Finite Group Databasebeta

← #26145001

Subgroups — #26145001

Order 12032.

1398 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×#15689030C211yesmaximal, characteristic
2C2×#15689030C211yesmaximal, characteristic
3#20241652C212yesmaximal
4#20241651C212yesmaximal
5#20241640C212yesmaximal
6#20241621C212yesmaximal
7#20241619C212yesmaximal
8Dic47⋊#95C212yesmaximal
9C2×C47×#391C211yesmaximal, characteristic
10C22×C4⋊Dic47C2211yescharacteristic
11C22×C4⋊Dic47C2211yescharacteristic
12C22×Dic47⋊C4C2212yes—
13C22×Dic47⋊C4C2212yes—
14C22×Dic47⋊C4C2212yes—
15C22×C4×Dic47C2211yescharacteristic
16C22×C4×Dic47C2211yescharacteristic
17#15689030—24no—
18#15689030C418yescomplemented
19C2×C47⋊#95C2212yes—
20C2×C47⋊#95C2212yes—
21C2×C47⋊#95C2212yes—
22C2×C47⋊#95C2212yes—
23C2×C47⋊#95C2212yes—
24C2×C47⋊#95C2212yes—
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
28C23×Q8×C47C2211yescharacteristic
29C22×C4⋊C4×C47C2211yescharacteristic
30C22×C42×C47C2211yescharacteristic
31C47×#391—24no—
32C2×#95×C47C2212yes—
33C2×#95×C47C2212yes—
34C23×Dic47C2312yes—
35C23×Dic47C2312yes—
36C23×Dic47C2312yes—
37C23×Dic47C2312yes—
38C2×C4⋊Dic47—22no—
39C2×C4⋊Dic47—41no—
40C2×C4⋊Dic47—41no—
41C2×C4⋊Dic47—41no—
42C2×C4⋊Dic47—41no—
43C2×C4⋊Dic47C2×C414yes—
44C2×Dic47⋊C4—24no—
45C2×Dic47⋊C4—42no—
46C2×Dic47⋊C4—42no—
47C2×Dic47⋊C4—42no—
48C2×Dic47⋊C4—42no—
49C2×Dic47⋊C4—42no—
50C2×Dic47⋊C4—42no—
51C2×Dic47⋊C4—42no—
52C2×Dic47⋊C4Q818yescomplemented
53C2×C4×Dic47—22no—
54C2×C4×Dic47—41no—
55C2×C4×Dic47—41no—
56C2×C4×Dic47—41no—
57C2×C4×Dic47—41no—
58C2×C4×Dic47C2×C414yes—
59C47⋊#95—24no—
60C47⋊#95—24no—
61C47⋊#95—24no—
62C47⋊#95—24no—
63C47⋊#95—42no—
64C47⋊#95—42no—
65C47⋊#95—42no—
66C47⋊#95C2×C418yes—
67C47⋊#95C2×C418yes—
68C22×Q8×C47—41no—
69C22×Q8×C47—41no—
70C22×Q8×C47—41no—
71C22×Q8×C47—41no—
72C22×Q8×C47C2×C414yes—
73C22×Q8×C47D414yes—
74C22×Q8×C47Q814yes—
75C23×C4×C47C2312yes—
76C23×C4×C47C2312yes—
77C23×C4×C47C2311yescharacteristic
78C23×C4×C47C2311yescharacteristic
79C23×C4×C47C2311yescharacteristic
80C2×C4⋊C4×C47—22no—
81C2×C4⋊C4×C47—41no—
82C2×C4⋊C4×C47—41no—
83C2×C42×C47—22no—
84C2×C42×C47—41no—
85C2×C42×C47—41no—
86#95×C47—24no—
87#95×C47—24no—
88C24×C47C2411yescharacteristic
89C2×Q8×C47—44no—
90C2×Q8×C47—44no—
91C2×Q8×C47—44no—
92C2×Q8×C47—44no—
93C2×Q8×C47—44no—
94C2×Q8×C47—44no—
95C2×Q8×C47C4⋊C4116yescomplemented
96C22×C4×C47—21no—
97C22×C4×C47—21no—
98C22×C4×C47—21no—
99C22×C4×C47—21no—
100C22×C4×C47—21no—
101C22×C4×C47—21no—
102C22×C4×C47—21no—
103C22×C4×C47—21no—
104C22×C4×C47—21no—
105C22×C4×C47—21no—
106C22×C4×C47—21no—
107C22×C4×C47—21no—
108C22×C4×C47—22no—
109C22×C4×C47—22no—
110C22×C4×C47—22no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47C22×C412yes—
122C22×C4×C47C22×C412yes—
123C22×C4×C47C22×C412yes—
124C22×C4×C47C2×D412yes—
125C22×C4×C47C2×D412yes—
126C22×C4×C47C2×D412yes—
127C22×C4×C47C2×Q814yes—
128C22×C4×C47C2×Q812yes—
129C22×C4×C47C2×Q812yes—
130C22×C4×C47C2×Q812yes—
131C22×Dic47—22no—
132C22×Dic47—22no—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—22no—
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47C22×C414yes—
155C22×Dic47C22×C414yes—
156C22×Dic47C2×Q814yes—
157C22×Dic47C2×Q814yes—
158C22×Dic47#4114yes—
159C4⋊C4×C47—44no—
160C42×C47—44no—
161C4⋊Dic47—44no—
162C4⋊Dic47—44no—
163Dic47⋊C4—48no—
164Dic47⋊C4—48no—
165Dic47⋊C4—48no—
166C4×Dic47—44no—
167C4×Dic47—44no—
168C23×C47C23×C411yescharacteristic
169C23×C47C22×D411yescharacteristic
170C23×C47C22×Q812yes—
171C23×C47C22×Q811yescharacteristic
172C23×C47C2×#4112yes—
173C23×C47H(5,𝔽2)11yescharacteristic
174C23×C47H(5,𝔽2)11yescharacteristic
175C23×C47H(5,𝔽2)11yescharacteristic
176C23×C47H(5,𝔽2)11yescharacteristic
177C23×C47H(5,𝔽2)12yes—
178C23×C472−1+412yes—
179Q8×C47—416no—
180C2×C4×C47—22no—
181C2×C4×C47—22no—
182C2×C4×C47—22no—
183C2×C4×C47—22no—
184C2×C4×C47—22no—
185C2×C4×C47—22no—
186C2×C4×C47—22no—
187C2×C4×C47—22no—
188C2×C4×C47—22no—
189C2×C4×C47—22no—
190C2×C4×C47—22no—
191C2×C4×C47—22no—
192C2×C4×C47—22no—
193C2×C4×C47—22no—
194C2×C4×C47—22no—
195C2×C4×C47—22no—
196C2×C4×C47—22no—
197C2×C4×C47—22no—
198C2×C4×C47—24no—
199C2×C4×C47—24no—
200C2×C4×C47—24no—
201C2×C4×C47—42no—
202C2×C4×C47—42no—
203C2×C4×C47—42no—
204C2×C4×C47—42no—
205C2×C4×C47—42no—
206C2×C4×C47—42no—
207C2×C4×C47—42no—
208C2×C4×C47—42no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47—42no—
212C2×C4×C47C2×C4⋊C414yes—
213C2×C4×C47C2×C4⋊C414yes—
214C2×C4×C47C2×C4⋊C414yes—
215C2×Dic47—24no—
216C2×Dic47—24no—
217C2×Dic47—24no—
218C2×Dic47—24no—
219C2×Dic47—24no—
220C2×Dic47—24no—
221C2×Dic47—24no—
222C2×Dic47—24no—
223C2×Dic47—24no—
224C2×Dic47—24no—
225C2×Dic47—24no—
226C2×Dic47—24no—
227C2×Dic47—42no—
228C2×Dic47—42no—
229C2×Dic47—42no—
230C2×Dic47—42no—
231C2×Dic47—42no—
232C2×Dic47—42no—
233C2×Dic47—42no—
234C2×Dic47—42no—
235C2×Dic47—42no—
236C2×Dic47—42no—
237C2×Dic47—42no—
238C2×Dic47—42no—
239C2×Dic47—42no—
240C2×Dic47—42no—
241C2×Dic47—42no—
242C2×Dic47C4×Q818yes—
243#14467—471nomaximal, Sylow
244C22×C47C22×C4⋊C411yescharacteristic
245C22×C47C2×C4×Q812yes—
246C22×C47#51811yescharacteristic
247C22×C47#51811yescharacteristic
248C22×C47#52012yes—
249C22×C47#53411yescharacteristic
250C22×C47#53411yescharacteristic
251C22×C47#53812yes—
252C22×C47#53912yes—
253C22×C47#54312yes—
254C22×C47#54311yescharacteristic
255C22×C47#54311yescharacteristic
256C22×C47#54412yes—
257C22×C47D4×Q812yes—
258C22×C47#55012yes—
259C22×C47#55412yes—
260C22×C47#55511yescharacteristic
261C22×C47#55712yes—
262C22×C47Q8211yescharacteristic
263C22×C47Q8212yes—
264C22×C47H(3,𝔽4)11yescharacteristic
265C22×C47#56212yes—
266C22×C47#56311yescharacteristic
267C4×C47—24no—
268C4×C47—24no—
269C4×C47—24no—
270C4×C47—44no—
271C4×C47—82no—
272Dic47—28no—
273Dic47—44no—
274Dic47—44no—
275Dic47—82no—
276C2×#391—471no—
277C2×#391—471no—
278C2×#391D4711yescharacteristic
279#1444—472no—
280#1444—472no—
281#1441—472no—
282#1436—472no—
283#1436—472no—
284C4⋊#95—472no—
285C2×C47#231511yescharacteristic
286C2×C47#232912yes—
287C2×C47Q8×C4⋊C412yes—
288C2×C47#236911yescharacteristic
289C2×C47#237911yescharacteristic
290C2×C47#273212yes—
291C2×C47#273311yescharacteristic
292C2×C47#273411yescharacteristic
293C2×C47#280612yes—
294C2×C47#280911yescharacteristic
295C2×C47#281111yescharacteristic
296C23×Q8C2×D4711yescharacteristic
297C22×C4⋊C4—472no—
298C22×C4⋊C4—471no—
299C22×C4⋊C4—472no—
300C22×C4⋊C4—471no—
301C22×C4⋊C4—472no—
302C22×C4⋊C4C2×D4711yescharacteristic
303C22×C42—471no—
304C22×C42—471no—
305C22×C42C2×D4711yescharacteristic
306#391—24no—
307#391—478no—
308#391—944no—
309C2×#95—472no—
310C2×#95—472no—
311C2×#95—472no—
312C2×#95—472no—
313C2×#95—472no—
314C2×#95—472no—
315C2×#95—472no—
316C2×#95—472no—
317C2×#95—472no—
318C2×#95C2×D4712yes—
319C2×#95C2×D4712yes—
320C47#1446711yescharacteristic, complemented, Sylow
321C22×Q8—41no—
322C22×Q8—41no—
323C22×Q8—41no—
324C22×Q8—41no—
325C22×Q8Dic9414yes—
326C22×Q8C4×D4714yes—
327C22×Q8C22⋊D4714yes—
328C23×C4—472no—
329C23×C4—472no—
330C23×C4—472no—
331C23×C4—472no—
332C23×C4C22×D4712yes—
333C23×C4C22×D4712yes—
334C23×C4C22×D4711yescharacteristic
335C23×C4C22×D4711yescharacteristic
336C23×C4C22×D4711yescharacteristic
337C2×C4⋊C4—22no—
338C2×C4⋊C4—41no—
339C2×C4⋊C4—41no—
340C2×C4⋊C4—474no—
341C2×C4⋊C4—478no—
342C2×C4⋊C4—944no—
343C2×C4⋊C4—942no—
344C2×C4⋊C4—1882no—
345C2×C4⋊C4—1882no—
346C2×C4⋊C4—1882no—
347C2×C4⋊C4—1882no—
348C2×C4⋊C4—1882no—
349C2×C4⋊C4—1882no—
350C2×C4⋊C4—1882no—
351C2×C4⋊C4—1881no—
352C2×C4⋊C4—1881no—
353C2×C4⋊C4—1881no—
354C2×C4⋊C4—1881no—
355C2×C42—22no—
356C2×C42—41no—
357C2×C42—41no—
358C2×C42—474no—
359C2×C42—942no—
360C2×C42—1881no—
361C2×C42—1881no—
362C2×C42—1881no—
363C2×C42—1881no—
364#95—24no—
365#95—24no—
366#95—478no—
367#95—478no—
368#95—944no—
369#95—944no—
370#95—944no—
371#95—944no—
372#95—1882no—
373#95—1882no—
374#95—1882no—
375C24C23×D4711yescharacteristic
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8Dic47⋊C4116yescomplemented
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—474no—
409C22×C4—474no—
410C22×C4—474no—
411C22×C4—474no—
412C22×C4—474no—
413C22×C4—942no—
414C22×C4—942no—
415C22×C4—942no—
416C22×C4—942no—
417C22×C4—942no—
418C22×C4—942no—
419C22×C4—942no—
420C22×C4—942no—
421C22×C4—942no—
422C22×C4—942no—
423C22×C4—942no—
424C22×C4—942no—
425C22×C4—942no—
426C22×C4—942no—
427C22×C4—942no—
428C22×C4—942no—
429C22×C4—942no—
430C22×C4—942no—
431C22×C4—942no—
432C22×C4—942no—
433C22×C4—942no—
434C22×C4—942no—
435C22×C4—942no—
436C22×C4C2×Dic9412yes—
437C22×C4C2×Dic9412yes—
438C22×C4C2×Dic9412yes—
439C22×C4C2×C4×D4712yes—
440C22×C4C2×C4×D4712yes—
441C22×C4C2×C4×D4712yes—
442C22×C4Q8×D4714yes—
443C22×C4C2×C22⋊D4712yes—
444C22×C4C2×C22⋊D4712yes—
445C22×C4C2×C22⋊D4712yes—
446C4⋊C4—44no—
447C4⋊C4—1888no—
448C4⋊C4—1884no—
449C4⋊C4—1884no—
450C4⋊C4—1888no—
451C4⋊C4—1888no—
452C42—44no—
453C42—1884no—
454C42—1884no—
455C23C22×C4×D4711yescharacteristic
456C23C22×C22⋊D4711yescharacteristic
457C23C2×Q8×D4712yes—
458C23C22×Dic9411yescharacteristic
459C23#41×D4712yes—
460C23#1309406911yescharacteristic
461C23#1309406911yescharacteristic
462C23#1309406911yescharacteristic
463C23#1309406911yescharacteristic
464C23#1309407012yes—
465C23#1309407212yes—
466Q8—416no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—478no—
500C2×C4—944no—
501C2×C4—944no—
502C2×C4—944no—
503C2×C4—944no—
504C2×C4—944no—
505C2×C4—944no—
506C2×C4—944no—
507C2×C4—944no—
508C2×C4—944no—
509C2×C4—944no—
510C2×C4—944no—
511C2×C4—944no—
512C2×C4—1882no—
513C2×C4—1882no—
514C2×C4—1882no—
515C2×C4—1882no—
516C2×C4—1882no—
517C2×C4—1882no—
518C2×C4—1882no—
519C2×C4—1882no—
520C2×C4—1882no—
521C2×C4—1882no—
522C2×C4—1882no—
523C2×C4—1882no—
524C2×C4—1882no—
525C2×C4—1882no—
526C2×C4—1882no—
527C2×C4C2×Dic47⋊C414yes—
528C2×C4C2×Dic47⋊C414yes—
529C2×C4C2×Dic47⋊C414yes—
530C22C22×Dic47⋊C411yescharacteristic
531C22C4×Q8×D4712yes—
532C22#1568959611yescharacteristic
533C22#1568959611yescharacteristic
534C22#1568960512yes—
535C22#1568968211yescharacteristic
536C22#1568968211yescharacteristic
537C22#1568970712yes—
538C22#1568971912yes—
539C22#1568974812yes—
540C22#1568974911yescharacteristic
541C22#1568974911yescharacteristic
542C22#1568975012yes—
543C22Q8×C22⋊D4712yes—
544C22#1568979212yes—
545C22#1568982712yes—
546C22#1568983511yescharacteristic
547C22#1568984912yes—
548C22#1568985311yescharacteristic
549C22Q8×Dic9412yes—
550C22#1568986511yescharacteristic
551C22#1568986912yes—
552C22#1568987211yescharacteristic
553C4—24no—
554C4—24no—
555C4—24no—
556C4—44no—
557C4—82no—
558C4—948no—
559C4—3762no—
560C4—1884no—
561C4—1884no—
562C2#2024600011yescharacteristic
563C2#2024612012yes—
564C2Q8×Dic47⋊C412yes—
565C2#2024647611yescharacteristic
566C2#2024656011yescharacteristic
567C2#2025040312yes—
568C2#2025041211yescharacteristic
569C2#2025041811yescharacteristic
570C2#2025134812yes—
571C2#2025137411yescharacteristic
572C2#2025138111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC2×C47×#391C2
socle, prime radical, cosocleC24×C47C24
hypercenterC2×#391D47
hypocenterC47#14467
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
18. #15689030C4semidirect
52. C2×Dic47⋊C4Q8semidirect
95. C2×Q8×C47C4⋊C4semidirect
320. C47#14467semidirect
382. C2×Q8Dic47⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#156890302
#202416522
#202416512
#202416402
#202416212
#202416192
Dic47⋊#952
C2×C47×#3912
#1446747