Finite Group Databasebeta

← #26377226

Subgroups — #26377226

Order 12032.

3106 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689605C211yesmaximal, characteristic
2C2×#15689605C211yesmaximal, characteristic
3C2×C4×#10625354C211yesmaximal, characteristic
4#20246309C211yesmaximal, characteristic, complemented
5#20246306C211yesmaximal, characteristic, complemented
6Q8×Dic47⋊C4C211yesmaximal, characteristic, complemented
7#20246299C213yesmaximal, complemented
8#20246297C213yesmaximal, complemented
9#20246289C213yesmaximal, complemented
10#20246288C213yesmaximal, complemented
11#20246009C213yesmaximal, complemented
12#20246003C213yesmaximal, complemented
13#20245868C213yesmaximal, complemented
14#389×D47C213yesmaximal, complemented
15Q8×C4⋊C4×C47C211yesmaximal, characteristic, complemented
16C22×#10625354C2211yescharacteristic
17#15689605—28no—
18#15689605C4116yescomplemented
19C4×#10625354—28no—
20C2×Q8×Dic47C2211yescharacteristic
21C2×Q8×Dic47C2211yescharacteristic
22C2×Q8×Dic47C2211yescharacteristic
23C2×C4×D188C2213yes—
24C2×C4×D188C2213yes—
25C2×C4×D188C2213yes—
26C2×#13093986C2213yes—
27C2×#13093986C2213yes—
28C2×#13093986C2213yes—
29C2×#13093982C2213yes—
30C2×#13093981C2213yes—
31C2×#13093981C2213yes—
32C2×C4⋊C4×D47C2213yes—
33C2×C4⋊C4×D47C2216yes—
34C2×C4⋊C4×D47C2213yes—
35C2×C42×D47C2213yes—
36#15689022C2213yes—
37#15689022C2213yes—
38#15689021C2213yes—
39#15689021C2213yes—
40#15689020C2213yes—
41#15689020C2213yes—
42#15689019C2213yes—
43#15689019C2213yes—
44#15689018C2216yes—
45#15689018C2211yescharacteristic
46#15689018C2213yes—
47#15688997C2213yes—
48#15688997C2213yes—
49#15688994C2213yes—
50#15688994C2216yes—
51#15688994C2216yes—
52#15688994C2213yes—
53#15688992C2213yes—
54#15688989C2213yes—
55#15688988C2213yes—
56#15688982C2216yes—
57#15688980C2216yes—
58C4×Dic47⋊C4C2213yes—
59C4×Dic47⋊C4C2213yes—
60C4×C4⋊Dic47C2213yes—
61#95×D47C2216yes—
62C2×C4×Q8×C47C2211yescharacteristic
63C2×C4×Q8×C47C2211yescharacteristic
64C2×C4×Q8×C47C2211yescharacteristic
65C47×#389C2213yes—
66C47×#389C2213yes—
67C47×#384C2213yes—
68C4×C4⋊C4×C47C2213yes—
69C2×#10625354C2×C418yes—
70C2×#10625354D418yes—
71C2×#10625354Q818yes—
72C22×D188C2313yes—
73C22×C4×D47C2313yes—
74C22×C4×D47C2313yes—
75C22×C4×D47C2316yes—
76C22×C4×D47C2316yes—
77C22×C4×D47C2313yes—
78Q8×Dic47—24no—
79Q8×Dic47—24no—
80Q8×Dic47C2×C418yescomplemented
81C4×D188—212no—
82C4×D188—212no—
83C4×D188C2×C4124yescomplemented
84#13093986—212no—
85#13093986—212no—
86#13093986C2×C4124yescomplemented
87#13093982—212no—
88#13093981—212no—
89#13093981C2×C4124yescomplemented
90C2×C4⋊Dic47C2313yes—
91C2×C4⋊Dic47C2313yes—
92C2×C4⋊Dic47C2313yes—
93C2×C4⋊Dic47C2313yes—
94C2×C4⋊Dic47C2313yes—
95C2×Dic47⋊C4C2311yescharacteristic
96C2×Dic47⋊C4C2313yes—
97C2×Dic47⋊C4C2313yes—
98C2×Dic47⋊C4C2311yescharacteristic
99C2×Dic47⋊C4C2313yes—
100C2×Dic47⋊C4C2313yes—
101C2×Dic47⋊C4C2316yes—
102C2×Dic47⋊C4C2311yescharacteristic
103C2×Dic47⋊C4C2313yes—
104C4⋊C4×D47—212no—
105C4⋊C4×D47—412no—
106C4⋊C4×D47C2×C4124yescomplemented
107C2×(C2×D47)⋊C4C2313yes—
108C2×(C2×D47)⋊C4C2316yes—
109C2×(C2×D47)⋊C4C2313yes—
110C2×(C2×D47)⋊C4C2313yes—
111C2×(C2×D47)⋊C4C2313yes—
112C2×C4×Dic47C2313yes—
113C2×C4×Dic47C2313yes—
114C2×C4×Dic47C2313yes—
115C2×C4×Dic47C2311yescharacteristic
116C2×C4×Dic47C2313yes—
117C42×D47—212no—
118C47⋊#95C2313yes—
119C47⋊#95C2316yes—
120C47⋊#95C2316yes—
121C47⋊#95C2311yescharacteristic
122C47⋊#95C2313yes—
123C47⋊#95C2311yescharacteristic
124C47⋊#95C2313yes—
125C47⋊#95C2313yes—
126C22×Q8×C47C2311yescharacteristic
127C4×Q8×C47—24no—
128C4×Q8×C47—24no—
129C4×Q8×C47C2×C418yescomplemented
130C2×C4⋊C4×C47C2313yes—
131C2×C4⋊C4×C47C2313yes—
132C2×C4⋊C4×C47C2311yescharacteristic
133C2×C4⋊C4×C47C2316yes—
134C2×C4⋊C4×C47C2313yes—
135C2×C4⋊C4×C47C2313yes—
136C2×C42×C47C2313yes—
137C2×C42×C47C2313yes—
138C2×C42×C47C2313yes—
139#95×C47C2313yes—
140#95×C47C2313yes—
141C23×D47C2413yes—
142C2×Q8×C47C22×C414yes—
143C2×Q8×C47C2×D414yes—
144C2×Q8×C47C2×Q814yes—
145C22×C4×C47C2411yescharacteristic
146C22×C4×C47C2413yes—
147C22×C4×C47C2413yes—
148C22×C4×C47C2411yescharacteristic
149C22×C4×C47C2413yes—
150C22×C4×C47C2411yescharacteristic
151C22×C4×C47C2413yes—
152C22×Dic47C2413yes—
153C22×Dic47C2411yescharacteristic
154C22×Dic47C2413yes—
155C22×Dic47C2411yescharacteristic
156C22×Dic47C2413yes—
157C22×Dic47C2411yescharacteristic
158C22×Dic47C2411yescharacteristic
159#10625354C4⋊C4164yescomplemented
160C2×D188C22×C4112yes—
161C2×D188C2×D4112yes—
162C2×D188C2×Q8112yes—
163C2×C4×D47—26no—
164C2×C4×D47—26no—
165C2×C4×D47—26no—
166C2×C4×D47—26no—
167C2×C4×D47—212no—
168C2×C4×D47—46no—
169C2×C4×D47—46no—
170C2×C4×D47—43no—
171C2×C4×D47—43no—
172C2×C4×D47—46no—
173C2×C4×D47—46no—
174C2×C4×D47C22×C4124yes—
175C2×C4×D47C22×C4112yes—
176C2×C4×D47C2×D4112yes—
177C2×C4×D47C2×Q8112yes—
178C4⋊C4×C47—26no—
179C4⋊C4×C47—26no—
180C4⋊C4×C47—22no—
181C4⋊C4×C47—43no—
182C4⋊C4×C47—46no—
183C4⋊C4×C47C22×C4112yes—
184C42×C47—26no—
185C42×C47—26no—
186C42×C47C22×C4112yes—
187(C2×D47)⋊C4—26no—
188(C2×D47)⋊C4—26no—
189(C2×D47)⋊C4—212no—
190(C2×D47)⋊C4C22×C4112yes—
191(C2×D47)⋊C4C22×C4112yes—
192C4⋊Dic47—26no—
193C4⋊Dic47—26no—
194C4⋊Dic47—26no—
195C4⋊Dic47—43no—
196C4⋊Dic47C22×C4112yes—
197Dic47⋊C4—22no—
198Dic47⋊C4—26no—
199Dic47⋊C4—22no—
200Dic47⋊C4—26no—
201Dic47⋊C4—46no—
202Dic47⋊C4—43no—
203Dic47⋊C4—43no—
204Dic47⋊C4C22×C4112yes—
205Dic47⋊C4C22×C414yes—
206C4×Dic47—26no—
207C4×Dic47—26no—
208C4×Dic47—22no—
209C4×Dic47—26no—
210C4×Dic47C22×C4112yes—
211C23×C47C2511yescharacteristic
212C22×D47—23no—
213C22×D47—23no—
214C22×D47—23no—
215C22×D47—23no—
216C22×D47C23×C416yes—
217C22×D47C22×D416yes—
218C22×D47C22×Q816yes—
219Q8×C47C2×C4⋊C4116yescomplemented
220C2×C4×C47—21no—
221C2×C4×C47—23no—
222C2×C4×C47—21no—
223C2×C4×C47—23no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—23no—
227C2×C4×C47—21no—
228C2×C4×C47—23no—
229C2×C4×C47—23no—
230C2×C4×C47—23no—
231C2×C4×C47—21no—
232C2×C4×C47—21no—
233C2×C4×C47—23no—
234C2×C4×C47—23no—
235C2×C4×C47C23×C412yes—
236C2×C4×C47C23×C416yes—
237C2×C4×C47C23×C416yes—
238C2×C4×C47C22×D416yes—
239C2×C4×C47C22×Q816yes—
240C2×C4×C472−1+412yes—
241C2×Dic47—21no—
242C2×Dic47—23no—
243C2×Dic47—21no—
244C2×Dic47—21no—
245C2×Dic47—23no—
246C2×Dic47—21no—
247C2×Dic47—21no—
248C2×Dic47—23no—
249C2×Dic47—23no—
250C2×Dic47—23no—
251C2×Dic47—21no—
252C2×Dic47—23no—
253C2×Dic47—23no—
254C2×Dic47—23no—
255C2×Dic47—21no—
256C2×Dic47C23×C412yes—
257C2×Dic47C23×C416yes—
258C2×Dic47C23×C412yes—
259C2×Dic47C22×D412yes—
260C2×Dic47C22×Q812yes—
261C2×Dic47H(5,𝔽2)12yes—
262D188C2×C4⋊C4148yescomplemented
263C4×D47—424no—
264C4×D47—412no—
265C4×D47—412no—
266C4×D47—812no—
267C4×D47C2×C4⋊C4148yescomplemented
268#34250—471nomaximal, Sylow
269C22×C47C24×C411yescharacteristic
270C22×C47C23×D411yescharacteristic
271C22×C47C23×Q811yescharacteristic
272C22×C47C2×H(5,𝔽2)11yescharacteristic
273C22×C47C2×2−1+411yescharacteristic
274C22×C47#58511yescharacteristic
275C22×C47#58511yescharacteristic
276C2×D47—26no—
277C2×D47—26no—
278C2×D47—26no—
279C2×D47—26no—
280C2×D47—26no—
281C2×D47—26no—
282C2×D47C22×C4⋊C4112yes—
283C4×C47—26no—
284C4×C47—22no—
285C4×C47—22no—
286C4×C47—43no—
287C4×C47—43no—
288C4×C47—41no—
289C4×C47C22×C4⋊C4112yes—
290Dic47—26no—
291Dic47—22no—
292Dic47—22no—
293Dic47—43no—
294Dic47—43no—
295Dic47—41no—
296Dic47C22×C4⋊C414yes—
297C2×#520—471no—
298C2×#520—471no—
299C2×C4×#41—471no—
300#2351—471no—
301#2351—471no—
302Q8×C4⋊C4—471no—
303Q8×C4⋊C4D4711yescharacteristic, complemented
304#2349—473no—
305#2349—473no—
306D4×C4⋊C4—473no—
307D4×C4⋊C4—473no—
308#2317—473no—
309#2316—473no—
310#2296—473no—
311C2×#389—473no—
312C2×C47C23×C4⋊C411yescharacteristic
313C2×C47C4×H(5,𝔽2)11yescharacteristic
314C2×C47C4×2−1+411yescharacteristic
315C2×C47#348811yescharacteristic
316C2×C47#348911yescharacteristic
317C2×C47#350311yescharacteristic
318C2×C47#350411yescharacteristic
319D47—212no—
320C22×#41—471no—
321#520—4716no—
322#520—948no—
323C4×#41—948no—
324C2×C4×Q8—471no—
325C2×C4×Q8—471no—
326C2×C4×Q8—471no—
327C2×C4×Q8C2×D4711yescharacteristic
328C2×C4×Q8C2×D4711yescharacteristic
329C2×C4×Q8C2×D4711yescharacteristic
330C2×C4×D4—473no—
331C2×C4×D4—473no—
332C2×C4×D4—473no—
333C2×C4×D4—473no—
334C2×C4×D4—473no—
335C2×C4×D4—473no—
336C2×#117—473no—
337C2×#117—473no—
338C2×#117—473no—
339C22×C4⋊C4—473no—
340C22×C4⋊C4—473no—
341C22×C4⋊C4—476no—
342C22×C42—473no—
343#389—476no—
344#389—473no—
345#389—473no—
346#389—473no—
347#389—473no—
348#389—473no—
349#389—471no—
350#389—473no—
351#389—473no—
352#389—473no—
353#389—473no—
354#389C2×D4713yes—
355#389C2×D4713yes—
356#385—476no—
357#385—476no—
358#385—473no—
359#385—473no—
360#385—473no—
361#385—473no—
362#384—473no—
363#384C2×D4713yes—
364#382—473no—
365#382—473no—
366#380—476no—
367#380—476no—
368C4×C4⋊C4—473no—
369C4×C4⋊C4—473no—
370C4×C4⋊C4—473no—
371C4×C4⋊C4C2×D4713yes—
372C2×#95—476no—
373C47#3425011yescharacteristic, complemented, Sylow
374C2×#41—478no—
375C2×#41—478no—
376C2×#41—478no—
377C22×Q8C22×D4711yescharacteristic
378C22×D4—473no—
379C23×C4—473no—
380C23×C4—476no—
381C23×C4—473no—
382C23×C4—473no—
383C23×C4—476no—
384C4×Q8—24no—
385C4×Q8—24no—
386C4×Q8—478no—
387C4×Q8—944no—
388C4×Q8—944no—
389C4×Q8C4×D4718yescomplemented
390C4×D4—4724no—
391C4×D4—4724no—
392C4×D4—9412no—
393C4×D4—9412no—
394C4×D4—9412no—
395C4×D4—9412no—
396#117—4724no—
397#117—9412no—
398#117—9412no—
399C2×C4⋊C4—473no—
400C2×C4⋊C4—473no—
401C2×C4⋊C4—471no—
402C2×C4⋊C4—4724no—
403C2×C4⋊C4—473no—
404C2×C4⋊C4—473no—
405C2×C4⋊C4—473no—
406C2×C4⋊C4—473no—
407C2×C4⋊C4—471no—
408C2×C4⋊C4—473no—
409C2×C4⋊C4—471no—
410C2×C4⋊C4—476no—
411C2×C4⋊C4—473no—
412C2×C4⋊C4—473no—
413C2×C4⋊C4—473no—
414C2×C4⋊C4—9412no—
415C2×C4⋊C4—18812no—
416C2×C4⋊C4C22×D4713yes—
417C2×C4⋊C4C22×D4713yes—
418C2×C4⋊C4C22×D4711yescharacteristic
419C2×C4⋊C4C22×D4716yes—
420C2×C4⋊C4C22×D4713yes—
421C2×C4⋊C4C22×D4713yes—
422C2×C22⋊C4—473no—
423C2×C22⋊C4—473no—
424C2×C22⋊C4—476no—
425C2×C22⋊C4—473no—
426C2×C22⋊C4—473no—
427C2×C42—473no—
428C2×C42—473no—
429C2×C42—473no—
430C2×C42—473no—
431C2×C42—471no—
432C2×C42—9412no—
433C2×C42C22×D4713yes—
434C2×C42C22×D4713yes—
435C2×C42C22×D4713yes—
436#95—473no—
437#95—476no—
438#95—476no—
439#95—471no—
440#95—471no—
441#95—473no—
442#95—473no—
443#95—473no—
444#95C22×D4713yes—
445#95C22×D4713yes—
446C24—473no—
447#41—4764no—
448C2×Q8C2×C4×D4714yes—
449C2×Q8D4×D4714yes—
450C2×Q8Q8×D4714yes—
451C2×D4—4712no—
452C2×D4—4712no—
453C2×D4—4712no—
454C22×C4—471no—
455C22×C4—4712no—
456C22×C4—4712no—
457C22×C4—4712no—
458C22×C4—471no—
459C22×C4—473no—
460C22×C4—471no—
461C22×C4—4724no—
462C22×C4—473no—
463C22×C4—471no—
464C22×C4—473no—
465C22×C4—946no—
466C22×C4—9412no—
467C22×C4—946no—
468C22×C4—946no—
469C22×C4—946no—
470C22×C4—1886no—
471C22×C4—1886no—
472C22×C4—1886no—
473C22×C4—1886no—
474C22×C4—1883no—
475C22×C4—1883no—
476C22×C4C23×D4713yes—
477C22×C4C23×D4711yescharacteristic
478C22×C4C23×D4713yes—
479C22×C4C23×D4711yescharacteristic
480C22×C4C23×D4713yes—
481C22×C4C23×D4711yescharacteristic
482C22×C4C23×D4713yes—
483C4⋊C4—26no—
484C4⋊C4—26no—
485C4⋊C4—22no—
486C4⋊C4—46no—
487C4⋊C4—43no—
488C4⋊C4—4712no—
489C4⋊C4—474no—
490C4⋊C4—4712no—
491C4⋊C4—946no—
492C4⋊C4—946no—
493C4⋊C4—942no—
494C4⋊C4—946no—
495C4⋊C4—946no—
496C4⋊C4—946no—
497C4⋊C4—942no—
498C4⋊C4—1883no—
499C4⋊C4—1883no—
500C4⋊C4—1886no—
501C4⋊C4—1883no—
502C4⋊C4C2×C4×D47112yes—
503C22⋊C4—4712no—
504C22⋊C4—4712no—
505C22⋊C4—946no—
506C22⋊C4—9412no—
507C22⋊C4—946no—
508C42—26no—
509C42—26no—
510C42—4712no—
511C42—942no—
512C42—946no—
513C42—946no—
514C42—946no—
515C42C2×C4×D47112yes—
516C23—476no—
517C23—476no—
518C23—476no—
519C23—943no—
520C23—943no—
521C23—943no—
522C23—943no—
523C23C24×D4711yescharacteristic
524Q8C4⋊C4×D47116yescomplemented
525D4—4748no—
526C2×C4—21no—
527C2×C4—23no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—23no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—23no—
534C2×C4—23no—
535C2×C4—23no—
536C2×C4—23no—
537C2×C4—21no—
538C2×C4—23no—
539C2×C4—23no—
540C2×C4—21no—
541C2×C4—472no—
542C2×C4—472no—
543C2×C4—472no—
544C2×C4—4748no—
545C2×C4—472no—
546C2×C4—476no—
547C2×C4—472no—
548C2×C4—941no—
549C2×C4—943no—
550C2×C4—941no—
551C2×C4—941no—
552C2×C4—941no—
553C2×C4—943no—
554C2×C4—941no—
555C2×C4—943no—
556C2×C4—943no—
557C2×C4—943no—
558C2×C4—941no—
559C2×C4—943no—
560C2×C4—943no—
561C2×C4—941no—
562C2×C4—943no—
563C2×C4—37612no—
564C2×C4—18824no—
565C2×C4—18812no—
566C2×C4—18812no—
567C2×C4C22×C4×D4712yes—
568C2×C4C22×C4×D4716yes—
569C2×C4C22×C4×D4716yes—
570C2×C4C2×D4×D4716yes—
571C2×C4C2×Q8×D4716yes—
572C2×C4#1309407212yes—
573C22—4712no—
574C22—946no—
575C22—946no—
576C22—946no—
577C22—946no—
578C22—946no—
579C22—946no—
580C22C23×C4×D4711yescharacteristic
581C22C22×D4×D4711yescharacteristic
582C22C22×Q8×D4711yescharacteristic
583C22H(5,𝔽2)×D4711yescharacteristic
584C22C2×#1309407211yescharacteristic
585C22#1568995911yescharacteristic
586C22#1568996011yescharacteristic
587C4—22no—
588C4—26no—
589C4—22no—
590C4—43no—
591C4—43no—
592C4—41no—
593C4—474no—
594C4—942no—
595C4—946no—
596C4—942no—
597C4—1883no—
598C4—1883no—
599C4—1881no—
600C4C2×C4⋊C4×D47112yes—
601C2—9412no—
602C2C22×C4⋊C4×D4711yescharacteristic
603C2#2025659411yescharacteristic
604C2C4×#1309407211yescharacteristic
605C2#2025705611yescharacteristic
606C2#2025706511yescharacteristic
607C2#2025720611yescharacteristic
608C2#2025721311yescharacteristic
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingQ8×C4⋊C4×C47C2
socle, cosocleC23×C47C25
prime radicalC22×#10625354C22
hypercenterQ8×C4⋊C4D47
hypocenterC47#34250
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #20246309C2semidirect
5. #20246306C2semidirect
6. Q8×Dic47⋊C4C2semidirect
7. #20246299C2semidirect
8. #20246297C2semidirect
9. #20246289C2semidirect
10. #20246288C2semidirect
11. #20246009C2semidirect
12. #20246003C2semidirect
13. #20245868C2semidirect
14. #389×D47C2semidirect
15. Q8×C4⋊C4×C47C2semidirect
18. #15689605C4semidirect
80. Q8×Dic47C2×C4semidirect
83. C4×D188C2×C4semidirect
86. #13093986C2×C4semidirect
89. #13093981C2×C4semidirect
106. C4⋊C4×D47C2×C4semidirect
129. C4×Q8×C47C2×C4semidirect
159. #10625354C4⋊C4semidirect
219. Q8×C47C2×C4⋊C4semidirect
262. D188C2×C4⋊C4semidirect
267. C4×D47C2×C4⋊C4semidirect
303. Q8×C4⋊C4D47semidirect
373. C47#34250semidirect
389. C4×Q8C4×D47semidirect
524. Q8C4⋊C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156896052
C2×C4×#106253542
#202463092
#202463062
Q8×Dic47⋊C42
#202462992
#202462972
#202462892
#202462882
#202460092
#202460032
#202458682
#389×D472
Q8×C4⋊C4×C472
#3425047