Finite Group Databasebeta

← #29646456

Subgroups — #29646456

Order 15616.

1214 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D61C211yesmaximal, characteristic
2#21454451C211yesmaximal, characteristic
3#21454447C211yesmaximal, characteristic
4#21444264C214yesmaximal, complemented
5#21444249C211yesmaximal, characteristic, complemented
6#21444248C211yesmaximal, characteristic, complemented
7#21444247C211yesmaximal, characteristic, complemented
8#21444212C214yesmaximal, complemented
9C61×#1487C211yesmaximal, characteristic, complemented
10C22×Q8×D61C411yescharacteristic
11C2×Q8×Dic61C411yescharacteristic
12C2×C4×Dic122—21no—
13C2×C4×Dic122C2211yescharacteristic
14C2×#14519742—21no—
15C2×#14519742C411yescharacteristic
16C4×Q8×D61—21no—
17C4×Q8×D61—21no—
18C4×Q8×D61—21no—
19C4×Q8×D61—21no—
20C4×Q8×D61C412yes—
21C4×Q8×D61C412yes—
22C4×Q8×D61C2212yes—
23C4×Q8×D61C2212yes—
24C2×C4⋊C4×D61—21no—
25C2×C4⋊C4×D61C411yescharacteristic
26C2×C42×D61—21no—
27C2×C42×D61C2211yescharacteristic
28#16839818—21no—
29#16839818—21no—
30#16839818C2212yes—
31#16839816—21no—
32#16839815C2211yescharacteristic
33#16839814—21no—
34#16839814C2212yes—
35#16839813C2212yes—
36#16839786C2211yescharacteristic
37#16839784—21no—
38#16839725—21no—
39#16839724—21no—
40M16×Dic61—21no—
41C4×#11758189C2212yes—
42#16839494C2214yescomplemented
43Q8⋊#92418C2214yescomplemented
44#16839492C2214yescomplemented
45C2×C4×Q8×C61C2211yescharacteristic
46C61×#431C2211yescharacteristic
47C61×#423C2211yescharacteristic
48C61×Q8⋊C8C2214yescomplemented
49C22×Dic122—21no—
50C22×Dic122C2×C411yescharacteristic
51C2×Q8×D61—28no—
52C2×Q8×D61—21no—
53C2×Q8×D61—21no—
54C2×Q8×D61C2×C412yes—
55C2×Q8×D61C2×C412yes—
56C22×C4×D61—21no—
57C22×C4×D61—21no—
58C22×C4×D61C2×C411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C22×C4×D61D411yescharacteristic
61C2×#11758194—21no—
62C2×#11758189—22no—
63Q8×Dic61—21no—
64Q8×Dic61—21no—
65Q8×Dic61C2×C412yes—
66Q8×Dic61C2×C412yes—
67C4×Dic122—21no—
68C4×Dic122—21no—
69C4×Dic122—22no—
70C4×Dic122—22no—
71C4×Dic122—22no—
72C4×Dic122—22no—
73C4×Dic122C2×C412yes—
74C4×Dic122C2312yes—
75#14519742—21no—
76#14519742—21no—
77#14519742—22no—
78#14519742—22no—
79#14519742—22no—
80#14519742—22no—
81#14519742C2×C412yes—
82#14519742C2×C412yes—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61C2×C411yescharacteristic
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C4⋊C4×D61—22no—
89C4⋊C4×D61—22no—
90C4⋊C4×D61—21no—
91C4⋊C4×D61—21no—
92C4⋊C4×D61—22no—
93C4⋊C4×D61—22no—
94C4⋊C4×D61C2×C412yes—
95C4⋊C4×D61C2×C412yes—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61C2×C411yescharacteristic
100C2×C4×Dic61D411yescharacteristic
101C2×C4×Dic61D411yescharacteristic
102C42×D61—22no—
103C42×D61—22no—
104C42×D61—22no—
105C42×D61—22no—
106C42×D61C2×C411yescharacteristic
107C42×D61C2×C411yescharacteristic
108C42×D61D411yescharacteristic
109C42×D61D411yescharacteristic
110C42×D61D411yescharacteristic
111C42×D61D411yescharacteristic
112C42×D61C2311yescharacteristic
113C42×D61C2311yescharacteristic
114C4⋊#92418—21no—
115C4⋊#92418—21no—
116C4⋊#92418C2312yes—
117#14519706—22no—
118C22⋊#92418—21no—
119#14519689—22no—
120#14519689—22no—
121#14519689—22no—
122#14519687—22no—
123#14519687—22no—
124C4×#92418C2312yes—
125C8×Dic61—22no—
126C22×Q8×C61C2×C411yescharacteristic
127C2×M16×C61—21no—
128C4×Q8×C61—21no—
129C4×Q8×C61—21no—
130C4×Q8×C61C2×C412yes—
131C4×Q8×C61C2312yes—
132C2×C4⋊C4×C61—21no—
133C2×C4⋊C4×C61C2×C411yescharacteristic
134C2×C42×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2312yes—
139C22⋊C8×C61—21no—
140C4×C8×C61C2312yes—
141C23×D61C22⋊C411yescharacteristic
142C2×Q8×C61—24no—
143C2×Q8×C61—21no—
144C2×Q8×C61C22×C412yes—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61C22×C411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61C22⋊C411yescharacteristic
153C22×Dic61C22⋊C411yescharacteristic
154C22×Dic61C22⋊C411yescharacteristic
155Q8×D61—28no—
156Q8×D61—28no—
157Q8×D61—24no—
158Q8×D61—24no—
159Q8×D61—24no—
160Q8×D61M1618yescomplemented
161C2×C4×D61—28no—
162C2×C4×D61—22no—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×C4×D61—24no—
167C2×C4×D61—22no—
168C2×C4×D61—22no—
169C2×C4×D61—42no—
170C2×C4×D61—44no—
171C2×C4×D61C22⋊C411yescharacteristic
172C2×C4×D61C22⋊C411yescharacteristic
173C2×C4×D61C22⋊C411yescharacteristic
174C2×C4×D61C22⋊C411yescharacteristic
175C2×C4×D61C22⋊C411yescharacteristic
176C2×C4×D61C22⋊C411yescharacteristic
177C2×C4×D61C22×C411yescharacteristic
178C2×C4×D61C22×C411yescharacteristic
179C2×C4×D61C2×D411yescharacteristic
180C2×C4×D61C2×D411yescharacteristic
181C2×C4×D61C2×D411yescharacteristic
182C2×C4×D61C2×D411yescharacteristic
183C2×Dic122—28no—
184C2×Dic122—21no—
185C2×Dic122—22no—
186C2×Dic122—22no—
187C2×Dic122—24no—
188C2×Dic122C22×C412yes—
189M16×C61—42no—
190C2×C8×C61—22no—
191C2×C8×C61—22no—
192C4⋊C4×C61—22no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—22no—
195C4⋊C4×C61C22×C412yes—
196C42×C61—22no—
197C42×C61—22no—
198C42×C61C22×C411yescharacteristic
199C42×C61C2×D411yescharacteristic
200C42×C61C2×D411yescharacteristic
201C42×C61C2411yescharacteristic
202C4⋊Dic61—21no—
203C4⋊Dic61—22no—
204C4⋊Dic61—22no—
205C4⋊Dic61C22×C412yes—
206#11758196—22no—
207#11758196—22no—
208#11758196—22no—
209#11758196—22no—
210#11758196—22no—
211#11758196—22no—
212C4×Dic61—22no—
213C4×Dic61—22no—
214C4×Dic61—22no—
215C4×Dic61—22no—
216C4×Dic61—22no—
217C4×Dic61—22no—
218C4×Dic61C22⋊C411yescharacteristic
219C4×Dic61C22⋊C411yescharacteristic
220C4×Dic61C22⋊C411yescharacteristic
221C4×Dic61C22⋊C411yescharacteristic
222C4×Dic61C22⋊C411yescharacteristic
223C4×Dic61C22⋊C411yescharacteristic
224C4×Dic61C22×C411yescharacteristic
225C4×Dic61C22×C411yescharacteristic
226C4×Dic61C2×D411yescharacteristic
227C4×Dic61C2×D411yescharacteristic
228C4×Dic61C2×D411yescharacteristic
229C4×Dic61C2×D411yescharacteristic
230#11758194—42no—
231C2×#92418—22no—
232C2×#92418—22no—
233#11758189—44no—
234C23×C61C2×C22⋊C411yescharacteristic
235C22×D61—22no—
236C22×D61—22no—
237C22×D61—22no—
238C22×D61C2×C22⋊C411yescharacteristic
239C22×D61C2×C22⋊C411yescharacteristic
240Q8×C61—24no—
241Q8×C61—22no—
242Q8×C61C2×M1614yes—
243C2×C4×C61—24no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—41no—
248C2×C4×C61—42no—
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61#13712yes—
253C2×C4×C61#13712yes—
254C2×C4×C61C23×C411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×Dic61—24no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—41no—
263C2×Dic61—42no—
264C2×Dic61C2×C22⋊C411yescharacteristic
265C2×Dic61C2×C22⋊C411yescharacteristic
266C2×Dic61C2×C22⋊C411yescharacteristic
267C2×Dic61C2×C22⋊C411yescharacteristic
268C2×Dic61C2×C22⋊C411yescharacteristic
269C2×Dic61C2×C22⋊C411yescharacteristic
270C4×D61—28no—
271C4×D61—28no—
272C4×D61—22no—
273C4×D61—22no—
274C4×D61—22no—
275C4×D61—22no—
276C4×D61—21no—
277C4×D61—21no—
278C4×D61—21no—
279C4×D61—21no—
280C4×D61—22no—
281C4×D61—22no—
282C4×D61—22no—
283C4×D61—41no—
284C4×D61—41no—
285C4×D61—44no—
286C4×D61—44no—
287C4×D61C2×M1614yes—
288Dic122—28no—
289Dic122—24no—
290Dic122—22no—
291Dic122—28no—
292Dic122C2×M1614yes—
293C8×C61—42no—
294C8×C61—42no—
295#92418—42no—
296#92418—42no—
297#17449—611nomaximal, Sylow
298C22×C61—21no—
299C22×C61—21no—
300C22×C61#41912yes—
301C22×C61C22×C22⋊C411yescharacteristic
302C2×D61—28no—
303C2×D61—21no—
304C2×D61—21no—
305C2×D61—21no—
306C2×D61—21no—
307C2×D61#40712yes—
308C2×D61#42112yes—
309C4×C61—24no—
310C4×C61—21no—
311C4×C61—21no—
312C4×C61—21no—
313C4×C61—42no—
314C4×C61#41912yes—
315C4×C61C22×M1612yes—
316C4×C61C2×#13711yescharacteristic
317C4×C61C2×#13711yescharacteristic
318Dic61—21no—
319Dic61—24no—
320Dic61—21no—
321Dic61—21no—
322Dic61—21no—
323Dic61—21no—
324Dic61—42no—
325Dic61#40712yes—
326C22×C4×Q8—611no—
327#2920—611no—
328#2920—611no—
329#1491—614no—
330#1487—611no—
331#1487—611no—
332#1487—611no—
333#1487D6111yescharacteristic, complemented
334#1479—614no—
335C2×C61—22no—
336C2×C61C2×#40711yescharacteristic
337C2×C61C2×#41911yescharacteristic
338C2×C61C2×#42111yescharacteristic
339D61—22no—
340D61—22no—
341C23×Q8—611no—
342C2×C4×Q8—611no—
343C2×C4×Q8—612no—
344C2×C4×Q8—612no—
345C2×C4×Q8—612no—
346C2×C4×Q8—612no—
347C2×C4×Q8—611no—
348C2×C4×Q8—611no—
349C2×C4×Q8—1221no—
350C2×C4×Q8—1221no—
351C2×C4×Q8—1221no—
352C2×C4×Q8—1221no—
353C2×C4×Q8—1221no—
354C2×C4×Q8—1221no—
355C2×C4×Q8C2×D6111yescharacteristic
356C22×C4⋊C4—611no—
357C22×C4⋊C4—1221no—
358C22×C42—611no—
359C22×C42—1221no—
360#432—612no—
361#432—1221no—
362#432—1221no—
363#432—1221no—
364#431—611no—
365#431—612no—
366#431—612no—
367#431—1221no—
368#431C2×D6111yescharacteristic
369#423—611no—
370#423—1221no—
371#423C2×D6111yescharacteristic
372#407—1221no—
373#407—1221no—
374C4×M16—612no—
375C4×M16—1221no—
376Q8⋊C8—614no—
377Q8⋊C8—614no—
378Q8⋊C8—614no—
379Q8⋊C8C2×D6114yescomplemented
380C61#1744911yescharacteristic, complemented, Sylow
381C22×Q8—612no—
382C22×Q8—612no—
383C22×Q8—611no—
384C22×Q8—1221no—
385C22×Q8—1221no—
386C22×Q8—1221no—
387C22×Q8—1228no—
388C22×Q8C4×D6111yescharacteristic
389C23×C4—611no—
390C23×C4—611no—
391C23×C4—611no—
392C23×C4—1221no—
393C23×C4—1221no—
394C2×M16—21no—
395C2×M16—1222no—
396C2×M16—1221no—
397C4×Q8—21no—
398C4×Q8—21no—
399C4×Q8—612no—
400C4×Q8—612no—
401C4×Q8—612no—
402C4×Q8—612no—
403C4×Q8—612no—
404C4×Q8—612no—
405C4×Q8—1222no—
406C4×Q8—1222no—
407C4×Q8—1222no—
408C4×Q8—1222no—
409C4×Q8—1222no—
410C4×Q8—1222no—
411C4×Q8—1222no—
412C4×Q8—1222no—
413C4×Q8—1221no—
414C4×Q8—1221no—
415C4×Q8—1221no—
416C4×Q8—1221no—
417C4×Q8—1221no—
418C4×Q8—1221no—
419C4×Q8C4×D6112yes—
420C4×Q8C22×D6112yes—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—612no—
424C2×C4⋊C4—612no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—1222no—
428C2×C4⋊C4—1222no—
429C2×C4⋊C4—1222no—
430C2×C4⋊C4—1222no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4C4×D6111yescharacteristic
436C2×C42—21no—
437C2×C42—611no—
438C2×C42—611no—
439C2×C42—611no—
440C2×C42—611no—
441C2×C42—611no—
442C2×C42—611no—
443C2×C42—611no—
444C2×C42—611no—
445C2×C42—611no—
446C2×C42—611no—
447C2×C42—611no—
448C2×C42—1222no—
449C2×C42—1222no—
450C2×C42—1222no—
451C2×C42—1222no—
452C2×C42—1221no—
453C2×C42—1221no—
454C2×C42—1221no—
455C2×C42C22×D6111yescharacteristic
456C4⋊C8—21no—
457C4⋊C8—21no—
458C4⋊C8—612no—
459C4⋊C8—1222no—
460C4⋊C8—1221no—
461C4⋊C8—1221no—
462C4⋊C8C22×D6112yes—
463C22⋊C8—21no—
464C22⋊C8—1222no—
465C22⋊C8—1222no—
466C22⋊C8—1222no—
467C22⋊C8—1221no—
468#97—1222no—
469#97—1222no—
470C4×C8—612no—
471C4×C8—1222no—
472C4×C8C22×D6112yes—
473C24—611no—
474C2×Q8—24no—
475C2×Q8—21no—
476C2×Q8—618no—
477C2×Q8—612no—
478C2×Q8—1222no—
479C2×Q8—1222no—
480C2×Q8—1224no—
481C2×Q8—1228no—
482C2×Q8—1224no—
483C2×Q8—1224no—
484C2×Q8—1228no—
485C2×Q8—1228no—
486C2×Q8—1224no—
487C2×Q8—1221no—
488C2×Q8C2×C4×D6112yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—611no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—611no—
499C22×C4—611no—
500C22×C4—611no—
501C22×C4—611no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—611no—
505C22×C4—611no—
506C22×C4—1224no—
507C22×C4—1222no—
508C22×C4—1222no—
509C22×C4—1228no—
510C22×C4—1222no—
511C22×C4—1222no—
512C22×C4—1221no—
513C22×C4—1222no—
514C22×C4—1222no—
515C22×C4—1221no—
516C22×C4—2442no—
517C22×C4—2444no—
518C22×C4C2×C4×D6111yescharacteristic
519C22×C4D4×D6111yescharacteristic
520C22×C4D4×D6111yescharacteristic
521M16—42no—
522M16—2444no—
523M16—2442no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—1222no—
527C2×C8—1222no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—21no—
531C4⋊C4—612no—
532C4⋊C4—1222no—
533C4⋊C4—1222no—
534C4⋊C4—1222no—
535C4⋊C4—1222no—
536C4⋊C4—1222no—
537C4⋊C4—1222no—
538C4⋊C4—1222no—
539C4⋊C4—1221no—
540C4⋊C4—1222no—
541C4⋊C4C2×C4×D6112yes—
542C42—22no—
543C42—22no—
544C42—611no—
545C42—611no—
546C42—611no—
547C42—611no—
548C42—611no—
549C42—611no—
550C42—611no—
551C42—611no—
552C42—611no—
553C42—611no—
554C42—611no—
555C42—611no—
556C42—1222no—
557C42—1222no—
558C42—1222no—
559C42—1222no—
560C42—1222no—
561C42—1222no—
562C42C2×C4×D6111yescharacteristic
563C42D4×D6111yescharacteristic
564C42D4×D6111yescharacteristic
565C42C23×D6111yescharacteristic
566C23—611no—
567C23—611no—
568C23—1222no—
569C23—1222no—
570C23—1222no—
571C23C22⋊C4×D6111yescharacteristic
572Q8—24no—
573Q8—22no—
574Q8—614no—
575Q8—1228no—
576Q8—1228no—
577Q8—1224no—
578Q8—1222no—
579Q8M16×D6114yes—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—611no—
587C2×C4—614no—
588C2×C4—611no—
589C2×C4—611no—
590C2×C4—611no—
591C2×C4—611no—
592C2×C4—611no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1228no—
598C2×C4—1228no—
599C2×C4—1222no—
600C2×C4—1224no—
601C2×C4—1222no—
602C2×C4—1222no—
603C2×C4—1222no—
604C2×C4—1222no—
605C2×C4—1222no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1222no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2444no—
614C2×C4—2444no—
615C2×C4—2441no—
616C2×C4—2442no—
617C2×C4C22⋊C4×D6111yescharacteristic
618C2×C4C22⋊C4×D6111yescharacteristic
619C2×C4C22⋊C4×D6111yescharacteristic
620C2×C4#1451980612yes—
621C2×C4#1451980912yes—
622C2×C4C22×C4×D6111yescharacteristic
623C2×C4C2×D4×D6111yescharacteristic
624C2×C4C2×D4×D6111yescharacteristic
625C8—42no—
626C8—42no—
627C8—2442no—
628C8—2442no—
629C22—21no—
630C22—21no—
631C22—612no—
632C22—612no—
633C22—1228no—
634C22—1221no—
635C22—1221no—
636C22—1221no—
637C22—1221no—
638C22#1683976912yes—
639C22C2×C22⋊C4×D6111yescharacteristic
640C4—21no—
641C4—21no—
642C4—24no—
643C4—21no—
644C4—42no—
645C4—612no—
646C4—1221no—
647C4—1221no—
648C4—1221no—
649C4—1224no—
650C4—1221no—
651C4—1221no—
652C4—2442no—
653C4#1683976912yes—
654C4C2×M16×D6112yes—
655C4C2×#1451980611yescharacteristic
656C4C2×#1451980911yescharacteristic
657C2—22no—
658C2—1222no—
659C2—1222no—
660C2#407×D6111yescharacteristic
661C2C2×#1683976911yescharacteristic
662C2#421×D6111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1487C2
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC42×C61C24
hypercenter#1487D61
hypocenterC61#17449
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
4. #21444264C2semidirect
5. #21444249C2semidirect
6. #21444248C2semidirect
7. #21444247C2semidirect
8. #21444212C2semidirect
9. C61×#1487C2semidirect
42. #16839494C22semidirect
43. Q8⋊#92418C22semidirect
44. #16839492C22semidirect
48. C61×Q8⋊C8C22semidirect
160. Q8×D61M16semidirect
333. #1487D61semidirect
379. Q8⋊C8C2×D61semidirect
380. C61#17449semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D612
#214544512
#214544472
#214442642
#214442492
#214442482
#214442472
#214442122
C61×#14872
#1744961