Finite Group Databasebeta

← #27080669

Subgroups — #27080669

Order 12032.

1810 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256791C211yesmaximal, characteristic
2#20252328C211yesmaximal, characteristic, complemented
3#20252326C212yesmaximal, complemented
4#20252324C218yesmaximal, complemented
5#20252323C211yesmaximal, characteristic, complemented
6#20252321C212yesmaximal, complemented
7#20252319C218yesmaximal, complemented
8#20252318C211yesmaximal, characteristic, complemented
9#20251950C211yesmaximal, characteristic
10#20251949C212yesmaximal
11C2×#15689080C211yesmaximal, characteristic
12#405×D47C212yesmaximal
13C47×#2961C211yesmaximal, characteristic, complemented
14C22×#10625324C2211yescharacteristic
15C2×M16×D47C2212yes—
16C22×C8×D47C2211yescharacteristic
17#15689753C414yes—
18#15689752C414yes—
19#15689750C414yes—
20#15689748C414yes—
21#15689680C2211yescharacteristic
22#15689677C2218yes—
23#15689676C2212yes—
24C2×#13093981C2211yescharacteristic
25#117×D47C2212yes—
26#15689254C2214yescomplemented
27#15689253C2218yescomplemented
28#15689252C2218yescomplemented
29#15689251C2218yescomplemented
30#15689251C2218yescomplemented
31#15689250C2214yescomplemented
32Q8×C47⋊C8C2214yescomplemented
33#15689247C2214yescomplemented
34#15689247C2218yescomplemented
35#15689151C2212yes—
36#15689150C2212yes—
37#15689149C2212yes—
38#15689148C2218yes—
39#15689148C2218yes—
40#15689147C2211yescharacteristic
41#15689146C2212yes—
42#15689145C2218yes—
43#15689081C2212yes—
44#15689080C2214yes—
45#15689080C2214yes—
46#15689080C2214yes—
47#15689079C2212yes—
48#15689078C2212yes—
49#15689077C2214yes—
50C2×#13093934C2211yescharacteristic
51#97×D47C2214yes—
52C2×C4×C47⋊C8C2211yescharacteristic
53C2×C8×Dic47C2211yescharacteristic
54C4×C8×D47C2214yes—
55C47×#533C2211yescharacteristic
56C47×#446C2218yescomplemented
57C47×#424C2212yes—
58C47×#423C2211yescharacteristic
59C47×#405C2212yes—
60C2×#97×C47C2211yescharacteristic
61C22×C4×D47C2311yescharacteristic
62C2×#10625329C2312yes—
63C2×#10625324—24no—
64C2×#10625324C2314yes—
65C2×#10625324C2314yes—
66M16×D47—28no—
67C22×C47⋊C8C2311yescharacteristic
68C22×C47⋊C8C2311yescharacteristic
69C2×C8×D47Q818yescomplemented
70C2×C8×D47C2314yes—
71C2×C8×D47C2312yes—
72C2×C8×D47C2312yes—
73#13094024C2×C412yes—
74#13094023C2×C414yes—
75#13094023C2×C418yes—
76#13094022C2×C418yes—
77#13094016C2×C412yes—
78#13094015C2×C414yes—
79#13094014C2×C418yes—
80#13094014C2×C418yes—
81#13094004C2×C418yes—
82#13094003C2×C414yes—
83#13094002C2×C418yes—
84#13094002C2×C418yes—
85#13094001C2×C418yes—
86#13094000C2×C418yes—
87C4×Dic94C2314yes—
88C4×Dic94C2318yes—
89#13093983C2312yes—
90#13093982C2×C414yes—
91#13093981C2×C412yes—
92#13093981C2×C412yes—
93#13093981C2314yes—
94#13093981C2314yes—
95#13093980C2×C414yes—
96C2×Dic47⋊C4C2×C412yes—
97C4⋊C4×D47C2×C414yes—
98C2×(C2×D47)⋊C4C2×C412yes—
99C22⋊C4×D47C2×C414yes—
100C2×C4×Dic47C2311yescharacteristic
101C42×D47C2314yes—
102C4⋊(C47⋊C8)C2314yes—
103C4⋊(C47⋊C8)C2318yes—
104Dic47⋊C8C2318yes—
105Dic47⋊C8C2318yes—
106Dic47⋊C8C2318yes—
107#13093935C2314yes—
108#13093934C2314yes—
109#13093934C2314yes—
110#13093934C2314yes—
111C4×C47⋊C8C2314yes—
112C4×C47⋊C8C2312yes—
113C4×C47⋊C8C2312yes—
114C8×Dic47C2314yes—
115C8×Dic47C2312yes—
116C8×Dic47C2314yes—
117C8×Dic47C2312yes—
118C2×M16×C47C2312yes—
119C22×C8×C47C2311yescharacteristic
120C22×C8×C47C2311yescharacteristic
121C4⋊Q8×C47C2×C412yes—
122#125×C47C2×C412yes—
123C22⋊Q8×C47C2×C414yes—
124C4×Q8×C47C2314yes—
125#117×C47C2312yes—
126C2×C42×C47C2311yescharacteristic
127C4⋊C8×C47C2314yes—
128C4⋊C8×C47C2318yes—
129#97×C47C2314yes—
130#97×C47C2314yes—
131C4×C8×C47C2314yes—
132C23×D47C22×C411yescharacteristic
133C2×Q8×C47C22×C414yes—
134C22×C4×C47C22×C412yes—
135C22×C4×C47C2411yescharacteristic
136C22×Dic47C22×C412yes—
137C22×Dic47C22×C411yescharacteristic
138C2×C4×D47C22×C418yes—
139C2×C4×D47C22×C412yes—
140C2×C4×D47C2×Q814yes—
141C2×C4×D47#4114yes—
142C2×C4×D47C2412yes—
143C2×Dic94C22×C418yes—
144C2×Dic94C22×C414yes—
145M16×C47—24no—
146C2×C8×C47—22no—
147C2×C8×C47C2×Q814yes—
148C2×C8×C47C2412yes—
149C2×C8×C47C2414yes—
150C2×C8×C47C2412yes—
151C4⋊C4×C47C22×C414yes—
152C4⋊C4×C47C22×C418yes—
153C4⋊C4×C47C22×C414yes—
154C22⋊C4×C47C22×C414yes—
155C42×C47C22×C411yescharacteristic
156C42×C47C22×C411yescharacteristic
157C42×C47C2412yes—
158C42×C47C2414yes—
159C22⋊Dic47C22×C414yes—
160(C2×D47)⋊C4C22×C418yes—
161(C2×D47)⋊C4C22×C414yes—
162(C2×D47)⋊C4C22×C414yes—
163C4⋊Dic47C22×C414yes—
164C4⋊Dic47C22×C418yes—
165C4⋊Dic47C22×C414yes—
166Dic47⋊C4C22×C418yes—
167Dic47⋊C4C22×C414yes—
168Dic47⋊C4C22×C414yes—
169Dic47⋊C4C22×C418yes—
170Dic47⋊C4C22×C418yes—
171Dic47⋊C4C22×C418yes—
172C4×Dic47C22×C418yes—
173C4×Dic47C22×C412yes—
174C4×Dic47C2414yes—
175C4×Dic47C2412yes—
176#10625329—24no—
177C2×C47⋊C8—22no—
178C2×C47⋊C8C2×Q814yes—
179C2×C47⋊C8C2414yes—
180C2×C47⋊C8C2412yes—
181C2×C47⋊C8C2411yescharacteristic
182C2×C47⋊C8C2411yescharacteristic
183#10625324—28no—
184#10625324—28no—
185C8×D47—28no—
186C8×D47C2×Q818yes—
187C8×D47C2×Q818yes—
188C23×C47C23×C411yescharacteristic
189C22×D47—42no—
190C22×D47C4×Q814yes—
191C22×D47C23×C412yes—
192Q8×C47—44no—
193C2×C4×C47—42no—
194C2×C4×C47C23×C418yes—
195C2×C4×C47C23×C412yes—
196C2×C4×C47C23×C412yes—
197C2×C4×C47C23×C411yescharacteristic
198C2×C4×C47C22×Q812yes—
199C2×C4×C47C2×#4112yes—
200C2×C4×C47C2511yescharacteristic
201C2×Dic47—42no—
202C2×Dic47C4×Q814yes—
203C2×Dic47C23×C414yes—
204C2×Dic47C23×C418yes—
205C2×Dic47C23×C412yes—
206C4×D47—48no—
207C4×D47C4×Q818yes—
208C4×D47C22×Q814yes—
209C4×D47C2×#4114yes—
210Dic94—44no—
211Dic94—48no—
212C8×C47—24no—
213C8×C47—22no—
214C8×C47C22×Q814yes—
215C47⋊C8—24no—
216C47⋊C8—22no—
217C47⋊C8C22×Q812yes—
218C47⋊C8C22×Q812yes—
219#60191—471nomaximal, Sylow
220C22×C47—22no—
221C22×C47C2×C4×Q812yes—
222C22×C47C24×C411yescharacteristic
223C2×D47—24no—
224C2×D47—44no—
225C2×D47C2×C4×Q814yes—
226C4×C47—22no—
227C4×C47—28no—
228C4×C47—22no—
229C4×C47C2×C4×Q812yes—
230C4×C47C23×Q811yescharacteristic
231C4×C47C22×#4111yescharacteristic
232Dic47—28no—
233Dic47—24no—
234Dic47C2×C4×Q814yes—
235#3458—471no—
236#2961—478no—
237#2961—472no—
238#2961—471no—
239#2961D4711yescharacteristic, complemented
240#2960—478no—
241#2960—472no—
242#2960—471no—
243#2960—471no—
244#2905—472no—
245#2905—471no—
246C2×#405—472no—
247C2×#405—471no—
248C2×C47—22no—
249C2×C47C22×C4×Q811yescharacteristic
250C2×C47#357312yes—
251D47—24no—
252C22×M16—472no—
253C22×M16—471no—
254C23×C8—471no—
255#544—474no—
256#544—474no—
257#544—474no—
258#543—474no—
259#533—478no—
260#533—472no—
261#533—471no—
262#533C2×D4711yescharacteristic
263C2×#117—472no—
264C2×#117—471no—
265#446—474no—
266#446—474no—
267#446—478no—
268#446—478no—
269#446—478no—
270#446—478no—
271#446C2×D4718yescomplemented
272C8×Q8—474no—
273C8×Q8—478no—
274C8×Q8—474no—
275#424—472no—
276#424—472no—
277#424—472no—
278#424—478no—
279#424—478no—
280#424C2×D4712yes—
281#423—472no—
282#423—471no—
283#423—478no—
284#423C2×D4711yescharacteristic
285#405—472no—
286#405—472no—
287#405—472no—
288#405—474no—
289#405—474no—
290#405—474no—
291#405—474no—
292#405C2×D4712yes—
293C2×#97—471no—
294C2×#97—474no—
295C2×#97C2×D4711yescharacteristic
296C2×C4×C8—471no—
297C2×C4×C8—471no—
298C2×C4×C8—474no—
299C47#6019111yescharacteristic, complemented, Sylow
300C23×C4—471no—
301C2×M16—472no—
302C2×M16—474no—
303C2×M16—474no—
304C2×M16—944no—
305C2×M16—948no—
306C2×M16C22×D4712yes—
307C22×C8—471no—
308C22×C8—478no—
309C22×C8—472no—
310C22×C8—472no—
311C22×C8—471no—
312C22×C8—474no—
313C22×C8C22×D4711yescharacteristic
314C22×C8C22×D4711yescharacteristic
315C4⋊Q8—478no—
316C4⋊Q8—478no—
317C4⋊Q8—474no—
318C4⋊Q8—472no—
319C4⋊Q8C4×D4712yes—
320#125—478no—
321#125—478no—
322#125—474no—
323#125—472no—
324#125C4×D4712yes—
325C22⋊Q8—478no—
326C22⋊Q8—474no—
327C22⋊Q8—478no—
328C22⋊Q8—478no—
329C22⋊Q8—478no—
330C22⋊Q8—478no—
331C22⋊Q8C4×D4714yes—
332C4×Q8—474no—
333C4×Q8—478no—
334C4×Q8C22×D4714yes—
335#117—472no—
336#117—474no—
337#117—474no—
338#117—472no—
339#117—472no—
340#117—474no—
341#117—474no—
342#117C22×D4712yes—
343C2×C4⋊C4—472no—
344C2×C4⋊C4—474no—
345C2×C22⋊C4—472no—
346C2×C22⋊C4—474no—
347C2×C42—471no—
348C2×C42—474no—
349C2×C42C22×D4711yescharacteristic
350C4⋊C8—478no—
351C4⋊C8—478no—
352C4⋊C8—478no—
353C4⋊C8—474no—
354C4⋊C8—478no—
355C4⋊C8C22×D4718yes—
356C4⋊C8C22×D4714yes—
357#97—474no—
358#97—474no—
359#97—474no—
360#97—474no—
361#97C22×D4714yes—
362#97C22×D4714yes—
363C4×C8—472no—
364C4×C8—474no—
365C4×C8—472no—
366C4×C8—474no—
367C4×C8—472no—
368C4×C8—474no—
369C4×C8—472no—
370C4×C8C22×D4714yes—
371C24—471no—
372C2×Q8—478no—
373C2×Q8—474no—
374C2×Q8C2×C4×D4714yes—
375C22×C4—471no—
376C22×C4—472no—
377C22×C4—474no—
378C22×C4—474no—
379C22×C4—472no—
380C22×C4—472no—
381C22×C4—478no—
382C22×C4C2×C4×D4712yes—
383C22×C4C23×D4711yescharacteristic
384M16—24no—
385M16—948no—
386M16—948no—
387M16—944no—
388C2×C8—22no—
389C2×C8—474no—
390C2×C8—471no—
391C2×C8—471no—
392C2×C8—478no—
393C2×C8—478no—
394C2×C8—474no—
395C2×C8—472no—
396C2×C8—948no—
397C2×C8—942no—
398C2×C8Q8×D4714yes—
399C2×C8C23×D4712yes—
400C2×C8C23×D4714yes—
401C2×C8C23×D4712yes—
402C4⋊C4—478no—
403C4⋊C4—474no—
404C4⋊C4—474no—
405C4⋊C4—478no—
406C4⋊C4—478no—
407C4⋊C4—478no—
408C4⋊C4—474no—
409C4⋊C4—478no—
410C4⋊C4—474no—
411C4⋊C4C2×C4×D4714yes—
412C4⋊C4C2×C4×D4714yes—
413C4⋊C4C2×C4×D4718yes—
414C22⋊C4—474no—
415C22⋊C4—478no—
416C22⋊C4—474no—
417C22⋊C4—474no—
418C22⋊C4C2×C4×D4714yes—
419C42—478no—
420C42—472no—
421C42—474no—
422C42—472no—
423C42C2×C4×D4711yescharacteristic
424C42C2×C4×D4711yescharacteristic
425C42C23×D4712yes—
426C42C23×D4714yes—
427C23—472no—
428C23—474no—
429C23—1882no—
430C23C22×C4×D4711yescharacteristic
431Q8—44no—
432Q8—1884no—
433Q8—1888no—
434C2×C4—42no—
435C2×C4—474no—
436C2×C4—472no—
437C2×C4—478no—
438C2×C4—474no—
439C2×C4—474no—
440C2×C4—474no—
441C2×C4—478no—
442C2×C4—1888no—
443C2×C4—1882no—
444C2×C4C22×C4×D4718yes—
445C2×C4C22×C4×D4712yes—
446C2×C4C22×C4×D4712yes—
447C2×C4C22×C4×D4711yescharacteristic
448C2×C4C2×Q8×D4712yes—
449C2×C4#41×D4712yes—
450C2×C4C24×D4711yescharacteristic
451C8—22no—
452C8—24no—
453C8—472no—
454C8—472no—
455C8—944no—
456C8—942no—
457C8C2×Q8×D4714yes—
458C22—22no—
459C22—474no—
460C22—944no—
461C22—1884no—
462C22C4×Q8×D4712yes—
463C22C23×C4×D4711yescharacteristic
464C4—22no—
465C4—22no—
466C4—28no—
467C4—474no—
468C4—948no—
469C4—944no—
470C4C4×Q8×D4712yes—
471C4C22×Q8×D4711yescharacteristic
472C4C2×#41×D4711yescharacteristic
473C2—22no—
474C2—944no—
475C2C2×C4×Q8×D4711yescharacteristic
476C2#2025810112yes—
subgroupG/H
center, FrattiniC2×C4C24×D47
commutator, socleC22×C47C24×C4
FittingC47×#2961C2
prime radicalC23×D47C22×C4
cosocleC2×C4×C47C25
hypercenter#2961D47
hypocenterC47#60191
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ #2961

Complements

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

NH ≅ G/Ntype
2. #20252328C2semidirect
3. #20252326C2semidirect
4. #20252324C2semidirect
5. #20252323C2semidirect
6. #20252321C2semidirect
7. #20252319C2semidirect
8. #20252318C2semidirect
13. C47×#2961C2semidirect
26. #15689254C22semidirect
27. #15689253C22semidirect
28. #15689252C22semidirect
29. #15689251C22semidirect
30. #15689251C22semidirect
31. #15689250C22semidirect
32. Q8×C47⋊C8C22semidirect
33. #15689247C22semidirect
34. #15689247C22semidirect
56. C47×#446C22semidirect
69. C2×C8×D47Q8semidirect
239. #2961D47semidirect
271. #446C2×D47semidirect
299. C47#60191semidirect

Maximal subgroups

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

subgroupindex
#202567912
#202523282
#202523262
#202523242
#202523232
#202523212
#202523192
#202523182
#202519502
#202519492
C2×#156890802
#405×D472
C47×#29612
#6019147