Finite Group Databasebeta

← #35412456

Subgroups — #35412456

Order 21248.

834 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#125×Dic83C211yesmaximal, characteristic, complemented
2#124×Dic83C211yesmaximal, characteristic
3#24908429C211yesmaximal, characteristic, complemented
4#24908275C211yesmaximal, characteristic
5#24908162C212yesmaximal
6#24908155C212yesmaximal, complemented
7#24907594C211yesmaximal, characteristic, complemented
8#24907295C212yesmaximal, complemented
9#24907290C212yesmaximal
10C4×#14581702C211yesmaximal, characteristic, complemented
11C2×#488×C83C211yesmaximal, characteristic
12C2×Q8×Dic83C2211yescharacteristic
13C2×D4×Dic83C2211yescharacteristic
14C2×#14581716C2212yes—
15C2×#14581713C2212yes—
16C2×#14581702C2212yes—
17C2×#14581702C2212yes—
18C2×C4×C83⋊C8C2211yescharacteristic
19#19127608C2211yescharacteristic
20#19127604C2211yescharacteristic
21#19127600C2212yes—
22#19127594—22no—
23#19127587—22no—
24#19127567C2211yescharacteristic
25#19127567C2211yescharacteristic
26C4⋊C4×Dic83—22no—
27C4⋊C4×Dic83C2212yes—
28C22⋊C4×Dic83—22no—
29C42×Dic83C2211yescharacteristic
30#19127464C2212yes—
31#19127464C2212yes—
32#19127463C2212yes—
33#19127463C2212yes—
34C2×#125×C83C2211yescharacteristic
35C2×#124×C83C2211yescharacteristic
36#488×C83C418yescomplemented
37C2×#103×C83C2212yes—
38C2×#102×C83C2212yes—
39C2×C4×C8×C83C2211yescharacteristic
40C23×Dic83—21no—
41C22×C83⋊C8C2312yes—
42Q8×Dic83—24no—
43D4×Dic83—24no—
44C2×C4⋊Dic83—22no—
45C2×C4⋊Dic83—21no—
46C2×C4⋊Dic83C2312yes—
47C2×C4⋊Dic83C2311yescharacteristic
48C2×C4⋊Dic83C2311yescharacteristic
49C2×C22⋊Dic83—21no—
50C2×C22⋊Dic83—22no—
51C2×C4×Dic83—22no—
52C2×C4×Dic83—21no—
53C2×C4×Dic83D412yes—
54C2×C4×Dic83D412yes—
55C2×C4×Dic83C2311yescharacteristic
56C2×C4×Dic83C2311yescharacteristic
57#14581716—24no—
58#14581713—24no—
59#14581702—24no—
60#14581702—24no—
61C4×C83⋊C8—22no—
62C83⋊#95—22no—
63C83⋊#95—22no—
64C83⋊#95—22no—
65C83⋊#95—22no—
66C83⋊#95—22no—
67C22×Q8×C83C2311yescharacteristic
68C22×D4×C83C2311yescharacteristic
69C22×C8×C83C2312yes—
70#125×C83—22no—
71#125×C83C2×C414yescomplemented
72#124×C83—22no—
73#124×C83C2×C414yes—
74C2×C4⋊C4×C83—22no—
75C2×C4⋊C4×C83C2312yes—
76C2×C22⋊C4×C83—22no—
77C2×C42×C83C2311yescharacteristic
78#103×C83C2×C418yescomplemented
79#102×C83C2×C418yes—
80C4×C8×C83C2×C414yescomplemented
81C22×Dic83—21no—
82C22×Dic83—22no—
83C22×Dic83—21no—
84C22×Dic83—44no—
85C22×Dic83—42no—
86C22×Dic83C2×D412yes—
87C22×Dic83C2×D411yescharacteristic
88C22×Dic83C2×D411yescharacteristic
89C2×C83⋊C8—22no—
90C2×C83⋊C8—22no—
91C2×C83⋊C8#4114yes—
92C4⋊Dic83—22no—
93C4⋊Dic83—22no—
94C4⋊Dic83—22no—
95C4⋊Dic83—22no—
96C4⋊Dic83—44no—
97C4⋊Dic83—42no—
98C22⋊Dic83—44no—
99C22⋊Dic83—42no—
100C4×Dic83—22no—
101C4×Dic83—42no—
102C4×Dic83—42no—
103C4×Dic83—44no—
104C4×Dic83—42no—
105C4×Dic83#4114yes—
106C24×C83—21no—
107C2×Q8×C83—21no—
108C2×Q8×C83—22no—
109C2×Q8×C83—22no—
110C2×Q8×C83C22×C412yes—
111C2×D4×C83—22no—
112C2×D4×C83—22no—
113C2×D4×C83—21no—
114C2×D4×C83C22×C412yes—
115C22×C4×C83—21no—
116C22×C4×C83—22no—
117C22×C4×C83C2×D412yes—
118C22×C4×C83C2411yescharacteristic
119C2×C8×C83—22no—
120C2×C8×C83—22no—
121C2×C8×C83C22×C414yes—
122C4⋊C4×C83—28no—
123C4⋊C4×C83—22no—
124C4⋊C4×C83C22×C414yes—
125C22⋊C4×C83—28no—
126C42×C83C22×C412yes—
127C42×C83#4112yes—
128C23×C83—22no—
129C23×C83—41no—
130C23×C83—41no—
131C23×C83—42no—
132C23×C83—42no—
133C23×C83C22×D411yescharacteristic
134Q8×C83—24no—
135Q8×C83—42no—
136D4×C83—24no—
137D4×C83—42no—
138C2×C4×C83—22no—
139C2×C4×C83—22no—
140C2×C4×C83—22no—
141C2×C4×C83—24no—
142C2×C4×C83—41no—
143C2×C4×C83—41no—
144C2×C4×C83—42no—
145C2×C4×C83—42no—
146C2×C4×C83C4×D414yes—
147C2×C4×C83C23×C411yescharacteristic
148C2×C4×C83C2×#4112yes—
149C2×C4×C83C2×#4112yes—
150C2×C4×C83C2×#4111yescharacteristic
151C2×Dic83—22no—
152C2×Dic83—22no—
153C2×Dic83—22no—
154C2×Dic83—22no—
155C2×Dic83—22no—
156C2×Dic83—41no—
157C2×Dic83—41no—
158C2×Dic83—42no—
159C2×Dic83—42no—
160C2×Dic83—42no—
161C2×Dic83—42no—
162C2×Dic83—42no—
163C2×Dic83—84no—
164C2×Dic83#12412yes—
165C2×Dic83#12412yes—
166C8×C83—24no—
167C83⋊C8—24no—
168C22×C83—44no—
169C22×C83—42no—
170C22×C83—81no—
171C22×C83—81no—
172C22×C83—82no—
173C22×C83C2×C4×D411yescharacteristic
174C22×C83C2×#12411yescharacteristic
175C22×C83#53211yescharacteristic
176C22×C83#57512yes—
177C22×C83#57512yes—
178C4×C83—24no—
179C4×C83—44no—
180C4×C83—42no—
181C4×C83C4×#4112yes—
182C4×C83#52912yes—
183Dic83—22no—
184Dic83—22no—
185Dic83—42no—
186Dic83—81no—
187Dic83—81no—
188Dic83—82no—
189#21648—831nomaximal, Sylow
190C2×C83—82no—
191C2×C83C4×#12411yescharacteristic
192C2×C83#294312yes—
193C2×C83#314011yescharacteristic
194C2×C83#314211yescharacteristic
195C2×C83#318312yes—
196C2×#488D8311yescharacteristic
197C4×#125—831no—
198C4×#124—831no—
199#1991—831no—
200#1958—831no—
201#1936—832no—
202#1935—832no—
203#1835—831no—
204#1763—832no—
205#1762—832no—
206C4×#97—831no—
207C83#2164811yescharacteristic, complemented, Sylow
208C2×#125C2×D8311yescharacteristic
209C2×#124C2×D8311yescharacteristic
210C2×C4×Q8—831no—
211C2×C4×D4—831no—
212#488Dic8318yescomplemented
213C2×#103—832no—
214C2×#103C2×D8312yes—
215C2×#102—832no—
216C2×#102C2×D8312yes—
217C2×#97—832no—
218C2×#97—832no—
219C2×C4×C8—831no—
220C2×C4×C8C2×D8311yescharacteristic
221#391—831no—
222#390—831no—
223#389—832no—
224#388—1662no—
225#387—1662no—
226#382—831no—
227#382—831no—
228C4×C4⋊C4—832no—
229C4×C4⋊C4—1662no—
230C4×C22⋊C4—1662no—
231C43—831no—
232#340—832no—
233#340—832no—
234#340—832no—
235#340—832no—
236C22×Q8C22×D8311yescharacteristic
237C22×D4C22×D8311yescharacteristic
238C23×C4—1661no—
239C22×C8—832no—
240C22×C8C22×D8312yes—
241#125—22no—
242#125C2×Dic8314yescomplemented
243#124—22no—
244#124C2×Dic8314yes—
245C4×Q8—1664no—
246C4×D4—1664no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—831no—
249C2×C4⋊C4—831no—
250C2×C4⋊C4—832no—
251C2×C4⋊C4—1661no—
252C2×C4⋊C4—1662no—
253C2×C4⋊C4C22×D8312yes—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—1661no—
256C2×C22⋊C4—1662no—
257C2×C42—832no—
258C2×C42—832no—
259C2×C42—831no—
260C2×C42—831no—
261C2×C42—1661no—
262C2×C42—1662no—
263C2×C42C22×D8311yescharacteristic
264#103—1664no—
265#103C2×Dic8318yescomplemented
266#102—1664no—
267#102C2×Dic8318yes—
268#97—1664no—
269#97—1664no—
270C4×C8—1662no—
271C4×C8C2×Dic8314yescomplemented
272#95—1662no—
273#95—1662no—
274#95—1662no—
275#95—1662no—
276#95—1662no—
277C24—21no—
278C2×Q8—21no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8C22×Dic8312yes—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—21no—
285C2×D4C22×Dic8312yes—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—3322no—
289C22×C4—3324no—
290C22×C4—831no—
291C22×C4—831no—
292C22×C4—832no—
293C22×C4—1662no—
294C22×C4—1661no—
295C22×C4—1661no—
296C22×C4D4×D8312yes—
297C22×C4C23×D8311yescharacteristic
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—834no—
301C2×C8—1662no—
302C2×C8—1662no—
303C2×C8C22×Dic8314yes—
304C4⋊C4—22no—
305C4⋊C4—28no—
306C4⋊C4—3322no—
307C4⋊C4—3324no—
308C4⋊C4—1662no—
309C4⋊C4—1662no—
310C4⋊C4—1662no—
311C4⋊C4—1662no—
312C4⋊C4C22×Dic8314yes—
313C22⋊C4—28no—
314C22⋊C4—3322no—
315C22⋊C4—3324no—
316C42—3322no—
317C42—3322no—
318C42—3322no—
319C42—3324no—
320C42—834no—
321C42—1662no—
322C42C22×Dic8312yes—
323C42#1305058912yes—
324C23—22no—
325C23—41no—
326C23—42no—
327C23—42no—
328C23—41no—
329C23C2×D4×D8311yescharacteristic
330Q8—24no—
331Q8—42no—
332D4—24no—
333D4—42no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—3321no—
343C2×C4—3321no—
344C2×C4—3322no—
345C2×C4—3322no—
346C2×C4—3322no—
347C2×C4—3322no—
348C2×C4—3322no—
349C2×C4—832no—
350C2×C4—832no—
351C2×C4—6644no—
352C2×C4—1662no—
353C2×C4—1662no—
354C2×C4—1662no—
355C2×C4—1662no—
356C2×C4—1662no—
357C2×C4D4×Dic8314yes—
358C2×C4C23×Dic8311yescharacteristic
359C2×C4#41×D8312yes—
360C2×C4#41×D8312yes—
361C2×C4C2×#1305058911yescharacteristic
362C8—24no—
363C8—1664no—
364C22—44no—
365C22—42no—
366C22—81no—
367C22—81no—
368C22—82no—
369C22C2×D4×Dic8311yescharacteristic
370C22#124×D8311yescharacteristic
371C22#1912825011yescharacteristic
372C22#1912849812yes—
373C22#1912849812yes—
374C4—24no—
375C4—44no—
376C4—42no—
377C4—3322no—
378C4—6641no—
379C4—6641no—
380C4—6642no—
381C4—1662no—
382C4—1662no—
383C4#41×Dic8312yes—
384C4#1912823012yes—
385C2—82no—
386C2#124×Dic8311yescharacteristic
387C2#2491626412yes—
388C2#2491784611yescharacteristic
389C2#2491786011yescharacteristic
390C2#2491819112yes—
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC2×#488×C83C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
cosocleC22×C4×C83C24
hypercenterC2×#488D83
hypocenterC83#21648
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#488

Complements

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

NH ≅ G/Ntype
1. #125×Dic83C2semidirect
3. #24908429C2semidirect
6. #24908155C2semidirect
7. #24907594C2semidirect
8. #24907295C2semidirect
10. C4×#14581702C2semidirect
36. #488×C83C4semidirect
71. #125×C83C2×C4semidirect
78. #103×C83C2×C4semidirect
80. C4×C8×C83C2×C4semidirect
207. C83#21648semidirect
212. #488Dic83semidirect
242. #125C2×Dic83semidirect
265. #103C2×Dic83semidirect
271. C4×C8C2×Dic83semidirect

Maximal subgroups

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

subgroupindex
#125×Dic832
#124×Dic832
#249084292
#249082752
#249081622
#249081552
#249075942
#249072952
#249072902
C4×#145817022
C2×#488×C832
#2164883