Finite Group Databasebeta

← #26184018

Subgroups — #26184018

Order 12032.

1018 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252412C211yesmaximal, characteristic, complemented
2#20246361C211yesmaximal, characteristic, complemented
3#20241952C211yesmaximal, characteristic, complemented
4#20241948C211yesmaximal, characteristic, complemented
5#20241944C214yesmaximal, complemented
6#20241887C211yesmaximal, characteristic, complemented
7#20241883C211yesmaximal, characteristic, complemented
8#20241879C214yesmaximal, complemented
9C47×#3506C211yesmaximal, characteristic
10D4×C47⋊C8—22no—
11D4×C47⋊C8—22no—
12#15689176C2211yescharacteristic
13#15689176C2211yescharacteristic
14C2×C22⋊(C47⋊C8)C2214yes—
15#15689026—21no—
16#15689016—22no—
17#15688989C2211yescharacteristic
18#15688986C2212yes—
19C4⋊C4×Dic47—21no—
20C4×C4⋊Dic47C2211yescharacteristic
21C4×C22⋊Dic47C2212yes—
22#15688860C2212yes—
23#15688860C2214yes—
24#15688860C2212yes—
25#15688857C2212yes—
26#15688857C2214yes—
27#15688857C2212yes—
28C47×#559—22no—
29C47×#556C412yescomplemented
30C47×#555C412yescomplemented
31C47×#553—22no—
32C47×#550C412yescomplemented
33C47×#547C412yescomplemented
34C47×#538—22no—
35C47×#535—21no—
36C47×#533C2211yescharacteristic
37C47×#532C2211yescharacteristic
38C47×#525C2214yes—
39C47×#520—21no—
40C2×C4×D4×C47C2211yescharacteristic
41C22×C47⋊C8—24no—
42C2×C4⋊Dic47—21no—
43C2×C4⋊Dic47—21no—
44C2×C22⋊Dic47—22no—
45C2×C22⋊Dic47—22no—
46C2×C4×Dic47—21no—
47C2×C4×Dic47—22no—
48C4⋊(C47⋊C8)—21no—
49C4⋊(C47⋊C8)—21no—
50C22⋊(C47⋊C8)—24no—
51C22⋊(C47⋊C8)C2312yes—
52C22⋊(C47⋊C8)C2312yes—
53C4×C47⋊C8—21no—
54C4×C47⋊C8—21no—
55C47⋊#95—22no—
56C47⋊#95—21no—
57C47⋊#95C2311yescharacteristic
58C47⋊#95C2312yes—
59C47⋊#95C2311yescharacteristic
60C2×#41×C47—21no—
61C2×#41×C47C2×C414yes—
62C22×D4×C47D411yescharacteristic
63C23×C4×C47C2312yes—
64C4⋊Q8×C47C2×C411yescharacteristic
65C4⋊Q8×C47C2×C411yescharacteristic
66C4⋊Q8×C47C2×C411yescharacteristic
67#127×C47—22no—
68#127×C47C2×C411yescharacteristic
69#126×C47—24no—
70#125×C47—21no—
71#124×C47—22no—
72#124×C47—24no—
73#124×C47—21no—
74#123×C47—24no—
75#123×C47—24no—
76C22⋊Q8×C47—21no—
77C22⋊Q8×C47C2×C414yes—
78C22⋊Q8×C47C2312yes—
79#121×C47—22no—
80#121×C47—22no—
81#121×C47—21no—
82#121×C47—24no—
83#121×C47C2×C414yes—
84#121×C47C2312yes—
85#120×C47—24no—
86C4×Q8×C47—22no—
87C4×Q8×C47—21no—
88C4×Q8×C47C2×C412yes—
89C4×D4×C47—24no—
90C4×D4×C47—24no—
91C4×D4×C47—22no—
92C4×D4×C47—21no—
93C4×D4×C47C2×C412yes—
94C4×D4×C47C2×C412yes—
95C4×D4×C47C2×C412yes—
96C4×D4×C47D412yes—
97C4×D4×C47D412yes—
98#117×C47—21no—
99#117×C47—22no—
100#117×C47C2×C414yes—
101C2×C4⋊C4×C47—22no—
102C2×C4⋊C4×C47D411yescharacteristic
103C2×C22⋊C4×C47D412yes—
104C2×C42×C47C2311yescharacteristic
105C24×C47C2×D412yes—
106#41×C47—42no—
107#41×C47—42no—
108#41×C47—84no—
109C2×Q8×C47—21no—
110C2×Q8×C47—21no—
111C2×Q8×C47C22×C412yes—
112C2×D4×C47—22no—
113C2×D4×C47—24no—
114C2×D4×C47—24no—
115C2×D4×C47—21no—
116C2×D4×C47—21no—
117C2×D4×C47—22no—
118C2×D4×C47—22no—
119C2×D4×C47C22⋊C412yes—
120C2×D4×C47C22⋊C412yes—
121C2×D4×C47C22×C412yes—
122C22×C4×C47—22no—
123C22×C4×C47—24no—
124C22×C4×C47—82no—
125C22×C4×C47C22×C414yes—
126C22×C4×C47C2×D411yescharacteristic
127C22×C4×C47C2×D412yes—
128C22×C4×C47C2×D414yes—
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2411yescharacteristic
131C22×Dic47—21no—
132C22×Dic47—22no—
133C22×Dic47—21no—
134C4⋊C4×C47—21no—
135C4⋊C4×C47—24no—
136C4⋊C4×C47—21no—
137C4⋊C4×C47—21no—
138C4⋊C4×C47—21no—
139C4⋊C4×C47—22no—
140C4⋊C4×C47—22no—
141C4⋊C4×C47C22⋊C412yes—
142C4⋊C4×C47C22⋊C412yes—
143C4⋊C4×C47C22×C412yes—
144C4⋊C4×C47C22×C412yes—
145C22⋊C4×C47—24no—
146C22⋊C4×C47—24no—
147C22⋊C4×C47—22no—
148C22⋊C4×C47—24no—
149C22⋊C4×C47—22no—
150C22⋊C4×C47—22no—
151C22⋊C4×C47C22⋊C414yes—
152C22⋊C4×C47C22⋊C414yes—
153C42×C47—24no—
154C42×C47—22no—
155C42×C47C22×C411yescharacteristic
156C42×C47C22×C411yescharacteristic
157C42×C47C2×D411yescharacteristic
158C42×C47C2×D411yescharacteristic
159C22⋊Dic47—44no—
160C22⋊Dic47—82no—
161C4⋊Dic47—42no—
162C4⋊Dic47—42no—
163C4×Dic47—42no—
164C4×Dic47—82no—
165C2×C47⋊C8—22no—
166C2×C47⋊C8—22no—
167C2×C47⋊C8—84no—
168C23×C47—22no—
169C23×C47—22no—
170C23×C47—22no—
171C23×C47—42no—
172C23×C47—42no—
173C23×C47C2×C22⋊C414yes—
174C23×C47C22×D411yescharacteristic
175Q8×C47—42no—
176Q8×C47—42no—
177Q8×C47—81no—
178D4×C47—24no—
179D4×C47—24no—
180D4×C47—42no—
181D4×C47—42no—
182D4×C47—42no—
183D4×C47—42no—
184D4×C47—81no—
185D4×C47—84no—
186D4×C47—84no—
187C2×C4×C47—22no—
188C2×C4×C47—22no—
189C2×C4×C47—22no—
190C2×C4×C47—44no—
191C2×C4×C47—42no—
192C2×C4×C47—41no—
193C2×C4×C47—41no—
194C2×C4×C47—44no—
195C2×C4×C47—44no—
196C2×C4×C47—84no—
197C2×C4×C47C2×C22⋊C414yes—
198C2×C4×C47C2×C22⋊C411yescharacteristic
199C2×C4×C47C2×C22⋊C411yescharacteristic
200C2×C4×C47C2×C22⋊C411yescharacteristic
201C2×C4×C47C2×C22⋊C411yescharacteristic
202C2×C4×C47#13512yes—
203C2×C4×C47#13512yes—
204C2×C4×C47C23×C411yescharacteristic
205C2×C4×C47C22×D411yescharacteristic
206C2×Dic47—22no—
207C2×Dic47—41no—
208C2×Dic47—41no—
209C2×Dic47—42no—
210C2×Dic47—41no—
211C2×Dic47—42no—
212C2×Dic47—41no—
213C2×Dic47—41no—
214C2×Dic47—42no—
215C47⋊C8—42no—
216C47⋊C8—42no—
217#17617—471nomaximal, Sylow
218C22×C47—21no—
219C22×C47—21no—
220C22×C47—24no—
221C22×C47—24no—
222C22×C47—24no—
223C22×C47—42no—
224C22×C47—42no—
225C22×C47—42no—
226C22×C47—82no—
227C22×C47—82no—
228C22×C47C22×C22⋊C411yescharacteristic
229C22×C47C2×#13511yescharacteristic
230C22×C47C2×#13511yescharacteristic
231C4×C47—21no—
232C4×C47—21no—
233C4×C47—24no—
234C4×C47—21no—
235C4×C47—21no—
236C4×C47—42no—
237C4×C47—42no—
238C4×C47—42no—
239C4×C47#41612yes—
240C4×C47#41612yes—
241Dic47—42no—
242Dic47—81no—
243Dic47—82no—
244#3506D4711yescharacteristic
245#2972—471no—
246#2357—471no—
247#1518—471no—
248#1517—471no—
249#1516—474no—
250#1503—471no—
251#1502—471no—
252#1501—474no—
253C2×C47—21no—
254C2×C47—21no—
255C2×C47—44no—
256C2×C47—82no—
257C2×C47#288311yescharacteristic
258C2×C47C2×#41611yescharacteristic
259C2×C47#290111yescharacteristic
260#559—22no—
261#556Dic4712yescomplemented
262#555Dic4712yescomplemented
263#553—22no—
264#550Dic4712yescomplemented
265#547Dic4712yescomplemented
266#538—22no—
267#535—21no—
268#533C2×D4711yescharacteristic
269#532C2×D4711yescharacteristic
270#525C2×D4714yes—
271#520—21no—
272C2×C4×D4C2×D4711yescharacteristic
273C8×D4—942no—
274C8×D4—942no—
275#431—471no—
276#431—471no—
277C2×C22⋊C8—474no—
278#390—941no—
279#388—942no—
280#382—471no—
281#381—472no—
282C4×C4⋊C4—471no—
283C4×C4⋊C4—941no—
284C4×C22⋊C4—472no—
285#328—472no—
286#328—474no—
287#328—472no—
288#327—472no—
289#327—474no—
290#327—472no—
291C47#1761711yescharacteristic, complemented, Sylow
292C2×#41—21no—
293C2×#41C2×Dic4714yes—
294C22×D4C22⋊D4711yescharacteristic
295C23×C4C22×D4712yes—
296C22×C8—944no—
297C4⋊Q8C2×Dic4711yescharacteristic
298C4⋊Q8C2×Dic4711yescharacteristic
299C4⋊Q8C2×Dic4711yescharacteristic
300#127—22no—
301#127C2×Dic4711yescharacteristic
302#126—24no—
303#125—21no—
304#124—22no—
305#124—24no—
306#124—21no—
307#123—24no—
308#123—24no—
309C22⋊Q8—21no—
310C22⋊Q8C2×Dic4714yes—
311C22⋊Q8C22×D4712yes—
312#121—22no—
313#121—22no—
314#121—21no—
315#121—24no—
316#121C2×Dic4714yes—
317#121C22×D4712yes—
318#120—24no—
319C4×Q8—22no—
320C4×Q8—21no—
321C4×Q8C2×Dic4712yes—
322C4×D4—24no—
323C4×D4—24no—
324C4×D4—22no—
325C4×D4—21no—
326C4×D4C2×Dic4712yes—
327C4×D4C2×Dic4712yes—
328C4×D4C2×Dic4712yes—
329C4×D4C22⋊D4712yes—
330C4×D4C22⋊D4712yes—
331#117—21no—
332#117—22no—
333#117C2×Dic4714yes—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—941no—
336C2×C4⋊C4—941no—
337C2×C4⋊C4C22⋊D4711yescharacteristic
338C2×C22⋊C4—942no—
339C2×C22⋊C4—942no—
340C2×C22⋊C4C22⋊D4712yes—
341C2×C42—942no—
342C2×C42—941no—
343C2×C42C22×D4711yescharacteristic
344C4⋊C8—941no—
345C4⋊C8—941no—
346C22⋊C8—472no—
347C22⋊C8—472no—
348C22⋊C8—944no—
349C4×C8—941no—
350C4×C8—941no—
351#95—471no—
352#95—472no—
353#95—471no—
354#95—942no—
355#95—941no—
356C24C2×C22⋊D4712yes—
357#41—42no—
358#41—42no—
359#41—84no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8C22×Dic4712yes—
363C2×D4—22no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4C22⋊Dic4712yes—
371C2×D4C22⋊Dic4712yes—
372C2×D4C22×Dic4712yes—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—82no—
376C22×C4—941no—
377C22×C4—942no—
378C22×C4—941no—
379C22×C4C22×Dic4714yes—
380C22×C4C2×C22⋊D4712yes—
381C22×C4C2×C22⋊D4711yescharacteristic
382C22×C4C2×C22⋊D4714yes—
383C22×C4C2×C22⋊D4711yescharacteristic
384C22×C4C23×D4711yescharacteristic
385C2×C8—942no—
386C2×C8—942no—
387C2×C8—3764no—
388C4⋊C4—21no—
389C4⋊C4—24no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—1882no—
396C4⋊C4—1882no—
397C4⋊C4C22⋊Dic4712yes—
398C4⋊C4C22⋊Dic4712yes—
399C4⋊C4C22×Dic4712yes—
400C4⋊C4C22×Dic4712yes—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—22no—
404C22⋊C4—24no—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—3762no—
408C22⋊C4—1884no—
409C22⋊C4C22⋊Dic4714yes—
410C22⋊C4C22⋊Dic4714yes—
411C42—24no—
412C42—22no—
413C42—3762no—
414C42—1882no—
415C42C22×Dic4711yescharacteristic
416C42C22×Dic4711yescharacteristic
417C42C2×C22⋊D4711yescharacteristic
418C42C2×C22⋊D4711yescharacteristic
419C23—22no—
420C23—22no—
421C23—22no—
422C23—42no—
423C23—42no—
424C23C2×C22⋊Dic4714yes—
425C23C22×C22⋊D4711yescharacteristic
426Q8—42no—
427Q8—42no—
428Q8—81no—
429D4—24no—
430D4—24no—
431D4—42no—
432D4—42no—
433D4—42no—
434D4—42no—
435D4—81no—
436D4—84no—
437D4—84no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—44no—
444C2×C4—41no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—84no—
448C2×C4—942no—
449C2×C4—1882no—
450C2×C4—1881no—
451C2×C4—1882no—
452C2×C4—1881no—
453C2×C4—1881no—
454C2×C4—1882no—
455C2×C4—1881no—
456C2×C4—1881no—
457C2×C4C2×C22⋊Dic4714yes—
458C2×C4C2×C22⋊Dic4711yescharacteristic
459C2×C4C2×C22⋊Dic4711yescharacteristic
460C2×C4C2×C22⋊Dic4711yescharacteristic
461C2×C4C2×C22⋊Dic4711yescharacteristic
462C2×C4#1309404412yes—
463C2×C4#1309404412yes—
464C2×C4C23×Dic4711yescharacteristic
465C2×C4C22×C22⋊D4711yescharacteristic
466C8—1882no—
467C8—1882no—
468C22—21no—
469C22—21no—
470C22—24no—
471C22—24no—
472C22—24no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—82no—
477C22—82no—
478C22C22×C22⋊Dic4711yescharacteristic
479C22C2×#1309404411yescharacteristic
480C22C2×#1309404411yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—24no—
486C4—42no—
487C4—42no—
488C4—42no—
489C4—3762no—
490C4—3761no—
491C4—1882no—
492C4#1568911912yes—
493C4#1568911912yes—
494C2—21no—
495C2—21no—
496C2—44no—
497C2—82no—
498C2#2025184911yescharacteristic
499C2C2×#1568911911yescharacteristic
500C2#2025193511yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
Fitting, prime radicalC47×#3506C2
socleC22×C47C22×C22⋊C4
cosocleC22×C4×C47C24
hypercenter#3506D47
hypocenterC47#17617
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #3506

Complements

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

NH ≅ G/Ntype
1. #20252412C2semidirect
2. #20246361C2semidirect
3. #20241952C2semidirect
4. #20241948C2semidirect
5. #20241944C2semidirect
6. #20241887C2semidirect
7. #20241883C2semidirect
8. #20241879C2semidirect
29. C47×#556C4semidirect
30. C47×#555C4semidirect
32. C47×#550C4semidirect
33. C47×#547C4semidirect
261. #556Dic47semidirect
262. #555Dic47semidirect
264. #550Dic47semidirect
265. #547Dic47semidirect
291. C47#17617semidirect

Maximal subgroups

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

subgroupindex
#202524122
#202463612
#202419522
#202419482
#202419442
#202418872
#202418832
#202418792
C47×#35062
#1761747