Finite Group Databasebeta

← #26218117

Subgroups — #26218117

Order 12032.

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×#15689890C211yesmaximal, characteristic, complemented
2C22×#13093941C212yesmaximal, complemented
3#20251833C211yesmaximal, characteristic, complemented
4#20243852C218yesmaximal, complemented
5#20243127C212yesmaximal, complemented
6C2×C47×#535C211yesmaximal, characteristic
7#15689890—28no—
8C2×#13094031—24no—
9C22×#10625329C2212yes—
10C22×#10625329C2212yes—
11C22×#10625329C2214yescomplemented
12C2×#13093941—28no—
13C2×#13093941C2218yes—
14#15689090—24no—
15#15689087C2214yescomplemented
16#15688892C2218yescomplemented
17C22×#41×C47C2211yescharacteristic
18C23×D4×C47C2211yescharacteristic
19C47×#535—24no—
20C47×#535—24no—
21C2×#127×C47C412yes—
22C2×#124×C47C412yes—
23C2×#121×C47C2214yes—
24C2×#120×C47C414yes—
25C2×#117×C47C2211yescharacteristic
26#13094031—232no—
27C2×#10625329—22no—
28C2×#10625329—24no—
29C2×#10625329—24no—
30C2×#10625329—216no—
31C2×#10625329—22no—
32C2×#10625329—24no—
33C2×#10625329—24no—
34C2×#10625329D418yescomplemented
35C2×#10625329C2314yes—
36C2×#10625329C2314yes—
37C22×C47⋊C8—22no—
38C22×C47⋊C8—24no—
39C22×C47⋊C8—22no—
40#13093941—216no—
41C22⋊(C47⋊C8)—28no—
42C25×C47C2×C412yes—
43C2×#41×C47—24no—
44C2×#41×C47—24no—
45C2×#41×C47D414yes—
46C2×#41×C47D414yes—
47C22×Q8×C47C2×C411yescharacteristic
48C22×D4×C47—22no—
49C22×D4×C47—22no—
50C22×D4×C47—24no—
51C22×D4×C47—24no—
52C22×D4×C47C2×C412yes—
53C22×D4×C47C2×C412yes—
54C22×D4×C47C2×C411yescharacteristic
55C22×D4×C47C2×C418yes—
56C22×D4×C47C2312yes—
57C23×C4×C47C2312yes—
58C23×C4×C47C2311yescharacteristic
59#127×C47—28no—
60#124×C47—28no—
61#121×C47—28no—
62#121×C47—28no—
63#120×C47—216no—
64#117×C47—22no—
65#117×C47—22no—
66C2×C4⋊C4×C47C2312yes—
67C2×C22⋊C4×C47C2×C412yes—
68C2×C22⋊C4×C47C2×C418yes—
69C2×C42×C47C2×C412yes—
70C24×C47—24no—
71C24×C47—28no—
72C24×C47—84no—
73C24×C47C22×C414yes—
74C24×C47C22×C412yes—
75C24×C47C22×C411yescharacteristic
76#41×C47—28no—
77#41×C47—28no—
78#41×C47—28no—
79#41×C47—28no—
80C2×Q8×C47—24no—
81C2×Q8×C47C22⋊C414yes—
82C2×D4×C47—28no—
83C2×D4×C47—216no—
84C2×D4×C47—24no—
85C2×D4×C47—24no—
86C2×D4×C47—28no—
87C2×D4×C47—22no—
88C2×D4×C47—22no—
89C2×D4×C47—24no—
90C2×D4×C47—24no—
91C2×D4×C47—44no—
92C2×D4×C47—44no—
93C2×D4×C47—44no—
94C2×D4×C47—44no—
95C2×D4×C47—88no—
96C2×D4×C47C22⋊C418yes—
97C2×D4×C47C22⋊C414yes—
98C22×C4×C47—24no—
99C22×C4×C47—24no—
100C22×C4×C47C22×C412yes—
101C22×C4×C47C22×C414yes—
102C22×C4×C47C22×C411yescharacteristic
103C22×C4×C47C2×D414yes—
104C22×C4×C47C2×D414yes—
105C22×C4×C47C2×D412yes—
106C22×C4×C47C2×D412yes—
107C22×C4×C47C2×D414yes—
108C22×C4×C47#4114yes—
109C22×C4×C47C2411yescharacteristic
110C4⋊C4×C47—22no—
111C4⋊C4×C47—22no—
112C22⋊C4×C47—24no—
113C22⋊C4×C47—216no—
114C42×C47—24no—
115#10625329—216no—
116#10625329—28no—
117#10625329—24no—
118#10625329—24no—
119#10625329—24no—
120#10625329—44no—
121#10625329C2×D418yes—
122C2×C47⋊C8—216no—
123C2×C47⋊C8—28no—
124C2×C47⋊C8—24no—
125C2×C47⋊C8—24no—
126C2×C47⋊C8—44no—
127C23×C47—24no—
128C23×C47—24no—
129C23×C47—28no—
130C23×C47—42no—
131C23×C47—44no—
132C23×C47—48no—
133C23×C47—48no—
134C23×C47—88no—
135C23×C47—816no—
136C23×C47C2×C22⋊C414yes—
137C23×C47C2×C22⋊C412yes—
138C23×C47C4×D414yes—
139C23×C47C23×C411yescharacteristic
140Q8×C47—28no—
141D4×C47—216no—
142D4×C47—28no—
143D4×C47—42no—
144D4×C47—44no—
145D4×C47—42no—
146D4×C47—48no—
147D4×C47—816no—
148C2×C4×C47—24no—
149C2×C4×C47—24no—
150C2×C4×C47—28no—
151C2×C4×C47—24no—
152C2×C4×C47—24no—
153C2×C4×C47—44no—
154C2×C4×C47C2×C22⋊C414yes—
155C2×C4×C47C2×C22⋊C412yes—
156C2×C4×C47C4×D414yes—
157C2×C4×C47#12014yes—
158C2×C4×C47#12114yes—
159C2×C4×C47#12114yes—
160C2×C4×C47#12314yes—
161C2×C4×C47C22×D412yes—
162C2×C4×C47C22×D411yescharacteristic
163C2×C4×C47C22×D411yescharacteristic
164C2×C4×C47C2×#4112yes—
165C47⋊C8—28no—
166C47⋊C8—44no—
167#20990—471nomaximal, Sylow
168C22×C47—22no—
169C22×C47—24no—
170C22×C47—28no—
171C22×C47—24no—
172C22×C47—44no—
173C22×C47—48no—
174C22×C47—44no—
175C22×C47—84no—
176C22×C47—88no—
177C22×C47—816no—
178C22×C47#38614yes—
179C22×C47C22×C22⋊C411yescharacteristic
180C22×C47C2×C4×D412yes—
181C4×C47—24no—
182C4×C47—44no—
183C4×C47#38614yes—
184C4×C47C2×#12011yescharacteristic
185C4×C47C2×#12112yes—
186C4×C47C2×#12311yescharacteristic
187C2×#568—471no—
188C2×#535D4711yescharacteristic
189C22×#100—472no—
190#2879—471no—
191#1900—478no—
192#1742—472no—
193C2×C47—22no—
194C2×C47—24no—
195C2×C47—44no—
196C2×C47—84no—
197C2×C47C2×#38611yescharacteristic
198C2×C47#288812yes—
199C22×#41C2×D4711yescharacteristic
200C23×D4C2×D4711yescharacteristic
201#568—948no—
202C2×#131—944no—
203C22×M16—472no—
204C22×M16—472no—
205C22×M16—474no—
206#535—24no—
207#535—24no—
208C2×#127Dic4712yes—
209C2×#124Dic4712yes—
210C2×#121C2×D4714yes—
211C2×#120Dic4714yes—
212C2×#117C2×D4711yescharacteristic
213C2×#100—478no—
214C2×#100—948no—
215#408—944no—
216#407—474no—
217#343—478no—
218C47#2099011yescharacteristic, complemented, Sylow
219C25C2×Dic4712yes—
220C2×#41—24no—
221C2×#41—24no—
222C2×#41C22⋊D4714yes—
223C2×#41C22⋊D4714yes—
224C22×Q8C2×Dic4711yescharacteristic
225C22×D4—24no—
226C22×D4—24no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4C2×Dic4712yes—
230C22×D4C2×Dic4712yes—
231C22×D4C2×Dic4711yescharacteristic
232C22×D4C2×Dic4718yes—
233C22×D4C22×D4712yes—
234C23×C4C22×D4712yes—
235C23×C4C22×D4711yescharacteristic
236#131—9432no—
237C2×M16—478no—
238C2×M16—474no—
239C2×M16—474no—
240C2×M16—9416no—
241C2×M16—944no—
242C2×M16—944no—
243C2×M16—942no—
244C2×M16—942no—
245C2×M16—944no—
246C2×M16—944no—
247C22×C8—944no—
248C22×C8—942no—
249C22×C8—942no—
250#127—28no—
251#124—28no—
252#121—28no—
253#121—28no—
254#120—216no—
255#117—22no—
256#117—22no—
257C2×C4⋊C4C22×D4712yes—
258C2×C22⋊C4C2×Dic4712yes—
259C2×C22⋊C4C2×Dic4718yes—
260C2×C42C2×Dic4712yes—
261#100—9416no—
262C22⋊C8—948no—
263C24—24no—
264C24—28no—
265C24—84no—
266C24C22×Dic4714yes—
267C24C22×Dic4712yes—
268C24C22×Dic4711yescharacteristic
269#41—28no—
270#41—28no—
271#41—28no—
272#41—28no—
273C2×Q8—24no—
274C2×Q8C22⋊Dic4714yes—
275C2×D4—28no—
276C2×D4—216no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—28no—
280C2×D4—24no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—24no—
284C2×D4—44no—
285C2×D4—44no—
286C2×D4—44no—
287C2×D4—44no—
288C2×D4—88no—
289C2×D4C22⋊Dic4718yes—
290C2×D4C22⋊Dic4714yes—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4D4×D4714yes—
294C22×C4#1062535214yes—
295C22×C4C22×Dic4712yes—
296C22×C4C22×Dic4714yes—
297C22×C4C22×Dic4711yescharacteristic
298C22×C4C2×C22⋊D4714yes—
299C22×C4C2×C22⋊D4714yes—
300C22×C4C2×C22⋊D4712yes—
301C22×C4C2×C22⋊D4712yes—
302C22×C4C23×D4711yescharacteristic
303M16—478no—
304M16—9416no—
305M16—948no—
306M16—944no—
307M16—944no—
308M16—944no—
309M16—1884no—
310C2×C8—9416no—
311C2×C8—948no—
312C2×C8—944no—
313C2×C8—944no—
314C2×C8—1884no—
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⋊Dic4714yes—
330C23C2×C22⋊Dic4712yes—
331C23D4×Dic4714yes—
332C23C23×Dic4711yescharacteristic
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⋊Dic4714yes—
348C2×C4C2×C22⋊Dic4712yes—
349C2×C4D4×Dic4714yes—
350C2×C4#1309399214yes—
351C2×C4#1309399614yes—
352C2×C4#1309399614yes—
353C2×C4#1309400514yes—
354C2×C4C2×D4×D4712yes—
355C2×C4C22×C22⋊D4711yescharacteristic
356C2×C4C22×C22⋊D4711yescharacteristic
357C2×C4C2×#1062535212yes—
358C8—948no—
359C8—1884no—
360C22—22no—
361C22—24no—
362C22—24no—
363C22—28no—
364C22—44no—
365C22—44no—
366C22—48no—
367C22—84no—
368C22—88no—
369C22—816no—
370C22#1568900314yes—
371C22C22×C22⋊Dic4711yescharacteristic
372C22C2×D4×Dic4712yes—
373C4—24no—
374C4—44no—
375C4#1568900314yes—
376C4C2×#1309399211yescharacteristic
377C4C2×#1309399612yes—
378C4C2×#1309400511yescharacteristic
379C2—22no—
380C2—24no—
381C2—44no—
382C2—84no—
383C2C2×#1568900311yescharacteristic
384C2#2025186712yes—
subgroupG/H
centerC22C22×C22⋊Dic47
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
Fitting, prime radicalC2×C47×#535C2
socleC22×C47C22×C22⋊C4
cosocleC22×C4×C47C24
hypercenterC2×#535D47
hypocenterC47#20990
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×#41 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
1. C2×#15689890C2semidirect
2. C22×#13093941C2semidirect
3. #20251833C2semidirect
4. #20243852C2semidirect
5. #20243127C2semidirect
11. C22×#10625329C22semidirect
15. #15689087C22semidirect
16. #15688892C22semidirect
34. C2×#10625329D4semidirect
218. C47#20990semidirect

Maximal subgroups

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

subgroupindex
C2×#156898902
C22×#130939412
#202518332
#202438522
#202431272
C2×C47×#5352
#2099047