Finite Group Databasebeta

← #29828837

Subgroups — #29828837

Order 15616.

1436 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D61C211yesmaximal, characteristic
2C2×#16839969C211yesmaximal, characteristic, complemented
3C2×#16839916C212yesmaximal
4#21454166C211yesmaximal, characteristic
5#21446162C212yesmaximal, complemented
6#21446157C212yesmaximal, complemented
7#21446123C212yesmaximal
8#21445825C211yesmaximal, characteristic, complemented
9#21445824C211yesmaximal, characteristic, complemented
10#21445796C211yesmaximal, characteristic
11C2×C61×#462C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2211yescharacteristic
13C22×#11758202C2212yes—
14C2×#14519753C2212yes—
15C2×Q8×Dic61C2211yescharacteristic
16C2×C4×Dic122—21no—
17C2×C4×Dic122C2211yescharacteristic
18C2×#14519742—21no—
19C2×#14519742C2211yescharacteristic
20C4×Q8×D61—24no—
21C4×Q8×D61—24no—
22C2×C4⋊C4×D61—21no—
23C2×C4⋊C4×D61C2211yescharacteristic
24C2×C42×D61—21no—
25C2×C42×D61C2211yescharacteristic
26#16839969—24no—
27#16839916—28no—
28C2×C4⋊#92418C2211yescharacteristic
29C2×#14519702C2212yes—
30C2×#14519701C2212yes—
31C2×#14519701C2212yes—
32C2×#14519689C2212yes—
33#16839669C2212yes—
34#16839658—21no—
35#16839658—21no—
36#16839632C2211yescharacteristic
37#16839619—21no—
38#16839619—21no—
39#16839525C2212yes—
40#16839524C2212yes—
41#16839516C2212yes—
42C22×Q16×C61C2212yes—
43C2×C4⋊Q8×C61C2211yescharacteristic
44C2×C4×Q8×C61C2211yescharacteristic
45C61×#462—24no—
46C2×C4⋊C8×C61C2211yescharacteristic
47C2×#103×C61C2212yes—
48C22×Dic122—21no—
49C22×Dic122C2311yescharacteristic
50C2×Q8×D61—22no—
51C2×Q8×D61—22no—
52C2×Q8×D61—22no—
53C2×Q8×D61—22no—
54C2×Q8×D61—22no—
55C2×Q8×D61D414yes—
56C22×C4×D61—21no—
57C22×C4×D61—21no—
58C22×C4×D61D411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C22×C4×D61C2311yescharacteristic
61C2×#11758202—24no—
62C2×#11758202—24no—
63C2×#11758202—24no—
64C22×#92418C2312yes—
65#14519753—24no—
66#14519753—24no—
67Q8×Dic61—22no—
68Q8×Dic61—22no—
69C4×Dic122—28no—
70C4×Dic122—22no—
71C4×Dic122D414yes—
72#14519742—28no—
73#14519742—22no—
74#14519742—22no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61C2312yes—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—22no—
84C4⋊C4×D61—28no—
85C4⋊C4×D61—22no—
86C4⋊C4×D61—22no—
87C2×#11758198—22no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61D411yescharacteristic
92C2×C4×Dic61D411yescharacteristic
93C2×C4×Dic61C2311yescharacteristic
94C42×D61—28no—
95C42×D61—22no—
96C42×D61—22no—
97C4⋊#92418—22no—
98#14519702—24no—
99#14519701—24no—
100#14519701—24no—
101#14519689—24no—
102#14519684—22no—
103#14519684—22no—
104C22×Q8×C61C2312yes—
105C22×Q8×C61C2311yescharacteristic
106C2×Q16×C61—24no—
107C2×Q16×C61—24no—
108C2×Q16×C61—24no—
109C22×C8×C61C2312yes—
110C4⋊Q8×C61—22no—
111C4⋊Q8×C61—22no—
112C4×Q8×C61—22no—
113C4×Q8×C61D414yes—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C42×C61—21no—
119C2×C42×C61C2311yescharacteristic
120C4⋊C8×C61—22no—
121#103×C61—24no—
122C23×D61C2×D411yescharacteristic
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61—22no—
127C2×Q8×C61—22no—
128C2×Q8×C61—22no—
129C2×Q8×C61—22no—
130C2×Q8×C61—22no—
131C2×Q8×C61—22no—
132C2×Q8×C61C2×D412yes—
133C2×Q8×C61C2×D412yes—
134C2×Q8×C61#4112yes—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—22no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2411yescharacteristic
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—22no—
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147Q8×D61—28no—
148Q8×D61—28no—
149Q8×D61—28no—
150Q8×D61—28no—
151C2×C4×D61—24no—
152C2×C4×D61—24no—
153C2×C4×D61—24no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—24no—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—41no—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
166C2×C4×D61Q1614yescomplemented
167C2×C4×D61C2×D412yes—
168C2×C4×D61#4112yes—
169C2×Dic122—24no—
170C2×Dic122—24no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—24no—
175C2×Dic122C2×D412yes—
176C2×Dic122C2×D412yes—
177C2×Dic122#4112yes—
178Q16×C61—48no—
179C2×C8×C61—22no—
180C2×C8×C61—22no—
181C2×C8×C61—22no—
182C4⋊C4×C61—24no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—42no—
185C4⋊C4×C61—42no—
186C4⋊C4×C61C2×D412yes—
187C42×C61—24no—
188C42×C61C2×D412yes—
189C42×C61C2×D412yes—
190#11758202—48no—
191#11758198—44no—
192C4⋊Dic61—24no—
193C4⋊Dic61—21no—
194C4⋊Dic61—22no—
195C4⋊Dic61—22no—
196C4⋊Dic61—42no—
197C4⋊Dic61—42no—
198C4⋊Dic61C2×D412yes—
199#11758196—24no—
200#11758196—24no—
201#11758196—24no—
202#11758196—44no—
203C4×Dic61—24no—
204C4×Dic61—24no—
205C4×Dic61—24no—
206C4×Dic61—22no—
207C4×Dic61—22no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C2×#92418—22no—
211C2×#92418—22no—
212C2×#92418—22no—
213C23×C61C22×D411yescharacteristic
214C22×D61—21no—
215C22×D61—21no—
216C22×D61—21no—
217C22×D61—21no—
218C22×D61#12112yes—
219C22×D61C2×Q1612yes—
220C22×D61#13712yes—
221Q8×C61—22no—
222Q8×C61—22no—
223Q8×C61—22no—
224Q8×C61—42no—
225Q8×C61—42no—
226Q8×C61—42no—
227Q8×C61—42no—
228Q8×C61#12114yes—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—22no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61#12012yes—
241C2×C4×C61#12012yes—
242C2×C4×C61C2×Q1612yes—
243C2×C4×C61#13712yes—
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—21no—
255C2×Dic61—22no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—22no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61#12112yes—
267C2×Dic61C2×Q1612yes—
268C4×D61—216no—
269C4×D61—22no—
270C4×D61—22no—
271C4×D61—22no—
272C4×D61—22no—
273C4×D61—28no—
274C4×D61—44no—
275C4×D61—48no—
276Dic122—216no—
277Dic122—22no—
278Dic122—22no—
279Dic122—22no—
280Dic122#12114yes—
281C8×C61—42no—
282#92418—42no—
283#32906—611nomaximal, Sylow
284C22×C61C2×#12011yescharacteristic
285C22×C61C2×#12111yescharacteristic
286C22×C61#52511yescharacteristic
287C22×C61C22×Q1611yescharacteristic
288C22×C61C2×#13711yescharacteristic
289C22×C61C2×#13711yescharacteristic
290C22×C61#57511yescharacteristic
291C2×D61—22no—
292C2×D61—22no—
293C2×D61—22no—
294C2×D61—22no—
295C2×D61—22no—
296C2×D61—22no—
297C2×D61#46214yescomplemented
298C4×C61—24no—
299C4×C61—22no—
300C4×C61—42no—
301C4×C61—82no—
302C4×C61#45114yescomplemented
303C4×C61C2×#12111yescharacteristic
304C4×C61C2×#12111yescharacteristic
305C4×C61#52511yescharacteristic
306C4×C61#52511yescharacteristic
307Dic61—22no—
308Dic61—24no—
309Dic61—22no—
310Dic61—41no—
311Dic61—42no—
312Dic61—82no—
313C22×C4×Q8—611no—
314C2×#462—611no—
315C2×#462D6111yescharacteristic, complemented
316C2×#451—612no—
317#2857—611no—
318#1878—612no—
319#1878—612no—
320#1872—612no—
321#1803—611no—
322#1803—611no—
323#1797—611no—
324C2×C61#284711yescharacteristic
325C2×C61C2×#45111yescharacteristic
326C2×C61#300711yescharacteristic
327C2×C61C2×#46211yescharacteristic
328C2×C61#303711yescharacteristic
329C2×C61#310411yescharacteristic
330C2×C61#311311yescharacteristic
331D61—24no—
332C23×Q8—611no—
333C22×Q16—612no—
334C22×Q16C2×D6112yes—
335C2×C4⋊Q8C2×D6111yescharacteristic
336C2×C22⋊Q8—612no—
337C2×C4×Q8—611no—
338C2×C4×Q8—611no—
339C2×C4×Q8—611no—
340C2×C4×Q8—1221no—
341C2×C4×Q8—1221no—
342C2×C4×Q8—1224no—
343C2×C4×Q8—1224no—
344C2×C4×Q8C2×D6111yescharacteristic
345C22×C4⋊C4—611no—
346C22×C4⋊C4—1221no—
347C22×C42—611no—
348C22×C42—1221no—
349#462—24no—
350#462—1224no—
351#451—1228no—
352C2×C4⋊C8—611no—
353C2×C4⋊C8C2×D6111yescharacteristic
354C2×#103—612no—
355C2×#103—612no—
356C2×#103—612no—
357C2×#103C2×D6112yes—
358C2×C22⋊C8—612no—
359#391—612no—
360#389—1221no—
361#389—1221no—
362#384—611no—
363#380—1221no—
364#380—1221no—
365#340—612no—
366#340—612no—
367#336—612no—
368C61#3290611yescharacteristic, complemented, Sylow
369C22×Q8—614no—
370C22×Q8—611no—
371C22×Q8—1221no—
372C22×Q8—1222no—
373C22×Q8—1222no—
374C22×Q8—1222no—
375C22×Q8—1222no—
376C22×Q8—1222no—
377C22×Q8C22×D6112yes—
378C22×Q8C22×D6111yescharacteristic
379C23×C4—611no—
380C23×C4—611no—
381C23×C4—611no—
382C23×C4—1221no—
383C23×C4—1221no—
384C2×Q16—24no—
385C2×Q16—24no—
386C2×Q16—24no—
387C2×Q16—1224no—
388C2×Q16—1224no—
389C2×Q16—1224no—
390C22×C8—612no—
391C22×C8C22×D6112yes—
392C4⋊Q8—22no—
393C4⋊Q8—22no—
394C22⋊Q8—1224no—
395C22⋊Q8—1224no—
396C4×Q8—22no—
397C4×Q8—614no—
398C4×Q8—1228no—
399C4×Q8—1222no—
400C4×Q8—1228no—
401C4×Q8—1222no—
402C4×Q8—1222no—
403C4×Q8—1222no—
404C4×Q8—1222no—
405C4×Q8C22⋊D6114yes—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—612no—
411C2×C4⋊C4—1222no—
412C2×C4⋊C4—1228no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1222no—
418C2×C4⋊C4—1222no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4C22×D6111yescharacteristic
422C2×C22⋊C4—1222no—
423C2×C42—21no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—1228no—
428C2×C42—1221no—
429C2×C42—1221no—
430C2×C42—1221no—
431C2×C42—1222no—
432C2×C42—1222no—
433C2×C42C22×D6111yescharacteristic
434C4⋊C8—22no—
435C4⋊C8—1222no—
436#103—24no—
437#103—1224no—
438#103—1224no—
439#103—1224no—
440C22⋊C8—1224no—
441#95—1222no—
442#95—1222no—
443C24—611no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—612no—
454C2×Q8—612no—
455C2×Q8—612no—
456C2×Q8—1224no—
457C2×Q8—1224no—
458C2×Q8—1224no—
459C2×Q8—1228no—
460C2×Q8—1228no—
461C2×Q8—1228no—
462C2×Q8—1228no—
463C2×Q8—1221no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8D4×D6112yes—
467C2×Q8#1175822412yes—
468C2×Q8C2×C22⋊D6112yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—612no—
475C22×C4—612no—
476C22×C4—614no—
477C22×C4—611no—
478C22×C4—1224no—
479C22×C4—1224no—
480C22×C4—1224no—
481C22×C4—1222no—
482C22×C4—1224no—
483C22×C4—1221no—
484C22×C4—1222no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1222no—
490C22×C4—1221no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4—2441no—
494C22×C4—2442no—
495C22×C4—2442no—
496C22×C4D4×D6111yescharacteristic
497C22×C4D4×D6111yescharacteristic
498C22×C4C23×D6111yescharacteristic
499Q16—48no—
500Q16—2448no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—1222no—
505C2×C8—1222no—
506C2×C8—1222no—
507C4⋊C4—21no—
508C4⋊C4—24no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—612no—
512C4⋊C4—1224no—
513C4⋊C4—1224no—
514C4⋊C4—1224no—
515C4⋊C4—1221no—
516C4⋊C4—1224no—
517C4⋊C4—1222no—
518C4⋊C4—1222no—
519C4⋊C4—2442no—
520C4⋊C4—2444no—
521C4⋊C4—2442no—
522C4⋊C4C2×C22⋊D6112yes—
523C22⋊C4—2444no—
524C42—24no—
525C42—1224no—
526C42—1224no—
527C42—1221no—
528C42—1221no—
529C42—1222no—
530C42—1222no—
531C42—1224no—
532C42D4×D6112yes—
533C42C2×C22⋊D6112yes—
534C23—612no—
535C23—612no—
536C23—612no—
537C23—1221no—
538C23—1221no—
539C23—1221no—
540C23—1221no—
541C23C2×D4×D6111yescharacteristic
542Q8—22no—
543Q8—22no—
544Q8—22no—
545Q8—42no—
546Q8—42no—
547Q8—42no—
548Q8—42no—
549Q8—614no—
550Q8—12216no—
551Q8—1222no—
552Q8—1222no—
553Q8—1222no—
554Q8#1451974814yes—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—612no—
567C2×C4—612no—
568C2×C4—12216no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1221no—
574C2×C4—1222no—
575C2×C4—1221no—
576C2×C4—1222no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—1228no—
581C2×C4—1222no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—2444no—
586C2×C4—2448no—
587C2×C4—2442no—
588C2×C4—2442no—
589C2×C4—2442no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4#1451974512yes—
593C2×C4#1451974512yes—
594C2×C4Q16×D6112yes—
595C2×C4#1451980912yes—
596C2×C4C2×D4×D6111yescharacteristic
597C2×C4C2×D4×D6111yescharacteristic
598C2×C4C22×C22⋊D6111yescharacteristic
599C2×C4#41×D6111yescharacteristic
600C2×C4#41×D6111yescharacteristic
601C2×C4C2×#1175822411yescharacteristic
602C8—42no—
603C8—2442no—
604C22—614no—
605C22—1222no—
606C22—1222no—
607C22—1222no—
608C22—1222no—
609C22—1222no—
610C22—1222no—
611C22C2×#1451974511yescharacteristic
612C22#121×D6111yescharacteristic
613C22#1684027211yescharacteristic
614C22C2×Q16×D6111yescharacteristic
615C22#137×D6111yescharacteristic
616C22C2×#1451980911yescharacteristic
617C22#1684056111yescharacteristic
618C4—24no—
619C4—22no—
620C4—42no—
621C4—82no—
622C4—1222no—
623C4—1224no—
624C4—1222no—
625C4—4882no—
626C4—2441no—
627C4—2442no—
628C4#1683991614yescomplemented
629C4#121×D6111yescharacteristic
630C4C2×#1451974811yescharacteristic
631C4#1684026911yescharacteristic
632C4#1684027211yescharacteristic
633C2—1224no—
634C2#2145409811yescharacteristic
635C2C2×#1683991611yescharacteristic
636C2#2145507511yescharacteristic
637C2#462×D6111yescharacteristic
638C2#2145529211yescharacteristic
639C2#2145585711yescharacteristic
640C2#2145594811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#462C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenterC2×#462D61
hypocenterC61#32906
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
2. C2×#16839969C2semidirect
5. #21446162C2semidirect
6. #21446157C2semidirect
8. #21445825C2semidirect
9. #21445824C2semidirect
11. C2×C61×#462C2semidirect
166. C2×C4×D61Q16semidirect
297. C2×D61#462semidirect
302. C4×C61#451semidirect
315. C2×#462D61semidirect
368. C61#32906semidirect
628. C4#16839916semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D612
C2×#168399692
C2×#168399162
#214541662
#214461622
#214461572
#214461232
#214458252
#214458242
#214457962
C2×C61×#4622
#3290661