Finite Group Databasebeta

← #26122762

Subgroups — #26122762

Order 12032.

978 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093942C212yesmaximal
2C2×#15689029C211yesmaximal, characteristic
3C2×#15688891C212yesmaximal
4#20243082C211yesmaximal, characteristic
5C2×C4×M16×C47C211yesmaximal, characteristic
6C23×Dic94C2211yescharacteristic
7C22×#10625329C2212yes—
8C22×C4⋊Dic47C2211yescharacteristic
9C2×#13093942—24no—
10C2×#13093942—24no—
11C2×#13093942C418yescomplemented
12#15689029—22no—
13#15689029—21no—
14#15689029—21no—
15C2×C47⋊#95C414yes—
16#15688891—24no—
17#15688878—22no—
18C22×M16×C47C2212yes—
19C22×C42×C47C2211yescharacteristic
20C4×M16×C47—24no—
21C2×#97×C47—21no—
22C2×C4×C8×C47—21no—
23C22×Dic94—21no—
24C22×Dic94—21no—
25C22×Dic94—22no—
26C22×Dic94—22no—
27C22×Dic94—22no—
28C22×Dic94—22no—
29C22×Dic94—22no—
30C22×Dic94C2×C414yes—
31C23×Dic47C2×C412yes—
32C23×Dic47C2×C412yes—
33C2×#10625329—22no—
34C2×#10625329—22no—
35C2×#10625329—22no—
36C2×#10625329—22no—
37C2×#10625329—22no—
38C2×#10625329C2×C414yes—
39C22×C47⋊C8—22no—
40C2×C4⋊Dic47—21no—
41C2×C4⋊Dic47—21no—
42C2×C4⋊Dic47—22no—
43C2×C4⋊Dic47—22no—
44#13093942—216no—
45C47⋊#95—24no—
46C47⋊#95—24no—
47C23×C4×C47C2×C412yes—
48C23×C4×C47C2311yescharacteristic
49C2×M16×C47—22no—
50C2×M16×C47—22no—
51C2×M16×C47—22no—
52C2×M16×C47C2×C414yes—
53C2×M16×C47D414yes—
54C2×M16×C47D414yes—
55C22×C8×C47—22no—
56C2×C42×C47—22no—
57C2×C42×C47—22no—
58C2×C42×C47D412yes—
59C2×C42×C47D411yescharacteristic
60C2×C42×C47D411yescharacteristic
61#97×C47—24no—
62C4×C8×C47—24no—
63C24×C47C22×C411yescharacteristic
64C22×C4×C47—24no—
65C22×C4×C47—24no—
66C22×C4×C47—42no—
67C22×C4×C47C22⋊C412yes—
68C22×C4×C47C22⋊C412yes—
69C22×C4×C47C22×C412yes—
70C22×C4×C47C2×D412yes—
71C22×C4×C47C2×D412yes—
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47C2×D411yescharacteristic
74C22×C4×C47#4112yes—
75C22×C4×C47#4112yes—
76C22×C4×C47#4112yes—
77C22×Dic47—22no—
78C22×Dic47—22no—
79C22×Dic47—24no—
80C22×Dic47—24no—
81C22×Dic47—22no—
82C22×Dic47—24no—
83C22×Dic47—44no—
84C22×Dic47C4214yes—
85C2×Dic94—24no—
86C2×Dic94—24no—
87C2×Dic94—24no—
88C2×Dic94—24no—
89C2×Dic94—24no—
90C2×Dic94—24no—
91C2×Dic94—41no—
92C2×Dic94—42no—
93C2×Dic94—42no—
94C2×Dic94—42no—
95C2×Dic94—42no—
96C2×Dic94—41no—
97C2×Dic94—41no—
98C2×Dic94—42no—
99C2×Dic94—42no—
100C2×Dic94—41no—
101M16×C47—24no—
102M16×C47—24no—
103M16×C47—24no—
104M16×C47C22⋊C418yes—
105C2×C8×C47—24no—
106C2×C8×C47—24no—
107C2×C8×C47—24no—
108C42×C47—21no—
109C42×C47—21no—
110C42×C47—24no—
111C42×C47—21no—
112C42×C47—21no—
113C4⋊Dic47—42no—
114C4⋊Dic47—41no—
115C4⋊Dic47—41no—
116#10625329—24no—
117#10625329—24no—
118#10625329—42no—
119#10625329—42no—
120C2×C47⋊C8—24no—
121C2×C47⋊C8—42no—
122C2×C47⋊C8—42no—
123C23×C47—22no—
124C23×C47—22no—
125C23×C47C2×C4211yescharacteristic
126C23×C47C2×C22⋊C411yescharacteristic
127C23×C47#11711yescharacteristic
128C23×C47C4×D412yes—
129C23×C47C4×D412yes—
130C2×C4×C47—22no—
131C2×C4×C47—22no—
132C2×C4×C47—22no—
133C2×C4×C47—22no—
134C2×C4×C47—22no—
135C2×C4×C47—48no—
136C2×C4×C47#10114yescomplemented
137C2×C4×C47C2×C22⋊C412yes—
138C2×C4×C47#11712yes—
139C2×C4×C47C4×D412yes—
140C2×C4×C47C4×D412yes—
141C2×C4×C47C4×D412yes—
142C2×C4×C47C4×D412yes—
143C2×C4×C47#12011yescharacteristic
144C2×C4×C47#12112yes—
145C2×C4×C47#12112yes—
146C2×C4×C47#12112yes—
147C2×C4×C47#12312yes—
148C2×C4×C47#12311yescharacteristic
149C2×C4×C47#12411yescharacteristic
150C2×C4×C47#12412yes—
151C2×C4×C47#12612yes—
152C2×C4×C47#12711yescharacteristic
153C2×Dic47—28no—
154C2×Dic47—24no—
155C2×Dic47—42no—
156C2×Dic47—44no—
157C2×Dic47—42no—
158C2×Dic47—42no—
159C2×Dic47—44no—
160C2×Dic47—84no—
161Dic94—44no—
162Dic94—44no—
163Dic94—44no—
164Dic94—44no—
165C8×C47—28no—
166C47⋊C8—44no—
167#10963—471nomaximal, Sylow
168C22×C47—24no—
169C22×C47—22no—
170C22×C47—22no—
171C22×C47—22no—
172C22×C47—22no—
173C22×C47C4×C22⋊C412yes—
174C22×C47C4×C22⋊C411yescharacteristic
175C22×C47#38611yescharacteristic
176C22×C47#38812yes—
177C22×C47#39011yescharacteristic
178C22×C47C2×#10112yes—
179C22×C47#41312yes—
180C4×C47—24no—
181C4×C47—42no—
182C4×C47#38612yes—
183C4×C47#38812yes—
184C4×C47#38812yes—
185C4×C47#39012yes—
186Dic47—44no—
187Dic47—82no—
188C22×#101—472no—
189C2×C4×M16D4711yescharacteristic
190C2×#391—471no—
191C2×#343—472no—
192#1728—471no—
193C2×C47—22no—
194C2×C47—22no—
195C2×C47#143711yescharacteristic
196C2×C47C4×#10112yes—
197C2×C47#190112yes—
198C2×C47#196611yescharacteristic
199C2×C47#196711yescharacteristic
200C23×Q8—471no—
201C22×M16—472no—
202C22×M16C2×D4712yes—
203C22×C4⋊C4—471no—
204C22×C42C2×D4711yescharacteristic
205C2×#101—478no—
206C2×#101—944no—
207C2×#101—944no—
208C4×M16—24no—
209C2×#97—21no—
210C2×C4×C8—21no—
211#391—942no—
212#391—941no—
213#391—941no—
214C2×#95—474no—
215#343—944no—
216#336—942no—
217C47#1096311yescharacteristic, complemented, Sylow
218C22×Q8—474no—
219C22×Q8—941no—
220C22×Q8—942no—
221C22×Q8—942no—
222C22×Q8—942no—
223C22×Q8—942no—
224C22×Q8—941no—
225C22×Q8—942no—
226C23×C4—472no—
227C23×C4—472no—
228C23×C4C4×D4712yes—
229C23×C4C22×D4711yescharacteristic
230C2×M16—22no—
231C2×M16—22no—
232C2×M16—22no—
233C2×M16—474no—
234C2×M16—942no—
235C2×M16—942no—
236C2×M16—942no—
237C2×M16—942no—
238C2×M16—942no—
239C2×M16C4×D4714yes—
240C2×M16D18814yes—
241C2×M16C22⋊D4714yes—
242C22×C8—22no—
243C22×C8—942no—
244C2×C4⋊C4—942no—
245C2×C4⋊C4—941no—
246C2×C4⋊C4—942no—
247C2×C4⋊C4—941no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42D18812yes—
251C2×C42C22⋊D4711yescharacteristic
252C2×C42C22⋊D4711yescharacteristic
253#101—9416no—
254#97—24no—
255C4×C8—24no—
256#95—944no—
257#95—944no—
258C24C2×C4×D4711yescharacteristic
259C2×Q8—944no—
260C2×Q8—944no—
261C2×Q8—944no—
262C2×Q8—944no—
263C2×Q8—944no—
264C2×Q8—944no—
265C2×Q8—1881no—
266C2×Q8—1882no—
267C2×Q8—1882no—
268C2×Q8—1882no—
269C2×Q8—1881no—
270C2×Q8—1882no—
271C2×Q8—1882no—
272C2×Q8—1881no—
273C2×Q8—1882no—
274C2×Q8—1881no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—42no—
278C22×C4—474no—
279C22×C4—942no—
280C22×C4—942no—
281C22×C4—944no—
282C22×C4—944no—
283C22×C4—944no—
284C22×C4—942no—
285C22×C4—1884no—
286C22×C4(C2×D47)⋊C412yes—
287C22×C4(C2×D47)⋊C412yes—
288C22×C4C2×C4×D4712yes—
289C22×C4C2×D18812yes—
290C22×C4C2×D18811yescharacteristic
291C22×C4#1062535012yes—
292C22×C4#1062535012yes—
293C22×C4#1062535012yes—
294C22×C4C2×C22⋊D4712yes—
295C22×C4C2×C22⋊D4711yescharacteristic
296M16—24no—
297M16—24no—
298M16—24no—
299M16—944no—
300M16—944no—
301M16—1882no—
302M16—1882no—
303M16(C2×D47)⋊C418yes—
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—944no—
308C2×C8—1882no—
309C2×C8—1882no—
310C4⋊C4—1882no—
311C4⋊C4—1881no—
312C4⋊C4—1881no—
313C42—21no—
314C42—21no—
315C42—24no—
316C42—21no—
317C42—21no—
318C23—22no—
319C23—22no—
320C23C42×D4711yescharacteristic
321C23C2×(C2×D47)⋊C411yescharacteristic
322C23#1309398111yescharacteristic
323C23C4×C22⋊D4712yes—
324C23C4×D18812yes—
325Q8—1884no—
326Q8—1884no—
327Q8—1884no—
328Q8—1884no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—48no—
335C2×C4—948no—
336C2×C4—944no—
337C2×C4—3764no—
338C2×C4—1882no—
339C2×C4—1884no—
340C2×C4—1882no—
341C2×C4—1884no—
342C2×C4—1882no—
343C2×C4#1309394214yescomplemented
344C2×C4C2×(C2×D47)⋊C412yes—
345C2×C4#1309398112yes—
346C2×C4C4×C22⋊D4712yes—
347C2×C4C4×C22⋊D4712yes—
348C2×C4C4×D18812yes—
349C2×C4C4×D18812yes—
350C2×C4#1309399311yescharacteristic
351C2×C4#1309399812yes—
352C2×C4#1309399812yes—
353C2×C4#1309399812yes—
354C2×C4#1309400712yes—
355C2×C4#1309400711yescharacteristic
356C2×C4#1309401211yescharacteristic
357C2×C4#1309401212yes—
358C2×C4#1309401812yes—
359C2×C4#1309402111yescharacteristic
360C8—28no—
361C8—1884no—
362C22—22no—
363C22—22no—
364C22—24no—
365C22—22no—
366C22—22no—
367C22C4×(C2×D47)⋊C412yes—
368C22C4×(C2×D47)⋊C411yescharacteristic
369C22#1568900211yescharacteristic
370C22#1568901412yes—
371C22#1568902511yescharacteristic
372C22C2×#1309394212yes—
373C22#1568910512yes—
374C4—24no—
375C4—42no—
376C4—3762no—
377C4—1884no—
378C4#1568900212yes—
379C4#1568901412yes—
380C4#1568901412yes—
381C4#1568902512yes—
382C2—22no—
383C2—22no—
384C2#2024162611yescharacteristic
385C2C4×#1309394212yes—
386C2#2024385612yes—
387C2#2024423211yescharacteristic
388C2#2024423711yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC23×C47C2×C42
FrattiniC23×C4C22×D47
FittingC2×C4×M16×C47C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22×C4
cosocleC23×C4×C47C23
hypercenterC2×C4×M16D47
hypocenterC47#10963
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×M16

Complements

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

NH ≅ G/Ntype
11. C2×#13093942C4semidirect
136. C2×C4×C47#101semidirect
217. C47#10963semidirect
343. C2×C4#13093942semidirect

Maximal subgroups

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

subgroupindex
C22×#130939422
C2×#156890292
C2×#156888912
#202430822
C2×C4×M16×C472
#1096347