Finite Group Databasebeta

← #26226689

Subgroups — #26226689

Order 12032.

754 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689151C211yesmaximal, characteristic
2#20245995C211yesmaximal, characteristic
3#20245995C211yesmaximal, characteristic
4#20244159C211yesmaximal, characteristic, complemented
5#20244157C211yesmaximal, characteristic, complemented
6#20244155C211yesmaximal, characteristic, complemented
7#20244153C211yesmaximal, characteristic, complemented
8#20244150C211yesmaximal, characteristic, complemented
9#20244148C211yesmaximal, characteristic, complemented
10#20244145C211yesmaximal, characteristic, complemented
11#20244143C211yesmaximal, characteristic, complemented
12#20243421C211yesmaximal, characteristic
13#20243415C211yesmaximal, characteristic
14#20243411C211yesmaximal, characteristic
15C47×#1809C211yesmaximal, characteristic
16C23×C47⋊C8C2211yescharacteristic
17C2×#13093983C2211yescharacteristic
18C2×#13093958C2211yescharacteristic
19C2×#13093957C2211yescharacteristic
20C2×#13093957C2211yescharacteristic
21C2×#13093956C2211yescharacteristic
22C2×#13093955C2211yescharacteristic
23C2×#13093955C2211yescharacteristic
24#15689151C418yescomplemented
25C2×Dic47⋊C8C2211yescharacteristic
26C2×Dic47⋊C8C2211yescharacteristic
27C2×Dic47⋊C8C2211yescharacteristic
28C2×Dic47⋊C8C2211yescharacteristic
29#15689017—21no—
30#15689017—21no—
31#15689011—21no—
32#15689011—21no—
33#15689008—21no—
34#15689008—21no—
35#15689006—21no—
36#15689006—21no—
37#15688992C2211yescharacteristic
38#15688992C2211yescharacteristic
39#15688988C2211yescharacteristic
40#15688988C2211yescharacteristic
41#15688983C2211yescharacteristic
42#15688983C2211yescharacteristic
43C4⋊C4×Dic47C2211yescharacteristic
44C4⋊C4×Dic47C2211yescharacteristic
45C4⋊C4×Dic47C2211yescharacteristic
46C4⋊C4×Dic47C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53C22×M16×C47C2211yescharacteristic
54C2×#107×C47C2211yescharacteristic
55C2×#106×C47C2211yescharacteristic
56C47×#380C2211yescharacteristic
57C47×#380C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C22×C47⋊C8—41no—
61C22×C47⋊C8—41no—
62C22×C47⋊C8C2×C414yes—
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65#13093983C2×C414yes—
66#13093983Q814yes—
67C2×C4⋊Dic47D411yescharacteristic
68C2×C4⋊Dic47D411yescharacteristic
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C2×C22⋊Dic47—21no—
78C2×C22⋊Dic47—21no—
79C2×C22⋊Dic47D411yescharacteristic
80C2×C22⋊Dic47D411yescharacteristic
81C2×C4×Dic47—21no—
82C2×C4×Dic47—21no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47C2311yescharacteristic
86C2×C4×Dic47C2311yescharacteristic
87#13093958—41no—
88#13093957—41no—
89#13093957—41no—
90#13093956—41no—
91#13093955—41no—
92#13093955—41no—
93Dic47⋊C8C2×C414yes—
94Dic47⋊C8C2×C414yes—
95Dic47⋊C8C2×C414yes—
96Dic47⋊C8C2×C414yes—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C23×C4×C47C2311yescharacteristic
108C2×M16×C47—41no—
109C2×M16×C47—41no—
110C2×M16×C47C2×C414yes—
111C22×C8×C47C2311yescharacteristic
112C22×C8×C47C2311yescharacteristic
113C2×C4⋊C4×C47C2311yescharacteristic
114C2×C4⋊C4×C47C2311yescharacteristic
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C22⋊C4×C47—21no—
118C2×C22⋊C4×C47—21no—
119#107×C47—41no—
120#106×C47—41no—
121#95×C47—21no—
122#95×C47—21no—
123C24×C47C2×D411yescharacteristic
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47C22×C412yes—
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×Q812yes—
135C22×C4×C47C2411yescharacteristic
136C22×Dic47—21no—
137C22×Dic47—21no—
138C22×Dic47—21no—
139C22×Dic47—21no—
140C22×Dic47C2×D411yescharacteristic
141C22×Dic47C2×D411yescharacteristic
142C22×Dic47C2×D411yescharacteristic
143C22×Dic47C2×D411yescharacteristic
144M16×C47—44no—
145C2×C8×C47—21no—
146C2×C8×C47—21no—
147C2×C8×C47—21no—
148C2×C8×C47—21no—
149C2×C8×C47C22×C412yes—
150C2×C8×C47C22×C412yes—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—21no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—21no—
155C4⋊C4×C47—21no—
156C4⋊C4×C47—21no—
157C4⋊C4×C47—21no—
158C4⋊C4×C47—21no—
159C22⋊C4×C47—81no—
160C22⋊C4×C47—81no—
161C22⋊Dic47—81no—
162C22⋊Dic47—81no—
163C22⋊Dic47C4⋊C414yes—
164C22⋊Dic47C4⋊C414yes—
165C4⋊Dic47—21no—
166C4⋊Dic47—21no—
167C4⋊Dic47—21no—
168C4⋊Dic47—21no—
169C4⋊Dic47—21no—
170C4⋊Dic47—21no—
171C4⋊Dic47—21no—
172C4⋊Dic47—21no—
173C4⋊Dic47C4⋊C414yes—
174C4⋊Dic47C4⋊C414yes—
175Dic47⋊C4—81no—
176Dic47⋊C4—81no—
177Dic47⋊C4—81no—
178Dic47⋊C4—81no—
179C4×Dic47—81no—
180C4×Dic47—81no—
181C4×Dic47—81no—
182C4×Dic47—81no—
183C4×Dic47C22×C412yes—
184C4×Dic47C22×C412yes—
185C4×Dic47C2×Q812yes—
186C4×Dic47C2×Q812yes—
187C2×C47⋊C8—21no—
188C2×C47⋊C8—21no—
189C2×C47⋊C8—21no—
190C2×C47⋊C8—21no—
191C2×C47⋊C8—44no—
192C2×C47⋊C8C22×C412yes—
193C2×C47⋊C8C22×C412yes—
194C23×C47—21no—
195C23×C47—21no—
196C23×C47—21no—
197C23×C47—21no—
198C23×C47—21no—
199C23×C47—21no—
200C23×C47C2×C4⋊C412yes—
201C23×C47C22×D411yescharacteristic
202C2×C4×C47—21no—
203C2×C4×C47—21no—
204C2×C4×C47—22no—
205C2×C4×C47—22no—
206C2×C4×C47—22no—
207C2×C4×C47—22no—
208C2×C4×C47—41no—
209C2×C4×C47—41no—
210C2×C4×C47—41no—
211C2×C4×C47—41no—
212C2×C4×C47—41no—
213C2×C4×C47—41no—
214C2×C4×C47—41no—
215C2×C4×C47—41no—
216C2×C4×C47—41no—
217C2×C4×C47—41no—
218C2×C4×C47—41no—
219C2×C4×C47—41no—
220C2×C4×C47C2×C4⋊C412yes—
221C2×C4×C47C23×C411yescharacteristic
222C2×C4×C47C22×Q811yescharacteristic
223C2×C4×C47H(5,𝔽2)11yescharacteristic
224C2×C4×C47H(5,𝔽2)11yescharacteristic
225C2×C4×C472−1+411yescharacteristic
226C2×C4×C472−1+411yescharacteristic
227C2×Dic47—21no—
228C2×Dic47—21no—
229C2×Dic47—21no—
230C2×Dic47—21no—
231C2×Dic47—21no—
232C2×Dic47—21no—
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—41no—
236C2×Dic47—41no—
237C2×Dic47—41no—
238C2×Dic47—41no—
239C2×Dic47—41no—
240C2×Dic47—41no—
241C2×Dic47—41no—
242C2×Dic47—41no—
243C2×Dic47—41no—
244C2×Dic47—41no—
245C2×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47C2×C4⋊C412yes—
248C2×Dic47C2×C4⋊C412yes—
249C2×Dic47C2×C4⋊C412yes—
250C2×Dic47C2×C4⋊C412yes—
251C8×C47—22no—
252C8×C47—22no—
253C47⋊C8—22no—
254C47⋊C8—22no—
255#21966—471nomaximal, Sylow
256C22×C47—22no—
257C22×C47—22no—
258C22×C47—22no—
259C22×C47—41no—
260C22×C47—41no—
261C22×C47—41no—
262C22×C47—41no—
263C22×C47C22×C4⋊C411yescharacteristic
264C22×C47#53711yescharacteristic
265C22×C47#53711yescharacteristic
266C22×C47#57611yescharacteristic
267C22×C47#57711yescharacteristic
268C22×C47#57711yescharacteristic
269C22×C47#57811yescharacteristic
270C4×C47—22no—
271C4×C47—81no—
272C4×C47—81no—
273C4×C47—81no—
274C4×C47—81no—
275C4×C47#51811yescharacteristic
276C4×C47#51911yescharacteristic
277C4×C47#54411yescharacteristic
278C4×C47#54411yescharacteristic
279Dic47—22no—
280Dic47—22no—
281Dic47—22no—
282Dic47—22no—
283Dic47—81no—
284Dic47—81no—
285Dic47—81no—
286Dic47—81no—
287C2×#424—471no—
288#2314—471no—
289#2314—471no—
290#1949—471no—
291#1949—471no—
292#1949—471no—
293#1949—471no—
294#1948—471no—
295#1948—471no—
296#1947—471no—
297#1947—471no—
298#1809—471no—
299#1809D4711yescharacteristic
300#1808—471no—
301#1807—471no—
302C2×C47—42no—
303C2×C47#231611yescharacteristic
304C2×C47#291211yescharacteristic
305C2×C47#291311yescharacteristic
306C2×C47#325711yescharacteristic
307C2×C47#325711yescharacteristic
308C2×C47#325811yescharacteristic
309C2×C47#325811yescharacteristic
310C22×M16C2×D4711yescharacteristic
311C23×C8—471no—
312C2×#117—471no—
313C2×#107—471no—
314C2×#107—471no—
315C2×#107—471no—
316C2×#107C2×D4711yescharacteristic
317C2×#106—471no—
318C2×#106—471no—
319C2×#106—471no—
320C2×#106C2×D4711yescharacteristic
321#424—478no—
322C2×C4⋊C8—471no—
323C2×C4⋊C8—471no—
324C2×C4⋊C8—471no—
325C2×C4⋊C8—471no—
326#388—941no—
327#388—941no—
328#388—941no—
329#388—941no—
330#387—941no—
331#387—941no—
332#387—941no—
333#387—941no—
334#384—471no—
335#384—471no—
336#382—471no—
337#382—471no—
338#380—471no—
339#380—471no—
340#380C2×D4711yescharacteristic
341#380C2×D4711yescharacteristic
342C4×C4⋊C4—471no—
343C4×C4⋊C4—471no—
344C4×C4⋊C4—471no—
345C4×C4⋊C4—471no—
346#340—471no—
347#340—471no—
348#340—471no—
349#340—471no—
350#340—471no—
351#340—471no—
352#340C2×D4711yescharacteristic
353#340C2×D4711yescharacteristic
354C47#2196611yescharacteristic, complemented, Sylow
355C23×C4C22×D4711yescharacteristic
356C2×M16—41no—
357C2×M16—41no—
358C2×M16C4×D4714yes—
359C22×C8—471no—
360C22×C8—471no—
361C22×C8—474no—
362C22×C8—1881no—
363C22×C8—1881no—
364C22×C8C22×D4711yescharacteristic
365C22×C8C22×D4711yescharacteristic
366#117—474no—
367#117—474no—
368C2×C4⋊C4—471no—
369C2×C4⋊C4—471no—
370C2×C4⋊C4—471no—
371C2×C4⋊C4—471no—
372C2×C4⋊C4—471no—
373C2×C4⋊C4—471no—
374C2×C4⋊C4—941no—
375C2×C4⋊C4—941no—
376C2×C4⋊C4—941no—
377C2×C4⋊C4—941no—
378C2×C4⋊C4C22×D4711yescharacteristic
379C2×C4⋊C4C22×D4711yescharacteristic
380C2×C4⋊C4C22×D4711yescharacteristic
381C2×C4⋊C4C22×D4711yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—471no—
385C2×C22⋊C4—471no—
386C2×C22⋊C4—941no—
387C2×C22⋊C4—941no—
388C2×C42—471no—
389C2×C42—471no—
390C2×C42—941no—
391C2×C42—941no—
392C2×C42—941no—
393C2×C42—941no—
394#107—41no—
395#107—1881no—
396#107—1881no—
397#107—1881no—
398#106—41no—
399#106—1881no—
400#106—1881no—
401#106—1881no—
402C4⋊C8—474no—
403C4⋊C8—474no—
404C4⋊C8—474no—
405C4⋊C8—474no—
406#95—21no—
407#95—21no—
408#95—941no—
409#95—941no—
410#95—941no—
411#95—941no—
412#95—941no—
413#95—941no—
414#95—941no—
415#95—941no—
416#95—941no—
417#95—941no—
418C24D4×D4711yescharacteristic
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—471no—
428C22×C4—471no—
429C22×C4—471no—
430C22×C4—471no—
431C22×C4—941no—
432C22×C4—941no—
433C22×C4—941no—
434C22×C4—941no—
435C22×C4C2×C4×D4712yes—
436C22×C4D4×D4711yescharacteristic
437C22×C4Q8×D4712yes—
438C22×C4C23×D4711yescharacteristic
439M16—44no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—472no—
445C2×C8—472no—
446C2×C8—941no—
447C2×C8—941no—
448C2×C8—941no—
449C2×C8—941no—
450C2×C8—1884no—
451C2×C8C2×C4×D4712yes—
452C2×C8C2×C4×D4712yes—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—474no—
462C4⋊C4—474no—
463C4⋊C4—941no—
464C4⋊C4—941no—
465C4⋊C4—941no—
466C4⋊C4—941no—
467C4⋊C4—941no—
468C4⋊C4—941no—
469C4⋊C4—941no—
470C4⋊C4—941no—
471C4⋊C4—3761no—
472C4⋊C4—3761no—
473C4⋊C4—3761no—
474C4⋊C4—3761no—
475C22⋊C4—81no—
476C22⋊C4—81no—
477C22⋊C4—474no—
478C22⋊C4—474no—
479C22⋊C4—3761no—
480C22⋊C4—3761no—
481C42—472no—
482C42—472no—
483C42—472no—
484C42—472no—
485C42—3761no—
486C42—3761no—
487C42—3761no—
488C42—3761no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23C4⋊C4×D4712yes—
496C23C2×D4×D4711yescharacteristic
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—472no—
516C2×C4—472no—
517C2×C4—472no—
518C2×C4—472no—
519C2×C4—941no—
520C2×C4—941no—
521C2×C4—941no—
522C2×C4—941no—
523C2×C4—941no—
524C2×C4—941no—
525C2×C4—941no—
526C2×C4—941no—
527C2×C4—1881no—
528C2×C4—1881no—
529C2×C4—1881no—
530C2×C4—1881no—
531C2×C4—1881no—
532C2×C4—1881no—
533C2×C4—1881no—
534C2×C4—1881no—
535C2×C4—1881no—
536C2×C4—1881no—
537C2×C4—1881no—
538C2×C4—1881no—
539C2×C4C4⋊C4×D4712yes—
540C2×C4C22×C4×D4711yescharacteristic
541C2×C4C2×Q8×D4711yescharacteristic
542C2×C4#1309406911yescharacteristic
543C2×C4#1309406911yescharacteristic
544C2×C4#1309407111yescharacteristic
545C2×C4#1309407111yescharacteristic
546C8—22no—
547C8—22no—
548C8—942no—
549C8—942no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22C2×C4⋊C4×D4711yescharacteristic
558C22#1568970011yescharacteristic
559C22#1568970011yescharacteristic
560C22#1568992711yescharacteristic
561C22#1568993211yescharacteristic
562C22#1568993311yescharacteristic
563C22#1568993911yescharacteristic
564C4—22no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—942no—
570C4—942no—
571C4—942no—
572C4—942no—
573C4—3761no—
574C4—3761no—
575C4—3761no—
576C4—3761no—
577C4#1568959611yescharacteristic
578C4#1568960011yescharacteristic
579C4#1568975211yescharacteristic
580C4#1568975211yescharacteristic
581C2—42no—
582C2#2024600511yescharacteristic
583C2#2025198211yescharacteristic
584C2#2025198711yescharacteristic
585C2#2025479411yescharacteristic
586C2#2025480011yescharacteristic
587C2#2025480911yescharacteristic
588C2#2025481511yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1809C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenter#1809D47
hypocenterC47#21966
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
4. #20244159C2semidirect
5. #20244157C2semidirect
6. #20244155C2semidirect
7. #20244153C2semidirect
8. #20244150C2semidirect
9. #20244148C2semidirect
10. #20244145C2semidirect
11. #20244143C2semidirect
24. #15689151C4semidirect
354. C47#21966semidirect

Maximal subgroups

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

subgroupindex
C2×#156891512
#202459952
#202441592
#202441572
#202441552
#202441532
#202441502
#202441482
#202441452
#202441432
#202434212
#202434152
#202434112
C47×#18092
#2196647