Finite Group Databasebeta

← #26104342

Subgroups — #26104342

Order 12032.

578 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20245918C211yesmaximal, characteristic
2#20243367C211yesmaximal, characteristic
3#336×D47C211yesmaximal, characteristic
4#20241213C211yesmaximal, characteristic, complemented
5#20241213C211yesmaximal, characteristic, complemented
6#20241211C211yesmaximal, characteristic, complemented
7C47×#1271C211yesmaximal, characteristic, complemented
8C22×C8×D47—21no—
9C2×#13094004C411yescharacteristic
10C2×#13093989C411yescharacteristic
11C2×C42×D47C2211yescharacteristic
12C2×Dic47⋊C8—21no—
13C2×(C2×D47)⋊C8—21no—
14#15689029C2211yescharacteristic
15#15689006—21no—
16#15688994—21no—
17#15688980C411yescharacteristic
18C4⋊C4×Dic47—21no—
19C4⋊C4×Dic47C411yescharacteristic
20#15688970—21no—
21#15688878C2211yescharacteristic
22#15688878C2211yescharacteristic
23#15688877—21no—
24C47×#389C2211yescharacteristic
25C47×#336C2211yescharacteristic
26C47×#336C2211yescharacteristic
27C22×Dic94C2×C411yescharacteristic
28C22×C4×D47C2×C411yescharacteristic
29C22×C4×D47D411yescharacteristic
30C22×C4×D47D411yescharacteristic
31C22×C47⋊C8—21no—
32C22×C47⋊C8—21no—
33C2×C8×D47—24no—
34C2×C8×D47—42no—
35C2×C8×D47—42no—
36#13094004—22no—
37#13094004C814yescomplemented
38#13093989—42no—
39C2×C4⋊Dic47—21no—
40C2×C4⋊Dic47C2×C411yescharacteristic
41C2×Dic47⋊C4—21no—
42C2×(C2×D47)⋊C4—21no—
43C2×(C2×D47)⋊C4—21no—
44C2×C4×Dic47—21no—
45C2×C4×Dic47—21no—
46C2×C4×Dic47C2×C411yescharacteristic
47C2×C4×Dic47D411yescharacteristic
48C2×C4×Dic47D411yescharacteristic
49C42×D47—42no—
50Dic47⋊C8—42no—
51(C2×D47)⋊C8—42no—
52C47⋊#95—21no—
53C47⋊#95—21no—
54C47⋊#95—21no—
55C47⋊#95—21no—
56C22×C8×C47—21no—
57C22×C8×C47—21no—
58C2×C4⋊C4×C47—21no—
59C2×C4⋊C4×C47C2×C411yescharacteristic
60C2×C4⋊C4×C47C2×C411yescharacteristic
61C2×C42×C47C2311yescharacteristic
62#95×C47—21no—
63C23×D47C22⋊C411yescharacteristic
64C22×C4×C47—21no—
65C22×C4×C47—21no—
66C22×C4×C47C22×C411yescharacteristic
67C22×C4×C47C2×D411yescharacteristic
68C22×C4×C47C2×D411yescharacteristic
69C22×Dic47—21no—
70C22×Dic47—21no—
71C22×Dic47C22⋊C411yescharacteristic
72C22×Dic47C22⋊C411yescharacteristic
73C22×Dic47C22⋊C411yescharacteristic
74C2×C4×D47—22no—
75C2×C4×D47—21no—
76C2×C4×D47—21no—
77C2×C4×D47—22no—
78C2×C4×D47—41no—
79C2×C4×D47—41no—
80C2×C4×D47—41no—
81C2×C4×D47—41no—
82C2×C4×D47C2×C812yes—
83C2×C4×D47M1612yes—
84C2×C4×D47SD1612yes—
85C2×C4×D47Q1612yes—
86C2×Dic94—21no—
87C2×Dic94—41no—
88C2×Dic94—41no—
89C2×Dic94C2×C812yescomplemented
90C2×C8×C47—22no—
91C2×C8×C47—41no—
92C2×C8×C47—41no—
93C2×C8×C47—41no—
94C2×C8×C47—41no—
95C2×C8×C47—41no—
96C4⋊C4×C47—21no—
97C4⋊C4×C47—41no—
98C4⋊C4×C47—41no—
99C4⋊C4×C47—41no—
100C4⋊C4×C47C2×C812yescomplemented
101C42×C47—41no—
102(C2×D47)⋊C4—24no—
103(C2×D47)⋊C4—42no—
104C4⋊Dic47—24no—
105C4⋊Dic47—41no—
106Dic47⋊C4—42no—
107C4×Dic47—22no—
108C4×Dic47—22no—
109C4×Dic47—41no—
110C4×Dic47—42no—
111C4×Dic47—42no—
112C2×C47⋊C8—22no—
113C2×C47⋊C8—41no—
114C2×C47⋊C8—41no—
115C2×C47⋊C8—41no—
116C2×C47⋊C8—41no—
117C2×C47⋊C8—41no—
118C8×D47—48no—
119C23×C47C2×C22⋊C411yescharacteristic
120C22×D47—21no—
121C22×D47—21no—
122C22×D47—21no—
123C22×D47—21no—
124C22×D47C22⋊C812yes—
125C22×D47#10312yes—
126C22×D47#10412yes—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—22no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47C22×C811yescharacteristic
140C2×C4×C47C2×M1611yescharacteristic
141C2×C4×C47C2×SD1611yescharacteristic
142C2×C4×C47C2×Q1611yescharacteristic
143C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C47#13711yescharacteristic
145C2×Dic47—21no—
146C2×Dic47—21no—
147C2×Dic47—21no—
148C2×Dic47—21no—
149C2×Dic47—21no—
150C2×Dic47—21no—
151C2×Dic47—22no—
152C2×Dic47—41no—
153C2×Dic47—41no—
154C2×Dic47—41no—
155C2×Dic47—41no—
156C2×Dic47—41no—
157C2×Dic47C22⋊C812yes—
158C2×Dic47#9912yes—
159C2×Dic47#10312yes—
160C4×D47—24no—
161C4×D47—24no—
162C4×D47—42no—
163C4×D47—42no—
164C4×D47—42no—
165C4×D47—42no—
166Dic94—42no—
167Dic94—42no—
168C8×C47—42no—
169C8×C47—81no—
170C47⋊C8—42no—
171C47⋊C8—81no—
172#7589—471nomaximal, Sylow
173C22×C47C2×C22⋊C811yescharacteristic
174C22×C47C2×#9911yescharacteristic
175C22×C47#41311yescharacteristic
176C22×C47C2×#10311yescharacteristic
177C22×C47#41711yescharacteristic
178C22×C47C2×#10411yescharacteristic
179C22×C47#42111yescharacteristic
180C2×D47—22no—
181C2×D47—22no—
182C2×D47—22no—
183C2×D47—22no—
184C2×D47—22no—
185C2×D47—22no—
186C2×D47Q8⋊C814yes—
187C4×C47—21no—
188C4×C47—21no—
189C4×C47—21no—
190C4×C47—21no—
191C4×C47—21no—
192C4×C47—21no—
193C4×C47—42no—
194C4×C47—81no—
195Dic47—22no—
196Dic47—22no—
197Dic47—22no—
198Dic47—42no—
199Dic47—81no—
200C4×C22⋊Q8—471no—
201#1797—471no—
202C2×#336—471no—
203#1272—471no—
204#1272—471no—
205#1271—471no—
206#1271D4711yescharacteristic, complemented
207C2×C47#145911yescharacteristic
208C2×C47C2×Q8⋊C811yescharacteristic
209C2×C47#147711yescharacteristic
210C2×C47#150411yescharacteristic
211C2×C47#152011yescharacteristic
212C2×C47#153211yescharacteristic
213C2×C47#155011yescharacteristic
214D47—24no—
215C23×C8—941no—
216C2×C22⋊Q8—471no—
217C2×C4×Q8—471no—
218C22×C42—471no—
219C2×C4⋊C8—941no—
220C2×C22⋊C8—941no—
221#391—471no—
222#389C2×D4711yescharacteristic
223#387—941no—
224#385—941no—
225#380—471no—
226C4×C4⋊C4—471no—
227C4×C4⋊C4—941no—
228C4×C22⋊C4—941no—
229#336—471no—
230#336—471no—
231#336—941no—
232#336C2×D4711yescharacteristic
233#336C2×D4711yescharacteristic
234C47#758911yescharacteristic, complemented, Sylow
235C22×Q8—471no—
236C23×C4—471no—
237C23×C4—471no—
238C23×C4—471no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—944no—
242C22×C8—941no—
243C22×C8—941no—
244C22×C8—1882no—
245C22×C8—1882no—
246C22⋊Q8—474no—
247C22⋊Q8—942no—
248C4×Q8—1882no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—471no—
251C2×C4⋊C4—941no—
252C2×C4⋊C4—941no—
253C2×C4⋊C4C4×D4711yescharacteristic
254C2×C4⋊C4C4×D4711yescharacteristic
255C2×C22⋊C4—941no—
256C2×C22⋊C4—941no—
257C2×C42—471no—
258C2×C42—471no—
259C2×C42—471no—
260C2×C42—941no—
261C2×C42—941no—
262C2×C42—1882no—
263C2×C42C22×D4711yescharacteristic
264C4⋊C8—1882no—
265C22⋊C8—1882no—
266#95—21no—
267#95—941no—
268#95—941no—
269#95—941no—
270#95—941no—
271C24—471no—
272C2×Q8—472no—
273C2×Q8—941no—
274C2×Q8—1881no—
275C2×Q8—1881no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—471no—
279C22×C4—471no—
280C22×C4—471no—
281C22×C4—472no—
282C22×C4—472no—
283C22×C4—472no—
284C22×C4—472no—
285C22×C4—942no—
286C22×C4—942no—
287C22×C4—941no—
288C22×C4—941no—
289C22×C4—941no—
290C22×C4—941no—
291C22×C4—1881no—
292C22×C4—1881no—
293C22×C4—1881no—
294C22×C4—1881no—
295C22×C4C2×C4×D4711yescharacteristic
296C22×C4D4×D4711yescharacteristic
297C22×C4D4×D4711yescharacteristic
298C2×C8—22no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—942no—
305C2×C8—1888no—
306C2×C8—1881no—
307C2×C8—1881no—
308C2×C8—1881no—
309C2×C8—1881no—
310C2×C8—1881no—
311C4⋊C4—21no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—944no—
316C4⋊C4—1881no—
317C4⋊C4—1882no—
318C4⋊C4C8×D4712yescomplemented
319C22⋊C4—944no—
320C22⋊C4—1882no—
321C42—41no—
322C42—942no—
323C42—942no—
324C42—1881no—
325C42—1882no—
326C42—1882no—
327C23—472no—
328C23—472no—
329C23—472no—
330C23—941no—
331C23—941no—
332C23—941no—
333C23—941no—
334C23C22⋊C4×D4711yescharacteristic
335Q8—1882no—
336Q8—1882no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—472no—
350C2×C4—472no—
351C2×C4—472no—
352C2×C4—944no—
353C2×C4—944no—
354C2×C4—941no—
355C2×C4—941no—
356C2×C4—941no—
357C2×C4—941no—
358C2×C4—941no—
359C2×C4—941no—
360C2×C4—942no—
361C2×C4—1882no—
362C2×C4—1882no—
363C2×C4—1882no—
364C2×C4—1882no—
365C2×C4—1881no—
366C2×C4—1881no—
367C2×C4—1881no—
368C2×C4—1881no—
369C2×C4—1881no—
370C2×C4C2×C8×D4711yescharacteristic
371C2×C4M16×D4711yescharacteristic
372C2×C4SD16×D4711yescharacteristic
373C2×C4Q16×D4711yescharacteristic
374C2×C4#1309404811yescharacteristic
375C2×C4#1309405311yescharacteristic
376C8—42no—
377C8—81no—
378C8—3761no—
379C8—1882no—
380C22—474no—
381C22—942no—
382C22—942no—
383C22—942no—
384C22—942no—
385C22—942no—
386C22—942no—
387C22C22⋊C8×D4711yescharacteristic
388C22#99×D4711yescharacteristic
389C22#1568910611yescharacteristic
390C22#103×D4711yescharacteristic
391C22#1568912111yescharacteristic
392C22#104×D4711yescharacteristic
393C22#1568913811yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—42no—
401C4—81no—
402C4—942no—
403C4—942no—
404C4—942no—
405C4—3761no—
406C4—1882no—
407C2—944no—
408C2#2024170511yescharacteristic
409C2Q8⋊C8×D4711yescharacteristic
410C2#2024177211yescharacteristic
411C2#2024188911yescharacteristic
412C2#2024195811yescharacteristic
413C2#2024200711yescharacteristic
414C2#2024207711yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC2×C4×C47C22×C8
FrattiniC2×C42C22×D47
FittingC47×#1271C2
socleC23×C47C2×C22⋊C4
prime radicalC23×D47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1271D47
hypocenterC47#7589
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
4. #20241213C2semidirect
5. #20241213C2semidirect
6. #20241211C2semidirect
7. C47×#1271C2semidirect
37. #13094004C8semidirect
89. C2×Dic94C2×C8semidirect
100. C4⋊C4×C47C2×C8semidirect
206. #1271D47semidirect
234. C47#7589semidirect
318. C4⋊C4C8×D47semidirect

Maximal subgroups

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

subgroupindex
#202459182
#202433672
#336×D472
#202412132
#202412112
C47×#12712
#758947