Finite Group Databasebeta

← #26376297

Subgroups — #26376297

Order 12032.

5430 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#10625354C211yesmaximal, characteristic
2C2×#15689597C212yesmaximal, complemented
3#20246217C2116yesmaximal, complemented
4#20245974C211yesmaximal, characteristic, complemented
5#20245965C214yesmaximal, complemented
6#20245767C212yesmaximal, complemented
7#381×D47C214yesmaximal, complemented
8C47×#2310C211yesmaximal, characteristic, complemented
9C22×#10625354—216no—
10C22×#10625354C4116yescomplemented
11C22×#10625354C2218yes—
12C23×D188C2212yes—
13C23×D188C2211yescharacteristic
14C23×C4×D47C2212yes—
15C23×C4×D47C2211yescharacteristic
16#15689597—216no—
17C2×C4×D188C2218yescomplemented
18C2×#13093986C2218yescomplemented
19C2×#13093983C2212yes—
20C2×C22⋊C4×D47C2218yes—
21#15689030C2218yescomplemented
22#15689001C22132yescomplemented
23#15688986C2214yes—
24#15688985C22116yescomplemented
25#15688984C2214yes—
26#15688980C2218yescomplemented
27#15688978C2214yes—
28C4×(C2×D47)⋊C4C2218yescomplemented
29#95×D47C22116yescomplemented
30C23×Q8×C47C2211yescharacteristic
31C2×#117×C47C2212yes—
32C47×#391C2218yescomplemented
33C47×#381C2214yes—
34C24×D47C2311yescharacteristic
35C24×D47C2312yes—
36C2×#10625354—2128no—
37C2×#10625354—232no—
38C2×#10625354—232no—
39C2×#10625354C2×C4164yescomplemented
40C22×D188—216no—
41C22×D188C2×C418yes—
42C22×D188C2×C4116yescomplemented
43C22×D188D4116yes—
44C22×D188C2318yes—
45C22×D188C2314yes—
46C23×Dic47C2311yescharacteristic
47C22×C4×D47—216no—
48C22×C4×D47—28no—
49C22×C4×D47C2×C4116yescomplemented
50C22×C4×D47C2×C418yescomplemented
51C22×C4×D47C2318yes—
52C22×C4×D47C2314yes—
53C22×C4×D47C23116yes—
54C4×D188—232no—
55#13093986—232no—
56#13093983—28no—
57C2×C4⋊Dic47C2314yes—
58C2×C22⋊Dic47C2314yes—
59C2×(C2×D47)⋊C4C23116yes—
60C2×(C2×D47)⋊C4C23116yes—
61C22⋊C4×D47—232no—
62C2×C4×Dic47C2314yes—
63C47⋊#95C23116yes—
64C47⋊#95C23116yes—
65C22×Q8×C47—28no—
66C22×Q8×C47C2×C418yescomplemented
67C22×Q8×C47C2314yes—
68C23×C4×C47C2312yes—
69C23×C4×C47C2311yescharacteristic
70#117×C47—28no—
71C2×C4⋊C4×C47C2314yes—
72C2×C22⋊C4×C47C2314yes—
73C2×C42×C47C2314yes—
74#95×C47C23116yes—
75C24×C47C2411yescharacteristic
76C23×D47—24no—
77C23×D47—48no—
78C23×D47—44no—
79C23×D47C22×C414yes—
80C23×D47C22×C418yes—
81C23×D47C2×D4116yes—
82C23×D47C2412yes—
83C23×D47C2414yes—
84C2×Q8×C47—232no—
85C2×Q8×C47—28no—
86C2×Q8×C47—28no—
87C2×Q8×C47C22×C4116yes—
88C22×C4×C47—28no—
89C22×C4×C47C22×C418yes—
90C22×C4×C47C22×C414yes—
91C22×C4×C47C2×D418yes—
92C22×C4×C47C2418yes—
93C22×C4×C47C2414yes—
94C22×C4×C47C2412yes—
95C22×Dic47—24no—
96C22×Dic47C22×C414yes—
97C22×Dic47C2418yes—
98C22×Dic47C2412yes—
99#10625354—2256no—
100C2×D188—264no—
101C2×D188—232no—
102C2×D188C22⋊C4164yescomplemented
103C2×D188C22×C4132yes—
104C2×D188C22×C4116yes—
105C2×D188C2×D4116yes—
106C2×D188C2×D4116yes—
107C2×C4×D47—264no—
108C2×C4×D47—232no—
109C2×C4×D47—232no—
110C2×C4×D47—28no—
111C2×C4×D47—28no—
112C2×C4×D47—232no—
113C2×C4×D47—432no—
114C2×C4×D47C22×C4132yes—
115C2×C4×D47C22×C4116yes—
116C4⋊C4×C47—28no—
117C22⋊C4×C47—28no—
118C42×C47—28no—
119C22⋊Dic47—28no—
120(C2×D47)⋊C4—232no—
121(C2×D47)⋊C4—416no—
122C4⋊Dic47—28no—
123C4×Dic47—28no—
124C23×C47—24no—
125C23×C47C23×C412yes—
126C23×C47C22×D414yes—
127C23×C47C2511yescharacteristic
128C22×D47—216no—
129C22×D47—28no—
130C22×D47—24no—
131C22×D47—28no—
132C22×D47—416no—
133C22×D47—48no—
134C22×D47—432no—
135C22×D47—416no—
136C22×D47C2×C22⋊C4132yes—
137C22×D47C23×C414yes—
138C22×D47C23×C418yes—
139C22×D47C22×D4116yes—
140Q8×C47—232no—
141C2×C4×C47—216no—
142C2×C4×C47—28no—
143C2×C4×C47—28no—
144C2×C4×C47—216no—
145C2×C4×C47C2×C22⋊C4116yes—
146C2×C4×C47C23×C418yes—
147C2×C4×C47C23×C414yes—
148C2×C4×C47C22×D414yes—
149C2×C4×C47C22×D414yes—
150C2×Dic47—28no—
151C2×Dic47—24no—
152C2×Dic47—28no—
153C2×Dic47—216no—
154C2×Dic47C23×C414yes—
155D188—264no—
156D188C2×C22⋊C4164yes—
157C4×D47—264no—
158C4×D47—232no—
159C4×D47—832no—
160#34172—471nomaximal, Sylow
161C22×C47—24no—
162C22×C47—28no—
163C22×C47C22×C22⋊C414yes—
164C22×C47C24×C411yescharacteristic
165C22×C47C23×D412yes—
166C22×C47#58514yes—
167C2×D47—216no—
168C2×D47—232no—
169C2×D47—24no—
170C2×D47—28no—
171C2×D47—432no—
172C2×D47—416no—
173C2×D47—48no—
174C2×D47C22×C22⋊C4116yes—
175C4×C47—28no—
176C4×C47—48no—
177C4×C47C22×C22⋊C418yes—
178Dic47—24no—
179Dic47—48no—
180C23×#41—471no—
181C2×#518—472no—
182#2339—4716no—
183#2310—471no—
184#2310D4711yescharacteristic, complemented
185#2308—474no—
186#2280—472no—
187C2×#381—474no—
188C2×C47—24no—
189C2×C47C23×C22⋊C411yescharacteristic
190C2×C47#342812yes—
191C2×C47#348514yes—
192D47—216no—
193D47—44no—
194C22×#41—4716no—
195C22×#41—478no—
196C22×#41—9416no—
197C23×Q8C2×D4711yescharacteristic
198C23×D4—471no—
199C23×D4—472no—
200C24×C4—472no—
201C24×C4—471no—
202#518—9416no—
203C2×C4×D4—478no—
204C2×C4×D4—478no—
205C2×#117—472no—
206C2×#117C2×D4712yes—
207C22×C22⋊C4—478no—
208#391—478no—
209#391C2×D4718yescomplemented
210#386—4732no—
211#381—474no—
212#381—474no—
213#381—4716no—
214#381C2×D4714yes—
215#380—478no—
216#379—474no—
217C4×C22⋊C4—478no—
218C2×#95—4716no—
219C47#3417211yescharacteristic, complemented, Sylow
220C25—472no—
221C25—471no—
222C2×#41—4764no—
223C2×#41—94128no—
224C2×#41—9432no—
225C2×#41—9432no—
226C22×Q8—28no—
227C22×Q8C4×D4718yescomplemented
228C22×Q8C22×D4714yes—
229C22×D4—478no—
230C22×D4—4716no—
231C22×D4—4716no—
232C22×D4—474no—
233C22×D4—478no—
234C22×D4—9416no—
235C23×C4—471no—
236C23×C4—4716no—
237C23×C4—478no—
238C23×C4—478no—
239C23×C4—474no—
240C23×C4—4716no—
241C23×C4—9416no—
242C23×C4—948no—
243C23×C4C22×D4712yes—
244C23×C4C22×D4711yescharacteristic
245C4×D4—9432no—
246C4×D4—9432no—
247#117—28no—
248#117—948no—
249C2×C4⋊C4—474no—
250C2×C4⋊C4C22×D4714yes—
251C2×C22⋊C4—474no—
252C2×C22⋊C4—4716no—
253C2×C22⋊C4—4716no—
254C2×C22⋊C4—9432no—
255C2×C22⋊C4C22×D4714yes—
256C2×C42—474no—
257C2×C42C22×D4714yes—
258#95—4716no—
259#95—4716no—
260#95C22×D47116yes—
261C24—474no—
262C24—478no—
263C24—4716no—
264C24—472no—
265C24—474no—
266C24—944no—
267C24—1888no—
268C24—1884no—
269C24C23×D4711yescharacteristic
270#41—94256no—
271C2×Q8—232no—
272C2×Q8—28no—
273C2×Q8—28no—
274C2×Q8C2×C4×D47116yes—
275C2×D4—4764no—
276C2×D4—4716no—
277C2×D4—4732no—
278C2×D4—4716no—
279C2×D4—4716no—
280C2×D4—9464no—
281C2×D4—9432no—
282C22×C4—28no—
283C22×C4—474no—
284C22×C4—472no—
285C22×C4—4732no—
286C22×C4—4716no—
287C22×C4—478no—
288C22×C4—9464no—
289C22×C4—9432no—
290C22×C4—9432no—
291C22×C4—944no—
292C22×C4—9432no—
293C22×C4—948no—
294C22×C4—948no—
295C22×C4—18832no—
296C22×C4C2×C4×D4718yes—
297C22×C4C2×C4×D4714yes—
298C22×C4D4×D4718yes—
299C22×C4C23×D4718yes—
300C22×C4C23×D4714yes—
301C22×C4C23×D4712yes—
302C4⋊C4—28no—
303C4⋊C4—948no—
304C22⋊C4—28no—
305C22⋊C4—9432no—
306C22⋊C4—948no—
307C22⋊C4—18816no—
308C42—28no—
309C42—948no—
310C23—24no—
311C23—478no—
312C23—4716no—
313C23—4732no—
314C23—474no—
315C23—9416no—
316C23—948no—
317C23—944no—
318C23—948no—
319C23—18832no—
320C23—18816no—
321C23—1888no—
322C23—18816no—
323C23C22×C4×D4712yes—
324C23C2×D4×D4714yes—
325C23C24×D4711yescharacteristic
326Q8—232no—
327D4—4764no—
328D4—9464no—
329C2×C4—216no—
330C2×C4—28no—
331C2×C4—28no—
332C2×C4—216no—
333C2×C4—474no—
334C2×C4—9464no—
335C2×C4—948no—
336C2×C4—944no—
337C2×C4—9432no—
338C2×C4—948no—
339C2×C4—9416no—
340C2×C4—37632no—
341C2×C4C22⋊C4×D47116yes—
342C2×C4C22×C4×D4718yes—
343C2×C4C22×C4×D4714yes—
344C2×C4C2×D4×D4714yes—
345C2×C4C2×D4×D4714yes—
346C22—28no—
347C22—24no—
348C22—4716no—
349C22—9432no—
350C22—9416no—
351C22—944no—
352C22—948no—
353C22—18832no—
354C22—18816no—
355C22—1888no—
356C22C2×C22⋊C4×D4714yes—
357C22C23×C4×D4711yescharacteristic
358C22C22×D4×D4712yes—
359C22#1568996014yes—
360C4—28no—
361C4—48no—
362C4—944no—
363C4—1888no—
364C4C2×C22⋊C4×D4718yes—
365C2—24no—
366C2—9416no—
367C2—1884no—
368C2C22×C22⋊C4×D4711yescharacteristic
369C2#2025659012yes—
370C2#2025703114yes—
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingC47×#2310C2
socle, cosocleC23×C47C25
prime radicalC23×#10625354C2
hypercenter#2310D47
hypocenterC47#34172
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ #2310

Complements

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

NH ≅ G/Ntype
2. C2×#15689597C2semidirect
3. #20246217C2semidirect
4. #20245974C2semidirect
5. #20245965C2semidirect
6. #20245767C2semidirect
7. #381×D47C2semidirect
8. C47×#2310C2semidirect
10. C22×#10625354C4semidirect
17. C2×C4×D188C22semidirect
18. C2×#13093986C22semidirect
21. #15689030C22semidirect
22. #15689001C22semidirect
24. #15688985C22semidirect
26. #15688980C22semidirect
28. C4×(C2×D47)⋊C4C22semidirect
29. #95×D47C22semidirect
32. C47×#391C22semidirect
39. C2×#10625354C2×C4semidirect
42. C22×D188C2×C4semidirect
49. C22×C4×D47C2×C4semidirect
50. C22×C4×D47C2×C4semidirect
66. C22×Q8×C47C2×C4semidirect
102. C2×D188C22⋊C4semidirect
184. #2310D47semidirect
209. #391C2×D47semidirect
219. C47#34172semidirect
227. C22×Q8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C23×#106253542
C2×#156895972
#202462172
#202459742
#202459652
#202457672
#381×D472
C47×#23102
#3417247