Finite Group Databasebeta

← #29689340

Subgroups — #29689340

Order 15616.

1374 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840311C211yesmaximal, characteristic
2C2×#16839813C211yesmaximal, characteristic, complemented
3#21448212C211yesmaximal, characteristic, complemented
4#21446744C212yesmaximal, complemented
5#21446740C212yesmaximal, complemented
6#21446631C211yesmaximal, characteristic, complemented
7#21446630C211yesmaximal, characteristic, complemented
8#21446628C211yesmaximal, characteristic, complemented
9#21445758C212yesmaximal, complemented
10#21445756C212yesmaximal, complemented
11C61×#1958C211yesmaximal, characteristic, complemented
12C22×#11758226C2211yescharacteristic
13C22×#11758224C2211yescharacteristic
14#16840311—24no—
15#16840311C418yescomplemented
16C2×#14519775C2211yescharacteristic
17C2×#14519773C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×#14519750—21no—
20C2×#14519750—21no—
21C2×#14519737—21no—
22C2×#14519737—21no—
23C2×C42×D61C2211yescharacteristic
24#16839813—24no—
25C2×#14519706C2212yes—
26C2×#14519702C2212yes—
27C2×#14519701C2212yes—
28C2×#14519700C2212yes—
29C2×#14519699C2212yes—
30C2×#14519698C2212yes—
31C2×#14519697C2212yes—
32C2×#14519689C2212yes—
33C2×C4×#92418C2211yescharacteristic
34#16839629—21no—
35#16839628C2211yescharacteristic
36#16839624—21no—
37#16839619C2212yes—
38C4⋊C4×Dic61C2212yes—
39#16839609—21no—
40#16839607—21no—
41C2×#124×C61C2211yescharacteristic
42C2×#103×C61C2212yes—
43C2×#102×C61C2212yes—
44C2×C4×C8×C61C2211yescharacteristic
45C61×#382C2211yescharacteristic
46C2×#11758226—22no—
47C2×#11758226—42no—
48C2×#11758226—42no—
49C2×#11758226C2×C414yes—
50C2×#11758224—22no—
51C2×#11758224—42no—
52C2×#11758224—42no—
53C2×#11758224C2×C414yes—
54C22×Dic122C2311yescharacteristic
55C22×D244—21no—
56C22×D244C2311yescharacteristic
57C22×C22⋊D61—21no—
58C22×C22⋊D61—21no—
59C22×C22⋊D61—21no—
60C23×Dic61—21no—
61C22×C4×D61—21no—
62C22×C4×D61D411yescharacteristic
63C22×C4×D61D411yescharacteristic
64C22×C4×D61C2311yescharacteristic
65C22×#92418C2312yes—
66#14519775—22no—
67#14519775C2×C414yes—
68#14519773—22no—
69#14519773C2×C414yes—
70#14519765—22no—
71#14519765C2×C414yes—
72#14519750—28no—
73#14519750—28no—
74#14519737—28no—
75#14519737—28no—
76C2×C4⋊Dic61C2312yes—
77C2×#11758196—21no—
78C2×#11758196—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—
88#14519706—24no—
89#14519702—42no—
90#14519701—42no—
91#14519700—42no—
92#14519699—42no—
93#14519698—42no—
94#14519697—42no—
95#14519689—24no—
96C4×#92418—22no—
97#14519684—21no—
98#14519684—21no—
99#14519684—22no—
100#14519684—22no—
101C22×Q8×C61C2311yescharacteristic
102C22×D4×C61C2311yescharacteristic
103C22×C8×C61C2312yes—
104#124×C61—22no—
105#124×C61C2×C414yes—
106C2×C4⋊C4×C61C2312yes—
107C2×C22⋊C4×C61—21no—
108C2×C22⋊C4×C61—21no—
109C2×C42×C61C2311yescharacteristic
110#103×C61—42no—
111#102×C61—42no—
112C4×C8×C61—22no—
113#95×C61—21no—
114#95×C61—21no—
115C24×C61—21no—
116C23×D61—21no—
117C23×D61C2×D411yescharacteristic
118C2×Q8×C61—21no—
119C2×Q8×C61—22no—
120C2×Q8×C61—22no—
121C2×Q8×C61C22×C412yes—
122C2×D4×C61—21no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61C22×C412yes—
126C22×C4×C61—21no—
127C22×C4×C61—22no—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2411yescharacteristic
131C2×C22⋊D61—24no—
132C2×C22⋊D61—24no—
133C2×C22⋊D61—24no—
134C2×C22⋊D61—44no—
135C2×C22⋊D61—42no—
136C2×C22⋊D61—42no—
137C2×C22⋊D61—44no—
138C22×Dic61—21no—
139C22×Dic61—24no—
140C22×Dic61—22no—
141C22×Dic61—42no—
142C22×Dic61—42no—
143C22×Dic61C2×D411yescharacteristic
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146#11758226—48no—
147#11758226—48no—
148#11758224—48no—
149#11758224—48no—
150C2×D244—22no—
151C2×D244—22no—
152C2×D244—24no—
153C2×D244—21no—
154C2×D244—42no—
155C2×D244—42no—
156C2×D244C22×C412yes—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—24no—
160C2×C4×D61—24no—
161C2×C4×D61—42no—
162C2×C4×D61—42no—
163C2×C4×D61—42no—
164C2×C4×D61C22⋊C414yes—
165C2×C4×D61C22⋊C414yes—
166C2×C4×D61C22×C412yes—
167C2×C4×D61C2×D412yes—
168C2×Dic122—22no—
169C2×Dic122—22no—
170C2×Dic122—21no—
171C2×Dic122C22×C412yes—
172C2×C8×C61—22no—
173C2×C8×C61—22no—
174C2×C8×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—22no—
177C22⋊C4×C61—24no—
178C22⋊C4×C61—24no—
179C42×C61C22×C412yes—
180C42×C61C2×D412yes—
181#11758198—24no—
182#11758198—24no—
183#11758198—44no—
184C4⋊Dic61—22no—
185C4⋊Dic61—22no—
186#11758196—24no—
187#11758196—24no—
188C4×Dic61—44no—
189C4×Dic61C22⋊C414yes—
190C4×Dic61C22⋊C414yes—
191C4×Dic61C22×C412yes—
192C4×Dic61C2×D412yes—
193C2×#92418—22no—
194C2×#92418—22no—
195C2×#92418—22no—
196C23×C61—22no—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61—42no—
200C23×C61—42no—
201C23×C61C22×D411yescharacteristic
202C22×D61—21no—
203C22×D61—21no—
204C22×D61—22no—
205C22×D61—22no—
206C22×D61—22no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—42no—
210C22×D61—42no—
211C22×D61C2×C22⋊C412yes—
212Q8×C61—24no—
213Q8×C61—42no—
214D4×C61—24no—
215D4×C61—42no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61C2×C22⋊C412yes—
225C2×C4×C61C2×C22⋊C412yes—
226C2×C4×C61C23×C411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61H(5,𝔽2)12yes—
229C2×C4×C61H(5,𝔽2)12yes—
230C22⋊D61—48no—
231C22⋊D61—48no—
232C22⋊D61—48no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—48no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61C2×C22⋊C412yes—
244C2×Dic61C2×C22⋊C412yes—
245C2×Dic61C2×C22⋊C412yes—
246C2×Dic61#13514yes—
247D244—24no—
248D244—48no—
249D244—42no—
250C4×D61—24no—
251C4×D61—24no—
252C4×D61—28no—
253C4×D61—44no—
254C4×D61—48no—
255Dic122—24no—
256Dic122—42no—
257C8×C61—42no—
258#92418—42no—
259#21807—611nomaximal, Sylow
260C22×C61—42no—
261C22×C61—42no—
262C22×C61—42no—
263C22×C61—82no—
264C22×C61—82no—
265C22×C61C22×C22⋊C411yescharacteristic
266C22×C61#53511yescharacteristic
267C22×C61#53511yescharacteristic
268C22×C61C2×#13512yes—
269C22×C61#57512yes—
270C2×D61—22no—
271C2×D61—22no—
272C2×D61—22no—
273C2×D61—42no—
274C2×D61—42no—
275C2×D61—42no—
276C2×D61—42no—
277C2×D61—42no—
278C2×D61—82no—
279C2×D61—82no—
280C4×C61—22no—
281C4×C61—22no—
282C4×C61—42no—
283C4×C61—82no—
284C4×C61#51812yes—
285C4×C61#53512yes—
286Dic61—22no—
287Dic61—22no—
288Dic61—42no—
289Dic61—82no—
290Dic61#41614yes—
291C2×#532—611no—
292C2×#431—611no—
293#2283—611no—
294#1982—612no—
295#1981—612no—
296#1958—611no—
297#1958—611no—
298#1958—611no—
299#1958D6111yescharacteristic, complemented
300#1788—612no—
301#1788—612no—
302C2×C61—82no—
303C2×C61#230911yescharacteristic
304C2×C61C2×#41611yescharacteristic
305C2×C61#289611yescharacteristic
306C2×C61#321212yes—
307C2×C61#321212yes—
308D61—42no—
309D61—82no—
310C22×#41—611no—
311C22×#41—611no—
312#532—618no—
313#532—1224no—
314C2×C4⋊Q8—611no—
315C2×#127—611no—
316C2×#124—611no—
317C2×#124C2×D6111yescharacteristic
318C2×#121—1221no—
319C2×#121—1221no—
320C2×C4×D4—1221no—
321C2×C4×D4—1221no—
322C22×C42—611no—
323#431—1224no—
324C2×C4⋊C8—612no—
325C2×#103—612no—
326C2×#103—612no—
327C2×#103—612no—
328C2×#103C2×D6112yes—
329C2×#102—612no—
330C2×#102—612no—
331C2×#102—612no—
332C2×#102C2×D6112yes—
333C2×C22⋊C8—612no—
334C2×C4×C8—611no—
335C2×C4×C8C2×D6111yescharacteristic
336#383—1221no—
337#382—611no—
338#382C2×D6111yescharacteristic
339#381—1221no—
340#380—612no—
341C4×C4⋊C4—612no—
342C4×C22⋊C4—1221no—
343#376—1221no—
344C61#2180711yescharacteristic, complemented, Sylow
345C2×#41—614no—
346C2×#41—614no—
347C2×#41—1222no—
348C2×#41—1222no—
349C2×#41—2442no—
350C2×#41—2442no—
351C2×#41—2442no—
352C2×#41—2442no—
353C22×Q8—611no—
354C22×Q8C22×D6111yescharacteristic
355C22×D4—611no—
356C22×D4—1221no—
357C22×D4—1221no—
358C22×D4—1221no—
359C22×D4—1221no—
360C22×D4C22×D6111yescharacteristic
361C23×C4—611no—
362C23×C4—611no—
363C23×C4—611no—
364C23×C4—1221no—
365C23×C4—1221no—
366C22×C8—612no—
367C22×C8C22×D6112yes—
368C4⋊Q8—614no—
369C4⋊Q8—1222no—
370#127—614no—
371#127—1222no—
372#124—22no—
373#124—614no—
374#124—1222no—
375#124C4×D6114yes—
376#121—1228no—
377#121—1228no—
378C4×D4—1228no—
379C4×D4—1228no—
380C2×C4⋊C4—612no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4C22×D6112yes—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1221no—
388C2×C22⋊C4—1222no—
389C2×C42—611no—
390C2×C42—611no—
391C2×C42—611no—
392C2×C42—614no—
393C2×C42—614no—
394C2×C42—1222no—
395C2×C42C22×D6111yescharacteristic
396C4⋊C8—1224no—
397#103—42no—
398#103—2442no—
399#103—2442no—
400#103—2442no—
401#102—42no—
402#102—2442no—
403#102—2442no—
404#102—2442no—
405C22⋊C8—1224no—
406C4×C8—22no—
407C4×C8—1222no—
408#95—21no—
409#95—21no—
410#95—1222no—
411#95—1222no—
412#95—1221no—
413#95—1221no—
414C24—21no—
415C24—611no—
416C24—1221no—
417#41—2448no—
418#41—2448no—
419#41—2448no—
420#41—2448no—
421C2×Q8—21no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—612no—
425C2×Q8—1221no—
426C2×Q8—1222no—
427C2×Q8—1222no—
428C2×Q8C2×C4×D6112yes—
429C2×D4—21no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—612no—
433C2×D4—1224no—
434C2×D4—1224no—
435C2×D4—1224no—
436C2×D4—1224no—
437C2×D4—1221no—
438C2×D4—1222no—
439C2×D4—1222no—
440C2×D4—2442no—
441C2×D4—2442no—
442C2×D4—2444no—
443C2×D4—2444no—
444C2×D4—2442no—
445C2×D4—2442no—
446C2×D4C2×C4×D6112yes—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—612no—
453C22×C4—612no—
454C22×C4—614no—
455C22×C4—614no—
456C22×C4—1224no—
457C22×C4—1224no—
458C22×C4—1224no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1221no—
462C22×C4—1222no—
463C22×C4—2442no—
464C22×C4—2442no—
465C22×C4—2442no—
466C22×C4—2442no—
467C22×C4—2442no—
468C22×C4D4×D6111yescharacteristic
469C22×C4D4×D6111yescharacteristic
470C22×C4C23×D6111yescharacteristic
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—1222no—
475C2×C8—1222no—
476C2×C8—1222no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—1224no—
480C4⋊C4—1224no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—1224no—
486C22⋊C4—1224no—
487C22⋊C4—2444no—
488C42—612no—
489C42—612no—
490C42—614no—
491C42—614no—
492C42—2444no—
493C42C2×C4×D6112yes—
494C42D4×D6112yes—
495C23—22no—
496C23—42no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23—612no—
501C23—1222no—
502C23—1222no—
503C23—1221no—
504C23—1221no—
505C23—1222no—
506C23—2442no—
507C23—2442no—
508C23—2441no—
509C23—2441no—
510C23C2×D4×D6111yescharacteristic
511Q8—24no—
512Q8—42no—
513Q8—1224no—
514Q8—2442no—
515D4—24no—
516D4—42no—
517D4—1224no—
518D4—2448no—
519D4—2448no—
520D4—2448no—
521D4—2448no—
522D4—2442no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—612no—
532C2×C4—614no—
533C2×C4—612no—
534C2×C4—612no—
535C2×C4—1228no—
536C2×C4—1222no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1224no—
540C2×C4—1224no—
541C2×C4—1222no—
542C2×C4—2444no—
543C2×C4—2448no—
544C2×C4—2448no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4C22⋊C4×D6112yes—
551C2×C4C22⋊C4×D6112yes—
552C2×C4C22×C4×D6111yescharacteristic
553C2×C4C2×D4×D6111yescharacteristic
554C2×C4#1451982212yes—
555C2×C4#1451982312yes—
556C8—42no—
557C8—2442no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—82no—
562C22—82no—
563C22—1222no—
564C22—1222no—
565C22—1222no—
566C22—4882no—
567C22—4882no—
568C22—2442no—
569C22—2442no—
570C22—2442no—
571C22—2442no—
572C22—2442no—
573C22C2×C22⋊C4×D6111yescharacteristic
574C22#1684032511yescharacteristic
575C22#1684033011yescharacteristic
576C22#135×D6112yes—
577C22#1684055912yes—
578C4—22no—
579C4—22no—
580C4—42no—
581C4—82no—
582C4—614no—
583C4—1222no—
584C4—1222no—
585C4—4882no—
586C4—2442no—
587C4#1684023412yes—
588C4#1684032812yes—
589C2—82no—
590C2—4882no—
591C2—2442no—
592C2#2144839811yescharacteristic
593C2#416×D6111yescharacteristic
594C2#2145433411yescharacteristic
595C2#2145679012yes—
596C2#2145679612yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1958C2
socleC23×C61C22×D4
prime radicalC2×#16840311C2
cosocleC22×C4×C61C24
hypercenter#1958D61
hypocenterC61#21807
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1958

Complements

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

NH ≅ G/Ntype
2. C2×#16839813C2semidirect
3. #21448212C2semidirect
4. #21446744C2semidirect
5. #21446740C2semidirect
6. #21446631C2semidirect
7. #21446630C2semidirect
8. #21446628C2semidirect
9. #21445758C2semidirect
10. #21445756C2semidirect
11. C61×#1958C2semidirect
15. #16840311C4semidirect
299. #1958D61semidirect
344. C61#21807semidirect

Maximal subgroups

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

subgroupindex
C2×#168403112
C2×#168398132
#214482122
#214467442
#214467402
#214466312
#214466302
#214466282
#214457582
#214457562
C61×#19582
#2180761