Finite Group Databasebeta

← #26107776

Subgroups — #26107776

Order 12032.

802 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
1C2×#15689680C211yesmaximal, characteristic
2#20243368C211yesmaximal, characteristic
3C2×#15688871C212yesmaximal, complemented
4C2×#15688870C212yesmaximal, complemented
5C47×#1797C211yesmaximal, characteristic
6#15689680—24no—
7#15689680—24no—
8C2×#13094024C2211yescharacteristic
9C2×#13094024C2211yescharacteristic
10C2×#13094016—21no—
11C2×#13094003—21no—
12C2×#13094003C412yes—
13C2×C4×Dic94—21no—
14C2×C4×Dic94C412yes—
15C2×#13093983—21no—
16C2×C4⋊(C47⋊C8)C2212yes—
17C2×C22⋊(C47⋊C8)—21no—
18#15688878—21no—
19#15688871—24no—
20#15688870—24no—
21C22×C42×C47C2211yescharacteristic
22C2×C4⋊C8×C47C2212yes—
23C2×C22⋊C8×C47—21no—
24C47×#336—21no—
25C22×Dic94—21no—
26C22×Dic94C2×C412yes—
27C22×C47⋊C8—22no—
28#13094024—22no—
29#13094024—22no—
30#13094024D414yes—
31#13094024D414yes—
32#13094016—28no—
33#13094003—28no—
34#13094003—24no—
35#13094003—24no—
36C4×Dic94—28no—
37C4×Dic94—24no—
38C4×Dic94—24no—
39#13093983—28no—
40C2×C4⋊Dic47—21no—
41C2×C4⋊Dic47—21no—
42C2×C4⋊Dic47—21no—
43C2×C4⋊Dic47C2×C412yes—
44C2×Dic47⋊C4—22no—
45C2×Dic47⋊C4—22no—
46C2×C22⋊Dic47—22no—
47C2×C4×Dic47—22no—
48C4⋊(C47⋊C8)—22no—
49C4⋊(C47⋊C8)—22no—
50C22⋊(C47⋊C8)—42no—
51C23×C4×C47C2×C411yescharacteristic
52C23×C4×C47D411yescharacteristic
53C23×C4×C47D411yescharacteristic
54C22×C8×C47—22no—
55C2×C42×C47—22no—
56C2×C42×C47—22no—
57C2×C42×C47C2×C411yescharacteristic
58C2×C42×C47D411yescharacteristic
59C2×C42×C47D411yescharacteristic
60C2×C42×C47C2311yescharacteristic
61C4⋊C8×C47—22no—
62C4⋊C8×C47—22no—
63C22⋊C8×C47—42no—
64C24×C47C22⋊C411yescharacteristic
65C22×C4×C47—22no—
66C22×C4×C47—22no—
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47—22no—
70C22×C4×C47—22no—
71C22×C4×C47—21no—
72C22×C4×C47—21no—
73C22×C4×C47—41no—
74C22×C4×C47—41no—
75C22×C4×C47C22⋊C411yescharacteristic
76C22×C4×C47C22⋊C411yescharacteristic
77C22×C4×C47C22⋊C411yescharacteristic
78C22×C4×C47SD1612yes—
79C22×C4×C47Q1612yes—
80C22×C4×C47C22×C411yescharacteristic
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47C2×D411yescharacteristic
83C22×Dic47—22no—
84C22×Dic47—22no—
85C2×Dic94—22no—
86C2×Dic94—24no—
87C2×Dic94—42no—
88C2×Dic94—41no—
89C2×Dic94—41no—
90C2×Dic94—42no—
91C2×Dic94—42no—
92C2×Dic94C22⋊C414yes—
93C2×C8×C47—42no—
94C2×C8×C47—42no—
95C2×C8×C47—42no—
96C42×C47—21no—
97C42×C47—21no—
98C42×C47—22no—
99C42×C47—22no—
100C42×C47C2×D412yes—
101C42×C47C2×D412yes—
102C22⋊Dic47—28no—
103C4⋊Dic47—22no—
104C4⋊Dic47—24no—
105C4⋊Dic47—24no—
106C4⋊Dic47—24no—
107C4⋊Dic47C22⋊C414yes—
108Dic47⋊C4—28no—
109Dic47⋊C4—28no—
110C4×Dic47—28no—
111C2×C47⋊C8—42no—
112C2×C47⋊C8—42no—
113C2×C47⋊C8—42no—
114C23×C47—21no—
115C23×C47—21no—
116C23×C47—21no—
117C23×C47—21no—
118C23×C47—21no—
119C23×C47—21no—
120C23×C47#10312yes—
121C23×C47C2×C22⋊C411yescharacteristic
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—24no—
127C2×C4×C47—24no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—21no—
134C2×C4×C47—42no—
135C2×C4×C47—42no—
136C2×C4×C47—42no—
137C2×C4×C47—42no—
138C2×C4×C47#10312yes—
139C2×C4×C47C2×C22⋊C411yescharacteristic
140C2×C4×C47C2×C22⋊C411yescharacteristic
141C2×C4×C47#12012yes—
142C2×C4×C47#12012yes—
143C2×C4×C47C2×SD1611yescharacteristic
144C2×C4×C47C2×Q1611yescharacteristic
145C2×C4×C47#13512yes—
146C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C47#13711yescharacteristic
148C2×C4×C47#13712yes—
149C2×Dic47—24no—
150C2×Dic47—24no—
151C2×Dic47—42no—
152C2×Dic47—42no—
153C2×Dic47—42no—
154C2×Dic47—42no—
155Dic94—44no—
156Dic94—44no—
157Dic94—44no—
158Dic94—44no—
159C8×C47—82no—
160C47⋊C8—82no—
161#8343—471nomaximal, Sylow
162C22×C47—22no—
163C22×C47—22no—
164C22×C47—22no—
165C22×C47—41no—
166C22×C47—41no—
167C22×C47—41no—
168C22×C47—41no—
169C22×C47#37911yescharacteristic
170C22×C47#41311yescharacteristic
171C22×C47#41311yescharacteristic
172C22×C47C2×#10311yescharacteristic
173C22×C47#41611yescharacteristic
174C22×C47#41711yescharacteristic
175C22×C47#41911yescharacteristic
176C4×C47—21no—
177C4×C47—21no—
178C4×C47—22no—
179C4×C47—41no—
180C4×C47—41no—
181C4×C47#44812yes—
182C4×C47#44912yes—
183C4×C47#45112yes—
184C4×C47#45212yes—
185Dic47—44no—
186Dic47—44no—
187C2×#533—471no—
188#1797—471no—
189#1797D4711yescharacteristic
190C2×#333—472no—
191C2×#332—472no—
192C2×C47—42no—
193C2×C47#154211yescharacteristic
194C2×C47#155011yescharacteristic
195C2×C47#155411yescharacteristic
196C2×C47#155911yescharacteristic
197C2×C47#178511yescharacteristic
198C2×C47#178711yescharacteristic
199C2×C47#178811yescharacteristic
200#533—944no—
201#533—944no—
202C2×C4⋊Q8—471no—
203C2×C4⋊Q8—471no—
204C2×#125—941no—
205C2×C22⋊Q8—472no—
206C2×C22⋊Q8—941no—
207C2×C4×Q8—472no—
208C2×C4×Q8—941no—
209C2×#117—941no—
210C22×C42C2×D4711yescharacteristic
211C2×C4⋊C8—472no—
212C2×C4⋊C8C2×D4712yes—
213C2×C22⋊C8—21no—
214C2×C22⋊C8—941no—
215#336—21no—
216#336—941no—
217#333—944no—
218#332—944no—
219C47#834311yescharacteristic, complemented, Sylow
220C22×Q8—472no—
221C22×Q8—941no—
222C23×C4C4×D4711yescharacteristic
223C23×C4D18811yescharacteristic
224C23×C4C22⋊D4711yescharacteristic
225C22×C8—22no—
226C22×C8—942no—
227C4⋊Q8—474no—
228C4⋊Q8—474no—
229C4⋊Q8—942no—
230C4⋊Q8—942no—
231#125—948no—
232C22⋊Q8—944no—
233C22⋊Q8—944no—
234C22⋊Q8—948no—
235C4×Q8—944no—
236C4×Q8—944no—
237C4×Q8—948no—
238#117—948no—
239C2×C4⋊C4—472no—
240C2×C4⋊C4—942no—
241C2×C4⋊C4—942no—
242C2×C4⋊C4—941no—
243C2×C4⋊C4—941no—
244C2×C4⋊C4—941no—
245C2×C22⋊C4—942no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—942no—
249C2×C42C4×D4711yescharacteristic
250C2×C42D18811yescharacteristic
251C2×C42C22⋊D4711yescharacteristic
252C2×C42C22×D4711yescharacteristic
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—942no—
256C4⋊C8—942no—
257C22⋊C8—42no—
258C22⋊C8—1882no—
259C24(C2×D47)⋊C411yescharacteristic
260C2×Q8—474no—
261C2×Q8—942no—
262C2×Q8—944no—
263C2×Q8—1882no—
264C2×Q8—1882no—
265C2×Q8—1882no—
266C2×Q8—1881no—
267C2×Q8—1881no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—942no—
279C22×C4—942no—
280C22×C4#1062532512yes—
281C22×C4Dic18812yes—
282C22×C4(C2×D47)⋊C411yescharacteristic
283C22×C4(C2×D47)⋊C411yescharacteristic
284C22×C4(C2×D47)⋊C411yescharacteristic
285C22×C4C2×C4×D4711yescharacteristic
286C22×C4C2×D18811yescharacteristic
287C22×C4C2×C22⋊D4711yescharacteristic
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—1882no—
292C2×C8—1882no—
293C2×C8—1882no—
294C4⋊C4—474no—
295C4⋊C4—948no—
296C4⋊C4—948no—
297C4⋊C4—942no—
298C4⋊C4—944no—
299C4⋊C4—944no—
300C4⋊C4—944no—
301C22⋊C4—948no—
302C42—21no—
303C42—21no—
304C42—22no—
305C42—22no—
306C42—948no—
307C42D4×D4712yes—
308C42D4×D4712yes—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23#1309394912yes—
316C23C2×(C2×D47)⋊C411yescharacteristic
317Q8—1884no—
318Q8—1884no—
319Q8—1884no—
320Q8—1884no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—944no—
338C2×C4—944no—
339C2×C4—1882no—
340C2×C4—1882no—
341C2×C4—1882no—
342C2×C4—1882no—
343C2×C4#1309394912yes—
344C2×C4C22⋊C4×D4711yescharacteristic
345C2×C4C22⋊C4×D4711yescharacteristic
346C2×C4#1309399212yes—
347C2×C4#1309399412yes—
348C2×C4C2×#1062532511yescharacteristic
349C2×C4C2×Dic18811yescharacteristic
350C2×C4#1309404412yes—
351C2×C4#1309405011yescharacteristic
352C2×C4#1309405511yescharacteristic
353C2×C4#1309405712yes—
354C8—82no—
355C8—3762no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22#1568897811yescharacteristic
364C22#1568910511yescharacteristic
365C22#1568910511yescharacteristic
366C22C2×#1309394911yescharacteristic
367C22#1568911711yescharacteristic
368C22#1568912211yescharacteristic
369C22#1568913311yescharacteristic
370C4—21no—
371C4—21no—
372C4—22no—
373C4—41no—
374C4—41no—
375C4—1884no—
376C4—1884no—
377C4#1568926412yes—
378C4#1568926612yes—
379C4#1568928112yes—
380C4#1568928312yes—
381C2—42no—
382C2#2024204311yescharacteristic
383C2#2024207611yescharacteristic
384C2#2024209911yescharacteristic
385C2#2024211611yescharacteristic
386C2#2024331411yescharacteristic
387C2#2024332411yescharacteristic
388C2#2024332611yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1797C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1797D47
hypocenterC47#8343
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
3. C2×#15688871C2semidirect
4. C2×#15688870C2semidirect
219. C47#8343semidirect

Maximal subgroups

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

subgroupindex
C2×#156896802
#202433682
C2×#156888712
C2×#156888702
C47×#17972
#834347