Finite Group Databasebeta

← #36268353

Subgroups — #36268353

Order 21248.

2706 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128467C212yesmaximal, complemented
2#24920796C211yesmaximal, characteristic
3#24916187C2116yesmaximal, complemented
4#24916180C214yesmaximal, complemented
5#24916174C214yesmaximal, complemented
6C2×#19127669C212yesmaximal, complemented
7C2×#19127657C211yesmaximal, characteristic, complemented
8C83×#2905C211yesmaximal, characteristic, complemented
9C22×#13050590C2212yes—
10#19128467—216no—
11C2×#14581800C2218yescomplemented
12C2×#14581798C2218yes—
13C22×#13050574C2211yescharacteristic
14C2×M16×D83C2218yescomplemented
15C23×C83⋊C8C2211yescharacteristic
16#19128282C414yes—
17#19128277C414yes—
18#19128270C414yes—
19#19128263C414yes—
20#19128251C2214yes—
21#19128213C2218yes—
22#19127778C22116yescomplemented
23#19127771C22132yescomplemented
24#19127766C22116yescomplemented
25#19127669C2218yes—
26#19127669C2214yescomplemented
27#19127669C2214yescomplemented
28C2×(C2×D83)⋊C8C2218yescomplemented
29#19127657C2218yescomplemented
30C2×#14581703C2212yescomplemented
31C2×C4×C83⋊C8C2212yescomplemented
32C22×M16×C83C2212yes—
33C2×#117×C83C2211yescharacteristic
34#424×C83C2218yescomplemented
35#423×C83C2214yescomplemented
36C2×#13050590—28no—
37C2×#13050590—28no—
38C2×#13050590C2×C414yes—
39C2×#13050590C2×C414yes—
40C2×#13050590C2314yes—
41C2×#13050590C2314yes—
42C22×Dic166C2×C412yes—
43C22×D332C2×C412yes—
44C22×C22⋊D83C2×C414yes—
45C22×C4×D83C2314yes—
46#14581800—232no—
47#14581798—232no—
48C2×#13050574D418yescomplemented
49C2×#13050574C2314yes—
50C2×#13050573C23116yes—
51M16×D83—232no—
52C22×C83⋊C8—24no—
53C22×C83⋊C8C2311yescharacteristic
54C22×C83⋊C8C2311yescharacteristic
55C22×C83⋊C8C2314yes—
56C22×C83⋊C8C2312yes—
57C2×C8×D83C23116yes—
58#14581792C2×C414yes—
59#14581789C2×C414yes—
60#14581780C2×C418yes—
61#14581776C2×C4116yes—
62#14581772C2×C4116yes—
63#14581767C2×C4116yes—
64#14581766C2×C4116yes—
65#14581762C2×C4116yes—
66C4×D332C23116yes—
67(C2×D83)⋊C8C23116yes—
68(C2×D83)⋊C8C23116yes—
69#14581703C2318yes—
70C4×C83⋊C8C2318yes—
71C23×C4×C83C2311yescharacteristic
72C2×M16×C83—28no—
73C2×M16×C83C2314yes—
74C2×M16×C83C2314yes—
75C22×C8×C83C2314yes—
76#117×C83C2×C414yes—
77#117×C83C2314yes—
78C2×C4⋊C4×C83C2×C412yes—
79C2×C22⋊C4×C83C2×C412yes—
80C2×C42×C83C2312yes—
81C4⋊C8×C83C23116yes—
82C23×D83C22×C414yes—
83#13050590—216no—
84#13050590—216no—
85#13050590—28no—
86#13050590—28no—
87#13050590—28no—
88#13050590—28no—
89C2×Dic166—28no—
90C2×Dic166C22×C414yes—
91C2×Dic166C22×C414yes—
92C2×D332—28no—
93C2×D332C22×C414yes—
94C2×D332C22×C414yes—
95C2×D332C22×C4116yes—
96C2×C22⋊D83—216no—
97C2×C22⋊D83C22×C418yes—
98C2×C22⋊D83C22×C418yes—
99C22×Dic83C22×C414yes—
100C2×C4×D83—28no—
101C2×C4×D83—28no—
102C2×C4×D83C22×C418yes—
103C2×C4×D83C2418yes—
104#13050574C2×D4116yes—
105#13050573—232no—
106C2×C83⋊C8—28no—
107C2×C83⋊C8—22no—
108C2×C83⋊C8—22no—
109C2×C83⋊C8C2×D418yes—
110C2×C83⋊C8C2412yes—
111C2×C83⋊C8C2412yes—
112C2×C83⋊C8C2414yes—
113C8×D83—232no—
114C4⋊Dic83C22×C4116yes—
115(C2×D83)⋊C4C22×C4132yes—
116C24×C83C22×C411yescharacteristic
117C22×C4×C83C22×C414yes—
118C22×C4×C83C22×C412yes—
119C22×C4×C83C22×C411yescharacteristic
120C22×C4×C83C2×D414yes—
121C22×C4×C83#4114yes—
122C22×C4×C83C2411yescharacteristic
123C22×C4×C83C2412yes—
124M16×C83—28no—
125M16×C83—28no—
126C2×C8×C83—28no—
127C2×C8×C83C2418yes—
128C4⋊C4×C83C22×C418yes—
129C22⋊C4×C83C22×C418yes—
130C42×C83C22×C414yes—
131C42×C83C2414yes—
132C23×C83—42no—
133C23×C83C4×D414yes—
134C23×C83C23×C412yes—
135C23×C83C23×C411yescharacteristic
136C22×D83—28no—
137C22×D83—48no—
138C22×D83C23×C418yes—
139C2×C4×C83—44no—
140C2×C4×C83C4×D418yes—
141C2×C4×C83C4×D414yes—
142C2×C4×C83C23×C418yes—
143C2×C4×C83C23×C411yescharacteristic
144C2×C4×C83C23×C412yes—
145C2×C4×C83C22×D414yes—
146C2×C4×C83C22×D412yes—
147C2×C4×C83C2×#4114yes—
148C2×C4×C83C2×#4112yes—
149C2×C4×C83C2511yescharacteristic
150C22⋊D83—216no—
151C22⋊D83—216no—
152C2×Dic83—28no—
153C2×Dic83C23×C418yes—
154D332—28no—
155D332—28no—
156D332—416no—
157C4×D83—216no—
158C4×D83—28no—
159C4×D83—28no—
160Dic166—28no—
161Dic166—28no—
162C8×C83—28no—
163C83⋊C8—22no—
164C83⋊C8—22no—
165C83⋊C8C22×D418yes—
166C22×C83—22no—
167C22×C83—24no—
168C22×C83—44no—
169C22×C83C2×C4×D414yes—
170C22×C83C2×C4×D412yes—
171C22×C83C24×C411yescharacteristic
172C2×D83—28no—
173C2×D83—216no—
174C2×D83—88no—
175C4×C83—28no—
176C4×C83C2×C4×D412yes—
177C4×C83C2×C4×D414yes—
178C4×C83C23×D411yescharacteristic
179C4×C83C22×#4111yescharacteristic
180Dic83—28no—
181#60143—831nomaximal, Sylow
182C2×C83—24no—
183C2×C83—22no—
184C2×C83C22×C4×D411yescharacteristic
185C2×C83#357312yes—
186D83—48no—
187C2×#568—832no—
188#3449—831no—
189#2933—8316no—
190#2932—834no—
191#2931—834no—
192#2905D8311yescharacteristic, complemented
193C2×#408—832no—
194C2×#405—831no—
195C83#6014311yescharacteristic, complemented, Sylow
196C22×#41—832no—
197#568—16616no—
198C2×#131—838no—
199C2×#131—838no—
200C22×M16—831no—
201C22×M16—838no—
202C22×M16C2×D8312yes—
203C23×C8—831no—
204#537—834no—
205#536—834no—
206#535—834no—
207#534—834no—
208#532—834no—
209#525—838no—
210C2×#117C2×D8311yescharacteristic
211#436—8316no—
212#435—8332no—
213C8×D4—8316no—
214#424C2×D8318yescomplemented
215#423C2×D8314yescomplemented
216#408—834no—
217#408—834no—
218#408—838no—
219C2×C22⋊C8—838no—
220#405—838no—
221C2×#97—832no—
222C2×C4×C8—832no—
223C2×#41—834no—
224C2×#41—834no—
225C2×#41—834no—
226C2×#41—834no—
227C2×#41—1668no—
228C2×#41—1668no—
229C22×Q8—832no—
230C22×D4—834no—
231C22×D4—832no—
232C23×C4—834no—
233C23×C4C22×D8311yescharacteristic
234#131—16632no—
235#131—16632no—
236C2×M16—28no—
237C2×M16—838no—
238C2×M16—834no—
239C2×M16—8316no—
240C2×M16—16632no—
241C2×M16C22×D8314yes—
242C2×M16C22×D8314yes—
243C22×C8—832no—
244C22×C8—831no—
245C22×C8—831no—
246C22×C8—834no—
247C22×C8—8316no—
248C22×C8—1664no—
249C22×C8C22×D8314yes—
250C4⋊Q8—834no—
251#127—834no—
252#124—838no—
253#123—8316no—
254C22⋊Q8—8316no—
255#121—8316no—
256#121—8316no—
257#120—8316no—
258C4×D4—8316no—
259#117C4×D8314yes—
260#117C22×D8314yes—
261C2×C4⋊C4C4×D8312yes—
262C2×C22⋊C4C4×D8312yes—
263C2×C42C22×D8312yes—
264C4⋊C8C22×D83116yes—
265C22⋊C8—8316no—
266C22⋊C8—8316no—
267#97—838no—
268C4×C8—838no—
269C24—834no—
270C24C2×C4×D8311yescharacteristic
271#41—1668no—
272#41—1668no—
273#41—1668no—
274#41—1668no—
275#41—16616no—
276#41—16616no—
277C2×Q8—834no—
278C2×Q8—834no—
279C2×Q8—1668no—
280C2×D4—838no—
281C2×D4—838no—
282C2×D4—8316no—
283C2×D4—834no—
284C2×D4—834no—
285C2×D4—16616no—
286C2×D4—1668no—
287C22×C4—834no—
288C22×C4—838no—
289C22×C4—838no—
290C22×C4—1668no—
291C22×C4—1668no—
292C22×C4C2×C4×D8314yes—
293C22×C4C2×C4×D8311yescharacteristic
294C22×C4C2×C4×D8312yes—
295C22×C4D4×D8314yes—
296C22×C4#1305059114yes—
297C22×C4C23×D8311yescharacteristic
298C22×C4C23×D8312yes—
299M16—28no—
300M16—28no—
301M16—8316no—
302M16—16632no—
303C2×C8—28no—
304C2×C8—838no—
305C2×C8—834no—
306C2×C8—832no—
307C2×C8—832no—
308C2×C8—16632no—
309C2×C8—1668no—
310C2×C8—1662no—
311C2×C8—1662no—
312C2×C8C23×D8318yes—
313C4⋊C4—8316no—
314C4⋊C4C2×C4×D8318yes—
315C22⋊C4—8332no—
316C22⋊C4C2×C4×D8318yes—
317C42C2×C4×D8314yes—
318C42C23×D8314yes—
319C23—42no—
320C23—3328no—
321C23—838no—
322C23—1668no—
323C23#1458175414yes—
324C23C22×C4×D8312yes—
325C23C22×C4×D8311yescharacteristic
326Q8—1668no—
327Q8—1668no—
328D4—33216no—
329D4—16616no—
330D4—16616no—
331D4—1668no—
332D4—1668no—
333C2×C4—44no—
334C2×C4—838no—
335C2×C4—1668no—
336C2×C4—16616no—
337C2×C4—1668no—
338C2×C4—1668no—
339C2×C4#1458175418yes—
340C2×C4#1458175414yes—
341C2×C4C22×C4×D8318yes—
342C2×C4C22×C4×D8311yescharacteristic
343C2×C4C22×C4×D8312yes—
344C2×C4C2×D4×D8314yes—
345C2×C4C2×D4×D8312yes—
346C2×C4C2×#1305059114yes—
347C2×C4C2×#1305059112yes—
348C2×C4C24×D8311yescharacteristic
349C8—28no—
350C8—838no—
351C8—1662no—
352C8—1662no—
353C22—22no—
354C22—24no—
355C22—44no—
356C22—6648no—
357C22—1668no—
358C22—16616no—
359C22C2×#1458175414yes—
360C22C2×#1458175412yes—
361C22C23×C4×D8311yescharacteristic
362C4—28no—
363C4—1668no—
364C4C2×#1458175414yes—
365C4C2×#1458175412yes—
366C4C22×D4×D8311yescharacteristic
367C4C22×#1305059111yescharacteristic
368C2—24no—
369C2—22no—
370C2—3328no—
371C2C22×#1458175411yescharacteristic
372C2#2492217612yes—
subgroupG/H
center, FrattiniC2×C4C24×D83
commutator, socleC22×C83C24×C4
FittingC83×#2905C2
prime radical#24920796C2
cosocleC2×C4×C83C25
hypercenter#2905D83
hypocenterC83#60143
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ #2905

Complements

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

NH ≅ G/Ntype
1. C2×#19128467C2semidirect
3. #24916187C2semidirect
4. #24916180C2semidirect
5. #24916174C2semidirect
6. C2×#19127669C2semidirect
7. C2×#19127657C2semidirect
8. C83×#2905C2semidirect
11. C2×#14581800C22semidirect
14. C2×M16×D83C22semidirect
22. #19127778C22semidirect
23. #19127771C22semidirect
24. #19127766C22semidirect
26. #19127669C22semidirect
27. #19127669C22semidirect
28. C2×(C2×D83)⋊C8C22semidirect
29. #19127657C22semidirect
30. C2×#14581703C22semidirect
31. C2×C4×C83⋊C8C22semidirect
34. #424×C83C22semidirect
35. #423×C83C22semidirect
48. C2×#13050574D4semidirect
192. #2905D83semidirect
195. C83#60143semidirect
214. #424C2×D83semidirect
215. #423C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191284672
#249207962
#249161872
#249161802
#249161742
C2×#191276692
C2×#191276572
C83×#29052
#6014383