C42C22×D8311yescharacteristic
257#107—1668no—
258#106—1668no—
259#102—1668no—
260#102—1668no—
261#95—24no—
262C24—3324no—
263C24—3324no—
264C24—3322no—
265C24—3322no—
266C24—1664no—
267C24—1662no—
268C24C2×C22⋊D8311yescharacteristic
269C2×D4—3322no—
270C2×D4—3328no—
271C2×D4—3322no—
272C2×D4—834no—
273C2×D4—1664no—
274C2×D4—16616no—
275C2×D4—1664no—
276C2×D4—1664no—
277C2×D4—1662no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—42no—
283C22×C4—3324no—
284C22×C4—3322no—
285C22×C4—1662no—
286C22×C4D4×D8314yes—
287C22×C4C2×C22⋊D8312yes—
288C22×C4C2×C22⋊D8312yes—
289C22×C4C2×C22⋊D8311yescharacteristic
290C22×C4Q8×D8312yes—
291C22×C4#1305059014yes—
292C22×C4#1305059112yes—
293C22×C4C23×D8311yescharacteristic
294M16—16616no—
295C2×C8—838no—
296C2×C8—1664no—
297C2×C8—1664no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—3322no—
305C4⋊C4—1664no—
306C22⋊C4—44no—
307C22⋊C4—3328no—
308C22⋊C4—3328no—
309C42—24no—
310C42—22no—
311C42—22no—
312C42—44no—
313C23—22no—
314C23—22no—
315C23—33216no—
316C23—3328no—
317C23—3322no—
318C23—3324no—
319C23—3322no—
320C23—3324no—
321C23—6648no—
322C23—6644no—
323C23—6644no—
324C23—1664no—
325C23#1458176812yes—
326C23#1458177512yes—
327C23#1458178812yes—
328C23C22×C22⋊D8311yescharacteristic
329D4—3328no—
330D4—3324no—
331D4—1668no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—48no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—3324no—
343C2×C4—3322no—
344C2×C4—6644no—
345C2×C4#1458176812yes—
346C2×C4#1458177018yes—
347C2×C4#1458177512yes—
348C2×C4#1458177718yes—
349C2×C4#1458178812yes—
350C2×C4#1458181714yes—
351C2×C4C2×D4×D8312yes—
352C2×C4C2×Q8×D8311yescharacteristic
353C2×C4C2×#1305059012yes—
354C2×C4C2×#1305059111yescharacteristic
355C8—1668no—
356C22—24no—
357C22—24no—
358C22—22no—
359C22—22no—
360C22—3328no—
361C22—3324no—
362C22—66416no—
363C22—6644no—
364C22—6642no—
365C22—6642no—
366C22#1912763614yes—
367C22C2×#1458176811yescharacteristic
368C22C2×#1458177511yescharacteristic
369C22C2×#1458178811yescharacteristic
370C22C2×#1458181712yes—
371C22#1912849912yes—
372C4—24no—
373C4—42no—
374C4—82no—
375C4—82no—
376C4—6642no—
377C4#1912763614yes—
378C4C2×#1458177012yes—
379C4C2×#1458177712yes—
380C2—24no—
381C2—6644no—
382C2C2×#1912763611yescharacteristic
383C2#2491725612yes—
384C2#2491740312yes—
385C2#2491791611yescharacteristic
386C2#2491793111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2285C2
socleC23×C83C22×D4
prime radicalC23×D332C22
hypercenter#2285D83
hypocenterC83#23731
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2285

Complements

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

NH ≅ G/Ntype
1. C2×#19127744C2semidirect
4. #24908997C2semidirect
5. #24908992C2semidirect
6. #24908987C2semidirect
7. #24907157C2semidirect
8. C83×#2285C2semidirect
14. C2×#14581725C22semidirect
15. C2×#14581723C22semidirect
22. #19127456C22semidirect
23. #19127456C22semidirect
26. #384×C83C22semidirect
27. #384×C83C22semidirect
28. #382×C83C22semidirect
197. #2285D83semidirect
202. C83#23731semidirect
218. #384C2×D83semidirect
219. #384C2×D83semidirect
220. #382C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191277442
C2×#191277052
C2×#191276032
#249089972
#249089922
#249089872
#249071572
C83×#22852
#2373183