Finite Group Databasebeta

← #26250237

Subgroups — #26250237

Order 12032.

1134 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689833C211yesmaximal, characteristic
2C2×#15689418C211yesmaximal, characteristic, complemented
3#20248306C211yesmaximal, characteristic
4#20244589C212yesmaximal, complemented
5#20244395C212yesmaximal, complemented
6#20244349C211yesmaximal, characteristic, complemented
7#20244182C211yesmaximal, characteristic
8#20243692C212yesmaximal, complemented
9#20243574C212yesmaximal, complemented
10#20243393C211yesmaximal, characteristic, complemented
11C2×C4×Q16×C47C211yesmaximal, characteristic, complemented
12#15689833—24no—
13#15689833—24no—
14C2×#13094024C2211yescharacteristic
15C2×#13094016—21no—
16C2×#13094002—22no—
17C2×#13094000C2212yes—
18C2×#13094000C2212yes—
19C2×C4×D188—21no—
20C2×C4×D188C2211yescharacteristic
21C2×C4⋊C4×D47C2212yes—
22#15689418—24no—
23C2×#13093957C2212yes—
24C2×C4⋊(C47⋊C8)C2211yescharacteristic
25C2×#13093948C2212yes—
26C2×#13093948C2212yes—
27C2×(C2×D47)⋊C8C2212yes—
28C2×(C2×D47)⋊C8C2212yes—
29#15689045—22no—
30#15689025—21no—
31#15689021—21no—
32#15688993C2211yescharacteristic
33#15688980C2212yes—
34#15688886C2212yes—
35#15688886C2212yes—
36#15688885C2212yes—
37C22×Q16×C47C2211yescharacteristic
38C2×C4×Q8×C47C2211yescharacteristic
39C2×C4×Q8×C47C2211yescharacteristic
40C4×Q16×C47—24no—
41C2×#107×C47C2211yescharacteristic
42C2×#103×C47C2211yescharacteristic
43C2×#103×C47C2211yescharacteristic
44C2×C4×C8×C47C2211yescharacteristic
45C22×Dic94—21no—
46C22×D188D411yescharacteristic
47C22×C4×D47—22no—
48C22×C4×D47C2312yes—
49C22×C47⋊C8C2312yes—
50#13094024—22no—
51#13094024D414yes—
52#13094016—28no—
53#13094002—216no—
54#13094000—24no—
55#13094000—24no—
56#13094000—24no—
57#13094000—42no—
58C4×D188—28no—
59C4×D188—22no—
60C4×D188—22no—
61C2×C4⋊Dic47—21no—
62C2×C4⋊Dic47—21no—
63C2×C4⋊Dic47—21no—
64C2×C4⋊Dic47D411yescharacteristic
65C2×C4⋊Dic47C2312yes—
66C2×C4⋊Dic47C2312yes—
67C2×Dic47⋊C4—22no—
68C2×Dic47⋊C4—22no—
69C2×Dic47⋊C4—22no—
70C4⋊C4×D47—24no—
71C4⋊C4×D47—24no—
72C2×(C2×D47)⋊C4—22no—
73C2×(C2×D47)⋊C4—22no—
74C2×(C2×D47)⋊C4—22no—
75C2×(C2×D47)⋊C4D412yes—
76#13093957—24no—
77C4⋊(C47⋊C8)—22no—
78#13093948—24no—
79#13093948—42no—
80(C2×D47)⋊C8—24no—
81(C2×D47)⋊C8—42no—
82C47⋊#95—22no—
83C22×Q8×C47C2311yescharacteristic
84C22×Q8×C47C2311yescharacteristic
85C2×Q16×C47—22no—
86C2×Q16×C47—22no—
87C2×Q16×C47—22no—
88C22×C8×C47C2311yescharacteristic
89C22×C8×C47C2311yescharacteristic
90C4×Q8×C47—22no—
91C4×Q8×C47—22no—
92C4×Q8×C47—22no—
93C4×Q8×C47D414yes—
94C2×C4⋊C4×C47—21no—
95C2×C4⋊C4×C47—21no—
96C2×C4⋊C4×C47C2311yescharacteristic
97C2×C4⋊C4×C47C2311yescharacteristic
98C2×C42×C47—21no—
99C2×C42×C47—21no—
100C2×C42×C47C2311yescharacteristic
101#107×C47—22no—
102#103×C47—22no—
103#103×C47—22no—
104C4×C8×C47—22no—
105C23×D47C2×D412yes—
106C2×Q8×C47—21no—
107C2×Q8×C47—21no—
108C2×Q8×C47—21no—
109C2×Q8×C47—21no—
110C2×Q8×C47—21no—
111C2×Q8×C47—21no—
112C2×Q8×C47—21no—
113C2×Q8×C47—21no—
114C2×Q8×C47—21no—
115C2×Q8×C47—21no—
116C2×Q8×C47C2×D412yes—
117C2×Q8×C47#4112yes—
118C22×C4×C47—21no—
119C22×C4×C47—21no—
120C22×C4×C47—21no—
121C22×C4×C47—21no—
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47C2×D411yescharacteristic
124C22×C4×C47C2411yescharacteristic
125C22×Dic47—22no—
126C22×Dic47—22no—
127C22×Dic47C2×D412yes—
128C2×D188—22no—
129C2×D188—22no—
130C2×D188Q1614yescomplemented
131C2×C4×D47—22no—
132C2×C4×D47—22no—
133C2×C4×D47—22no—
134C2×C4×D47—22no—
135C2×C4×D47—22no—
136C2×C4×D47—22no—
137C2×C4×D47—28no—
138C2×C4×D47—82no—
139C2×C4×D47—82no—
140C2×Dic94—24no—
141C2×Dic94—81no—
142C2×Dic94—81no—
143Q16×C47—28no—
144C2×C8×C47—21no—
145C2×C8×C47—21no—
146C2×C8×C47—21no—
147C2×C8×C47—21no—
148C2×C8×C47—21no—
149C2×C8×C47#4112yes—
150C4⋊C4×C47—24no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—21no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—42no—
155C4⋊C4×C47C2×D412yes—
156C42×C47—24no—
157C42×C47—42no—
158C42×C47C2×D412yes—
159C42×C47C2×D412yes—
160(C2×D47)⋊C4—28no—
161(C2×D47)⋊C4—24no—
162(C2×D47)⋊C4—44no—
163(C2×D47)⋊C4—82no—
164C4⋊Dic47—24no—
165C4⋊Dic47—22no—
166C4⋊Dic47—22no—
167C4⋊Dic47—22no—
168C4⋊Dic47—22no—
169C4⋊Dic47—42no—
170C4⋊Dic47—81no—
171C4⋊Dic47C2×D414yes—
172Dic47⋊C4—28no—
173Dic47⋊C4—28no—
174Dic47⋊C4—44no—
175C2×C47⋊C8—22no—
176C2×C47⋊C8—22no—
177C2×C47⋊C8—22no—
178C23×C47C22×D411yescharacteristic
179C22×D47—22no—
180C22×D47—22no—
181C22×D47—22no—
182C22×D47—22no—
183C22×D47—22no—
184C22×D47—22no—
185C22×D47C2×Q1614yes—
186Q8×C47—22no—
187Q8×C47—22no—
188Q8×C47—22no—
189Q8×C47—22no—
190Q8×C47—41no—
191Q8×C47—41no—
192Q8×C47—41no—
193Q8×C47—41no—
194C2×C4×C47—21no—
195C2×C4×C47—21no—
196C2×C4×C47—22no—
197C2×C4×C47—22no—
198C2×C4×C47—22no—
199C2×C4×C47—41no—
200C2×C4×C47—41no—
201C2×C4×C47—41no—
202C2×C4×C47—41no—
203C2×C4×C47—41no—
204C2×C4×C47—41no—
205C2×C4×C47—41no—
206C2×C4×C47—41no—
207C2×C4×C47—41no—
208C2×C4×C47#12012yes—
209C2×C4×C47#12012yes—
210C2×C4×C47C2×Q1612yes—
211C2×C4×C47#13712yes—
212C2×C4×C47C22×D411yescharacteristic
213C2×C4×C47C22×D411yescharacteristic
214C2×C4×C47C2×#4111yescharacteristic
215C2×C4×C47C2×#4111yescharacteristic
216C2×C4×C47H(5,𝔽2)11yescharacteristic
217C2×C4×C472−1+411yescharacteristic
218C2×Dic47—22no—
219C2×Dic47—24no—
220C2×Dic47—22no—
221C2×Dic47—22no—
222C2×Dic47—42no—
223C2×Dic47—42no—
224C2×Dic47—42no—
225C2×Dic47—42no—
226C2×Dic47—42no—
227D188—28no—
228C4×D47—42no—
229C4×D47—42no—
230C4×D47—42no—
231C4×D47—42no—
232C4×D47—88no—
233Dic94—84no—
234C8×C47—22no—
235C8×C47—41no—
236C47⋊C8—42no—
237#24122—471nomaximal, Sylow
238C22×C47C2×#12011yescharacteristic
239C22×C47#54611yescharacteristic
240C22×C47#54711yescharacteristic
241C22×C47C22×Q1611yescharacteristic
242C22×C47C2×#13711yescharacteristic
243C22×C47#57611yescharacteristic
244C22×C47#57711yescharacteristic
245C2×D47—24no—
246C2×D47—24no—
247C2×D47—24no—
248C2×D47—42no—
249C2×D47—42no—
250C2×D47—42no—
251C2×D47—42no—
252C4×C47—22no—
253C4×C47—42no—
254C4×C47—42no—
255C4×C47—42no—
256C4×C47—81no—
257C4×C47#45114yes—
258C4×C47#52511yescharacteristic
259C4×C47#53711yescharacteristic
260C4×C47#54811yescharacteristic
261C4×C47#55011yescharacteristic
262Dic47—42no—
263Dic47—44no—
264Dic47—82no—
265C2×#554—471no—
266C2×#477—471no—
267C2×C4×Q16D4711yescharacteristic, complemented
268#2538—471no—
269#2036—472no—
270#2001—472no—
271#1990—471no—
272#1954—471no—
273#1872—472no—
274#1852—472no—
275#1803—471no—
276C2×C47#240711yescharacteristic
277C2×C47C2×#45111yescharacteristic
278C2×C47#301211yescharacteristic
279C2×C47#329911yescharacteristic
280C2×C47#333311yescharacteristic
281C2×C47#333811yescharacteristic
282C2×C47#335611yescharacteristic
283D47—44no—
284C22×Q16C2×D4711yescharacteristic
285#554—944no—
286#554—944no—
287C2×C4⋊Q8—471no—
288C2×#125—941no—
289C2×C22⋊Q8—472no—
290C2×C22⋊Q8—472no—
291C2×C22⋊Q8—942no—
292C2×C4×Q8C2×D4711yescharacteristic
293C2×C4×Q8C2×D4711yescharacteristic
294C2×C4×D4—471no—
295C2×C4×D4—941no—
296C22×C4⋊C4—472no—
297#477—944no—
298C4×Q16—24no—
299C2×#107—472no—
300C2×#107C2×D4711yescharacteristic
301C2×C4⋊C8—471no—
302C2×#103—472no—
303C2×#103—472no—
304C2×#103C2×D4711yescharacteristic
305C2×#103C2×D4711yescharacteristic
306C2×C22⋊C8—472no—
307C2×C22⋊C8—472no—
308C2×C4×C8C2×D4711yescharacteristic
309#396—942no—
310#390—941no—
311#389—941no—
312#384—471no—
313#380—472no—
314#340—472no—
315#340—472no—
316#340—472no—
317C47#2412211yescharacteristic, complemented, Sylow
318C22×Q8—941no—
319C22×Q8C22×D4711yescharacteristic
320C22×Q8C22×D4711yescharacteristic
321C22×D4—471no—
322C23×C4—472no—
323C23×C4—942no—
324C2×Q16—22no—
325C2×Q16—22no—
326C2×Q16—22no—
327C22×C8—472no—
328C22×C8C22×D4711yescharacteristic
329C22×C8C22×D4711yescharacteristic
330C4⋊Q8—474no—
331C4⋊Q8—942no—
332#125—948no—
333C22⋊Q8—9416no—
334C22⋊Q8—944no—
335C22⋊Q8—944no—
336C22⋊Q8—944no—
337C22⋊Q8—1882no—
338C4×Q8—22no—
339C4×Q8—22no—
340C4×Q8—22no—
341C4×Q8C22⋊D4714yes—
342C4×D4—948no—
343C4×D4—942no—
344C4×D4—942no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—472no—
348C2×C4⋊C4—471no—
349C2×C4⋊C4—472no—
350C2×C4⋊C4—944no—
351C2×C4⋊C4—942no—
352C2×C4⋊C4—942no—
353C2×C4⋊C4—941no—
354C2×C4⋊C4—944no—
355C2×C4⋊C4—941no—
356C2×C4⋊C4—942no—
357C2×C4⋊C4—941no—
358C2×C4⋊C4C22×D4711yescharacteristic
359C2×C4⋊C4C22×D4711yescharacteristic
360C2×C22⋊C4—472no—
361C2×C22⋊C4—942no—
362C2×C22⋊C4—942no—
363C2×C22⋊C4—942no—
364C2×C42—21no—
365C2×C42—21no—
366C2×C42C22×D4711yescharacteristic
367#107—22no—
368#107—944no—
369C4⋊C8—942no—
370#103—22no—
371#103—22no—
372#103—944no—
373#103—1882no—
374C22⋊C8—944no—
375C22⋊C8—1882no—
376C4×C8—22no—
377#95—942no—
378C24—472no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—944no—
390C2×Q8—3761no—
391C2×Q8—3761no—
392C2×Q8#1062535012yes—
393C2×Q8C2×C22⋊D4712yes—
394C2×D4—474no—
395C2×D4—942no—
396C2×D4—942no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—472no—
402C22×C4—948no—
403C22×C4—942no—
404C22×C4—942no—
405C22×C4—942no—
406C22×C4—942no—
407C22×C4—942no—
408C22×C4—942no—
409C22×C4—942no—
410C22×C4—942no—
411C22×C4—3762no—
412C22×C4—3762no—
413C22×C4D4×D4711yescharacteristic
414C22×C4D4×D4711yescharacteristic
415C22×C4C23×D4711yescharacteristic
416Q16—28no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—942no—
423C2×C8—942no—
424C2×C8—942no—
425C2×C8#1062535012yes—
426C4⋊C4—24no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—42no—
431C4⋊C4—474no—
432C4⋊C4—948no—
433C4⋊C4—948no—
434C4⋊C4—942no—
435C4⋊C4—944no—
436C4⋊C4—942no—
437C4⋊C4—942no—
438C4⋊C4—942no—
439C4⋊C4—3761no—
440C4⋊C4—1884no—
441C4⋊C4—1882no—
442C4⋊C4C2×C22⋊D4712yes—
443C22⋊C4—948no—
444C22⋊C4—944no—
445C22⋊C4—3762no—
446C22⋊C4—1884no—
447C42—24no—
448C42—42no—
449C42D4×D4712yes—
450C42C2×C22⋊D4712yes—
451C23—474no—
452C23—942no—
453C23—942no—
454C23—942no—
455C23—942no—
456C23—942no—
457C23—942no—
458C23C2×D4×D4711yescharacteristic
459Q8—22no—
460Q8—22no—
461Q8—22no—
462Q8—22no—
463Q8—41no—
464Q8—41no—
465Q8—41no—
466Q8—41no—
467Q8—3764no—
468D4—948no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—942no—
484C2×C4—942no—
485C2×C4—944no—
486C2×C4—942no—
487C2×C4—3768no—
488C2×C4—1882no—
489C2×C4—1882no—
490C2×C4—1882no—
491C2×C4—1882no—
492C2×C4—1882no—
493C2×C4—1882no—
494C2×C4—1882no—
495C2×C4—1882no—
496C2×C4—1882no—
497C2×C4#1309399212yes—
498C2×C4#1309399212yes—
499C2×C4Q16×D4712yes—
500C2×C4#1309405612yes—
501C2×C4C2×D4×D4711yescharacteristic
502C2×C4C22×C22⋊D4711yescharacteristic
503C2×C4C2×#1062535011yescharacteristic
504C2×C4C2×#1062535011yescharacteristic
505C2×C4#1309407011yescharacteristic
506C2×C4#1309407111yescharacteristic
507C8—22no—
508C8—41no—
509C8—1882no—
510C22—944no—
511C22—944no—
512C22—944no—
513C22—1882no—
514C22—1882no—
515C22—1882no—
516C22—1882no—
517C22C2×#1309399211yescharacteristic
518C22#1568976911yescharacteristic
519C22#1568977811yescharacteristic
520C22C2×Q16×D4711yescharacteristic
521C22C2×#1309405611yescharacteristic
522C22#1568992511yescharacteristic
523C22#1568993011yescharacteristic
524C4—22no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—81no—
529C4—3762no—
530C4—1884no—
531C4—1882no—
532C4#1568927714yes—
533C4#1568963411yescharacteristic
534C4#1568970111yescharacteristic
535C4#1568978211yescharacteristic
536C4#1568979411yescharacteristic
537C2—1884no—
538C2#2024673311yescharacteristic
539C2C2×#1568927711yescharacteristic
540C2#2025268811yescharacteristic
541C2#2025526611yescharacteristic
542C2#2025572811yescharacteristic
543C2#2025578211yescharacteristic
544C2#2025600511yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C4×Q16×C47C2
socleC23×C47C22×D4
prime radicalC22×D188D4
hypercenterC2×C4×Q16D47
hypocenterC47#24122
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
2. C2×#15689418C2semidirect
4. #20244589C2semidirect
5. #20244395C2semidirect
6. #20244349C2semidirect
8. #20243692C2semidirect
9. #20243574C2semidirect
10. #20243393C2semidirect
11. C2×C4×Q16×C47C2semidirect
130. C2×D188Q16semidirect
267. C2×C4×Q16D47semidirect
317. C47#24122semidirect

Maximal subgroups

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

subgroupindex
C2×#156898332
C2×#156894182
#202483062
#202445892
#202443952
#202443492
#202441822
#202436922
#202435742
#202433932
C2×C4×Q16×C472
#2412247