Finite Group Databasebeta

← #26279394

Subgroups — #26279394

Order 12032.

828 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689430C211yesmaximal, characteristic, complemented
2C2×C4×#10625336C211yesmaximal, characteristic
3#20247260C211yesmaximal, characteristic
4#20244917C211yesmaximal, characteristic
5#20244865C211yesmaximal, characteristic, complemented
6#20244773C211yesmaximal, characteristic
7#20244726C211yesmaximal, characteristic, complemented
8#20244118C211yesmaximal, characteristic, complemented
9#20243947C211yesmaximal, characteristic, complemented
10#20243743C211yesmaximal, characteristic
11#20243743C211yesmaximal, characteristic
12#20243648C211yesmaximal, characteristic
13#20243642C211yesmaximal, characteristic, complemented
14#20243642C211yesmaximal, characteristic, complemented
15Q8×C4⋊C4×C47C211yesmaximal, characteristic, complemented
16C22×#10625336C2211yescharacteristic
17C2×#13094016C2211yescharacteristic
18C2×C4×D188C2211yescharacteristic
19#15689430—24no—
20C4×#10625336—24no—
21C2×#13093957C2211yescharacteristic
22C2×#13093957C2211yescharacteristic
23C2×#13093957C2211yescharacteristic
24C2×#13093955C2211yescharacteristic
25C2×#13093955C2211yescharacteristic
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27C2×#13093948C2211yescharacteristic
28C2×#13093948C2211yescharacteristic
29C2×#13093948C2211yescharacteristic
30C2×#13093944C2211yescharacteristic
31C2×#13093944C2211yescharacteristic
32C2×#13093944C2211yescharacteristic
33C2×C4×C47⋊C8C2211yescharacteristic
34#15689058—21no—
35#15689047—21no—
36#15689025C2211yescharacteristic
37#15689025C2211yescharacteristic
38#15689021C2211yescharacteristic
39#15689021C2211yescharacteristic
40#15688994—21no—
41#15688982—21no—
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688878C2211yescharacteristic
47#15688878C2211yescharacteristic
48#15688878C2211yescharacteristic
49#15688878C2211yescharacteristic
50C2×C4×Q8×C47C2211yescharacteristic
51C2×C4×Q8×C47C2211yescharacteristic
52C2×C4×Q8×C47C2211yescharacteristic
53C47×#389—21no—
54C47×#389—21no—
55C47×#389C2211yescharacteristic
56C47×#389C2211yescharacteristic
57C47×#384—21no—
58C47×#384C2211yescharacteristic
59C4×C4⋊C4×C47—21no—
60C4×C4⋊C4×C47C2211yescharacteristic
61C22×D188C2311yescharacteristic
62C22×C4×D47—21no—
63C2×#10625336—22no—
64C2×#10625336—22no—
65C2×#10625336Q814yes—
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68C22×C47⋊C8C2311yescharacteristic
69C22×C47⋊C8C2311yescharacteristic
70#13094016—22no—
71#13094016—41no—
72C4×D188—22no—
73C4×D188—41no—
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2311yescharacteristic
76C2×C4⋊Dic47C2311yescharacteristic
77C2×Dic47⋊C4—21no—
78C2×Dic47⋊C4—21no—
79C2×Dic47⋊C4—21no—
80C2×Dic47⋊C4—21no—
81C2×(C2×D47)⋊C4—21no—
82C2×(C2×D47)⋊C4—21no—
83C2×(C2×D47)⋊C4—21no—
84C2×(C2×D47)⋊C4—21no—
85C2×(C2×D47)⋊C4—21no—
86#13093957—22no—
87#13093957—22no—
88#13093957—41no—
89#13093955—22no—
90#13093955—22no—
91C4⋊(C47⋊C8)—22no—
92#13093948—22no—
93#13093948—22no—
94#13093948D414yes—
95#13093944—22no—
96#13093944—22no—
97#13093944—41no—
98C4×C47⋊C8—22no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C22×Q8×C47C2311yescharacteristic
102C4×Q8×C47—22no—
103C4×Q8×C47—22no—
104C4×Q8×C47—22no—
105C4×Q8×C47—22no—
106C4×Q8×C47—41no—
107C4×Q8×C47—41no—
108C2×C4⋊C4×C47—21no—
109C2×C4⋊C4×C47—21no—
110C2×C4⋊C4×C47—21no—
111C2×C4⋊C4×C47—21no—
112C2×C4⋊C4×C47—21no—
113C2×C4⋊C4×C47—21no—
114C2×C4⋊C4×C47—21no—
115C2×C4⋊C4×C47D411yescharacteristic
116C2×C4⋊C4×C47D411yescharacteristic
117C2×C4⋊C4×C47C2311yescharacteristic
118C2×C4⋊C4×C47C2311yescharacteristic
119C2×C4⋊C4×C47C2311yescharacteristic
120C2×C42×C47—21no—
121C2×C42×C47—21no—
122C2×C42×C47—21no—
123C2×C42×C47C2311yescharacteristic
124C2×C42×C47C2311yescharacteristic
125C2×C42×C47C2311yescharacteristic
126#95×C47—21no—
127#95×C47—21no—
128#95×C47D411yescharacteristic
129#95×C47D411yescharacteristic
130C23×D47—21no—
131C2×Q8×C47—21no—
132C2×Q8×C47—21no—
133C2×Q8×C47—21no—
134C2×Q8×C47C2×D412yes—
135C2×Q8×C47C2×Q812yes—
136C2×Q8×C47#4112yes—
137C22×C4×C47—21no—
138C22×C4×C47—21no—
139C22×C4×C47—21no—
140C22×C4×C47—21no—
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47C2×D411yescharacteristic
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2411yescharacteristic
148C22×Dic47—21no—
149C22×Dic47—21no—
150C22×Dic47—21no—
151C2×D188—21no—
152C2×D188—21no—
153C2×D188—21no—
154C2×D188—21no—
155C2×D188—21no—
156C2×D188C2×Q812yes—
157C2×C4×D47—42no—
158C2×C4×D47—42no—
159C2×C4×D47—81no—
160C4⋊C4×C47—24no—
161C4⋊C4×C47—22no—
162C4⋊C4×C47—22no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—21no—
168C4⋊C4×C47—42no—
169C4⋊C4×C47—42no—
170C4⋊C4×C47—42no—
171C4⋊C4×C47—81no—
172C4⋊C4×C47—81no—
173C4⋊C4×C47—81no—
174C4⋊C4×C47#4112yes—
175C42×C47—24no—
176C42×C47—21no—
177C42×C47—21no—
178C42×C47—21no—
179C42×C47—21no—
180C42×C47—21no—
181C42×C47—21no—
182C42×C47—42no—
183C42×C47—42no—
184#10625336—28no—
185(C2×D47)⋊C4—42no—
186(C2×D47)⋊C4—42no—
187(C2×D47)⋊C4—81no—
188(C2×D47)⋊C4—81no—
189(C2×D47)⋊C4—81no—
190C4⋊Dic47—21no—
191C4⋊Dic47—21no—
192C4⋊Dic47—21no—
193C4⋊Dic47—21no—
194C4⋊Dic47—21no—
195C4⋊Dic47C2×D412yes—
196Dic47⋊C4—42no—
197Dic47⋊C4—42no—
198Dic47⋊C4—81no—
199Dic47⋊C4—81no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8—21no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—21no—
206C2×C47⋊C8—21no—
207C2×C47⋊C8—21no—
208C2×C47⋊C8—21no—
209C2×C47⋊C8—21no—
210C2×C47⋊C8C2×D412yes—
211C2×C47⋊C8C2×Q812yes—
212C23×C47C22×D411yescharacteristic
213C22×D47—22no—
214C22×D47—41no—
215C22×D47—41no—
216C22×D47—41no—
217C22×D47—41no—
218C22×D47—41no—
219C22×D47—41no—
220Q8×C47—22no—
221Q8×C47—22no—
222Q8×C47—22no—
223Q8×C47C22⋊Q814yes—
224C2×C4×C47—21no—
225C2×C4×C47—22no—
226C2×C4×C47—21no—
227C2×C4×C47—22no—
228C2×C4×C47—22no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—21no—
232C2×C4×C47—22no—
233C2×C4×C47—21no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—21no—
242C2×C4×C47—21no—
243C2×C4×C47—21no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—41no—
247C2×C4×C47—41no—
248C2×C4×C47—41no—
249C2×C4×C47—41no—
250C2×C4×C47—41no—
251C2×C4×C47—41no—
252C2×C4×C47#13512yes—
253C2×C4×C47#13712yes—
254C2×C4×C47C22×D411yescharacteristic
255C2×C4×C47C22×Q811yescharacteristic
256C2×C4×C47C2×#4111yescharacteristic
257C2×C4×C47C2×#4111yescharacteristic
258C2×C4×C47H(5,𝔽2)11yescharacteristic
259C2×C4×C472−1+411yescharacteristic
260C2×Dic47—22no—
261C2×Dic47—41no—
262C2×Dic47—41no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269D188—22no—
270D188—22no—
271D188—41no—
272D188—41no—
273C4×D47—84no—
274C47⋊C8—22no—
275C47⋊C8—22no—
276C47⋊C8—41no—
277C47⋊C8—41no—
278#26379—471nomaximal, Sylow
279C22×C47#52511yescharacteristic
280C22×C47D4211yescharacteristic
281C22×C47D4×Q811yescharacteristic
282C22×C47C2×#13511yescharacteristic
283C22×C47C2×#13711yescharacteristic
284C22×C47#57611yescharacteristic
285C22×C47#57711yescharacteristic
286C2×D47—42no—
287C2×D47—42no—
288C2×D47—42no—
289C2×D47—81no—
290C2×D47—81no—
291C2×D47—81no—
292C2×D47—81no—
293C4×C47—22no—
294C4×C47—24no—
295C4×C47—22no—
296C4×C47—41no—
297C4×C47—42no—
298C4×C47—41no—
299C4×C47—41no—
300C4×C47—41no—
301C4×C47—81no—
302C4×C47—81no—
303C4×C47C2×C22⋊Q811yescharacteristic
304C4×C47#53911yescharacteristic
305C4×C47#54711yescharacteristic
306C4×C47#55411yescharacteristic
307Dic47—42no—
308Dic47—81no—
309Dic47—81no—
310C2×#479—471no—
311C2×C4×SD16—471no—
312#2454—471no—
313Q8×C4⋊C4D4711yescharacteristic, complemented
314#2093—471no—
315#2084—471no—
316#2067—471no—
317#2059—471no—
318#1942—471no—
319#1916—471no—
320#1880—471no—
321#1880—471no—
322#1864—471no—
323#1863—471no—
324#1863—471no—
325C2×C47#244911yescharacteristic
326C2×C47#310811yescharacteristic
327C2×C47#311911yescharacteristic
328C2×C47#326611yescharacteristic
329C2×C47#328711yescharacteristic
330C2×C47#338111yescharacteristic
331C2×C47#338511yescharacteristic
332D47—82no—
333C22×SD16—471no—
334C2×#125—471no—
335C2×C4×Q8C2×D4711yescharacteristic
336C2×C4×Q8C2×D4711yescharacteristic
337C2×C4×Q8C2×D4711yescharacteristic
338C2×C4×D4—471no—
339#479—944no—
340C4×SD16—944no—
341C2×#107—471no—
342C2×#107—471no—
343C2×#107—471no—
344C2×#106—471no—
345C2×#106—471no—
346C2×C4⋊C8—471no—
347C2×#103—471no—
348C2×#103—471no—
349C2×#103—471no—
350C2×#102—471no—
351C2×#102—471no—
352C2×#102—471no—
353C2×C4×C8—471no—
354#399—941no—
355#396—941no—
356#390—471no—
357#390—471no—
358#389—21no—
359#389—21no—
360#389—471no—
361#389—471no—
362#389C2×D4711yescharacteristic
363#389C2×D4711yescharacteristic
364#385—941no—
365#384—21no—
366#384C2×D4711yescharacteristic
367#380—941no—
368C4×C4⋊C4—21no—
369C4×C4⋊C4C2×D4711yescharacteristic
370#340—471no—
371#340—471no—
372#340—471no—
373#340—471no—
374#336—471no—
375#336—471no—
376#336—471no—
377#336—471no—
378C47#2637911yescharacteristic, complemented, Sylow
379C22×Q8C22×D4711yescharacteristic
380C22×D4—471no—
381C23×C4—941no—
382C2×SD16—474no—
383C2×SD16—942no—
384C2×SD16—942no—
385C22×C8—471no—
386C22×C8—471no—
387C22×C8—471no—
388C22×C8—471no—
389#125—942no—
390#125—1881no—
391C4×Q8—22no—
392C4×Q8—22no—
393C4×Q8—22no—
394C4×Q8—22no—
395C4×Q8—41no—
396C4×Q8—41no—
397C4×D4—942no—
398C4×D4—1881no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—471no—
407C2×C4⋊C4—471no—
408C2×C4⋊C4—471no—
409C2×C4⋊C4—941no—
410C2×C4⋊C4—941no—
411C2×C4⋊C4—941no—
412C2×C4⋊C4—941no—
413C2×C4⋊C4C22⋊D4711yescharacteristic
414C2×C4⋊C4C22⋊D4711yescharacteristic
415C2×C4⋊C4C22×D4711yescharacteristic
416C2×C4⋊C4C22×D4711yescharacteristic
417C2×C4⋊C4C22×D4711yescharacteristic
418C2×C22⋊C4—941no—
419C2×C22⋊C4—941no—
420C2×C22⋊C4—941no—
421C2×C22⋊C4—941no—
422C2×C22⋊C4—941no—
423C2×C42—21no—
424C2×C42—21no—
425C2×C42—21no—
426C2×C42C22×D4711yescharacteristic
427C2×C42C22×D4711yescharacteristic
428C2×C42C22×D4711yescharacteristic
429#107—942no—
430#107—942no—
431#107—1881no—
432#106—942no—
433#106—942no—
434C4⋊C8—942no—
435#103—474no—
436#103—942no—
437#103—942no—
438#102—942no—
439#102—942no—
440#102—1881no—
441C4×C8—942no—
442#95—21no—
443#95—21no—
444#95—941no—
445#95—941no—
446#95C22⋊D4711yescharacteristic
447#95C22⋊D4711yescharacteristic
448C24—941no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8#1062535012yes—
453C2×Q8D4×D4712yes—
454C2×Q8Q8×D4712yes—
455C2×D4—472no—
456C2×D4—941no—
457C2×D4—941no—
458C2×D4—941no—
459C2×D4—941no—
460C2×D4—941no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—941no—
466C22×C4—941no—
467C22×C4—941no—
468C22×C4—3761no—
469C22×C4—1882no—
470C22×C4—1882no—
471C22×C4C2×C22⋊D4711yescharacteristic
472C22×C4C2×C22⋊D4711yescharacteristic
473C22×C4C2×C22⋊D4711yescharacteristic
474C22×C4C2×C22⋊D4711yescharacteristic
475C22×C4C2×C22⋊D4711yescharacteristic
476C22×C4C2×C22⋊D4711yescharacteristic
477C22×C4C23×D4711yescharacteristic
478SD16—948no—
479C2×C8—472no—
480C2×C8—472no—
481C2×C8—941no—
482C2×C8—941no—
483C2×C8—941no—
484C2×C8—941no—
485C2×C8—941no—
486C2×C8—941no—
487C2×C8—941no—
488C2×C8—941no—
489C2×C8—941no—
490C2×C8—941no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—81no—
503C4⋊C4—81no—
504C4⋊C4—81no—
505C4⋊C4—472no—
506C4⋊C4—941no—
507C4⋊C4—941no—
508C4⋊C4—941no—
509C4⋊C4—941no—
510C4⋊C4—941no—
511C4⋊C4—3761no—
512C4⋊C4—3761no—
513C4⋊C4—1882no—
514C4⋊C4—1882no—
515C4⋊C4#1062535012yes—
516C22⋊C4—3761no—
517C22⋊C4—3761no—
518C22⋊C4—3761no—
519C22⋊C4—1882no—
520C22⋊C4—1882no—
521C42—24no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—42no—
529C42—42no—
530C23—942no—
531C23—1881no—
532C23—1881no—
533C23—1881no—
534C23—1881no—
535C23—1881no—
536C23—1881no—
537C23C22×C22⋊D4711yescharacteristic
538Q8—22no—
539Q8—22no—
540Q8—22no—
541Q8#1309400214yes—
542D4—942no—
543D4—942no—
544D4—1881no—
545D4—1881no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—942no—
575C2×C4—3764no—
576C2×C4—1881no—
577C2×C4—1881no—
578C2×C4—1881no—
579C2×C4—1881no—
580C2×C4—1881no—
581C2×C4—1881no—
582C2×C4—1881no—
583C2×C4—1881no—
584C2×C4#1309404412yes—
585C2×C4#1309405512yes—
586C2×C4C2×D4×D4711yescharacteristic
587C2×C4C2×Q8×D4711yescharacteristic
588C2×C4C2×#1062535011yescharacteristic
589C2×C4C2×#1062535011yescharacteristic
590C2×C4#1309406911yescharacteristic
591C2×C4#1309407211yescharacteristic
592C8—942no—
593C8—942no—
594C8—1881no—
595C8—1881no—
596C22—3761no—
597C22—3761no—
598C22—3761no—
599C22—3761no—
600C22—1882no—
601C22—1882no—
602C22—1882no—
603C22#1568963411yescharacteristic
604C22D4×C22⋊D4711yescharacteristic
605C22Q8×C22⋊D4711yescharacteristic
606C22C2×#1309404411yescharacteristic
607C22C2×#1309405511yescharacteristic
608C22#1568992611yescharacteristic
609C22#1568993111yescharacteristic
610C4—22no—
611C4—24no—
612C4—22no—
613C4—41no—
614C4—42no—
615C4—41no—
616C4—41no—
617C4—41no—
618C4—81no—
619C4—81no—
620C4—3761no—
621C4—3761no—
622C4—1882no—
623C4C2×#1309400211yescharacteristic
624C4#1568972011yescharacteristic
625C4#1568977411yescharacteristic
626C4#1568982811yescharacteristic
627C2—3762no—
628C2#2024719711yescharacteristic
629C2#2025346811yescharacteristic
630C2#2025358711yescharacteristic
631C2#2025489911yescharacteristic
632C2#2025514411yescharacteristic
633C2#2025625911yescharacteristic
634C2#2025629411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingQ8×C4⋊C4×C47C2
socleC23×C47C22×D4
prime radicalC22×D188C23
hypercenterQ8×C4⋊C4D47
hypocenterC47#26379
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#15689430C2semidirect
5. #20244865C2semidirect
7. #20244726C2semidirect
8. #20244118C2semidirect
9. #20243947C2semidirect
13. #20243642C2semidirect
14. #20243642C2semidirect
15. Q8×C4⋊C4×C47C2semidirect
313. Q8×C4⋊C4D47semidirect
378. C47#26379semidirect

Maximal subgroups

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

subgroupindex
C2×#156894302
C2×C4×#106253362
#202472602
#202449172
#202448652
#202447732
#202447262
#202441182
#202439472
#202437432
#202436482
#202436422
Q8×C4⋊C4×C472
#2637947