Finite Group Databasebeta

← #35406243

Subgroups — #35406243

Order 21248.

1866 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128468C211yesmaximal, characteristic, complemented
2C22×#14581709C212yesmaximal, complemented
3#24915908C211yesmaximal, characteristic, complemented
4#24907927C218yesmaximal, complemented
5#24907202C212yesmaximal, complemented
6C2×#535×C83C211yesmaximal, characteristic
7#19128468—28no—
8C2×#14581799—24no—
9C22×#13050574C2212yes—
10C22×#13050574C2212yes—
11C22×#13050574C2214yescomplemented
12C2×#14581709—28no—
13C2×#14581709C2218yes—
14#19127668—24no—
15#19127665C2214yescomplemented
16#19127470C2218yescomplemented
17C22×#41×C83C2211yescharacteristic
18C23×D4×C83C2211yescharacteristic
19#535×C83—24no—
20#535×C83—24no—
21C2×#127×C83C412yes—
22C2×#124×C83C412yes—
23C2×#121×C83C2214yes—
24C2×#120×C83C414yes—
25C2×#117×C83C2211yescharacteristic
26#14581799—232no—
27C2×#13050574—216no—
28C2×#13050574—22no—
29C2×#13050574—24no—
30C2×#13050574—24no—
31C2×#13050574—22no—
32C2×#13050574—24no—
33C2×#13050574—24no—
34C2×#13050574D418yescomplemented
35C2×#13050574C2314yes—
36C2×#13050574C2314yes—
37C22×C83⋊C8—22no—
38C22×C83⋊C8—24no—
39C22×C83⋊C8—22no—
40#14581709—216no—
41C22⋊(C83⋊C8)—28no—
42C25×C83C2×C412yes—
43C2×#41×C83—24no—
44C2×#41×C83—24no—
45C2×#41×C83D414yes—
46C2×#41×C83D414yes—
47C22×Q8×C83C2×C411yescharacteristic
48C22×D4×C83—22no—
49C22×D4×C83—22no—
50C22×D4×C83—24no—
51C22×D4×C83—24no—
52C22×D4×C83C2×C412yes—
53C22×D4×C83C2×C412yes—
54C22×D4×C83C2×C411yescharacteristic
55C22×D4×C83C2×C418yes—
56C22×D4×C83C2312yes—
57C23×C4×C83C2312yes—
58C23×C4×C83C2311yescharacteristic
59#127×C83—28no—
60#124×C83—28no—
61#121×C83—28no—
62#121×C83—28no—
63#120×C83—216no—
64#117×C83—22no—
65#117×C83—22no—
66C2×C4⋊C4×C83C2312yes—
67C2×C22⋊C4×C83C2×C412yes—
68C2×C22⋊C4×C83C2×C418yes—
69C2×C42×C83C2×C412yes—
70#13050574—216no—
71#13050574—28no—
72#13050574—24no—
73#13050574—24no—
74#13050574—24no—
75#13050574—44no—
76#13050574C2×D418yes—
77C2×C83⋊C8—216no—
78C2×C83⋊C8—28no—
79C2×C83⋊C8—24no—
80C2×C83⋊C8—24no—
81C2×C83⋊C8—44no—
82C24×C83—24no—
83C24×C83—28no—
84C24×C83—84no—
85C24×C83C22×C414yes—
86C24×C83C22×C412yes—
87C24×C83C22×C411yescharacteristic
88#41×C83—28no—
89#41×C83—28no—
90#41×C83—28no—
91#41×C83—28no—
92C2×Q8×C83—24no—
93C2×Q8×C83C22⋊C414yes—
94C2×D4×C83—28no—
95C2×D4×C83—216no—
96C2×D4×C83—24no—
97C2×D4×C83—24no—
98C2×D4×C83—28no—
99C2×D4×C83—22no—
100C2×D4×C83—22no—
101C2×D4×C83—24no—
102C2×D4×C83—24no—
103C2×D4×C83—44no—
104C2×D4×C83—44no—
105C2×D4×C83—44no—
106C2×D4×C83—44no—
107C2×D4×C83—88no—
108C2×D4×C83C22⋊C418yes—
109C2×D4×C83C22⋊C414yes—
110C22×C4×C83—24no—
111C22×C4×C83—24no—
112C22×C4×C83C22×C412yes—
113C22×C4×C83C22×C414yes—
114C22×C4×C83C22×C411yescharacteristic
115C22×C4×C83C2×D414yes—
116C22×C4×C83C2×D414yes—
117C22×C4×C83C2×D414yes—
118C22×C4×C83C2×D412yes—
119C22×C4×C83C2×D412yes—
120C22×C4×C83#4114yes—
121C22×C4×C83C2411yescharacteristic
122C4⋊C4×C83—22no—
123C4⋊C4×C83—22no—
124C22⋊C4×C83—24no—
125C22⋊C4×C83—216no—
126C42×C83—24no—
127C23×C83—24no—
128C23×C83—24no—
129C23×C83—28no—
130C23×C83—42no—
131C23×C83—44no—
132C23×C83—48no—
133C23×C83—48no—
134C23×C83—88no—
135C23×C83—816no—
136C23×C83C2×C22⋊C414yes—
137C23×C83C2×C22⋊C412yes—
138C23×C83C4×D414yes—
139C23×C83C23×C411yescharacteristic
140Q8×C83—28no—
141D4×C83—216no—
142D4×C83—28no—
143D4×C83—42no—
144D4×C83—44no—
145D4×C83—42no—
146D4×C83—48no—
147D4×C83—816no—
148C2×C4×C83—24no—
149C2×C4×C83—24no—
150C2×C4×C83—28no—
151C2×C4×C83—24no—
152C2×C4×C83—24no—
153C2×C4×C83—44no—
154C2×C4×C83C2×C22⋊C414yes—
155C2×C4×C83C2×C22⋊C412yes—
156C2×C4×C83C4×D414yes—
157C2×C4×C83#12014yes—
158C2×C4×C83#12114yes—
159C2×C4×C83#12114yes—
160C2×C4×C83#12314yes—
161C2×C4×C83C22×D412yes—
162C2×C4×C83C22×D411yescharacteristic
163C2×C4×C83C22×D411yescharacteristic
164C2×C4×C83C2×#4112yes—
165C83⋊C8—28no—
166C83⋊C8—44no—
167C22×C83—22no—
168C22×C83—24no—
169C22×C83—28no—
170C22×C83—24no—
171C22×C83—44no—
172C22×C83—48no—
173C22×C83—44no—
174C22×C83—84no—
175C22×C83—88no—
176C22×C83—816no—
177C22×C83#38614yes—
178C22×C83C22×C22⋊C411yescharacteristic
179C22×C83C2×C4×D412yes—
180C4×C83—24no—
181C4×C83—44no—
182C4×C83#38614yes—
183C4×C83C2×#12011yescharacteristic
184C4×C83C2×#12112yes—
185C4×C83C2×#12311yescharacteristic
186#20990—831nomaximal, Sylow
187C2×C83—22no—
188C2×C83—24no—
189C2×C83—44no—
190C2×C83—84no—
191C2×C83C2×#38611yescharacteristic
192C2×C83#288812yes—
193C2×#568—831no—
194C2×#535D8311yescharacteristic
195C22×#100—832no—
196#2879—831no—
197#1900—838no—
198#1742—832no—
199C83#2099011yescharacteristic, complemented, Sylow
200C22×#41C2×D8311yescharacteristic
201C23×D4C2×D8311yescharacteristic
202#568—1668no—
203C2×#131—1664no—
204C22×M16—832no—
205C22×M16—832no—
206C22×M16—834no—
207#535—24no—
208#535—24no—
209C2×#127Dic8312yes—
210C2×#124Dic8312yes—
211C2×#121C2×D8314yes—
212C2×#120Dic8314yes—
213C2×#117C2×D8311yescharacteristic
214C2×#100—838no—
215C2×#100—1668no—
216#408—1664no—
217#407—834no—
218#343—838no—
219C25C2×Dic8312yes—
220C2×#41—24no—
221C2×#41—24no—
222C2×#41C22⋊D8314yes—
223C2×#41C22⋊D8314yes—
224C22×Q8C2×Dic8311yescharacteristic
225C22×D4—24no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—24no—
229C22×D4C2×Dic8312yes—
230C22×D4C2×Dic8312yes—
231C22×D4C2×Dic8311yescharacteristic
232C22×D4C2×Dic8318yes—
233C22×D4C22×D8312yes—
234C23×C4C22×D8312yes—
235C23×C4C22×D8311yescharacteristic
236#131—16632no—
237C2×M16—838no—
238C2×M16—834no—
239C2×M16—834no—
240C2×M16—1664no—
241C2×M16—1664no—
242C2×M16—1662no—
243C2×M16—16616no—
244C2×M16—1664no—
245C2×M16—1664no—
246C2×M16—1662no—
247C22×C8—1662no—
248C22×C8—1664no—
249C22×C8—1662no—
250#127—28no—
251#124—28no—
252#121—28no—
253#121—28no—
254#120—216no—
255#117—22no—
256#117—22no—
257C2×C4⋊C4C22×D8312yes—
258C2×C22⋊C4C2×Dic8312yes—
259C2×C22⋊C4C2×Dic8318yes—
260C2×C42C2×Dic8312yes—
261#100—16616no—
262C22⋊C8—1668no—
263C24—24no—
264C24—28no—
265C24—84no—
266C24C22×Dic8314yes—
267C24C22×Dic8312yes—
268C24C22×Dic8311yescharacteristic
269#41—28no—
270#41—28no—
271#41—28no—
272#41—28no—
273C2×Q8—24no—
274C2×Q8C22⋊Dic8314yes—
275C2×D4—28no—
276C2×D4—216no—
277C2×D4—24no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—28no—
284C2×D4—44no—
285C2×D4—44no—
286C2×D4—44no—
287C2×D4—44no—
288C2×D4—88no—
289C2×D4C22⋊Dic8318yes—
290C2×D4C22⋊Dic8314yes—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4C22×Dic8312yes—
294C22×C4C22×Dic8314yes—
295C22×C4C22×Dic8311yescharacteristic
296C22×C4D4×D8314yes—
297C22×C4C2×C22⋊D8314yes—
298C22×C4C2×C22⋊D8314yes—
299C22×C4C2×C22⋊D8312yes—
300C22×C4C2×C22⋊D8312yes—
301C22×C4#1305058914yes—
302C22×C4C23×D8311yescharacteristic
303M16—3324no—
304M16—838no—
305M16—1664no—
306M16—1664no—
307M16—16616no—
308M16—1668no—
309M16—1664no—
310C2×C8—3324no—
311C2×C8—1664no—
312C2×C8—16616no—
313C2×C8—1668no—
314C2×C8—1664no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C22⋊C4—24no—
318C22⋊C4—216no—
319C42—24no—
320C23—24no—
321C23—24no—
322C23—28no—
323C23—42no—
324C23—44no—
325C23—48no—
326C23—48no—
327C23—88no—
328C23—816no—
329C23C2×C22⋊Dic8314yes—
330C23C2×C22⋊Dic8312yes—
331C23D4×Dic8314yes—
332C23C23×Dic8311yescharacteristic
333Q8—28no—
334D4—216no—
335D4—28no—
336D4—48no—
337D4—42no—
338D4—44no—
339D4—42no—
340D4—816no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—28no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—44no—
347C2×C4C2×C22⋊Dic8314yes—
348C2×C4C2×C22⋊Dic8312yes—
349C2×C4D4×Dic8314yes—
350C2×C4#1458176014yes—
351C2×C4#1458176414yes—
352C2×C4#1458176414yes—
353C2×C4#1458177314yes—
354C2×C4C2×D4×D8312yes—
355C2×C4C22×C22⋊D8311yescharacteristic
356C2×C4C22×C22⋊D8311yescharacteristic
357C2×C4C2×#1305058912yes—
358C8—3324no—
359C8—1668no—
360C22—22no—
361C22—24no—
362C22—24no—
363C22—28no—
364C22—44no—
365C22—44no—
366C22—48no—
367C22—84no—
368C22—88no—
369C22—816no—
370C22#1912758114yes—
371C22C22×C22⋊Dic8311yescharacteristic
372C22C2×D4×Dic8312yes—
373C4—24no—
374C4—44no—
375C4#1912758114yes—
376C4C2×#1458176011yescharacteristic
377C4C2×#1458176412yes—
378C4C2×#1458177311yescharacteristic
379C2—22no—
380C2—24no—
381C2—44no—
382C2—84no—
383C2C2×#1912758111yescharacteristic
384C2#2491594212yes—
subgroupG/H
centerC22C22×C22⋊Dic83
commutatorC23×C83C23×C4
FrattiniC22×C4C23×D83
Fitting, prime radicalC2×#535×C83C2
socleC22×C83C22×C22⋊C4
cosocleC22×C4×C83C24
hypercenterC2×#535D83
hypocenterC83#20990
LCSG ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×#41 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
1. C2×#19128468C2semidirect
2. C22×#14581709C2semidirect
3. #24915908C2semidirect
4. #24907927C2semidirect
5. #24907202C2semidirect
11. C22×#13050574C22semidirect
15. #19127665C22semidirect
16. #19127470C22semidirect
34. C2×#13050574D4semidirect
199. C83#20990semidirect

Maximal subgroups

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

subgroupindex
C2×#191284682
C22×#145817092
#249159082
#249079272
#249072022
C2×#535×C832
#2099083