Finite Group Databasebeta

← D61×#3195 #30550619

Subgroups — D61×#3195 #30550619

Order 15616.

2322 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊Q8×D61C212yesmaximal
2#21456643C211yesmaximal, characteristic, complemented
3#21456642C214yesmaximal, complemented
4#21456641C214yesmaximal, complemented
5#21456640C212yesmaximal, complemented
6#21456639C214yesmaximal, complemented
7#483×D61C214yesmaximal, complemented
8#470×D61C214yesmaximal, complemented
9#451×D61C214yesmaximal, complemented
10#407×D61C211yesmaximal, characteristic
11C61×#3195C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2212yes—
13C23×C4×D61C2211yescharacteristic
14C2×Q16×D61C2214yescomplemented
15C2×M16×D61C2212yes—
16C2×#14519757C2212yes—
17C2×#14519756C2212yes—
18C2×#14519755—22no—
19C2×#14519754—22no—
20C2×#14519753C2212yes—
21C22⋊Q8×D61—22no—
22C22⋊Q8×D61—22no—
23C22⋊Q8×D61—22no—
24C22⋊Q8×D61—22no—
25C22⋊Q8×D61C2214yes—
26C22⋊Q8×D61C2214yes—
27C22⋊Q8×D61C2214yes—
28C22⋊Q8×D61C2214yes—
29C2×C4⋊C4×D61—22no—
30C2×C4⋊C4×D61C2212yes—
31C2×C22⋊C4×D61—22no—
32#16840094C2218yescomplemented
33#16840093C2214yescomplemented
34#16840092C2214yescomplemented
35#16840091C2218yescomplemented
36#16840090C2214yescomplemented
37#16840021C2214yescomplemented
38#16840020C2214yescomplemented
39#16840019C2218yescomplemented
40#16840018C2218yescomplemented
41#16840017C2214yescomplemented
42#16839920C2214yescomplemented
43#16839919C2218yescomplemented
44#16839918C2214yescomplemented
45#16839917C2214yescomplemented
46#16839916C2218yescomplemented
47#106×D61C2214yescomplemented
48#103×D61C2218yescomplemented
49#16839726C2211yescharacteristic
50#16839725C2212yes—
51#16839724C2214yes—
52C22⋊C8×D61C2214yes—
53C2×C22⋊Q8×C61C2212yes—
54C61×#483C2214yescomplemented
55C61×#470C2214yescomplemented
56C61×#451C2214yescomplemented
57C61×#407C2211yescharacteristic
58C24×D61D411yescharacteristic
59C22×Dic122—22no—
60C22×Dic122C2312yes—
61C2×Q8×D61—22no—
62C2×Q8×D61—22no—
63C2×Q8×D61—48no—
64C2×Q8×D61C2314yes—
65C2×Q8×D61C2314yes—
66C23×Dic61D411yescharacteristic
67C22×C4×D61—22no—
68C22×C4×D61—22no—
69C22×C4×D61—44no—
70C22×C4×D61D412yes—
71C22×C4×D61D411yescharacteristic
72C22×C4×D61D411yescharacteristic
73C22×C4×D61D412yes—
74C22×C4×D61C2312yes—
75C22×C4×D61C2311yescharacteristic
76C22×C4×D61C2312yes—
77C22×C4×D61C2311yescharacteristic
78C2×Dic244C2314yescomplemented
79C2×#11758202C2318yescomplemented
80Q16×D61—48no—
81C2×#11758194C2312yes—
82C2×#11758189C2314yes—
83M16×D61—44no—
84C2×C8×D61C2314yes—
85#14519757—22no—
86#14519757—22no—
87#14519757C2314yes—
88#14519757C2314yes—
89#14519756—22no—
90#14519756—22no—
91#14519756C2314yes—
92#14519756C2314yes—
93#14519755—24no—
94#14519755—24no—
95#14519755—24no—
96#14519755—24no—
97#14519754—24no—
98#14519754—24no—
99#14519754—24no—
100#14519754—24no—
101#14519753—22no—
102#14519753—22no—
103#14519753C2314yes—
104#14519753C2314yes—
105C2×C4⋊Dic61—22no—
106C2×C4⋊Dic61C2312yes—
107C2×#11758196—22no—
108C2×#11758196—22no—
109C2×#11758196—22no—
110C4⋊C4×D61—24no—
111C4⋊C4×D61—24no—
112C4⋊C4×D61—24no—
113C4⋊C4×D61—24no—
114C4⋊C4×D61—22no—
115C4⋊C4×D61—22no—
116C4⋊C4×D61C2314yes—
117C4⋊C4×D61C2314yes—
118C2×C22⋊Dic61—22no—
119C2×#11758198—22no—
120C2×#11758198—22no—
121C22⋊C4×D61—24no—
122C22⋊C4×D61—24no—
123C22⋊C4×D61—24no—
124C22⋊C4×D61—24no—
125#14519709C2314yescomplemented
126#14519708C2318yescomplemented
127#14519702C2318yescomplemented
128#14519701C2318yescomplemented
129#14519700C2318yescomplemented
130C22⋊#92418C2314yes—
131#14519689C2318yes—
132C22×Q8×C61C2312yes—
133C23×C4×C61C2311yescharacteristic
134C2×Q16×C61C2314yescomplemented
135C2×M16×C61C2312yes—
136C22⋊Q8×C61—22no—
137C22⋊Q8×C61—22no—
138C22⋊Q8×C61C2314yes—
139C22⋊Q8×C61C2314yes—
140C2×C4⋊C4×C61—22no—
141C2×C4⋊C4×C61C2312yes—
142C2×C22⋊C4×C61—22no—
143#106×C61C2314yescomplemented
144#103×C61C2318yescomplemented
145C22⋊C8×C61C2314yes—
146C24×C61C2×D411yescharacteristic
147C23×D61—28no—
148C23×D61—42no—
149C23×D61C2×D411yescharacteristic
150C23×D61C2×D411yescharacteristic
151C23×D61C2×D412yes—
152C23×D61C2×D412yes—
153C2×Q8×C61—22no—
154C2×Q8×C61—44no—
155C2×Q8×C61C2414yes—
156C22×C4×C61—22no—
157C22×C4×C61—22no—
158C22×C4×C61—42no—
159C22×C4×C61C2×D412yes—
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2412yes—
162C22×C4×C61C2411yescharacteristic
163C22×Dic61—22no—
164C22×Dic61—22no—
165C22×Dic61—42no—
166C22×Dic61C2×D412yes—
167C22×Dic61C2×D411yescharacteristic
168C22×Dic61C2×D412yes—
169C22×Dic61C2×D411yescharacteristic
170Q8×D61—216no—
171Q8×D61—48no—
172Q8×D61—48no—
173Q8×D61—48no—
174C2×C4×D61—28no—
175C2×C4×D61—24no—
176C2×C4×D61—24no—
177C2×C4×D61—24no—
178C2×C4×D61—24no—
179C2×C4×D61—24no—
180C2×C4×D61—44no—
181C2×C4×D61—42no—
182C2×C4×D61—44no—
183C2×C4×D61—42no—
184C2×C4×D61—44no—
185C2×C4×D61—42no—
186C2×C4×D61—48no—
187C2×C4×D61—48no—
188C2×C4×D61C2×D412yes—
189C2×C4×D61C2×D412yes—
190C2×C4×D61C2×D411yescharacteristic
191C2×C4×D61C2×D411yescharacteristic
192C2×C4×D61C2×D412yes—
193C2×C4×D61C2×D412yes—
194C2×C4×D61C2×D412yes—
195C2×C4×D61C2412yes—
196C2×C4×D61C2411yescharacteristic
197C2×C4×D61C2411yescharacteristic
198C2×Dic122—22no—
199C2×Dic122—24no—
200C2×Dic122—24no—
201C2×Dic122—44no—
202C2×Dic122—48no—
203C2×Dic122C2414yes—
204Q16×C61—44no—
205M16×C61—42no—
206C2×C8×C61C2414yes—
207C4⋊C4×C61—24no—
208C4⋊C4×C61—24no—
209C4⋊C4×C61—22no—
210C4⋊C4×C61C2414yes—
211C22⋊C4×C61—24no—
212C22⋊C4×C61—24no—
213C22⋊Dic61—24no—
214C22⋊Dic61—24no—
215#11758202—48no—
216#11758198—24no—
217#11758198—24no—
218#11758198—24no—
219#11758198—24no—
220C4⋊Dic61—22no—
221C4⋊Dic61—24no—
222C4⋊Dic61—24no—
223C4⋊Dic61C2414yes—
224#11758196—24no—
225#11758196—24no—
226#11758196—24no—
227#11758196—24no—
228#11758196—24no—
229#11758196—24no—
230#11758194—42no—
231C2×#92418C2414yes—
232Dic244—44no—
233#11758189—44no—
234C8×D61—28no—
235C23×C61—24no—
236C23×C61—41no—
237C23×C61C22×D411yescharacteristic
238C23×C61C22×D412yes—
239C22×D61—24no—
240C22×D61—24no—
241C22×D61—24no—
242C22×D61—22no—
243C22×D61—24no—
244C22×D61—24no—
245C22×D61—24no—
246C22×D61—41no—
247C22×D61—42no—
248C22×D61—416no—
249C22×D61#13718yes—
250C22×D61C22×D411yescharacteristic
251C22×D61C22×D412yes—
252C22×D61C22×D411yescharacteristic
253Q8×C61—28no—
254Q8×C61—44no—
255C2×C4×C61—22no—
256C2×C4×C61—24no—
257C2×C4×C61—24no—
258C2×C4×C61—24no—
259C2×C4×C61—41no—
260C2×C4×C61—42no—
261C2×C4×C61—44no—
262C2×C4×C61—44no—
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C22×D412yes—
265C2×C4×C61C2511yescharacteristic
266C2×Dic61—24no—
267C2×Dic61—22no—
268C2×Dic61—24no—
269C2×Dic61—24no—
270C2×Dic61—41no—
271C2×Dic61—42no—
272C2×Dic61—44no—
273C2×Dic61—44no—
274C2×Dic61C22×D411yescharacteristic
275C2×Dic61C22×D412yes—
276C2×Dic61C22×D411yescharacteristic
277C4×D61—22no—
278C4×D61—24no—
279C4×D61—22no—
280C4×D61—24no—
281C4×D61—22no—
282C4×D61—24no—
283C4×D61—42no—
284C4×D61—42no—
285C4×D61—42no—
286C4×D61—48no—
287C4×D61—48no—
288C4×D61—48no—
289C4×D61—48no—
290C4×D61H(5,𝔽2)14yes—
291Dic122—28no—
292Dic122—48no—
293Dic122—44no—
294Dic122—48no—
295C8×C61—24no—
296#92418—24no—
297C2×#3195—611nomaximal, Sylow
298C22×C61—22no—
299C22×C61—22no—
300C22×C61—21no—
301C22×C61—22no—
302C22×C61—44no—
303C22×C61C2×#13714yes—
304C22×C61C23×D411yescharacteristic
305C2×D61—22no—
306C2×D61—22no—
307C2×D61—216no—
308C2×D61—21no—
309C2×D61—22no—
310C2×D61—44no—
311C2×D61—48no—
312C2×D61—44no—
313C2×D61#53412yes—
314C2×D61C2×#13714yes—
315C2×D61C2×#13714yes—
316C4×C61—22no—
317C4×C61—21no—
318C4×C61—44no—
319C4×C61—44no—
320C4×C61C2×H(5,𝔽2)12yes—
321Dic61—22no—
322Dic61—21no—
323Dic61—44no—
324Dic61—44no—
325Dic61#53412yes—
326C22×C22⋊Q8—612no—
327#3195—614no—
328#3195—614no—
329#3195—614no—
330#3195—611no—
331#3195—612no—
332#3195D6111yescharacteristic, direct product factor
333C2×#483—614no—
334C2×#470—614no—
335C2×#451—614no—
336C2×#407—611no—
337C2×C61—24no—
338C2×C61—41no—
339C2×C61C2×#53411yescharacteristic
340C2×C61C22×#13712yes—
341D61—24no—
342D61—41no—
343D61#319514yesdirect product factor
344C23×Q8—612no—
345C24×C4—611no—
346C22×Q16—614no—
347C22×M16—612no—
348C2×C22⋊Q8—612no—
349C2×C22⋊Q8—612no—
350C2×C22⋊Q8—614no—
351C2×C22⋊Q8—614no—
352C2×C22⋊Q8—614no—
353C2×C22⋊Q8—614no—
354C2×C22⋊Q8—612no—
355C2×C22⋊Q8—1222no—
356C2×C22⋊Q8—1222no—
357C2×C22⋊Q8—1222no—
358C2×C22⋊Q8—1222no—
359C2×C22⋊Q8—1222no—
360C2×C22⋊Q8—1222no—
361C2×C22⋊Q8C2×D6112yes—
362C22×C4⋊C4—612no—
363C22×C4⋊C4—1222no—
364C22×C22⋊C4—1222no—
365#483—614no—
366#483—614no—
367#483—618no—
368#483—618no—
369#483—614no—
370#483C2×D6114yescomplemented
371#470—614no—
372#470—614no—
373#470—618no—
374#470—614no—
375#470—618no—
376#470C2×D6114yescomplemented
377#451—614no—
378#451—618no—
379#451—614no—
380#451—618no—
381#451—614no—
382#451C2×D6114yescomplemented
383C2×#106—614no—
384C2×#103—618no—
385#407—611no—
386#407—612no—
387#407—614no—
388#407C2×D6111yescharacteristic
389C2×C22⋊C8—614no—
390C61C2×#319511yescharacteristic, complemented, Sylow
391C25—611no—
392C22×Q8—614no—
393C22×Q8—614no—
394C22×Q8—612no—
395C22×Q8—1222no—
396C22×Q8—1222no—
397C22×Q8—1222no—
398C22×Q8—2448no—
399C22×Q8C22×D6112yes—
400C23×C4—611no—
401C23×C4—612no—
402C23×C4—611no—
403C23×C4—612no—
404C23×C4—612no—
405C23×C4—611no—
406C23×C4—611no—
407C23×C4—612no—
408C23×C4—611no—
409C23×C4—1222no—
410C23×C4—1222no—
411C23×C4—2444no—
412C23×C4C22×D6111yescharacteristic
413C2×Q16—618no—
414C2×Q16—614no—
415C2×Q16—2448no—
416C2×Q16C22×D6114yescomplemented
417C2×M16—612no—
418C2×M16—614no—
419C2×M16—2444no—
420C2×M16C22×D6112yes—
421C22×C8—614no—
422C22⋊Q8—22no—
423C22⋊Q8—22no—
424C22⋊Q8—614no—
425C22⋊Q8—614no—
426C22⋊Q8—614no—
427C22⋊Q8—614no—
428C22⋊Q8—614no—
429C22⋊Q8—614no—
430C22⋊Q8—1224no—
431C22⋊Q8—1224no—
432C22⋊Q8—1224no—
433C22⋊Q8—1224no—
434C22⋊Q8—1224no—
435C22⋊Q8—1224no—
436C22⋊Q8—1224no—
437C22⋊Q8—1224no—
438C22⋊Q8—1222no—
439C22⋊Q8—1222no—
440C22⋊Q8—1222no—
441C22⋊Q8—1222no—
442C22⋊Q8—1222no—
443C22⋊Q8—1222no—
444C22⋊Q8C22×D6114yes—
445C22⋊Q8C22×D6114yes—
446C2×C4⋊C4—22no—
447C2×C4⋊C4—612no—
448C2×C4⋊C4—614no—
449C2×C4⋊C4—614no—
450C2×C4⋊C4—1222no—
451C2×C4⋊C4—1222no—
452C2×C4⋊C4—1224no—
453C2×C4⋊C4—1224no—
454C2×C4⋊C4—1224no—
455C2×C4⋊C4—1224no—
456C2×C4⋊C4—1222no—
457C2×C4⋊C4—1222no—
458C2×C4⋊C4—1222no—
459C2×C4⋊C4—1222no—
460C2×C4⋊C4C22×D6112yes—
461C2×C22⋊C4—22no—
462C2×C22⋊C4—1224no—
463C2×C22⋊C4—1224no—
464C2×C22⋊C4—1224no—
465C2×C22⋊C4—1224no—
466C2×C22⋊C4—1222no—
467C2×C22⋊C4—1222no—
468C2×C22⋊C4—1222no—
469#106—614no—
470#106—618no—
471#106C22×D6114yescomplemented
472#103—618no—
473#103—618no—
474#103—618no—
475#103C22×D6118yescomplemented
476C22⋊C8—614no—
477C22⋊C8—618no—
478C22⋊C8C22×D6114yes—
479C24—612no—
480C24—611no—
481C24—611no—
482C24—612no—
483C24—1228no—
484C24—2442no—
485C24D4×D6111yescharacteristic
486C2×Q8—22no—
487C2×Q8—44no—
488C2×Q8—614no—
489C2×Q8—1224no—
490C2×Q8—1224no—
491C2×Q8—12216no—
492C2×Q8—1222no—
493C2×Q8—2444no—
494C2×Q8—2448no—
495C2×Q8—2448no—
496C2×Q8—2448no—
497C2×Q8—2448no—
498C2×Q8C23×D6114yes—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—42no—
502C22×C4—611no—
503C22×C4—612no—
504C22×C4—611no—
505C22×C4—612no—
506C22×C4—611no—
507C22×C4—611no—
508C22×C4—612no—
509C22×C4—612no—
510C22×C4—611no—
511C22×C4—612no—
512C22×C4—611no—
513C22×C4—612no—
514C22×C4—612no—
515C22×C4—612no—
516C22×C4—1224no—
517C22×C4—1228no—
518C22×C4—1224no—
519C22×C4—1224no—
520C22×C4—1224no—
521C22×C4—1224no—
522C22×C4—1222no—
523C22×C4—1222no—
524C22×C4—2442no—
525C22×C4—2444no—
526C22×C4—2444no—
527C22×C4—2442no—
528C22×C4—2444no—
529C22×C4—2442no—
530C22×C4—2448no—
531C22×C4—2448no—
532C22×C4—2442no—
533C22×C4D4×D6112yes—
534C22×C4D4×D6111yescharacteristic
535C22×C4C23×D6112yes—
536C22×C4C23×D6111yescharacteristic
537Q16—44no—
538Q16—2444no—
539Q16—2448no—
540M16—42no—
541M16—2444no—
542M16—2442no—
543C2×C8—614no—
544C2×C8—1228no—
545C2×C8C23×D6114yes—
546C4⋊C4—24no—
547C4⋊C4—24no—
548C4⋊C4—22no—
549C4⋊C4—614no—
550C4⋊C4—1224no—
551C4⋊C4—1224no—
552C4⋊C4—1224no—
553C4⋊C4—1224no—
554C4⋊C4—1224no—
555C4⋊C4—1224no—
556C4⋊C4—1222no—
557C4⋊C4—1224no—
558C4⋊C4—1224no—
559C4⋊C4C23×D6114yes—
560C22⋊C4—24no—
561C22⋊C4—24no—
562C22⋊C4—1224no—
563C22⋊C4—1224no—
564C22⋊C4—1224no—
565C22⋊C4—1224no—
566C22⋊C4—1224no—
567C22⋊C4—1224no—
568C23—24no—
569C23—41no—
570C23—611no—
571C23—618no—
572C23—612no—
573C23—611no—
574C23—1224no—
575C23—1224no—
576C23—1222no—
577C23—1224no—
578C23—1224no—
579C23—1224no—
580C23—1224no—
581C23—2442no—
582C23—2441no—
583C23—24416no—
584C23C2×D4×D6111yescharacteristic
585C23C2×D4×D6112yes—
586Q8—28no—
587Q8—44no—
588Q8—1228no—
589Q8—2444no—
590Q8—2448no—
591Q8—2448no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—611no—
601C2×C4—614no—
602C2×C4—611no—
603C2×C4—612no—
604C2×C4—1222no—
605C2×C4—1224no—
606C2×C4—1222no—
607C2×C4—1224no—
608C2×C4—1222no—
609C2×C4—1224no—
610C2×C4—1222no—
611C2×C4—1224no—
612C2×C4—1224no—
613C2×C4—1224no—
614C2×C4—2442no—
615C2×C4—2442no—
616C2×C4—2442no—
617C2×C4—2448no—
618C2×C4—2448no—
619C2×C4—2448no—
620C2×C4—2448no—
621C2×C4—2442no—
622C2×C4—2441no—
623C2×C4—2444no—
624C2×C4—2444no—
625C2×C4C2×D4×D6111yescharacteristic
626C2×C4C2×D4×D6112yes—
627C2×C4C24×D6111yescharacteristic
628C8—24no—
629C8—1224no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—21no—
634C22—44no—
635C22—614no—
636C22—612no—
637C22—614no—
638C22—12216no—
639C22—1222no—
640C22—1222no—
641C22—1222no—
642C22—1221no—
643C22—2448no—
644C22—2444no—
645C22—2444no—
646C22#137×D6114yes—
647C22C22×D4×D6111yescharacteristic
648C4—21no—
649C4—22no—
650C4—44no—
651C4—44no—
652C4—612no—
653C4—1221no—
654C4—1222no—
655C4—2444no—
656C4—2444no—
657C4H(5,𝔽2)×D6112yes—
658C2—24no—
659C2—41no—
660C2—614no—
661C2—1224no—
662C2—2441no—
663C2#534×D6111yescharacteristic
664C2C2×#137×D6112yes—
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3195C2
socleC22×C61C23×D4
prime radicalC24×D61D4
hypercenter#3195D61
hypocenterC61C2×#3195
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3195

Complements

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

NH ≅ G/Ntype
2. #21456643C2semidirect
3. #21456642C2semidirect
4. #21456641C2semidirect
5. #21456640C2semidirect
6. #21456639C2semidirect
7. #483×D61C2semidirect
8. #470×D61C2semidirect
9. #451×D61C2semidirect
11. C61×#3195C2semidirect
14. C2×Q16×D61C22semidirect
32. #16840094C22semidirect
33. #16840093C22semidirect
34. #16840092C22semidirect
35. #16840091C22semidirect
36. #16840090C22semidirect
37. #16840021C22semidirect
38. #16840020C22semidirect
39. #16840019C22semidirect
40. #16840018C22semidirect
41. #16840017C22semidirect
42. #16839920C22semidirect
43. #16839919C22semidirect
44. #16839918C22semidirect
45. #16839917C22semidirect
46. #16839916C22semidirect
47. #106×D61C22semidirect
48. #103×D61C22semidirect
54. C61×#483C22semidirect
55. C61×#470C22semidirect
56. C61×#451C22semidirect
78. C2×Dic244C23semidirect
79. C2×#11758202C23semidirect
125. #14519709C23semidirect
126. #14519708C23semidirect
127. #14519702C23semidirect
128. #14519701C23semidirect
129. #14519700C23semidirect
134. C2×Q16×C61C23semidirect
143. #106×C61C23semidirect
144. #103×C61C23semidirect
332. #3195D61direct (N × H)
343. D61#3195direct (N × H)
370. #483C2×D61semidirect
376. #470C2×D61semidirect
382. #451C2×D61semidirect
390. C61C2×#3195semidirect
416. C2×Q16C22×D61semidirect
471. #106C22×D61semidirect
475. #103C22×D61semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊Q8×D612
#214566432
#214566422
#214566412
#214566402
#214566392
#483×D612
#470×D612
#451×D612
#407×D612
C61×#31952
C2×#319561