Finite Group Databasebeta

← #35459783

Subgroups — #35459783

Order 21248.

1774 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
1C23×D664C211yesmaximal, characteristic
2C2×#19128085C212yesmaximal, complemented
3C2×#19127942C211yesmaximal, characteristic, complemented
4C2×#19127854C212yesmaximal, complemented
5#24911114C212yesmaximal, complemented
6#24908543C214yesmaximal, complemented
7#24908329C212yesmaximal, complemented
8C8×C22⋊C4×C83C211yesmaximal, characteristic, complemented
9C23×D332C2212yes—
10C22×Dic332C2211yescharacteristic, complemented
11C22×#13050579C2214yescomplemented
12C22×D664—24no—
13C22×D664—28no—
14C22×D664C2212yes—
15C22×D664C2212yes—
16C2×#14581780C2214yescomplemented
17C2×#14581771C2212yes—
18#19128085—28no—
19#19127942—24no—
20#19127854—28no—
21C2×#14581726C2211yescharacteristic, complemented
22C2×#14581717C2212yescomplemented
23C2×#14581714C2214yescomplemented
24#19127619—24no—
25#19127603C2214yescomplemented
26#19127556—24no—
27C23×C8×C83C2211yescharacteristic
28C2×C22⋊C8×C83C2211yescharacteristic
29C2×C4×C8×C83C2212yescomplemented
30C4×C22⋊C4×C83C2211yescharacteristic
31#336×C83C2212yescomplemented
32C24×D83—22no—
33C22×Dic166C2312yes—
34C22×D332—28no—
35C22×D332—24no—
36C22×D332—24no—
37C22×D332—22no—
38C22×D332—44no—
39C22×D332C2314yes—
40C2×Dic332—24no—
41C2×Dic332—22no—
42C2×#13050579—28no—
43C2×#13050579—28no—
44C2×#13050579—44no—
45C2×D664—232no—
46C2×D664—28no—
47C2×D664—24no—
48C2×D664—24no—
49C2×D664—24no—
50C2×D664D418yes—
51#14581780—28no—
52#14581780—44no—
53#14581771—24no—
54#14581771—24no—
55C2×C4⋊Dic83C2312yes—
56C2×Dic83⋊C4—22no—
57C2×C22⋊Dic83—22no—
58C2×(C2×D83)⋊C4—28no—
59C2×(C2×D83)⋊C4—28no—
60#14581726—22no—
61#14581717—24no—
62#14581714—28no—
63C23×C4×C83C2311yescharacteristic
64C22×C8×C83—24no—
65C22×C8×C83D414yes—
66C22×C8×C83C2311yescharacteristic
67C22×C8×C83C2312yes—
68C22×C8×C83C2311yescharacteristic
69C2×C22⋊C4×C83D412yes—
70C2×C42×C83C2312yes—
71C22⋊C8×C83D414yes—
72C4×C8×C83—24no—
73#95×C83D412yes—
74C23×D83—24no—
75C23×D83—48no—
76C23×D83—44no—
77C23×D83—48no—
78C23×D83—84no—
79C2×Dic166—22no—
80C2×Dic166—22no—
81C2×Dic166—24no—
82C2×Dic166—24no—
83C2×D332—216no—
84C2×D332—24no—
85C2×D332—24no—
86C2×D332—24no—
87C2×D332—24no—
88C2×D332—24no—
89C2×D332—416no—
90C2×D332—48no—
91C2×D332—48no—
92C2×D332—44no—
93C2×D332C2×D418yes—
94C22×Dic83—22no—
95C22×Dic83—22no—
96Dic332—44no—
97#13050579—416no—
98D664—216no—
99D664—216no—
100C4⋊Dic83—22no—
101C4⋊Dic83—22no—
102Dic83⋊C4—44no—
103C22⋊Dic83—44no—
104(C2×D83)⋊C4—416no—
105(C2×D83)⋊C4—88no—
106C24×C83C2×D411yescharacteristic
107C22×C4×C83—22no—
108C22×C4×C83—22no—
109C22×C4×C83C2×D414yes—
110C22×C4×C83C2×D411yescharacteristic
111C22×C4×C83C2×D412yes—
112C22×C4×C83C2×D412yes—
113C22×C4×C83C2411yescharacteristic
114C2×C8×C83—22no—
115C2×C8×C83—22no—
116C2×C8×C83—22no—
117C2×C8×C83—28no—
118C2×C8×C83C2×D414yes—
119C2×C8×C83C2×D412yes—
120C2×C8×C83C2×D414yes—
121C2×C8×C83C2×D412yes—
122C22⋊C4×C83—24no—
123C42×C83—22no—
124C42×C83—22no—
125C23×C83—22no—
126C23×C83—22no—
127C23×C83#12712yes—
128C23×C83#13514yes—
129C23×C83C22×D411yescharacteristic
130C22×D83—28no—
131C22×D83—416no—
132C22×D83—44no—
133C22×D83—44no—
134C22×D83—48no—
135C22×D83—48no—
136C22×D83—832no—
137C22×D83—88no—
138C22×D83—88no—
139C2×C4×C83—22no—
140C2×C4×C83—22no—
141C2×C4×C83—22no—
142C2×C4×C83—24no—
143C2×C4×C83—24no—
144C2×C4×C83—24no—
145C2×C4×C83—24no—
146C2×C4×C83#12712yes—
147C2×C4×C83C22×D412yes—
148C2×C4×C83C22×D411yescharacteristic
149C2×C4×C83C22×D411yescharacteristic
150C2×C4×C83H(5,𝔽2)12yes—
151C2×Dic83—42no—
152C2×Dic83—42no—
153C2×Dic83—44no—
154C2×Dic83—44no—
155D332—28no—
156D332—28no—
157D332—416no—
158D332—44no—
159D332—44no—
160Dic166—42no—
161Dic166—42no—
162Dic166—44no—
163C8×C83—22no—
164C8×C83—24no—
165C8×C83#12014yes—
166C22×C83—22no—
167C22×C83—22no—
168C22×C83—28no—
169C22×C83#49514yescomplemented
170C22×C83C2×#12711yescharacteristic
171C22×C83D4212yes—
172C22×C83C2×#13512yes—
173C22×C83#57612yes—
174C2×D83—48no—
175C2×D83—416no—
176C2×D83—832no—
177C2×D83—84no—
178C2×D83—84no—
179C2×D83—88no—
180C4×C83—22no—
181C4×C83—44no—
182C4×C83C2×#12011yescharacteristic
183C4×C83#53511yescharacteristic
184C4×C83D4212yes—
185Dic83—82no—
186Dic83—82no—
187#25781—831nomaximal, Sylow
188C2×C83—24no—
189C2×C83#243311yescharacteristic
190C2×C83C2×#49511yescharacteristic
191C2×C83#315511yescharacteristic
192C2×C83#328014yes—
193D83—88no—
194C23×D8—831no—
195C2×#495—832no—
196C2×#467—831no—
197C2×#450—832no—
198#2435—832no—
199#2012—834no—
200#1970—832no—
201C8×C22⋊C4D8311yescharacteristic, complemented
202C83#2578111yescharacteristic, complemented, Sylow
203C23×D4—832no—
204C22×Q16—831no—
205C22×SD16—834no—
206C22×D8—832no—
207C22×D8—832no—
208C22×D8—1668no—
209C22×D8—1664no—
210C23×C8C2×D8311yescharacteristic
211C2×#124—834no—
212C2×C22⋊Q8—832no—
213#495—1668no—
214#467—1664no—
215#450—1668no—
216C2×#107—831no—
217C2×#103—832no—
218C2×#102—834no—
219C2×C22⋊C8C2×D8311yescharacteristic
220C2×C4×C8C2×D8312yescomplemented
221#394—1664no—
222#390—834no—
223#379—1664no—
224C4×C22⋊C4C2×D8311yescharacteristic
225#336C2×D8312yescomplemented
226C25—1662no—
227C22×Q8—832no—
228C22×D4—3324no—
229C22×D4—834no—
230C22×D4—1668no—
231C22×D4—1662no—
232C22×D4—1664no—
233C22×D4—1664no—
234C23×C4C22×D8311yescharacteristic
235C2×Q16—1664no—
236C2×Q16—1662no—
237C2×SD16—3324no—
238C2×SD16—1668no—
239C2×SD16—1668no—
240C2×D8—838no—
241C2×D8—1668no—
242C2×D8—16632no—
243C2×D8—1664no—
244C2×D8—1664no—
245C2×D8—1664no—
246C22×C8—24no—
247C22×C8D33214yes—
248C22×C8C22×D8311yescharacteristic
249C22×C8C22×D8312yes—
250C22×C8C22×D8311yescharacteristic
251#124—3324no—
252#124—1668no—
253C22⋊Q8—1664no—
254C22⋊Q8—1664no—
255C2×C4⋊C4—832no—
256C2×C4⋊C4—1662no—
257C2×C22⋊C4—1662no—
258C2×C22⋊C4—1668no—
259C2×C22⋊C4—1668no—
260C2×C22⋊C4D33212yes—
261C2×C42C22×D8312yes—
262#107—1662no—
263#103—1664no—
264#102—1668no—
265C22⋊C8D33214yes—
266C4×C8—24no—
267#95D33212yes—
268C24—3328no—
269C24—3324no—
270C24—3328no—
271C24—6644no—
272C24—1664no—
273C24C2×D33211yescharacteristic
274C2×Q8—1664no—
275C2×Q8—1662no—
276C2×Q8—1662no—
277C2×Q8—1664no—
278C2×D4—3328no—
279C2×D4—3328no—
280C2×D4—33216no—
281C2×D4—3324no—
282C2×D4—838no—
283C2×D4—16616no—
284C2×D4—1664no—
285C2×D4—1664no—
286C2×D4—1664no—
287C2×D4—1664no—
288C2×D4—1664no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—1662no—
292C22×C4—1662no—
293C22×C4C2×D33214yes—
294C22×C4C2×D33211yescharacteristic
295C22×C4C2×D33212yes—
296C22×C4C2×D33212yes—
297C22×C4C23×D8311yescharacteristic
298Q16—3324no—
299SD16—33216no—
300D8—16616no—
301D8—16616no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—28no—
306C2×C8D4×D8314yes—
307C2×C8C2×D33214yes—
308C2×C8C2×D33212yes—
309C2×C8C2×D33212yes—
310C4⋊C4—3324no—
311C4⋊C4—1662no—
312C4⋊C4—1662no—
313C22⋊C4—24no—
314C22⋊C4—33216no—
315C22⋊C4—3324no—
316C22⋊C4—6648no—
317C42—22no—
318C42—22no—
319C23—22no—
320C23—22no—
321C23—33216no—
322C23—3324no—
323C23—3328no—
324C23—3324no—
325C23—3328no—
326C23—66432no—
327C23—6648no—
328C23—6648no—
329C23—1668no—
330C23#1458178912yes—
331C23#1458181414yes—
332C23C22×D33211yescharacteristic
333Q8—3324no—
334Q8—3322no—
335Q8—3322no—
336D4—33216no—
337D4—3324no—
338D4—3324no—
339D4—1668no—
340D4—1668no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—3324no—
349C2×C4—3322no—
350C2×C4—3322no—
351C2×C4—3324no—
352C2×C4#1458178912yes—
353C2×C4C2×D4×D8312yes—
354C2×C4C22×D33211yescharacteristic
355C2×C4C22×D33211yescharacteristic
356C2×C4#1458183812yes—
357C8—24no—
358C8—22no—
359C8#1458176214yes—
360C22—28no—
361C22—22no—
362C22—22no—
363C22—33216no—
364C22—3328no—
365C22—66432no—
366C22—6648no—
367C22—6644no—
368C22—6644no—
369C22#1912808514yescomplemented
370C22C2×#1458178911yescharacteristic
371C22D4×D33212yes—
372C22C2×#1458181412yes—
373C22#1912850612yes—
374C4—22no—
375C4—44no—
376C4—6642no—
377C4—6642no—
378C4C2×#1458176211yescharacteristic
379C4#1912827011yescharacteristic
380C4D4×D33212yes—
381C2—24no—
382C2—6648no—
383C2#2491110011yescharacteristic
384C2C2×#1912808511yescharacteristic
385C2#2491794911yescharacteristic
386C2#2491915014yes—
subgroupG/H
centerC23C22×D332
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC8×C22⋊C4×C83C2
socleC23×C83C22×D4
prime radicalC23×D664C2
hypercenterC8×C22⋊C4D83
hypocenterC83#25781
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#19128085C2semidirect
3. C2×#19127942C2semidirect
4. C2×#19127854C2semidirect
5. #24911114C2semidirect
6. #24908543C2semidirect
7. #24908329C2semidirect
8. C8×C22⋊C4×C83C2semidirect
10. C22×Dic332C22semidirect
11. C22×#13050579C22semidirect
16. C2×#14581780C22semidirect
21. C2×#14581726C22semidirect
22. C2×#14581717C22semidirect
23. C2×#14581714C22semidirect
25. #19127603C22semidirect
29. C2×C4×C8×C83C22semidirect
31. #336×C83C22semidirect
169. C22×C83#495semidirect
201. C8×C22⋊C4D83semidirect
202. C83#25781semidirect
220. C2×C4×C8C2×D83semidirect
225. #336C2×D83semidirect
369. C22#19128085semidirect

Maximal subgroups

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

subgroupindex
C23×D6642
C2×#191280852
C2×#191279422
C2×#191278542
#249111142
#249085432
#249083292
C8×C22⋊C4×C832
#2578183