Finite Group Databasebeta

← #29680140

Subgroups — #29680140

Order 15616.

1336 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840230C211yesmaximal, characteristic
2#21454448C211yesmaximal, characteristic, complemented
3#21448213C211yesmaximal, characteristic, complemented
4#21445918C211yesmaximal, characteristic, complemented
5#21445838C214yesmaximal, complemented
6#21445764C211yesmaximal, characteristic
7#21445762C211yesmaximal, characteristic
8#21445628C214yesmaximal, complemented
9C61×#1826C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11#16840230—21no—
12#16840230—21no—
13#16840230—21no—
14#16840230—21no—
15#16840230C412yescomplemented
16#16840230C412yescomplemented
17#16840230C2212yes—
18#16840230C2212yes—
19C2×#14519742C411yescharacteristic, complemented
20C2×#14519742C411yescharacteristic, complemented
21C2×#14519739C411yescharacteristic, complemented
22C2×#14519739C411yescharacteristic, complemented
23C2×#14519737—21no—
24C2×#14519737—21no—
25C2×#14519734—21no—
26C22×C4×Dic61C2211yescharacteristic
27C2×C42×D61—21no—
28#16839816—21no—
29#16839816—21no—
30#16839812—21no—
31#16839812—21no—
32#16839785C2212yescomplemented
33C2×#14519705C2214yes—
34C2×#14519703C2214yes—
35#16839726C2211yescharacteristic
36M16×Dic61C2212yescomplemented
37#16839629—21no—
38#16839626—21no—
39#16839623—21no—
40#16839620—21no—
41C4×#11758196C2212yescomplemented
42C4×C22⋊Dic61C2211yescharacteristic
43#16839608C2212yescomplemented
44#16839521C2214yescomplemented
45C2×#117×C61C2211yescharacteristic
46C61×#407C2211yescharacteristic
47C4×C22⋊C4×C61C2211yescharacteristic
48C61×#339C2214yescomplemented
49C2×#11758222—21no—
50C2×#11758222—21no—
51C2×#11758222—28no—
52C2×#11758222C2×C412yes—
53C2×#11758222C2312yes—
54C22×Dic122C2×C411yescharacteristic
55C22×D244C2×C411yescharacteristic
56C22×C22⋊D61—21no—
57C23×Dic61D411yescharacteristic
58C23×Dic61D411yescharacteristic
59C22×C4×D61—21no—
60C22×C4×D61—21no—
61C22×C4×D61—21no—
62C2×#11758194C2312yes—
63#14519742—21no—
64#14519742—21no—
65#14519742—21no—
66#14519742—21no—
67#14519742C2×C412yes—
68#14519742C2×C412yes—
69#14519742C2×C412yes—
70#14519742C2×C412yes—
71#14519739—21no—
72#14519739—21no—
73#14519739—21no—
74#14519739—21no—
75#14519739C2×C412yes—
76#14519739C2×C412yes—
77#14519739C2×C412yes—
78#14519739C2×C412yes—
79#14519737—22no—
80#14519737—22no—
81#14519737—22no—
82#14519737—22no—
83#14519737—22no—
84#14519737—22no—
85#14519737—22no—
86#14519737—22no—
87#14519734—22no—
88#14519734—22no—
89#14519734—22no—
90#14519734—22no—
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—22no—
94C2×C22⋊Dic61—21no—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61C2×C411yescharacteristic
100C2×C4×Dic61C2×C411yescharacteristic
101C2×C4×Dic61C2×C411yescharacteristic
102C2×C4×Dic61C2×C411yescharacteristic
103C2×C4×Dic61D411yescharacteristic
104C2×C4×Dic61D411yescharacteristic
105C2×C4×Dic61D411yescharacteristic
106C2×C4×Dic61D411yescharacteristic
107C2×C4×Dic61D411yescharacteristic
108C2×C4×Dic61D411yescharacteristic
109C2×C4×Dic61C2312yes—
110C2×C4×Dic61C2311yescharacteristic
111C2×C4×Dic61C2311yescharacteristic
112C42×D61—22no—
113C42×D61—22no—
114C42×D61—22no—
115C42×D61—22no—
116#14519706—22no—
117#14519706—22no—
118#14519705—28no—
119#14519703—28no—
120C22⋊#92418—21no—
121C22⋊#92418—21no—
122#14519687—22no—
123C8×Dic61—22no—
124#14519684—21no—
125#14519684—22no—
126#14519684—22no—
127#14519684—22no—
128C23×C4×C61C2311yescharacteristic
129C2×M16×C61C2312yes—
130#117×C61—21no—
131#117×C61—21no—
132#117×C61C2×C412yes—
133#117×C61C2312yes—
134C2×C4⋊C4×C61C2×C411yescharacteristic
135C2×C4⋊C4×C61C2×C411yescharacteristic
136C2×C22⋊C4×C61—21no—
137C2×C22⋊C4×C61—21no—
138C2×C42×C61—21no—
139C2×C42×C61C2312yes—
140C22⋊C8×C61—21no—
141C22⋊C8×C61—21no—
142#95×C61—21no—
143C24×C61C2×D411yescharacteristic
144C23×D61—21no—
145C22×C4×C61—24no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—22no—
149C22×C4×C61C22×C411yescharacteristic
150C22×C4×C61C22×C411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2411yescharacteristic
155C2×C22⋊D61—28no—
156C2×C22⋊D61—22no—
157C2×C22⋊D61—22no—
158C22×Dic61—21no—
159C22×Dic61—22no—
160C22×Dic61—42no—
161C22×Dic61—42no—
162C22×Dic61C22⋊C411yescharacteristic
163C22×Dic61C22⋊C411yescharacteristic
164C22×Dic61C22⋊C411yescharacteristic
165C22×Dic61C22⋊C411yescharacteristic
166C22×Dic61C22⋊C411yescharacteristic
167C22×Dic61C22⋊C411yescharacteristic
168C22×Dic61C22⋊C411yescharacteristic
169C22×Dic61C22⋊C411yescharacteristic
170C22×Dic61C2×D411yescharacteristic
171C22×Dic61C2×D411yescharacteristic
172C22×Dic61C2×D411yescharacteristic
173C22×Dic61C2×D411yescharacteristic
174#11758222—28no—
175#11758222—28no—
176#11758222—24no—
177#11758222—24no—
178#11758222—24no—
179#11758222#4118yes—
180C2×D244—21no—
181C2×D244—24no—
182C2×D244C22×C412yes—
183C2×C4×D61—28no—
184C2×C4×D61—22no—
185C2×C4×D61—22no—
186C2×C4×D61—22no—
187C2×C4×D61—22no—
188C2×C4×D61—22no—
189C2×C4×D61—22no—
190C2×C4×D61—44no—
191C2×C4×D61—44no—
192C2×Dic122—21no—
193C2×Dic122—24no—
194C2×Dic122C22×C412yes—
195M16×C61—24no—
196C2×C8×C61—22no—
197C4⋊C4×C61—21no—
198C4⋊C4×C61—21no—
199C4⋊C4×C61C22×C412yes—
200C4⋊C4×C61C22×C412yes—
201C22⋊C4×C61—22no—
202C22⋊C4×C61—22no—
203C22⋊C4×C61—42no—
204C42×C61—22no—
205C42×C61—22no—
206C42×C61—24no—
207C22⋊Dic61—42no—
208#11758198—22no—
209#11758198—22no—
210#11758198—22no—
211#11758198—22no—
212#11758196—22no—
213#11758196—22no—
214#11758196—22no—
215#11758196—22no—
216#11758196—44no—
217C4×Dic61—22no—
218C4×Dic61—22no—
219C4×Dic61—22no—
220C4×Dic61—22no—
221C4×Dic61—24no—
222C4×Dic61C22⋊C411yescharacteristic
223C4×Dic61C22⋊C411yescharacteristic
224C4×Dic61C22⋊C411yescharacteristic
225C4×Dic61C22⋊C411yescharacteristic
226C4×Dic61C22⋊C411yescharacteristic
227C4×Dic61C22⋊C411yescharacteristic
228C4×Dic61C22⋊C411yescharacteristic
229C4×Dic61C22⋊C411yescharacteristic
230C4×Dic61C22×C411yescharacteristic
231C4×Dic61C22×C411yescharacteristic
232C4×Dic61C22×C411yescharacteristic
233C4×Dic61C22×C411yescharacteristic
234C4×Dic61C2×D411yescharacteristic
235C4×Dic61C2×D411yescharacteristic
236C4×Dic61C2×D411yescharacteristic
237C4×Dic61C2×D411yescharacteristic
238#11758194—24no—
239C2×#92418—22no—
240C23×C61—22no—
241C23×C61—22no—
242C23×C61—41no—
243C23×C61C2×C22⋊C411yescharacteristic
244C23×C61C2×C22⋊C411yescharacteristic
245C23×C61C22×D411yescharacteristic
246C22×D61—24no—
247C22×D61—22no—
248C22×D61—42no—
249C22×D61—42no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
257C2×C4×C61—42no—
258C2×C4×C61—42no—
259C2×C4×C61—42no—
260C2×C4×C61—42no—
261C2×C4×C61—42no—
262C2×C4×C61C2×C22⋊C411yescharacteristic
263C2×C4×C61C2×C22⋊C411yescharacteristic
264C2×C4×C61C23×C411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C2×#4114yes—
267C22⋊D61—28no—
268C22⋊D61—28no—
269C2×Dic61—24no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—22no—
274C2×Dic61—22no—
275C2×Dic61—22no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—42no—
284C2×Dic61—42no—
285C2×Dic61—42no—
286C2×Dic61C2×C22⋊C411yescharacteristic
287C2×Dic61C2×C22⋊C411yescharacteristic
288C2×Dic61C2×C22⋊C411yescharacteristic
289C2×Dic61C2×C22⋊C411yescharacteristic
290C2×Dic61C2×C22⋊C411yescharacteristic
291C2×Dic61C2×C22⋊C411yescharacteristic
292C2×Dic61C2×C22⋊C411yescharacteristic
293C2×Dic61C2×C22⋊C411yescharacteristic
294D244—24no—
295D244—22no—
296D244#11714yes—
297C4×D61—28no—
298C4×D61—28no—
299C4×D61—44no—
300C4×D61—44no—
301C4×D61—44no—
302C4×D61—44no—
303Dic122—24no—
304Dic122—22no—
305Dic122#11714yes—
306C8×C61—42no—
307#92418—42no—
308#20748—611nomaximal, Sylow
309C22×C61—21no—
310C22×C61—21no—
311C22×C61—21no—
312C22×C61—21no—
313C22×C61—21no—
314C22×C61—21no—
315C22×C61—21no—
316C22×C61—44no—
317C22×C61#42112yes—
318C22×C61C22×C22⋊C411yescharacteristic
319C22×C61#52512yes—
320C2×D61—24no—
321C2×D61—48no—
322C2×D61—42no—
323C2×D61—42no—
324C4×C61—21no—
325C4×C61—21no—
326C4×C61—42no—
327C4×C61—42no—
328C4×C61—82no—
329C4×C61C2×#11712yes—
330C4×C61#52512yes—
331Dic61—21no—
332Dic61—21no—
333Dic61—21no—
334Dic61—21no—
335Dic61—21no—
336Dic61—21no—
337Dic61—21no—
338Dic61—24no—
339Dic61—82no—
340Dic61#42112yes—
341C2×C4×#41—611no—
342#2920—611no—
343#2283—611no—
344#1826—611no—
345#1826D6111yescharacteristic, complemented
346#1806—614no—
347#1789—611no—
348#1789—611no—
349#1759—614no—
350C2×C61—22no—
351C2×C61—22no—
352C2×C61—41no—
353C2×C61#228011yescharacteristic
354C2×C61C2×#42111yescharacteristic
355C2×C61C2×#42111yescharacteristic
356D61—44no—
357C22×#41—611no—
358C4×#41—612no—
359C4×#41—612no—
360C4×#41—612no—
361C4×#41—612no—
362C4×#41—1221no—
363C4×#41—1221no—
364C4×#41—1221no—
365C4×#41—1221no—
366C2×C4×Q8—611no—
367C2×C4×Q8—611no—
368C2×C4×D4—611no—
369C2×C4×D4—611no—
370C2×C4×D4—1221no—
371C2×C4×D4—1221no—
372C2×#117—1221no—
373C2×#117C2×D6111yescharacteristic
374C22×C42—611no—
375C22×C42—1221no—
376#432—1221no—
377#432—1221no—
378#431—1221no—
379#431—1221no—
380#423—612no—
381C2×#104—614no—
382C2×#104—614no—
383#407—611no—
384#407C2×D6111yescharacteristic
385C4×M16—612no—
386#383—1221no—
387#382—1221no—
388#381—1221no—
389#380—1221no—
390C4×C4⋊C4—612no—
391C4×C22⋊C4—611no—
392C4×C22⋊C4C2×D6111yescharacteristic
393#376—612no—
394#339—614no—
395#339C2×D6114yescomplemented
396C61#2074811yescharacteristic, complemented, Sylow
397C2×#41—612no—
398C2×#41—612no—
399C2×#41—1228no—
400C2×#41—1221no—
401C2×#41—1221no—
402C22×Q8—611no—
403C22×D4—611no—
404C22×D4—1221no—
405C23×C4—611no—
406C23×C4—611no—
407C23×C4—1221no—
408C23×C4—1221no—
409C23×C4—1221no—
410C23×C4C22×D6111yescharacteristic
411C2×M16—612no—
412C2×M16C22×D6112yes—
413C4×Q8—612no—
414C4×Q8—612no—
415C4×Q8—612no—
416C4×Q8—612no—
417C4×Q8—1221no—
418C4×Q8—1221no—
419C4×Q8—1221no—
420C4×Q8—1221no—
421C4×D4—612no—
422C4×D4—612no—
423C4×D4—612no—
424C4×D4—612no—
425C4×D4—1221no—
426C4×D4—1221no—
427C4×D4—1221no—
428C4×D4—1221no—
429C4×D4—1222no—
430C4×D4—1222no—
431C4×D4—1222no—
432C4×D4—1222no—
433C4×D4—1222no—
434C4×D4—1222no—
435C4×D4—1222no—
436C4×D4—1222no—
437#117—21no—
438#117—21no—
439#117—1222no—
440#117—1222no—
441#117—1222no—
442#117—1222no—
443#117C4×D6112yes—
444#117C22×D6112yes—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4—1222no—
448C2×C4⋊C4C4×D6111yescharacteristic
449C2×C4⋊C4C4×D6111yescharacteristic
450C2×C22⋊C4—21no—
451C2×C22⋊C4—21no—
452C2×C22⋊C4—1221no—
453C2×C22⋊C4—1221no—
454C2×C22⋊C4—1221no—
455C2×C42—21no—
456C2×C42—611no—
457C2×C42—611no—
458C2×C42—611no—
459C2×C42—611no—
460C2×C42—611no—
461C2×C42—611no—
462C2×C42—611no—
463C2×C42—611no—
464C2×C42—611no—
465C2×C42—611no—
466C2×C42—611no—
467C2×C42—611no—
468C2×C42—612no—
469C2×C42—1222no—
470C2×C42—1222no—
471C2×C42—1222no—
472C2×C42—1222no—
473C2×C42—1221no—
474C2×C42—1221no—
475C2×C42C22×D6112yes—
476C4⋊C8—1222no—
477C4⋊C8—1222no—
478#104—1228no—
479#104—1228no—
480C22⋊C8—21no—
481C22⋊C8—21no—
482C22⋊C8—1221no—
483C22⋊C8—1221no—
484#97—1222no—
485C4×C8—1222no—
486#95—21no—
487#95—1222no—
488#95—1222no—
489#95—1222no—
490#95—1221no—
491C24—1221no—
492C24D4×D6111yescharacteristic
493#41—618no—
494#41—1224no—
495#41—1224no—
496#41—1228no—
497#41—1228no—
498#41—1224no—
499C2×Q8—612no—
500C2×Q8—1224no—
501C2×Q8—1221no—
502C2×D4—612no—
503C2×D4—1224no—
504C2×D4—1228no—
505C2×D4—1222no—
506C2×D4—1222no—
507C2×D4—1221no—
508C22×C4—24no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—22no—
512C22×C4—611no—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—611no—
516C22×C4—611no—
517C22×C4—611no—
518C22×C4—611no—
519C22×C4—611no—
520C22×C4—611no—
521C22×C4—611no—
522C22×C4—611no—
523C22×C4—611no—
524C22×C4—1222no—
525C22×C4—1222no—
526C22×C4—1222no—
527C22×C4—1222no—
528C22×C4—1228no—
529C22×C4—1221no—
530C22×C4—1222no—
531C22×C4—1222no—
532C22×C4—1222no—
533C22×C4—2444no—
534C22×C4—2444no—
535C22×C4—2442no—
536C22×C4—2442no—
537C22×C4C2×C4×D6111yescharacteristic
538C22×C4C2×C4×D6111yescharacteristic
539C22×C4D4×D6111yescharacteristic
540C22×C4D4×D6111yescharacteristic
541C22×C4D4×D6111yescharacteristic
542C22×C4C23×D6111yescharacteristic
543M16—24no—
544M16—1224no—
545C2×C8—22no—
546C2×C8—1222no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—1222no—
550C4⋊C4—1222no—
551C4⋊C4—1222no—
552C4⋊C4—1222no—
553C4⋊C4—2444no—
554C4⋊C4C2×C4×D6112yes—
555C4⋊C4C2×C4×D6112yes—
556C22⋊C4—22no—
557C22⋊C4—22no—
558C22⋊C4—42no—
559C22⋊C4—1222no—
560C22⋊C4—1222no—
561C22⋊C4—1222no—
562C22⋊C4—1222no—
563C22⋊C4—2442no—
564C42—22no—
565C42—22no—
566C42—24no—
567C42—611no—
568C42—611no—
569C42—611no—
570C42—611no—
571C42—611no—
572C42—611no—
573C42—611no—
574C42—611no—
575C42—611no—
576C42—611no—
577C42—611no—
578C42—611no—
579C42—611no—
580C42—611no—
581C42—611no—
582C42—611no—
583C42—1222no—
584C42—1222no—
585C42—1222no—
586C42—1222no—
587C42—1224no—
588C23—22no—
589C23—22no—
590C23—41no—
591C23—1224no—
592C23—1222no—
593C23—2442no—
594C23—2442no—
595C23C22⋊C4×D6111yescharacteristic
596C23C22⋊C4×D6111yescharacteristic
597C23C2×D4×D6111yescharacteristic
598Q8—614no—
599Q8—1224no—
600Q8—1222no—
601D4—614no—
602D4—1224no—
603D4—1228no—
604D4—1228no—
605D4—1222no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—611no—
619C2×C4—611no—
620C2×C4—611no—
621C2×C4—611no—
622C2×C4—611no—
623C2×C4—611no—
624C2×C4—611no—
625C2×C4—611no—
626C2×C4—1228no—
627C2×C4—1222no—
628C2×C4—1222no—
629C2×C4—1222no—
630C2×C4—1224no—
631C2×C4—1222no—
632C2×C4—1222no—
633C2×C4—1222no—
634C2×C4—1228no—
635C2×C4—2444no—
636C2×C4—2444no—
637C2×C4—2444no—
638C2×C4—2444no—
639C2×C4—2441no—
640C2×C4—2441no—
641C2×C4—2441no—
642C2×C4—2441no—
643C2×C4—2441no—
644C2×C4—2441no—
645C2×C4—2441no—
646C2×C4—2442no—
647C2×C4—2442no—
648C2×C4—2442no—
649C2×C4C22⋊C4×D6111yescharacteristic
650C2×C4C22⋊C4×D6111yescharacteristic
651C2×C4C22×C4×D6111yescharacteristic
652C2×C4C2×D4×D6111yescharacteristic
653C2×C4#41×D6114yes—
654C8—42no—
655C8—2442no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—44no—
664C22—1224no—
665C22—2448no—
666C22—2442no—
667C22—2442no—
668C22#1683977712yes—
669C22C2×C22⋊C4×D6111yescharacteristic
670C22#1684027012yes—
671C4—21no—
672C4—21no—
673C4—42no—
674C4—42no—
675C4—82no—
676C4—612no—
677C4—1224no—
678C4—1221no—
679C4—1221no—
680C4—1221no—
681C4—1221no—
682C4—1221no—
683C4—1221no—
684C4—1221no—
685C4—4882no—
686C4#117×D6112yes—
687C4#1684027012yes—
688C2—22no—
689C2—22no—
690C2—41no—
691C2—2444no—
692C2#2144819911yescharacteristic
693C2#421×D6111yescharacteristic
694C2C2×#1683977711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1826C2
socleC22×C61C22×C22⋊C4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenter#1826D61
hypocenterC61#20748
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1826

Complements

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

NH ≅ G/Ntype
2. #21454448C2semidirect
3. #21448213C2semidirect
4. #21445918C2semidirect
5. #21445838C2semidirect
8. #21445628C2semidirect
9. C61×#1826C2semidirect
15. #16840230C4semidirect
16. #16840230C4semidirect
19. C2×#14519742C4semidirect
20. C2×#14519742C4semidirect
21. C2×#14519739C4semidirect
22. C2×#14519739C4semidirect
32. #16839785C22semidirect
36. M16×Dic61C22semidirect
41. C4×#11758196C22semidirect
43. #16839608C22semidirect
44. #16839521C22semidirect
48. C61×#339C22semidirect
345. #1826D61semidirect
395. #339C2×D61semidirect
396. C61#20748semidirect

Maximal subgroups

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

subgroupindex
C2×#168402302
#214544482
#214482132
#214459182
#214458382
#214457642
#214457622
#214456282
C61×#18262
#2074861