Finite Group Databasebeta

← #29689939

Subgroups — #29689939

Order 15616.

1162 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840299C211yesmaximal, characteristic
2C2×#16839784C211yesmaximal, characteristic, complemented
3#21448224C211yesmaximal, characteristic, complemented
4#21446775C212yesmaximal, complemented
5#21446748C212yesmaximal, complemented
6#21446256C212yesmaximal
7#21446249C212yesmaximal, complemented
8#21445824C211yesmaximal, characteristic, complemented
9#21445820C211yesmaximal, characteristic, complemented
10#21445816C211yesmaximal, characteristic, complemented
11C61×#1803C211yesmaximal, characteristic, complemented
12C22×#11758189C2212yes—
13#16840299—24no—
14#16840299C418yescomplemented
15C2×#14519772—21no—
16C2×#14519768C2211yescharacteristic
17C2×#14519766C2211yescharacteristic
18C2×#14519762—21no—
19C2×#14519753C2211yescharacteristic
20C2×#14519752C2211yescharacteristic
21C2×C4×D244—21no—
22C2×C4×D244C2211yescharacteristic
23C2×#14519735—21no—
24C2×#14519734—21no—
25C2×C42×D61C2211yescharacteristic
26#16839784—24no—
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519706C2212yes—
29C2×#14519701C2212yes—
30C2×#14519699C2212yes—
31C2×#14519697C2212yes—
32#16839632C2211yescharacteristic
33#16839627—21no—
34#16839619—21no—
35#16839619C2212yes—
36C4⋊C4×Dic61—21no—
37C4⋊C4×Dic61C2212yes—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C4×Q8×C61C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#103×C61C2212yes—
44C61×#384C2211yescharacteristic
45C61×#340C2212yes—
46C22×D244—21no—
47C22×D244C2311yescharacteristic
48C22×C4×D61—21no—
49C22×C4×D61D411yescharacteristic
50C22×C4×D61D411yescharacteristic
51C22×C4×D61C2311yescharacteristic
52C2×#11758189—24no—
53C2×#11758189—42no—
54C2×#11758189—42no—
55C22×#92418C2312yes—
56#14519772—28no—
57#14519768—22no—
58#14519768C2×C414yes—
59#14519766—22no—
60#14519766C2×C414yes—
61#14519762—28no—
62#14519753—22no—
63#14519753C2×C414yes—
64#14519752—22no—
65#14519752C2×C414yes—
66C4×D244—28no—
67C4×D244—22no—
68C4×D244C2×C414yes—
69#14519735—44no—
70#14519734—28no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61C2312yes—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—22no—
84C2×C4×Dic61—22no—
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88C42×D61C2×C414yes—
89C42×D61D414yes—
90C4⋊#92418—22no—
91#14519706—24no—
92#14519701—42no—
93#14519699—42no—
94#14519697—42no—
95#14519684—22no—
96#14519684—22no—
97C22×Q8×C61C2311yescharacteristic
98C22×C8×C61C2312yes—
99C4×Q8×C61—22no—
100C4×Q8×C61C2×C414yes—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61C2312yes—
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C42×C61—21no—
107C2×C42×C61C2311yescharacteristic
108C4⋊C8×C61—22no—
109#103×C61—42no—
110C23×D61—21no—
111C23×D61C2×D411yescharacteristic
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61—21no—
117C2×Q8×C61C22×C412yes—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—22no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2411yescharacteristic
124C22×Dic61—21no—
125C22×Dic61—22no—
126C22×Dic61C2×D411yescharacteristic
127C22×Dic61C2×D411yescharacteristic
128C22×Dic61C2×D411yescharacteristic
129C2×D244—21no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244—24no—
134C2×D244—21no—
135C2×D244—42no—
136C2×D244—42no—
137C2×D244C22×C412yes—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—21no—
141C2×C4×D61—21no—
142C2×C4×D61—21no—
143C2×C4×D61—21no—
144C2×C4×D61—24no—
145C2×C4×D61—41no—
146C2×C4×D61—41no—
147C2×C4×D61—42no—
148C2×C4×D61—42no—
149C2×C4×D61C22⋊C414yes—
150C2×C4×D61C22⋊C414yes—
151C2×C4×D61C22×C412yes—
152C2×C4×D61C2×D412yes—
153C2×C8×C61—22no—
154C2×C8×C61—22no—
155C2×C8×C61—22no—
156C4⋊C4×C61—24no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—42no—
161C4⋊C4×C61—42no—
162C4⋊C4×C61C22×C412yes—
163C42×C61—24no—
164C42×C61C22×C412yes—
165C42×C61C2×D412yes—
166#11758198—24no—
167#11758198—24no—
168#11758198—24no—
169#11758198—24no—
170#11758198—24no—
171#11758198—44no—
172C4⋊Dic61—21no—
173C4⋊Dic61—24no—
174C4⋊Dic61—22no—
175C4⋊Dic61—22no—
176C4⋊Dic61—42no—
177C4⋊Dic61—42no—
178C4⋊Dic61C22×C412yes—
179#11758196—24no—
180#11758196—24no—
181C4×Dic61—44no—
182C4×Dic61C22⋊C414yes—
183C4×Dic61C22⋊C414yes—
184C4×Dic61C22×C412yes—
185C4×Dic61C2×D412yes—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418—22no—
189#11758189—48no—
190C23×C61C22×D411yescharacteristic
191C22×D61—21no—
192C22×D61—21no—
193C22×D61—21no—
194C22×D61—21no—
195C22×D61—21no—
196C22×D61—21no—
197C22×D61—22no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61C2×C22⋊C412yes—
205Q8×C61—22no—
206Q8×C61—22no—
207Q8×C61—41no—
208Q8×C61—41no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61C2×C22⋊C412yes—
221C2×C4×C61C2×C22⋊C412yes—
222C2×C4×C61AGL(1,ℤ/8)12yes—
223C2×C4×C61#13712yes—
224C2×C4×C61C23×C411yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61H(5,𝔽2)11yescharacteristic
227C2×C4×C61H(5,𝔽2)11yescharacteristic
228C2×C4×C612−1+411yescharacteristic
229C2×C4×C612−1+411yescharacteristic
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—22no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61C2×C22⋊C412yes—
243C2×Dic61C2×C22⋊C412yes—
244C2×Dic61C2×C22⋊C412yes—
245D244—22no—
246D244—22no—
247D244—48no—
248D244—41no—
249D244—41no—
250C4×D61—22no—
251C4×D61—22no—
252C4×D61—22no—
253C4×D61—22no—
254C4×D61—28no—
255C4×D61—44no—
256C4×D61—48no—
257C8×C61—42no—
258#92418—42no—
259#21890—611nomaximal, Sylow
260C22×C61C22×C22⋊C411yescharacteristic
261C22×C61#53711yescharacteristic
262C22×C61#53711yescharacteristic
263C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
264C22×C61C2×#13711yescharacteristic
265C22×C61#57511yescharacteristic
266C22×C61#57511yescharacteristic
267C2×D61—22no—
268C2×D61—22no—
269C2×D61—22no—
270C2×D61—41no—
271C2×D61—41no—
272C2×D61—41no—
273C2×D61—41no—
274C2×D61—42no—
275C2×D61—42no—
276C2×D61—42no—
277C2×D61—81no—
278C2×D61—81no—
279C2×D61—81no—
280C2×D61—81no—
281C4×C61—22no—
282C4×C61—42no—
283C4×C61—42no—
284C4×C61—82no—
285C4×C61#41714yes—
286C4×C61#52011yescharacteristic
287C4×C61#52011yescharacteristic
288C4×C61#53511yescharacteristic
289C4×C61#53611yescharacteristic
290Dic61—22no—
291Dic61—22no—
292Dic61—22no—
293Dic61—42no—
294Dic61—82no—
295C2×#529—611no—
296C2×#423—611no—
297#2285—611no—
298#1989—612no—
299#1983—612no—
300#1895—612no—
301#1894—612no—
302#1803—611no—
303#1803D6111yescharacteristic, complemented
304#1802—611no—
305#1802—611no—
306C2×C61#231111yescharacteristic
307C2×C61C2×#41711yescharacteristic
308C2×C61#289611yescharacteristic
309C2×C61#325111yescharacteristic
310C2×C61#325111yescharacteristic
311C2×C61#325211yescharacteristic
312C2×C61#325211yescharacteristic
313D61—42no—
314D61—82no—
315C22×M16—612no—
316#529—618no—
317#529—1224no—
318C2×#126—1221no—
319C2×#125—611no—
320C2×#124—611no—
321C2×#123—1221no—
322C2×C22⋊Q8—611no—
323C2×#121—611no—
324C2×C4×Q8C2×D6111yescharacteristic
325C2×C4×D4—611no—
326C2×C4×D4—1221no—
327C2×#117—1221no—
328C2×#117—1221no—
329C22×C42—611no—
330#423—1224no—
331C2×C4⋊C8—612no—
332C2×C4⋊C8—611no—
333C2×C4⋊C8C2×D6111yescharacteristic
334C2×#103—612no—
335C2×#103C2×D6112yes—
336C2×#102—612no—
337C2×#102—612no—
338#384—611no—
339#384C2×D6111yescharacteristic
340#382—1221no—
341#380—612no—
342#380—1221no—
343C4×C4⋊C4—612no—
344C4×C4⋊C4—1221no—
345#340—612no—
346#340—612no—
347#340—612no—
348#340C2×D6112yes—
349C61#2189011yescharacteristic, complemented, Sylow
350C22×Q8C22×D6111yescharacteristic
351C22×D4—611no—
352C22×D4—1221no—
353C23×C4—611no—
354C23×C4—611no—
355C23×C4—611no—
356C23×C4—1221no—
357C2×M16—1224no—
358C2×M16—2442no—
359C2×M16—2442no—
360C22×C8—612no—
361C22×C8C22×D6112yes—
362#126—1228no—
363#125—614no—
364#125—1222no—
365#124—614no—
366#124—1222no—
367#123—1228no—
368C22⋊Q8—614no—
369C22⋊Q8—1222no—
370#121—614no—
371#121—1222no—
372C4×Q8—22no—
373C4×Q8C4×D6114yes—
374C4×D4—614no—
375C4×D4—1228no—
376C4×D4—1222no—
377#117—1228no—
378#117—2444no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—612no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4C22×D6112yes—
390C2×C4⋊C4C22×D6111yescharacteristic
391C2×C22⋊C4—1221no—
392C2×C22⋊C4—1221no—
393C2×C22⋊C4—1221no—
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—1221no—
396C2×C22⋊C4—1222no—
397C2×C42—21no—
398C2×C42—611no—
399C2×C42—611no—
400C2×C42—611no—
401C2×C42—614no—
402C2×C42—614no—
403C2×C42—1222no—
404C2×C42C22×D6111yescharacteristic
405C4⋊C8—22no—
406C4⋊C8—1224no—
407C4⋊C8—1222no—
408#103—42no—
409#103—2442no—
410#102—2442no—
411#102—2442no—
412#95—1222no—
413#95—1222no—
414C24—611no—
415C24—1221no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8C2×C4×D6112yes—
422C2×D4—612no—
423C2×D4—1224no—
424C2×D4—1221no—
425C2×D4—1221no—
426C2×D4—1221no—
427C2×D4—1221no—
428C2×D4—1221no—
429C2×D4—2442no—
430C2×D4—2442no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—611no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—612no—
438C22×C4—612no—
439C22×C4—614no—
440C22×C4—614no—
441C22×C4—1222no—
442C22×C4—1221no—
443C22×C4—1221no—
444C22×C4—1221no—
445C22×C4—1221no—
446C22×C4—1222no—
447C22×C4—1221no—
448C22×C4—1224no—
449C22×C4—1222no—
450C22×C4—2441no—
451C22×C4—2441no—
452C22×C4—2442no—
453C22×C4—2442no—
454C22×C4D4×D6111yescharacteristic
455C22×C4D4×D6111yescharacteristic
456C22×C4C23×D6111yescharacteristic
457M16—2448no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—1222no—
462C2×C8—1222no—
463C2×C8—1222no—
464C4⋊C4—21no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—612no—
471C4⋊C4—1224no—
472C4⋊C4—1224no—
473C4⋊C4—1224no—
474C4⋊C4—1221no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4—2442no—
478C4⋊C4—2442no—
479C4⋊C4C2×C4×D6112yes—
480C22⋊C4—1224no—
481C22⋊C4—1224no—
482C22⋊C4—1224no—
483C22⋊C4—1224no—
484C22⋊C4—1224no—
485C22⋊C4—2444no—
486C42—24no—
487C42—612no—
488C42—612no—
489C42—614no—
490C42—614no—
491C42—2444no—
492C42C2×C4×D6112yes—
493C42D4×D6112yes—
494C23—612no—
495C23—1221no—
496C23—1221no—
497C23—1221no—
498C23—1221no—
499C23—1221no—
500C23—1221no—
501C23—1222no—
502C23—2441no—
503C23—2441no—
504C23—2441no—
505C23—2441no—
506C23—2441no—
507C23—2441no—
508C23C2×D4×D6111yescharacteristic
509Q8—22no—
510Q8—22no—
511Q8—41no—
512Q8—41no—
513D4—1222no—
514D4—1222no—
515D4—2448no—
516D4—2441no—
517D4—2441no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—612no—
530C2×C4—612no—
531C2×C4—612no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1228no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—2444no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2448no—
551C2×C4C22⋊C4×D6112yes—
552C2×C4C22⋊C4×D6112yes—
553C2×C4#1451980412yes—
554C2×C4#1451980912yes—
555C2×C4C22×C4×D6111yescharacteristic
556C2×C4C2×D4×D6111yescharacteristic
557C2×C4#1451982211yescharacteristic
558C2×C4#1451982311yescharacteristic
559C2×C4#1451982411yescharacteristic
560C2×C4#1451982511yescharacteristic
561C8—42no—
562C8—2442no—
563C22—1222no—
564C22—1222no—
565C22—1222no—
566C22—4881no—
567C22—4881no—
568C22—4881no—
569C22—4881no—
570C22—2442no—
571C22—2442no—
572C22—2442no—
573C22—2441no—
574C22—2441no—
575C22—2441no—
576C22—2441no—
577C22C2×C22⋊C4×D6111yescharacteristic
578C22#1684034111yescharacteristic
579C22#1684034211yescharacteristic
580C22C2×#1451980411yescharacteristic
581C22C2×#1451980911yescharacteristic
582C22#1684055811yescharacteristic
583C22#1684056111yescharacteristic
584C4—22no—
585C4—42no—
586C4—42no—
587C4—82no—
588C4—1222no—
589C4—1222no—
590C4—1222no—
591C4—4882no—
592C4—2442no—
593C4#1683976214yes—
594C4#1684024211yescharacteristic
595C4#1684024311yescharacteristic
596C4#1684032811yescharacteristic
597C4#1684033511yescharacteristic
598C2—4882no—
599C2—2442no—
600C2#2144840811yescharacteristic
601C2C2×#1683976211yescharacteristic
602C2#2145433511yescharacteristic
603C2#2145717911yescharacteristic
604C2#2145718411yescharacteristic
605C2#2145719011yescharacteristic
606C2#2145719511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1803C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
cosocleC22×C4×C61C24
hypercenter#1803D61
hypocenterC61#21890
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. C2×#16839784C2semidirect
3. #21448224C2semidirect
4. #21446775C2semidirect
5. #21446748C2semidirect
7. #21446249C2semidirect
8. #21445824C2semidirect
9. #21445820C2semidirect
10. #21445816C2semidirect
11. C61×#1803C2semidirect
14. #16840299C4semidirect
303. #1803D61semidirect
349. C61#21890semidirect

Maximal subgroups

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

subgroupindex
C2×#168402992
C2×#168397842
#214482242
#214467752
#214467482
#214462562
#214462492
#214458242
#214458202
#214458162
C61×#18032
#2189061