Finite Group Databasebeta

← #26369127

Subgroups — #26369127

Order 12032.

1220 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689107C212yesmaximal, complemented
2#20245083C212yesmaximal, complemented
3#20245080C212yesmaximal, complemented
4#20245077C214yesmaximal, complemented
5#20245073C214yesmaximal, complemented
6C47×#3449C211yesmaximal, characteristic
7C22×#10625329—22no—
8#15689107—22no—
9#15689107—22no—
10#15689107C2214yes—
11#15689107C2214yescomplemented
12C2×#13093943C2214yescomplemented
13C2×#13093941—22no—
14#15688922C2218yescomplemented
15#15688920C2218yescomplemented
16C22×#41×C47C411yescharacteristic
17C22×#41×C47C2211yescharacteristic
18C47×#537—22no—
19C47×#536C411yescharacteristic
20C47×#536C2211yescharacteristic
21C47×#536C2212yes—
22C47×#535C412yes—
23C47×#535C411yescharacteristic
24C47×#535C2211yescharacteristic
25C47×#534—22no—
26C47×#532C412yes—
27C47×#532C2212yes—
28C47×#525—22no—
29C47×#525—22no—
30C2×#117×C47C411yescharacteristic
31C2×#10625329—24no—
32C2×#10625329—24no—
33C2×#10625329—24no—
34C2×#10625329—42no—
35C2×#10625329—42no—
36C2×#10625329—42no—
37C22×C47⋊C8—42no—
38#13093943—24no—
39#13093943C2318yes—
40#13093941—24no—
41#13093941—24no—
42C2×#41×C47—21no—
43C2×#41×C47—21no—
44C2×#41×C47—42no—
45C2×#41×C47—42no—
46C2×#41×C47—42no—
47C2×#41×C47—42no—
48C2×#41×C47C2×C412yes—
49C2×#41×C47C2×C412yes—
50C2×#41×C47C2×C411yescharacteristic
51C2×#41×C47C2×C411yescharacteristic
52C2×#41×C47D412yes—
53C2×#41×C47D412yes—
54C2×#41×C47C2311yescharacteristic
55C2×#41×C47C2311yescharacteristic
56C22×Q8×C47C2×C411yescharacteristic
57C22×Q8×C47C2311yescharacteristic
58C22×D4×C47—21no—
59C22×D4×C47C2×C411yescharacteristic
60C22×D4×C47C2×C411yescharacteristic
61C22×D4×C47D412yes—
62C23×C4×C47—21no—
63C23×C4×C47C2×C411yescharacteristic
64C23×C4×C47D412yes—
65C4⋊Q8×C47C2×C412yes—
66C4⋊Q8×C47C2312yes—
67#127×C47C2×C412yes—
68#127×C47C2×C412yes—
69#124×C47C2×C412yes—
70#124×C47C2×C412yes—
71#124×C47C2×C412yes—
72#124×C47C2312yes—
73#123×C47—22no—
74#123×C47—22no—
75#123×C47—24no—
76C22⋊Q8×C47—24no—
77C22⋊Q8×C47—22no—
78C22⋊Q8×C47—22no—
79#121×C47—24no—
80#121×C47—24no—
81#121×C47—24no—
82#121×C47—22no—
83#121×C47—22no—
84#120×C47—24no—
85#120×C47—22no—
86#120×C47—22no—
87C4×D4×C47—24no—
88C4×D4×C47—24no—
89#117×C47—21no—
90#117×C47—21no—
91#117×C47C2×C411yescharacteristic
92#117×C47C2×C412yes—
93#117×C47C2×C411yescharacteristic
94C2×C4⋊C4×C47—21no—
95C2×C22⋊C4×C47—21no—
96C2×C42×C47C2×C412yes—
97C24×C47—21no—
98C24×C47C22⋊C411yescharacteristic
99C24×C47C22⋊C412yes—
100#41×C47—42no—
101#41×C47—41no—
102#41×C47—41no—
103#41×C47—42no—
104#41×C47—42no—
105#41×C47—42no—
106#41×C47—42no—
107#41×C47—41no—
108#41×C47—42no—
109#41×C47—41no—
110#41×C47—44no—
111#41×C47—44no—
112#41×C47—42no—
113#41×C47—42no—
114#41×C47—42no—
115#41×C47—42no—
116C2×Q8×C47—21no—
117C2×Q8×C47—42no—
118C2×Q8×C47—42no—
119C2×Q8×C47C22×C412yes—
120C2×Q8×C47C22×C411yescharacteristic
121C2×Q8×C47C2×D412yes—
122C2×Q8×C47C2411yescharacteristic
123C2×D4×C47—22no—
124C2×D4×C47—22no—
125C2×D4×C47—24no—
126C2×D4×C47—24no—
127C2×D4×C47—21no—
128C2×D4×C47—42no—
129C2×D4×C47—42no—
130C2×D4×C47—42no—
131C2×D4×C47—42no—
132C2×D4×C47—44no—
133C2×D4×C47C22⋊C412yes—
134C2×D4×C47C22⋊C412yes—
135C2×D4×C47C22⋊C412yes—
136C2×D4×C47C22×C412yes—
137C2×D4×C47C22×C411yescharacteristic
138C2×D4×C47C22×C411yescharacteristic
139C2×D4×C47C2×D412yes—
140C2×D4×C47C2×D412yes—
141C22×C4×C47—22no—
142C22×C4×C47—22no—
143C22×C4×C47—22no—
144C22×C4×C47—22no—
145C22×C4×C47—22no—
146C22×C4×C47—22no—
147C22×C4×C47—21no—
148C22×C4×C47—22no—
149C22×C4×C47—22no—
150C22×C4×C47—22no—
151C22×C4×C47—42no—
152C22×C4×C47—42no—
153C22×C4×C47C22⋊C411yescharacteristic
154C22×C4×C47C22⋊C411yescharacteristic
155C22×C4×C47C22⋊C412yes—
156C22×C4×C47C22⋊C412yes—
157C22×C4×C47C22⋊C411yescharacteristic
158C22×C4×C47C22×C411yescharacteristic
159C22×C4×C47C22×C411yescharacteristic
160C22×C4×C47C22×C411yescharacteristic
161C22×C4×C47C2×D412yes—
162C22×C4×C47C2×D412yes—
163C22×C4×C47C2×D412yes—
164C4⋊C4×C47—22no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—24no—
168C4⋊C4×C47—24no—
169C22⋊C4×C47—21no—
170C22⋊C4×C47—21no—
171C22⋊C4×C47—22no—
172C22⋊C4×C47—24no—
173C22⋊C4×C47—24no—
174C22⋊C4×C47—24no—
175C22⋊C4×C47—24no—
176C42×C47—22no—
177C42×C47C22×C412yes—
178C42×C47C22×C412yes—
179#10625329—28no—
180#10625329—44no—
181#10625329—44no—
182#10625329—44no—
183C2×C47⋊C8—44no—
184C2×C47⋊C8—44no—
185C2×C47⋊C8—44no—
186C23×C47—22no—
187C23×C47—22no—
188C23×C47—22no—
189C23×C47—22no—
190C23×C47—42no—
191C23×C47—82no—
192C23×C47—81no—
193C23×C47—82no—
194C23×C47C2×C22⋊C411yescharacteristic
195C23×C47C2×C22⋊C412yes—
196C23×C47C2×C22⋊C412yes—
197C23×C47C2×C22⋊C411yescharacteristic
198C23×C47C2×C22⋊C411yescharacteristic
199Q8×C47—41no—
200Q8×C47—42no—
201Q8×C47—41no—
202Q8×C47—42no—
203Q8×C47—42no—
204D4×C47—42no—
205D4×C47—42no—
206D4×C47—42no—
207D4×C47—41no—
208D4×C47—41no—
209D4×C47—42no—
210D4×C47—42no—
211D4×C47—42no—
212D4×C47—44no—
213D4×C47—44no—
214D4×C47—42no—
215D4×C47—84no—
216D4×C47—84no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—22no—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—22no—
223C2×C4×C47—22no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—22no—
227C2×C4×C47—22no—
228C2×C4×C47—22no—
229C2×C4×C47—22no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—21no—
233C2×C4×C47—21no—
234C2×C4×C47—22no—
235C2×C4×C47—22no—
236C2×C4×C47—22no—
237C2×C4×C47—44no—
238C2×C4×C47—42no—
239C2×C4×C47—42no—
240C2×C4×C47—42no—
241C2×C4×C47—82no—
242C2×C4×C47#9912yes—
243C2×C4×C47#9912yes—
244C2×C4×C47C2×C22⋊C412yes—
245C2×C4×C47C2×C22⋊C411yescharacteristic
246C2×C4×C47C2×C22⋊C411yescharacteristic
247C2×C4×C47C2×C22⋊C411yescharacteristic
248C2×C4×C47C23×C411yescharacteristic
249C2×C4×C47C22×D412yes—
250C47⋊C8—48no—
251#33369—471nomaximal, Sylow
252C22×C47—21no—
253C22×C47—21no—
254C22×C47—22no—
255C22×C47—22no—
256C22×C47—41no—
257C22×C47—41no—
258C22×C47—42no—
259C22×C47—42no—
260C22×C47—41no—
261C22×C47—44no—
262C22×C47—44no—
263C22×C47—42no—
264C22×C47—82no—
265C22×C47—82no—
266C22×C47—84no—
267C22×C47C2×#9912yes—
268C22×C47C22×C22⋊C411yescharacteristic
269C4×C47—22no—
270C4×C47—21no—
271C4×C47—22no—
272C4×C47—21no—
273C4×C47—42no—
274C4×C47—42no—
275C4×C47—42no—
276C4×C47C2×#9911yescharacteristic
277C4×C47C2×#9912yes—
278C4×C47C2×#9911yescharacteristic
279#3449D4711yescharacteristic
280C2×#413—472no—
281#2135—472no—
282#2134—472no—
283#2133—474no—
284#2132—474no—
285C2×C47—21no—
286C2×C47—22no—
287C2×C47—42no—
288C2×C47—42no—
289C2×C47—41no—
290C2×C47—41no—
291C2×C47—82no—
292C2×C47C22×#9911yescharacteristic
293C22×#41Dic4711yescharacteristic
294C22×#41C2×D4711yescharacteristic
295C22×M16—942no—
296#537—22no—
297#536Dic4711yescharacteristic
298#536C2×D4711yescharacteristic
299#536C2×D4712yes—
300#535Dic4712yes—
301#535Dic4711yescharacteristic
302#535C2×D4711yescharacteristic
303#534—22no—
304#532Dic4712yes—
305#532C2×D4712yes—
306#525—22no—
307#525—22no—
308C2×#117Dic4711yescharacteristic
309#413—474no—
310#413—474no—
311#413—942no—
312#413—942no—
313C2×#101—474no—
314C2×#100—942no—
315#356—478no—
316#355—478no—
317C47#3336911yescharacteristic, complemented, Sylow
318C2×#41—21no—
319C2×#41—21no—
320C2×#41—42no—
321C2×#41—42no—
322C2×#41—42no—
323C2×#41—42no—
324C2×#41C2×Dic4712yes—
325C2×#41C2×Dic4712yes—
326C2×#41C2×Dic4711yescharacteristic
327C2×#41C2×Dic4711yescharacteristic
328C2×#41C22⋊D4712yes—
329C2×#41C22⋊D4712yes—
330C2×#41C22×D4711yescharacteristic
331C2×#41C22×D4711yescharacteristic
332C22×Q8C2×Dic4711yescharacteristic
333C22×Q8C22×D4711yescharacteristic
334C22×D4—21no—
335C22×D4C2×Dic4711yescharacteristic
336C22×D4C2×Dic4711yescharacteristic
337C22×D4C22⋊D4712yes—
338C23×C4—21no—
339C23×C4C2×Dic4711yescharacteristic
340C23×C4C22⋊D4712yes—
341C2×M16—944no—
342C2×M16—944no—
343C2×M16—944no—
344C2×M16—1882no—
345C2×M16—1882no—
346C2×M16—1882no—
347C22×C8—1882no—
348C4⋊Q8C2×Dic4712yes—
349C4⋊Q8C22×D4712yes—
350#127C2×Dic4712yes—
351#127C2×Dic4712yes—
352#124C2×Dic4712yes—
353#124C2×Dic4712yes—
354#124C2×Dic4712yes—
355#124C22×D4712yes—
356#123—22no—
357#123—22no—
358#123—24no—
359C22⋊Q8—24no—
360C22⋊Q8—22no—
361C22⋊Q8—22no—
362#121—24no—
363#121—24no—
364#121—24no—
365#121—22no—
366#121—22no—
367#120—24no—
368#120—22no—
369#120—22no—
370C4×D4—24no—
371C4×D4—24no—
372#117—21no—
373#117—21no—
374#117C2×Dic4711yescharacteristic
375#117C2×Dic4712yes—
376#117C2×Dic4711yescharacteristic
377C2×C4⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C42C2×Dic4712yes—
380#101—478no—
381#101—944no—
382#100—944no—
383#100—944no—
384C24—21no—
385C24C22⋊Dic4711yescharacteristic
386C24C22⋊Dic4712yes—
387#41—42no—
388#41—41no—
389#41—41no—
390#41—42no—
391#41—42no—
392#41—42no—
393#41—42no—
394#41—42no—
395#41—41no—
396#41—41no—
397#41—44no—
398#41—44no—
399#41—42no—
400#41—42no—
401#41—42no—
402#41—42no—
403C2×Q8—21no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8C22×Dic4712yes—
407C2×Q8C22×Dic4711yescharacteristic
408C2×Q8C2×C22⋊D4712yes—
409C2×Q8C23×D4711yescharacteristic
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—21no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—44no—
420C2×D4C22⋊Dic4712yes—
421C2×D4C22⋊Dic4712yes—
422C2×D4C22⋊Dic4712yes—
423C2×D4C22×Dic4712yes—
424C2×D4C22×Dic4711yescharacteristic
425C2×D4C22×Dic4711yescharacteristic
426C2×D4C2×C22⋊D4712yes—
427C2×D4C2×C22⋊D4712yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4C22⋊Dic4711yescharacteristic
441C22×C4C22⋊Dic4711yescharacteristic
442C22×C4C22⋊Dic4712yes—
443C22×C4C22⋊Dic4712yes—
444C22×C4C22⋊Dic4711yescharacteristic
445C22×C4C22×Dic4711yescharacteristic
446C22×C4C22×Dic4711yescharacteristic
447C22×C4C22×Dic4711yescharacteristic
448C22×C4C2×C22⋊D4712yes—
449C22×C4C2×C22⋊D4712yes—
450C22×C4C2×C22⋊D4712yes—
451M16—948no—
452M16—1884no—
453M16—1884no—
454M16—1884no—
455C2×C8—1884no—
456C2×C8—1884no—
457C2×C8—1884no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C22⋊C4—21no—
464C22⋊C4—21no—
465C22⋊C4—22no—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C42—22no—
471C42C22×Dic4712yes—
472C42C22×Dic4712yes—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—42no—
478C23—82no—
479C23—81no—
480C23—82no—
481C23C2×C22⋊Dic4711yescharacteristic
482C23C2×C22⋊Dic4712yes—
483C23C2×C22⋊Dic4712yes—
484C23C2×C22⋊Dic4711yescharacteristic
485C23C2×C22⋊Dic4711yescharacteristic
486Q8—41no—
487Q8—42no—
488Q8—41no—
489Q8—42no—
490Q8—42no—
491D4—42no—
492D4—42no—
493D4—42no—
494D4—41no—
495D4—41no—
496D4—42no—
497D4—42no—
498D4—42no—
499D4—44no—
500D4—44no—
501D4—42no—
502D4—84no—
503D4—84no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—82no—
529C2×C4#1309393912yes—
530C2×C4#1309393912yes—
531C2×C4C2×C22⋊Dic4712yes—
532C2×C4C2×C22⋊Dic4711yescharacteristic
533C2×C4C2×C22⋊Dic4711yescharacteristic
534C2×C4C2×C22⋊Dic4711yescharacteristic
535C2×C4C23×Dic4711yescharacteristic
536C2×C4C22×C22⋊D4712yes—
537C8—1888no—
538C22—21no—
539C22—21no—
540C22—22no—
541C22—22no—
542C22—41no—
543C22—41no—
544C22—42no—
545C22—42no—
546C22—41no—
547C22—44no—
548C22—44no—
549C22—42no—
550C22—82no—
551C22—82no—
552C22—84no—
553C22C2×#1309393912yes—
554C22C22×C22⋊Dic4711yescharacteristic
555C4—22no—
556C4—21no—
557C4—22no—
558C4—21no—
559C4—42no—
560C4—42no—
561C4—42no—
562C4C2×#1309393911yescharacteristic
563C4C2×#1309393912yes—
564C4C2×#1309393911yescharacteristic
565C2—21no—
566C2—22no—
567C2—42no—
568C2—42no—
569C2—41no—
570C2—41no—
571C2—82no—
572C2C22×#1309393911yescharacteristic
subgroupG/H
centerC4C2×#13093939
commutatorC2×C4×C47C23×C4
FrattiniC2×Q8C23×D47
Fitting, prime radicalC47×#3449C2
socleC2×C47C22×#99
cosocleC2×Q8×C47C24
hypercenter#3449D47
hypocenterC47#33369
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ #3449

Complements

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

NH ≅ G/Ntype
1. C2×#15689107C2semidirect
2. #20245083C2semidirect
3. #20245080C2semidirect
4. #20245077C2semidirect
5. #20245073C2semidirect
11. #15689107C22semidirect
12. C2×#13093943C22semidirect
14. #15688922C22semidirect
15. #15688920C22semidirect
317. C47#33369semidirect

Maximal subgroups

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

subgroupindex
C2×#156891072
#202450832
#202450802
#202450772
#202450732
C47×#34492
#3336947