Finite Group Databasebeta

← #29689713

Subgroups — #29689713

Order 15616.

1138 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840299C211yesmaximal, characteristic
2C2×#16839814C211yesmaximal, characteristic, complemented
3#21448224C211yesmaximal, characteristic, complemented
4#21446782C212yesmaximal, complemented
5#21446627C212yesmaximal, complemented
6#21446140C212yesmaximal
7#21446128C212yesmaximal, complemented
8#21445824C211yesmaximal, characteristic, complemented
9#21445820C211yesmaximal, characteristic, complemented
10#21445816C211yesmaximal, characteristic, complemented
11C61×#1803C211yesmaximal, characteristic, complemented
12#16840299—24no—
13#16840299C418yescomplemented
14C2×#14519772—21no—
15C2×#14519768C2211yescharacteristic
16C2×#14519766C2211yescharacteristic
17C2×#14519762—21no—
18C2×#14519753C2211yescharacteristic
19C2×#14519752C2211yescharacteristic
20C2×C4×D244—21no—
21C2×C4×D244C2211yescharacteristic
22C2×#14519735C2212yes—
23C2×#14519734—21no—
24C2×C42×D61C2211yescharacteristic
25#16839814—24no—
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519701C2212yes—
28C2×#14519699C2212yes—
29C2×#14519697C2212yes—
30C2×#14519689C2212yes—
31C2×C8×Dic61C2212yes—
32#16839632C2211yescharacteristic
33#16839627C2212yes—
34#16839619—21no—
35#16839619—21no—
36C4⋊C4×Dic61—21no—
37C4⋊C4×Dic61—21no—
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
52C22×#92418C2312yes—
53#14519772—28no—
54#14519768—22no—
55#14519768C2×C414yes—
56#14519766—22no—
57#14519766C2×C414yes—
58#14519762—28no—
59#14519753—22no—
60#14519753C2×C414yes—
61#14519752—22no—
62#14519752C2×C414yes—
63C4×D244—28no—
64C4×D244—22no—
65C4×D244C2×C414yes—
66#14519735—42no—
67#14519735—42no—
68#14519734—28no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2312yes—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—22no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86C42×D61C2×C414yes—
87C42×D61D414yes—
88C4⋊#92418—22no—
89#14519701—42no—
90#14519699—42no—
91#14519697—42no—
92#14519689—24no—
93C8×Dic61—24no—
94#14519684—22no—
95#14519684—22no—
96C22×Q8×C61C2311yescharacteristic
97C22×C8×C61C2312yes—
98C4×Q8×C61—22no—
99C4×Q8×C61C2×C414yes—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61C2312yes—
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C42×C61—21no—
106C2×C42×C61C2311yescharacteristic
107C4⋊C8×C61—22no—
108#103×C61—42no—
109C23×D61—21no—
110C23×D61C2×D411yescharacteristic
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61C22×C412yes—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2411yescharacteristic
123C22×Dic61—21no—
124C22×Dic61—22no—
125C22×Dic61C2×D411yescharacteristic
126C22×Dic61C2×D411yescharacteristic
127C22×Dic61C2×D411yescharacteristic
128C2×D244—21no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—24no—
133C2×D244—21no—
134C2×D244—42no—
135C2×D244—42no—
136C2×D244C22×C412yes—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—21no—
140C2×C4×D61—21no—
141C2×C4×D61—21no—
142C2×C4×D61—21no—
143C2×C4×D61—24no—
144C2×C4×D61—41no—
145C2×C4×D61—41no—
146C2×C4×D61—42no—
147C2×C4×D61—42no—
148C2×C4×D61C22⋊C414yes—
149C2×C4×D61C22⋊C414yes—
150C2×C4×D61C22×C412yes—
151C2×C4×D61C2×D412yes—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C2×C8×C61—22no—
155C4⋊C4×C61—24no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—42no—
160C4⋊C4×C61—42no—
161C4⋊C4×C61C22×C412yes—
162C42×C61—24no—
163C42×C61C22×C412yes—
164C42×C61C2×D412yes—
165#11758198—24no—
166#11758198—24no—
167#11758198—24no—
168#11758198—24no—
169#11758198—24no—
170#11758198—44no—
171C4⋊Dic61—21no—
172C4⋊Dic61—24no—
173C4⋊Dic61—22no—
174C4⋊Dic61—22no—
175C4⋊Dic61—42no—
176C4⋊Dic61—42no—
177C4⋊Dic61C22×C412yes—
178#11758196—24no—
179#11758196—24no—
180C4×Dic61—44no—
181C4×Dic61C22⋊C414yes—
182C4×Dic61C22⋊C414yes—
183C4×Dic61C22×C412yes—
184C4×Dic61C2×D412yes—
185C2×#92418—22no—
186C2×#92418—22no—
187C2×#92418—22no—
188C23×C61C22×D411yescharacteristic
189C22×D61—21no—
190C22×D61—21no—
191C22×D61—21no—
192C22×D61—21no—
193C22×D61—21no—
194C22×D61—21no—
195C22×D61—22no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61C2×C22⋊C412yes—
203Q8×C61—22no—
204Q8×C61—22no—
205Q8×C61—41no—
206Q8×C61—41no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61C2×C22⋊C412yes—
219C2×C4×C61C2×C22⋊C412yes—
220C2×C4×C61#13512yes—
221C2×C4×C61#13512yes—
222C2×C4×C61C23×C411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61H(5,𝔽2)11yescharacteristic
225C2×C4×C61H(5,𝔽2)11yescharacteristic
226C2×C4×C612−1+411yescharacteristic
227C2×C4×C612−1+411yescharacteristic
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—22no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61C2×C22⋊C412yes—
241C2×Dic61C2×C22⋊C412yes—
242C2×Dic61C2×C22⋊C412yes—
243D244—22no—
244D244—22no—
245D244—48no—
246D244—41no—
247D244—41no—
248C4×D61—22no—
249C4×D61—22no—
250C4×D61—22no—
251C4×D61—22no—
252C4×D61—28no—
253C4×D61—44no—
254C4×D61—48no—
255C8×C61—42no—
256#92418—42no—
257#21862—611nomaximal, Sylow
258C22×C61C22×C22⋊C411yescharacteristic
259C22×C61#53511yescharacteristic
260C22×C61#53611yescharacteristic
261C22×C61C2×#13511yescharacteristic
262C22×C61C2×#13511yescharacteristic
263C22×C61#57511yescharacteristic
264C22×C61#57511yescharacteristic
265C2×D61—22no—
266C2×D61—22no—
267C2×D61—22no—
268C2×D61—41no—
269C2×D61—41no—
270C2×D61—41no—
271C2×D61—41no—
272C2×D61—42no—
273C2×D61—42no—
274C2×D61—42no—
275C2×D61—81no—
276C2×D61—81no—
277C2×D61—81no—
278C2×D61—81no—
279C4×C61—22no—
280C4×C61—42no—
281C4×C61—42no—
282C4×C61—82no—
283C4×C61#41614yes—
284C4×C61#52011yescharacteristic
285C4×C61#52011yescharacteristic
286C4×C61#53711yescharacteristic
287C4×C61#53711yescharacteristic
288Dic61—22no—
289Dic61—22no—
290Dic61—22no—
291Dic61—42no—
292Dic61—82no—
293C2×#529—611no—
294C2×#431—611no—
295#2285—611no—
296#1991—612no—
297#1958—612no—
298#1875—612no—
299#1873—612no—
300#1803—611no—
301#1803D6111yescharacteristic, complemented
302#1802—611no—
303#1802—611no—
304C2×C61#231111yescharacteristic
305C2×C61C2×#41611yescharacteristic
306C2×C61#289611yescharacteristic
307C2×C61#321211yescharacteristic
308C2×C61#321211yescharacteristic
309C2×C61#323311yescharacteristic
310C2×C61#323311yescharacteristic
311D61—42no—
312D61—82no—
313#529—618no—
314#529—1224no—
315C2×#126—1221no—
316C2×#125—611no—
317C2×#124—611no—
318C2×#123—1221no—
319C2×C22⋊Q8—611no—
320C2×#121—611no—
321C2×C4×Q8C2×D6111yescharacteristic
322C2×C4×D4—611no—
323C2×C4×D4—1221no—
324C2×#117—612no—
325C2×#117—1221no—
326C22×C42—611no—
327#431—1224no—
328C2×C4⋊C8—611no—
329C2×C4⋊C8C2×D6111yescharacteristic
330C2×#103—612no—
331C2×#103C2×D6112yes—
332C2×#102—612no—
333C2×#102—612no—
334C2×C22⋊C8—612no—
335C2×C4×C8—612no—
336#384—611no—
337#384C2×D6111yescharacteristic
338#382—612no—
339#380—1221no—
340#380—1221no—
341C4×C4⋊C4—1221no—
342C4×C4⋊C4—1221no—
343#340—612no—
344#340—612no—
345#340—612no—
346#340C2×D6112yes—
347C61#2186211yescharacteristic, complemented, Sylow
348C22×Q8C22×D6111yescharacteristic
349C22×D4—611no—
350C22×D4—1221no—
351C23×C4—611no—
352C23×C4—611no—
353C23×C4—611no—
354C23×C4—1221no—
355C22×C8—612no—
356C22×C8C22×D6112yes—
357#126—1228no—
358#125—614no—
359#125—1222no—
360#124—614no—
361#124—1222no—
362#123—1228no—
363C22⋊Q8—614no—
364C22⋊Q8—1222no—
365#121—614no—
366#121—1222no—
367C4×Q8—22no—
368C4×Q8C4×D6114yes—
369C4×D4—614no—
370C4×D4—1228no—
371C4×D4—1222no—
372#117—1228no—
373#117—2442no—
374#117—2442no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—612no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4C22×D6112yes—
386C2×C4⋊C4C22×D6111yescharacteristic
387C2×C22⋊C4—1221no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1221no—
391C2×C22⋊C4—1221no—
392C2×C22⋊C4—1222no—
393C2×C42—21no—
394C2×C42—611no—
395C2×C42—611no—
396C2×C42—611no—
397C2×C42—614no—
398C2×C42—614no—
399C2×C42—1222no—
400C2×C42C22×D6111yescharacteristic
401C4⋊C8—22no—
402C4⋊C8—1222no—
403#103—42no—
404#103—2442no—
405#102—2442no—
406#102—2442no—
407C22⋊C8—1224no—
408C4×C8—1224no—
409#95—1222no—
410#95—1222no—
411C24—611no—
412C24—1221no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8C2×C4×D6112yes—
419C2×D4—612no—
420C2×D4—1224no—
421C2×D4—1221no—
422C2×D4—1221no—
423C2×D4—1221no—
424C2×D4—1221no—
425C2×D4—1221no—
426C2×D4—2442no—
427C2×D4—2442no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—611no—
434C22×C4—612no—
435C22×C4—612no—
436C22×C4—614no—
437C22×C4—614no—
438C22×C4—1221no—
439C22×C4—1221no—
440C22×C4—1221no—
441C22×C4—1221no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1221no—
445C22×C4—1224no—
446C22×C4—1222no—
447C22×C4—2441no—
448C22×C4—2441no—
449C22×C4—2442no—
450C22×C4—2442no—
451C22×C4D4×D6111yescharacteristic
452C22×C4D4×D6111yescharacteristic
453C22×C4C23×D6111yescharacteristic
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—1222no—
458C2×C8—1222no—
459C2×C8—1222no—
460C4⋊C4—21no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—612no—
467C4⋊C4—1224no—
468C4⋊C4—1224no—
469C4⋊C4—1224no—
470C4⋊C4—1221no—
471C4⋊C4—1222no—
472C4⋊C4—1222no—
473C4⋊C4—2442no—
474C4⋊C4—2442no—
475C4⋊C4C2×C4×D6112yes—
476C22⋊C4—1224no—
477C22⋊C4—1224no—
478C22⋊C4—1224no—
479C22⋊C4—1224no—
480C22⋊C4—1224no—
481C22⋊C4—2444no—
482C42—24no—
483C42—612no—
484C42—612no—
485C42—614no—
486C42—614no—
487C42—2444no—
488C42C2×C4×D6112yes—
489C42D4×D6112yes—
490C23—612no—
491C23—1221no—
492C23—1221no—
493C23—1221no—
494C23—1221no—
495C23—1221no—
496C23—1221no—
497C23—1222no—
498C23—2441no—
499C23—2441no—
500C23—2441no—
501C23—2441no—
502C23—2441no—
503C23—2441no—
504C23C2×D4×D6111yescharacteristic
505Q8—22no—
506Q8—22no—
507Q8—41no—
508Q8—41no—
509D4—1222no—
510D4—1222no—
511D4—2448no—
512D4—2441no—
513D4—2441no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—612no—
526C2×C4—612no—
527C2×C4—612no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1222no—
535C2×C4—1222no—
536C2×C4—1228no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—2444no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2448no—
547C2×C4C22⋊C4×D6112yes—
548C2×C4C22⋊C4×D6112yes—
549C2×C4#1451979812yes—
550C2×C4#1451980012yes—
551C2×C4C22×C4×D6111yescharacteristic
552C2×C4C2×D4×D6111yescharacteristic
553C2×C4#1451982211yescharacteristic
554C2×C4#1451982311yescharacteristic
555C2×C4#1451982411yescharacteristic
556C2×C4#1451982511yescharacteristic
557C8—42no—
558C8—2442no—
559C22—1222no—
560C22—1222no—
561C22—1222no—
562C22—4881no—
563C22—4881no—
564C22—4881no—
565C22—4881no—
566C22—2442no—
567C22—2442no—
568C22—2442no—
569C22—2441no—
570C22—2441no—
571C22—2441no—
572C22—2441no—
573C22C2×C22⋊C4×D6111yescharacteristic
574C22#1684032811yescharacteristic
575C22#1684033511yescharacteristic
576C22C2×#1451979811yescharacteristic
577C22C2×#1451980011yescharacteristic
578C22#1684055811yescharacteristic
579C22#1684056111yescharacteristic
580C4—22no—
581C4—42no—
582C4—42no—
583C4—82no—
584C4—1222no—
585C4—1222no—
586C4—1222no—
587C4—4882no—
588C4—2442no—
589C4#1683975714yes—
590C4#1684024211yescharacteristic
591C4#1684024311yescharacteristic
592C4#1684034111yescharacteristic
593C4#1684034211yescharacteristic
594C2—4882no—
595C2—2442no—
596C2#2144840811yescharacteristic
597C2C2×#1683975711yescharacteristic
598C2#2145433511yescharacteristic
599C2#2145679911yescharacteristic
600C2#2145680011yescharacteristic
601C2#2145700211yescharacteristic
602C2#2145700311yescharacteristic
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#21862
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×#16839814C2semidirect
3. #21448224C2semidirect
4. #21446782C2semidirect
5. #21446627C2semidirect
7. #21446128C2semidirect
8. #21445824C2semidirect
9. #21445820C2semidirect
10. #21445816C2semidirect
11. C61×#1803C2semidirect
13. #16840299C4semidirect
301. #1803D61semidirect
347. C61#21862semidirect

Maximal subgroups

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

subgroupindex
C2×#168402992
C2×#168398142
#214482242
#214467822
#214466272
#214461402
#214461282
#214458242
#214458202
#214458162
C61×#18032
#2186261