Finite Group Databasebeta

← #26220930

Subgroups — #26220930

Order 12032.

1142 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689752C211yesmaximal, characteristic
2#20246005C211yesmaximal, characteristic
3#20245743C211yesmaximal, characteristic
4#20244374C211yesmaximal, characteristic
5#20244373C211yesmaximal, characteristic
6#20244372C211yesmaximal, characteristic
7#20244371C214yesmaximal
8#20243119C214yesmaximal
9C47×#1996C211yesmaximal, characteristic
10#15689752—22no—
11#15689752—22no—
12#15689752—22no—
13#15689752—22no—
14C2×#13094023C412yescomplemented
15C2×#13094022C412yescomplemented
16C2×#13094014C412yescomplemented
17C2×#13094014C412yescomplemented
18C2×#13094003C2211yescharacteristic
19C2×#13094001C2212yes—
20C2×#13093980C2212yes—
21C2×#13093980C2212yes—
22C2×#13093980C2212yes—
23C22×Dic47⋊C4C2211yescharacteristic
24#15689020—22no—
25#15689018—22no—
26#15688983C2211yescharacteristic
27#15688981C2212yes—
28C4×Dic47⋊C4—22no—
29#15688974—22no—
30C22⋊C4×Dic47C2212yes—
31C4×C22⋊Dic47C2211yescharacteristic
32#15688890C2214yes—
33#15688889C2214yes—
34#15688889C2214yes—
35C2×C22⋊Q8×C47C2211yescharacteristic
36C47×#380C2211yescharacteristic
37C4×C22⋊C4×C47C2211yescharacteristic
38C47×#342C2214yes—
39C22×Dic94C2×C412yes—
40C22×Dic94C2×C411yescharacteristic
41C23×Dic47C2311yescharacteristic
42C23×Dic47C2311yescharacteristic
43#13094023—24no—
44#13094023—24no—
45#13094022—24no—
46#13094022—24no—
47#13094014—24no—
48#13094014—24no—
49#13094014—24no—
50#13094014—24no—
51#13094003—21no—
52#13094003—21no—
53#13094003—21no—
54#13094003—21no—
55#13094001—24no—
56#13094001—24no—
57#13093980—24no—
58#13093980—24no—
59#13093980—24no—
60#13093980—28no—
61#13093980Q818yescomplemented
62C2×C4⋊Dic47—21no—
63C2×C4⋊Dic47C2×C412yes—
64C2×C4⋊Dic47C2×C411yescharacteristic
65C2×Dic47⋊C4—22no—
66C2×Dic47⋊C4—22no—
67C2×Dic47⋊C4—22no—
68C2×Dic47⋊C4C2×C412yes—
69C2×Dic47⋊C4C2×C412yes—
70C2×Dic47⋊C4C2×C412yes—
71C2×Dic47⋊C4C2×C412yes—
72C2×Dic47⋊C4C2×C412yes—
73C2×Dic47⋊C4C2×C412yes—
74C2×Dic47⋊C4C2×C412yes—
75C2×Dic47⋊C4D412yes—
76C2×Dic47⋊C4D412yes—
77C2×Dic47⋊C4D412yes—
78C2×Dic47⋊C4D412yes—
79C2×C22⋊Dic47C2312yes—
80C2×C22⋊Dic47C2312yes—
81C2×C22⋊Dic47C2312yes—
82C2×C4×Dic47—22no—
83C2×C4×Dic47—22no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47—22no—
86C2×C4×Dic47C2×C412yes—
87C2×C4×Dic47C2×C412yes—
88C47⋊#95—21no—
89C47⋊#95—21no—
90C47⋊#95—22no—
91C47⋊#95—22no—
92C22×Q8×C47C2×C411yescharacteristic
93C23×C4×C47C2311yescharacteristic
94C22⋊Q8×C47—21no—
95C22⋊Q8×C47—21no—
96C22⋊Q8×C47—21no—
97C22⋊Q8×C47—21no—
98C2×C4⋊C4×C47—21no—
99C2×C4⋊C4×C47C2×C411yescharacteristic
100C2×C4⋊C4×C47C2×C412yes—
101C2×C22⋊C4×C47C2312yes—
102C2×C22⋊C4×C47C2312yes—
103C2×C22⋊C4×C47C2312yes—
104C2×C42×C47—21no—
105#95×C47—21no—
106#95×C47—21no—
107C24×C47C2411yescharacteristic
108C2×Q8×C47—21no—
109C2×Q8×C47—21no—
110C2×Q8×C47—81no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—81no—
114C22×C4×C47C22×C412yes—
115C22×C4×C47C22×C412yes—
116C22×C4×C47C22×C411yescharacteristic
117C22×C4×C47C22×C411yescharacteristic
118C22×C4×C47C2×D411yescharacteristic
119C22×C4×C47C2×D411yescharacteristic
120C22×C4×C47C2×D411yescharacteristic
121C22×C4×C47C2×D411yescharacteristic
122C22×Dic47—22no—
123C22×Dic47—22no—
124C22×Dic47—81no—
125C22×Dic47C22×C412yes—
126C22×Dic47C22×C412yes—
127C22×Dic47C22×C412yes—
128C22×Dic47C22×C412yes—
129C22×Dic47C2×D412yes—
130C22×Dic47C2×D412yes—
131C22×Dic47C2×D412yes—
132C22×Dic47C2×D412yes—
133C22×Dic47C2×Q814yes—
134C22×Dic47#4114yes—
135C2×Dic94—21no—
136C2×Dic94—21no—
137C2×Dic94—24no—
138C2×Dic94—82no—
139C2×Dic94—81no—
140C4⋊C4×C47—21no—
141C4⋊C4×C47—21no—
142C4⋊C4×C47—22no—
143C4⋊C4×C47—22no—
144C4⋊C4×C47—81no—
145C22⋊C4×C47—22no—
146C22⋊C4×C47—22no—
147C22⋊C4×C47—22no—
148C22⋊C4×C47—24no—
149C22⋊C4×C47C2×Q814yes—
150C42×C47—81no—
151C22⋊Dic47—22no—
152C22⋊Dic47—22no—
153C22⋊Dic47—22no—
154C22⋊Dic47—24no—
155C22⋊Dic47C2×Q814yes—
156C4⋊Dic47—21no—
157C4⋊Dic47—21no—
158C4⋊Dic47—24no—
159C4⋊Dic47—81no—
160Dic47⋊C4—22no—
161Dic47⋊C4—22no—
162Dic47⋊C4—24no—
163Dic47⋊C4—24no—
164Dic47⋊C4—24no—
165Dic47⋊C4—24no—
166Dic47⋊C4—28no—
167Dic47⋊C4—44no—
168Dic47⋊C4—82no—
169Dic47⋊C4C22⋊C414yes—
170Dic47⋊C4C22⋊C414yes—
171Dic47⋊C4C22⋊C414yes—
172Dic47⋊C4C22⋊C414yes—
173C4×Dic47—24no—
174C4×Dic47—24no—
175C4×Dic47—28no—
176C4×Dic47—44no—
177C4×Dic47—82no—
178C4×Dic47—81no—
179C23×C47—41no—
180C23×C47—41no—
181C23×C47C23×C411yescharacteristic
182C23×C47C22×D411yescharacteristic
183C23×C47C22×D411yescharacteristic
184C23×C47C22×Q812yes—
185C23×C47C2×#4112yes—
186Q8×C47—81no—
187Q8×C47—81no—
188C2×C4×C47—22no—
189C2×C4×C47—22no—
190C2×C4×C47—24no—
191C2×C4×C47—41no—
192C2×C4×C47—41no—
193C2×C4×C47—41no—
194C2×C4×C47—41no—
195C2×C4×C47—41no—
196C2×C4×C47—41no—
197C2×C4×C47—42no—
198C2×C4×C47—42no—
199C2×C4×C47—42no—
200C2×C4×C47—42no—
201C2×C4×C47—42no—
202C2×C4×C47—42no—
203C2×C4×C47—42no—
204C2×C4×C47C2×C22⋊C411yescharacteristic
205C2×C4×C47C2×C22⋊C411yescharacteristic
206C2×C4×C47C2×C22⋊C411yescharacteristic
207C2×C4×C47C2×C22⋊C411yescharacteristic
208C2×Dic47—22no—
209C2×Dic47—22no—
210C2×Dic47—24no—
211C2×Dic47—42no—
212C2×Dic47—42no—
213C2×Dic47—42no—
214C2×Dic47—42no—
215C2×Dic47—42no—
216C2×Dic47—42no—
217C2×Dic47—42no—
218C2×Dic47—42no—
219C2×Dic47—42no—
220C2×Dic47—42no—
221C2×Dic47C2×C22⋊C414yes—
222C2×Dic47C2×C22⋊C414yes—
223C2×Dic47C4×Q818yes—
224C2×Dic47C22⋊Q814yes—
225C2×Dic47C22⋊Q814yes—
226C2×Dic47#12414yes—
227C2×Dic47C4⋊Q814yes—
228Dic94—84no—
229Dic94—81no—
230Dic94—81no—
231#21334—471nomaximal, Sylow
232C22×C47—21no—
233C22×C47—21no—
234C22×C47—21no—
235C22×C47—21no—
236C22×C47—21no—
237C22×C47—21no—
238C22×C47—42no—
239C22×C47—42no—
240C22×C47C22×C22⋊C411yescharacteristic
241C22×C47C2×C4×Q812yes—
242C22×C47C2×C22⋊Q812yes—
243C22×C47C2×#12411yescharacteristic
244C22×C47C2×C4⋊Q811yescharacteristic
245C4×C47—21no—
246C4×C47—21no—
247C4×C47—21no—
248C4×C47—21no—
249C4×C47—44no—
250C4×C47—42no—
251C4×C47—42no—
252C4×C47—82no—
253Dic47—24no—
254Dic47—44no—
255Dic47—42no—
256Dic47—42no—
257Dic47—82no—
258Dic47#39118yes—
259C2×#544—471no—
260#2316—471no—
261#2274—471no—
262#1996—471no—
263#1996—471no—
264#1996—474no—
265#1996—471no—
266#1996D4711yescharacteristic
267#1740—474no—
268C2×C47—22no—
269C2×C47—21no—
270C2×C47—22no—
271C2×C47—21no—
272C2×C47C2×#39111yescharacteristic
273C2×C47#288311yescharacteristic
274C2×C47#288411yescharacteristic
275#544—942no—
276#544—942no—
277#544—942no—
278#544—942no—
279C2×C4⋊Q8—472no—
280C2×C4⋊Q8—472no—
281C2×#125—472no—
282C2×#125—472no—
283C2×C22⋊Q8—471no—
284C2×C22⋊Q8—472no—
285C2×C22⋊Q8C2×D4711yescharacteristic
286C2×#117—472no—
287C2×#117—472no—
288C2×#117—472no—
289C22×C4⋊C4—471no—
290#389—942no—
291#389—942no—
292#380—471no—
293#380—472no—
294#380C2×D4711yescharacteristic
295C4×C4⋊C4—942no—
296C4×C4⋊C4—942no—
297C4×C22⋊C4—471no—
298C4×C22⋊C4—472no—
299C4×C22⋊C4C2×D4711yescharacteristic
300#342—474no—
301#342—474no—
302#342—474no—
303#342C2×D4714yes—
304C47#2133411yescharacteristic, complemented, Sylow
305C22×Q8—472no—
306C22×Q8—471no—
307C22×Q8C4×D4711yescharacteristic
308C23×C4—471no—
309C23×C4—471no—
310C23×C4C22×D4711yescharacteristic
311C4⋊Q8—944no—
312C4⋊Q8—944no—
313C4⋊Q8—944no—
314C4⋊Q8—944no—
315#125—944no—
316#125—944no—
317#125—944no—
318#125—944no—
319C22⋊Q8—21no—
320C22⋊Q8—21no—
321C22⋊Q8—21no—
322C22⋊Q8—21no—
323C22⋊Q8—944no—
324C22⋊Q8—944no—
325C22⋊Q8—941no—
326C22⋊Q8—941no—
327C22⋊Q8—941no—
328C22⋊Q8—941no—
329#117—478no—
330#117—944no—
331#117—944no—
332#117—944no—
333#117—948no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—472no—
336C2×C4⋊C4—472no—
337C2×C4⋊C4—472no—
338C2×C4⋊C4—472no—
339C2×C4⋊C4—472no—
340C2×C4⋊C4—472no—
341C2×C4⋊C4—472no—
342C2×C4⋊C4—472no—
343C2×C4⋊C4—472no—
344C2×C4⋊C4—471no—
345C2×C4⋊C4—472no—
346C2×C4⋊C4—472no—
347C2×C4⋊C4—472no—
348C2×C4⋊C4—942no—
349C2×C4⋊C4—942no—
350C2×C4⋊C4—941no—
351C2×C4⋊C4—942no—
352C2×C4⋊C4C4×D4711yescharacteristic
353C2×C4⋊C4C4×D4712yes—
354C2×C22⋊C4—472no—
355C2×C22⋊C4—472no—
356C2×C22⋊C4—472no—
357C2×C22⋊C4C22×D4712yes—
358C2×C22⋊C4C22×D4712yes—
359C2×C22⋊C4C22×D4712yes—
360C2×C42—21no—
361C2×C42—472no—
362C2×C42—472no—
363C2×C42—942no—
364C2×C42—942no—
365C2×C42—941no—
366C2×C42—942no—
367#95—21no—
368#95—21no—
369#95—942no—
370#95—942no—
371#95—941no—
372#95—941no—
373C24C23×D4711yescharacteristic
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—81no—
377C2×Q8—944no—
378C2×Q8—941no—
379C2×Q8—941no—
380C2×Q8—3761no—
381C2×Q8—3762no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—81no—
385C22×C4—472no—
386C22×C4—472no—
387C22×C4—472no—
388C22×C4—474no—
389C22×C4—474no—
390C22×C4—472no—
391C22×C4—472no—
392C22×C4—472no—
393C22×C4—472no—
394C22×C4—472no—
395C22×C4—942no—
396C22×C4—942no—
397C22×C4—3761no—
398C22×C4C2×C4×D4712yes—
399C22×C4C2×C4×D4712yes—
400C22×C4C2×C4×D4711yescharacteristic
401C22×C4C2×C4×D4711yescharacteristic
402C22×C4D4×D4711yescharacteristic
403C22×C4D4×D4711yescharacteristic
404C22×C4D4×D4711yescharacteristic
405C22×C4D4×D4711yescharacteristic
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—81no—
411C4⋊C4—474no—
412C4⋊C4—474no—
413C4⋊C4—474no—
414C4⋊C4—474no—
415C4⋊C4—944no—
416C4⋊C4—944no—
417C4⋊C4—944no—
418C4⋊C4—944no—
419C4⋊C4—948no—
420C4⋊C4—944no—
421C4⋊C4—942no—
422C4⋊C4—942no—
423C4⋊C4—941no—
424C4⋊C4—941no—
425C4⋊C4—3762no—
426C4⋊C4—3761no—
427C4⋊C4—1884no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—24no—
432C22⋊C4—474no—
433C22⋊C4—942no—
434C22⋊C4—942no—
435C22⋊C4—942no—
436C22⋊C4—944no—
437C22⋊C4Q8×D4714yes—
438C42—81no—
439C42—944no—
440C42—944no—
441C42—948no—
442C42—3761no—
443C42—3762no—
444C42—1884no—
445C23—41no—
446C23—41no—
447C23C22×C4×D4711yescharacteristic
448C23C2×D4×D4711yescharacteristic
449C23C2×D4×D4711yescharacteristic
450C23C2×Q8×D4712yes—
451C23#41×D4712yes—
452Q8—81no—
453Q8—81no—
454Q8—3761no—
455Q8—3761no—
456Q8—3764no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—474no—
474C2×C4—478no—
475C2×C4—474no—
476C2×C4—474no—
477C2×C4—474no—
478C2×C4—474no—
479C2×C4—474no—
480C2×C4—942no—
481C2×C4—942no—
482C2×C4—944no—
483C2×C4—1882no—
484C2×C4—1882no—
485C2×C4—1882no—
486C2×C4—1882no—
487C2×C4—1882no—
488C2×C4—1882no—
489C2×C4—1882no—
490C2×C4—1882no—
491C2×C4—1882no—
492C2×C4—1882no—
493C2×C4C22⋊C4×D4711yescharacteristic
494C2×C4C22⋊C4×D4711yescharacteristic
495C2×C4C22⋊C4×D4711yescharacteristic
496C2×C4C22⋊C4×D4711yescharacteristic
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—42no—
504C22—42no—
505C22C2×C22⋊C4×D4711yescharacteristic
506C22C4×Q8×D4712yes—
507C22C22⋊Q8×D4712yes—
508C22#124×D4711yescharacteristic
509C22C4⋊Q8×D4711yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—44no—
515C4—42no—
516C4—42no—
517C4—82no—
518C4—478no—
519C4—944no—
520C4—3762no—
521C4—1884no—
522C4—1882no—
523C4—1882no—
524C2—22no—
525C2—21no—
526C2—22no—
527C2—21no—
528C2#391×D4711yescharacteristic
529C2#2025184611yescharacteristic
530C2#2025185011yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC47×#1996C2
socleC22×C47C22×C22⋊C4
prime radical, cosocleC24×C47C24
hypercenter#1996D47
hypocenterC47#21334
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
14. C2×#13094023C4semidirect
15. C2×#13094022C4semidirect
16. C2×#13094014C4semidirect
17. C2×#13094014C4semidirect
61. #13093980Q8semidirect
304. C47#21334semidirect

Maximal subgroups

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

subgroupindex
C2×#156897522
#202460052
#202457432
#202443742
#202443732
#202443722
#202443712
#202431192
C47×#19962
#2133447