Finite Group Databasebeta

← #35412857

Subgroups — #35412857

Order 21248.

1398 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128077C211yesmaximal, characteristic
2C2×#19127811C214yesmaximal, complemented
3C4×#14581789C211yesmaximal, characteristic, complemented
4#24908338C214yesmaximal, complemented
5#24908270C211yesmaximal, characteristic, complemented
6#24908268C211yesmaximal, characteristic, complemented
7#24907619C211yesmaximal, characteristic, complemented
8C4×#14581703C211yesmaximal, characteristic, complemented
9C4×#127×C83C211yesmaximal, characteristic, complemented
10C22×#13050575C2214yes—
11C2×#14581789C2211yescharacteristic
12C2×C4×D332—22no—
13C2×C4×D332C2212yes—
14#19128077C418yescomplemented
15#19127811—216no—
16C2×#14581723C2214yes—
17C2×#14581713C2214yes—
18C2×#14581712C2214yes—
19C2×#14581703C2212yes—
20C2×#14581703C2212yes—
21C2×C4×C83⋊C8C2211yescharacteristic
22C2×C4×C83⋊C8C2211yescharacteristic
23#19127603—22no—
24#19127603C2212yes—
25#19127571C2211yescharacteristic
26C2×#127×C83C2211yescharacteristic
27C2×C4×D4×C83—22no—
28C2×C4×D4×C83C2212yes—
29#390×C83—22no—
30#390×C83C2212yes—
31#384×C83C2211yescharacteristic
32C43×C83C2211yescharacteristic
33C22×D332—22no—
34C22×D332C2312yes—
35C22×C4×D83—22no—
36C2×#13050575—28no—
37C2×#13050575—28no—
38C2×#13050575C2×C4116yescomplemented
39C22×C83⋊C8C2312yes—
40C22×C83⋊C8C2312yes—
41#14581789—22no—
42#14581789C2×C414yescomplemented
43C4×D332—28no—
44C4×D332—48no—
45C2×C4⋊Dic83—22no—
46C2×C4⋊Dic83C2312yes—
47C2×(C2×D83)⋊C4—22no—
48C2×(C2×D83)⋊C4—24no—
49#14581723—28no—
50#14581713—28no—
51#14581712—28no—
52#14581703—24no—
53#14581703D418yescomplemented
54C4×C83⋊C8—22no—
55C4×C83⋊C8C2×C414yescomplemented
56C22×D4×C83—22no—
57C22×D4×C83C2312yes—
58C23×C4×C83—22no—
59#127×C83—22no—
60#127×C83C2×C414yescomplemented
61C4×D4×C83—28no—
62C4×D4×C83—48no—
63C2×C4⋊C4×C83—22no—
64C2×C4⋊C4×C83C2312yes—
65C2×C22⋊C4×C83—24no—
66C2×C22⋊C4×C83—22no—
67C2×C42×C83D412yes—
68C2×C42×C83D412yes—
69C2×C42×C83C2311yescharacteristic
70C2×C42×C83C2311yescharacteristic
71C2×C42×C83C2311yescharacteristic
72C23×D83—22no—
73C2×D332—24no—
74C2×D332—24no—
75C2×D332—28no—
76C2×D332—22no—
77C2×D332—44no—
78C2×D332—44no—
79C2×D332C22×C414yes—
80C22×Dic83—22no—
81C2×C4×D83—48no—
82C2×C4×D83—44no—
83#13050575—232no—
84C2×C83⋊C8—22no—
85C2×C83⋊C8—22no—
86C2×C83⋊C8C22×C414yes—
87C2×C83⋊C8C2×D414yes—
88C2×C83⋊C8C2×D414yes—
89C2×C83⋊C8#4114yes—
90C4⋊Dic83—24no—
91C4⋊Dic83—44no—
92(C2×D83)⋊C4—48no—
93(C2×D83)⋊C4—44no—
94C24×C83—22no—
95C2×D4×C83—28no—
96C2×D4×C83—24no—
97C2×D4×C83—24no—
98C2×D4×C83—22no—
99C2×D4×C83—44no—
100C2×D4×C83—44no—
101C2×D4×C83C22×C414yes—
102C22×C4×C83—22no—
103C22×C4×C83—48no—
104C22×C4×C83—44no—
105C22×C4×C83C2×D412yes—
106C22×C4×C83C2×D411yescharacteristic
107C22×C4×C83C2×D411yescharacteristic
108C22×C4×C83C2×D412yes—
109C22×C4×C83C2411yescharacteristic
110C4⋊C4×C83—24no—
111C4⋊C4×C83—44no—
112C22⋊C4×C83—48no—
113C22⋊C4×C83—44no—
114C42×C83—24no—
115C42×C83—24no—
116C42×C83—22no—
117C42×C83C22×C412yes—
118C42×C83C2×D414yes—
119C42×C83#4112yes—
120C23×C83—24no—
121C23×C83—42no—
122C23×C83—42no—
123C23×C83—44no—
124C23×C83—44no—
125C23×C83C22×D411yescharacteristic
126C22×D83—24no—
127C22×D83—42no—
128C22×D83—42no—
129C22×D83—44no—
130C22×D83—44no—
131D4×C83—28no—
132D4×C83—44no—
133D4×C83—416no—
134C2×C4×C83—22no—
135C2×C4×C83—22no—
136C2×C4×C83—22no—
137C2×C4×C83—22no—
138C2×C4×C83—22no—
139C2×C4×C83—42no—
140C2×C4×C83—44no—
141C2×C4×C83—88no—
142C2×C4×C83C4×D414yes—
143C2×C4×C83#12712yes—
144C2×C4×C83#12712yes—
145C2×C4×C83AGL(1,ℤ/8)14yes—
146C2×C4×C83AGL(1,ℤ/8)14yes—
147C2×C4×C83C23×C411yescharacteristic
148C2×C4×C83C22×D412yes—
149C2×C4×C83C2×#4112yes—
150C2×C4×C83C2×#4111yescharacteristic
151C2×Dic83—42no—
152C2×Dic83—44no—
153D332—28no—
154D332—44no—
155D332—416no—
156C4×D83—88no—
157C83⋊C8—24no—
158C83⋊C8C4×D418yescomplemented
159C22×C83—44no—
160C22×C83—48no—
161C22×C83—82no—
162C22×C83—82no—
163C22×C83—84no—
164C22×C83C2×C4×D411yescharacteristic
165C22×C83C2×#12711yescharacteristic
166C22×C83#53211yescharacteristic
167C22×C83C2×AGL(1,ℤ/8)12yes—
168C22×C83C2×AGL(1,ℤ/8)12yes—
169C2×D83—44no—
170C2×D83—48no—
171C2×D83—82no—
172C2×D83—82no—
173C2×D83—84no—
174C4×C83—22no—
175C4×C83—22no—
176C4×C83—42no—
177C4×C83—82no—
178C4×C83#44218yescomplemented
179C4×C83C2×C4×D412yes—
180C4×C83#53212yes—
181Dic83—82no—
182#21690—831nomaximal, Sylow
183C2×C83—84no—
184C2×C83C4×#12711yescharacteristic
185C2×C83C2×#44212yes—
186C2×C83#315111yescharacteristic
187C2×C83#315111yescharacteristic
188C2×C83#318412yes—
189D83—84no—
190C2×#493—831no—
191C2×#442—834no—
192C4×#127—831no—
193C4×#127D8311yescharacteristic, complemented
194#1972—834no—
195#1957—831no—
196#1957—831no—
197#1844—831no—
198C4×#97—831no—
199C83#2169011yescharacteristic, complemented, Sylow
200C22×D8—834no—
201C2×#127—831no—
202C2×#127C2×D8311yescharacteristic
203C2×C4×D4—22no—
204C2×C4×D4—832no—
205C2×C4×D4—1662no—
206C2×C4×D4C2×D8312yes—
207#493—838no—
208#442—16616no—
209C2×#106—834no—
210C2×#102—834no—
211C2×#102—834no—
212C2×#97—832no—
213C2×#97—832no—
214C2×C4×C8—831no—
215C2×C4×C8—831no—
216#390—22no—
217#390—832no—
218#390—1662no—
219#390C2×D8312yes—
220#384—831no—
221#384C2×D8311yescharacteristic
222C43C2×D8311yescharacteristic
223C22×D4—22no—
224C22×D4—832no—
225C22×D4—1662no—
226C22×D4C22×D8312yes—
227C23×C4—22no—
228C23×C4—1662no—
229C2×D8—8316no—
230C2×D8—1668no—
231C2×D8—1668no—
232C22×C8—832no—
233C22×C8—832no—
234#127—22no—
235#127—834no—
236#127—1662no—
237#127C4×D8314yescomplemented
238C4×D4—28no—
239C4×D4—48no—
240C4×D4—3328no—
241C4×D4—1668no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—832no—
244C2×C4⋊C4—1662no—
245C2×C4⋊C4C22×D8312yes—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—1662no—
249C2×C22⋊C4—1664no—
250C2×C42C22⋊D8312yes—
251C2×C42C22⋊D8312yes—
252C2×C42C22×D8311yescharacteristic
253C2×C42C22×D8311yescharacteristic
254C2×C42C22×D8311yescharacteristic
255#106—1668no—
256#102—1668no—
257#102—1668no—
258#97—838no—
259#97—1664no—
260C4×C8—834no—
261C4×C8—1662no—
262C24—22no—
263C24—1662no—
264C2×D4—24no—
265C2×D4—24no—
266C2×D4—22no—
267C2×D4—28no—
268C2×D4—44no—
269C2×D4—44no—
270C2×D4—3324no—
271C2×D4—3324no—
272C2×D4—834no—
273C2×D4—1664no—
274C2×D4—1664no—
275C2×D4—1662no—
276C2×D4—1668no—
277C2×D4C2×C4×D8314yes—
278C22×C4—22no—
279C22×C4—48no—
280C22×C4—44no—
281C22×C4—3328no—
282C22×C4—3324no—
283C22×C4—1662no—
284C22×C4C2×C22⋊D8312yes—
285C22×C4C2×C22⋊D8311yescharacteristic
286C22×C4C2×C22⋊D8311yescharacteristic
287C22×C4C2×C22⋊D8312yes—
288C22×C4C23×D8311yescharacteristic
289D8—16632no—
290C2×C8—834no—
291C2×C8—834no—
292C2×C8—834no—
293C2×C8—834no—
294C2×C8—1662no—
295C2×C8—1662no—
296C4⋊C4—24no—
297C4⋊C4—44no—
298C4⋊C4—3324no—
299C4⋊C4—1664no—
300C22⋊C4—48no—
301C22⋊C4—44no—
302C22⋊C4—3328no—
303C22⋊C4—3324no—
304C42—24no—
305C42—24no—
306C42—22no—
307C42C2×C4×D8312yes—
308C42D4×D8314yes—
309C42#1305059012yes—
310C23—24no—
311C23—44no—
312C23—44no—
313C23—42no—
314C23—42no—
315C23—3324no—
316C23—3324no—
317C23—3322no—
318C23—3322no—
319C23—1664no—
320C23C22×C22⋊D8311yescharacteristic
321D4—28no—
322D4—416no—
323D4—44no—
324D4—33216no—
325D4—3324no—
326D4—1668no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—88no—
335C2×C4—3324no—
336C2×C4—3322no—
337C2×C4—6648no—
338C2×C4C4×C22⋊D8314yes—
339C2×C4#1458178812yes—
340C2×C4#1458178812yes—
341C2×C4#1458181714yes—
342C2×C4#1458181814yes—
343C2×C4C22×C4×D8311yescharacteristic
344C2×C4C2×D4×D8312yes—
345C2×C4#41×D8312yes—
346C2×C4C2×#1305059011yescharacteristic
347C8—838no—
348C8—1664no—
349C22—48no—
350C22—44no—
351C22—82no—
352C22—82no—
353C22—84no—
354C22—3328no—
355C22—3324no—
356C22—6642no—
357C22—6642no—
358C22—6644no—
359C22C2×C4×C22⋊D8311yescharacteristic
360C22C2×#1458178811yescharacteristic
361C22#1912824711yescharacteristic
362C22C2×#1458181712yes—
363C22C2×#1458181812yes—
364C4—22no—
365C4—22no—
366C4—42no—
367C4—82no—
368C4—6642no—
369C4#1912781118yescomplemented
370C4C4×D4×D8312yes—
371C4#1912824812yes—
372C2—84no—
373C2—6644no—
374C2C4×#1458178811yescharacteristic
375C2C2×#1912781112yes—
376C2#2491791311yescharacteristic
377C2#2491791611yescharacteristic
378C2#2491819712yes—
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC4×#127×C83C2
socleC23×C83C22×D4
prime radicalC2×#19128077C2
cosocleC22×C4×C83C24
hypercenterC4×#127D83
hypocenterC83#21690
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C43 ◁ C4×#127

Complements

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

NH ≅ G/Ntype
2. C2×#19127811C2semidirect
3. C4×#14581789C2semidirect
4. #24908338C2semidirect
5. #24908270C2semidirect
6. #24908268C2semidirect
7. #24907619C2semidirect
8. C4×#14581703C2semidirect
9. C4×#127×C83C2semidirect
14. #19128077C4semidirect
38. C2×#13050575C2×C4semidirect
42. #14581789C2×C4semidirect
53. #14581703D4semidirect
55. C4×C83⋊C8C2×C4semidirect
60. #127×C83C2×C4semidirect
158. C83⋊C8C4×D4semidirect
178. C4×C83#442semidirect
193. C4×#127D83semidirect
199. C83#21690semidirect
237. #127C4×D83semidirect
369. C4#19127811semidirect

Maximal subgroups

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

subgroupindex
C2×#191280772
C2×#191278112
C4×#145817892
#249083382
#249082702
#249082682
#249076192
C4×#145817032
C4×#127×C832
#2169083