Finite Group Databasebeta

← #29712212

Subgroups — #29712212

Order 15616.

1304 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×#16840460C211yesmaximal, characteristic
2C2×#16839901C211yesmaximal, characteristic, complemented
3#21449178C211yesmaximal, characteristic, complemented
4#21446968C212yesmaximal, complemented
5#21446821C212yesmaximal, complemented
6#21446247C211yesmaximal, characteristic, complemented
7#21446150C212yesmaximal, complemented
8#21446128C212yesmaximal, complemented
9#21446111C211yesmaximal, characteristic, complemented
10#21445909C211yesmaximal, characteristic, complemented
11C22×#103×C61C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13#16840460—24no—
14#16840460—24no—
15C2×#14519772—21no—
16C2×#14519768C2211yescharacteristic
17C2×#14519762—21no—
18C2×#14519753C2212yes—
19C2×#14519752C2211yescharacteristic
20C2×#14519751—21no—
21C2×#14519751C2211yescharacteristic
22C2×#14519739C2211yescharacteristic
23C2×C4×C22⋊D61—21no—
24C2×C4×C22⋊D61C2211yescharacteristic
25C2×#14519735C2211yescharacteristic
26#16839901—24no—
27C2×#14519701C2212yes—
28C2×#14519699C2212yes—
29C2×C22⋊#92418C2211yescharacteristic
30C2×#14519689C2212yes—
31#16839681—21no—
32#16839673—21no—
33#16839669C2212yes—
34#16839641—21no—
35#16839635—21no—
36#16839622C2211yescharacteristic
37#16839525C2212yes—
38#16839524C2212yes—
39#16839524C2212yes—
40#16839516C2212yes—
41C23×Q8×C61C2211yescharacteristic
42C23×C8×C61C2211yescharacteristic
43C22×C4⋊C4×C61C2211yescharacteristic
44C2×#103×C61—24no—
45C2×#103×C61C2212yes—
46C2×#103×C61C2212yes—
47C22×D244—21no—
48C22×D244C2311yescharacteristic
49C22×C22⋊D61—21no—
50C22×C22⋊D61D411yescharacteristic
51C22×C4×D61—21no—
52C22×C4×D61C2311yescharacteristic
53C2×#11758201—24no—
54C2×#11758201—24no—
55C2×#11758201—42no—
56C22×#92418C2312yes—
57#14519772—28no—
58#14519768—22no—
59#14519768—22no—
60#14519762—28no—
61#14519753—24no—
62#14519753—42no—
63#14519752—22no—
64#14519752—22no—
65#14519751—28no—
66#14519751—22no—
67#14519751D414yes—
68#14519739—22no—
69#14519739—22no—
70C4×C22⋊D61—28no—
71C4×C22⋊D61—22no—
72C4×C22⋊D61—22no—
73#14519735—22no—
74#14519735—22no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61C2312yes—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×#11758196—22no—
80C2×#11758196D411yescharacteristic
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61D411yescharacteristic
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—22no—
88C2×#11758198D411yescharacteristic
89C2×C4×Dic61—21no—
90C2×C4×Dic61C2311yescharacteristic
91#14519701—24no—
92#14519699—42no—
93C22⋊#92418—22no—
94#14519689—42no—
95#14519684—21no—
96#14519684—22no—
97#14519684—22no—
98C22×Q8×C61—21no—
99C22×Q8×C61—24no—
100C22×Q8×C61—22no—
101C22×Q8×C61—22no—
102C22×Q8×C61—42no—
103C22×Q8×C61C2312yes—
104C23×C4×C61—21no—
105C23×C4×C61—21no—
106C23×C4×C61C2311yescharacteristic
107C22×C8×C61—22no—
108C22×C8×C61—42no—
109C22×C8×C61C2312yes—
110C2×C4⋊C4×C61—22no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61D414yes—
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115#103×C61—24no—
116#103×C61—24no—
117C24×C61C2×D411yescharacteristic
118C23×D61—21no—
119C23×D61C2×D411yescharacteristic
120C2×Q8×C61—22no—
121C2×Q8×C61—28no—
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C2×Q8×C61—22no—
126C2×Q8×C61—44no—
127C2×Q8×C61—48no—
128C2×Q8×C61—44no—
129C2×Q8×C61—42no—
130C2×Q8×C61#4114yes—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—22no—
136C22×C4×C61—24no—
137C22×C4×C61—44no—
138C22×C4×C61—42no—
139C22×C4×C61—82no—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D412yes—
142C22×C4×C61C2×D412yes—
143C22×C4×C61C2411yescharacteristic
144C2×C22⋊D61—24no—
145C2×C22⋊D61—22no—
146C2×C22⋊D61—22no—
147C2×C22⋊D61—22no—
148C2×C22⋊D61—42no—
149C2×C22⋊D61—42no—
150C22×Dic61—21no—
151C22×Dic61—22no—
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C2×D244—22no—
156C2×D244—22no—
157C2×D244—21no—
158C2×D244—24no—
159C2×D244—82no—
160C2×D244C2×D412yes—
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—24no—
166C2×C4×D61—82no—
167C2×C8×C61—22no—
168C2×C8×C61—22no—
169C2×C8×C61—22no—
170C2×C8×C61—44no—
171C4⋊C4×C61—24no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61C2×D412yes—
175C4⋊C4×C61C2×D412yes—
176C22⋊Dic61—22no—
177C22⋊Dic61—42no—
178#11758201—48no—
179#11758198—24no—
180#11758198—24no—
181#11758198—24no—
182#11758198—24no—
183#11758198—22no—
184#11758198—44no—
185C4⋊Dic61—24no—
186C4⋊Dic61—21no—
187C4⋊Dic61—22no—
188C4⋊Dic61—22no—
189C4⋊Dic61C2×D412yes—
190#11758196—24no—
191#11758196—22no—
192#11758196—44no—
193C4×Dic61—24no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C2×#92418—22no—
197C2×#92418—22no—
198C2×#92418—22no—
199C23×C61—22no—
200C23×C61—22no—
201C23×C61#12012yes—
202C23×C61#13512yes—
203C23×C61#13712yes—
204C23×C61C22×D411yescharacteristic
205C22×D61—21no—
206C22×D61—22no—
207C22×D61—22no—
208C22×D61—21no—
209C22×D61—21no—
210C22×D61—22no—
211C22×D61—42no—
212C22×D61—41no—
213C22×D61—41no—
214C22×D61—42no—
215Q8×C61—24no—
216Q8×C61—24no—
217Q8×C61—48no—
218Q8×C61—42no—
219Q8×C61—42no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—24no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—24no—
227C2×C4×C61—48no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—84no—
233C2×C4×C61#12012yes—
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C2×#4112yes—
237C2×C4×C61H(5,𝔽2)12yes—
238C22⋊D61—44no—
239C22⋊D61—48no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—22no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253D244—24no—
254D244—41no—
255D244—41no—
256D244—84no—
257C4×D61—41no—
258C4×D61—41no—
259C4×D61—42no—
260C4×D61—84no—
261C8×C61—42no—
262#92418—42no—
263#23964—611nomaximal, Sylow
264C22×C61—22no—
265C22×C61—22no—
266C22×C61—24no—
267C22×C61—24no—
268C22×C61#45214yes—
269C22×C61C2×#12011yescharacteristic
270C22×C61#54612yes—
271C22×C61C2×#13511yescharacteristic
272C22×C61C2×#13711yescharacteristic
273C22×C61#57712yes—
274C2×D61—41no—
275C2×D61—42no—
276C2×D61—41no—
277C2×D61—42no—
278C2×D61—42no—
279C2×D61—41no—
280C2×D61—42no—
281C2×D61—42no—
282C2×D61—82no—
283C2×D61—82no—
284C4×C61—22no—
285C4×C61—44no—
286C4×C61—42no—
287C4×C61—42no—
288C4×C61#52511yescharacteristic
289C4×C61#53511yescharacteristic
290C4×C61#54612yes—
291Dic61—41no—
292Dic61—41no—
293Dic61—42no—
294Dic61—41no—
295Dic61—82no—
296C2×#553—611no—
297C2×#448—611no—
298C22×#103D6111yescharacteristic, complemented
299#2409—611no—
300#2026—612no—
301#2000—612no—
302#1893—611no—
303#1877—612no—
304#1873—612no—
305#1869—611no—
306#1824—611no—
307C2×C61—24no—
308C2×C61#240411yescharacteristic
309C2×C61C2×#45211yescharacteristic
310C2×C61#300911yescharacteristic
311C2×C61#330112yes—
312C2×C61#332012yes—
313D61—81no—
314D61—82no—
315C23×Q8C2×D6111yescharacteristic
316C22×SD16—612no—
317C23×C8C2×D6111yescharacteristic
318#553—1224no—
319#553—1224no—
320C2×#126—1221no—
321C2×#125—611no—
322C2×#123—1221no—
323C2×C22⋊Q8—612no—
324C2×#121—611no—
325C2×#121—611no—
326C2×#121—1221no—
327C2×C4×D4—611no—
328C2×C4×D4—611no—
329C2×C4×D4—1221no—
330C2×#117—611no—
331C22×C4⋊C4C2×D6111yescharacteristic
332#448—1224no—
333C2×#103—24no—
334C2×#103—612no—
335C2×#103C2×D6112yes—
336C2×#103C2×D6112yes—
337C2×#102—612no—
338C2×C22⋊C8—611no—
339C2×C22⋊C8—612no—
340#395—1221no—
341#393—1221no—
342#391—612no—
343#386—1221no—
344#385—1221no—
345#380—611no—
346#340—612no—
347#340—612no—
348#340—612no—
349#336—612no—
350C61#2396411yescharacteristic, complemented, Sylow
351C22×Q8—21no—
352C22×Q8—24no—
353C22×Q8—22no—
354C22×Q8—22no—
355C22×Q8—42no—
356C22×Q8C22×D6112yes—
357C22×D4—611no—
358C22×D4—611no—
359C22×D4—1221no—
360C22×D4—1221no—
361C23×C4—21no—
362C23×C4—21no—
363C23×C4—611no—
364C23×C4—1221no—
365C23×C4C22×D6111yescharacteristic
366C2×SD16—1224no—
367C2×SD16—1224no—
368C2×SD16—2442no—
369C22×C8—22no—
370C22×C8—42no—
371C22×C8—612no—
372C22×C8C22×D6112yes—
373#126—1228no—
374#125—1222no—
375#125—1222no—
376#123—1228no—
377C22⋊Q8—1224no—
378C22⋊Q8—2442no—
379#121—614no—
380#121—1222no—
381#121—1228no—
382#121—1222no—
383#121—1222no—
384C4×D4—1222no—
385C4×D4—1228no—
386C4×D4—1222no—
387C4×D4—1222no—
388C4×D4—1222no—
389#117—1222no—
390#117—1222no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—612no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1222no—
399C2×C4⋊C4C22⋊D6114yes—
400C2×C4⋊C4C22×D6111yescharacteristic
401C2×C4⋊C4C22×D6111yescharacteristic
402C2×C22⋊C4—611no—
403C2×C22⋊C4—611no—
404C2×C22⋊C4—1221no—
405C2×C22⋊C4—1221no—
406C2×C22⋊C4—1221no—
407C2×C22⋊C4—1221no—
408C2×C22⋊C4—1222no—
409C2×C22⋊C4—1221no—
410C2×C42—611no—
411C2×C42—1221no—
412#103—24no—
413#103—24no—
414#103—1224no—
415#102—2442no—
416C22⋊C8—1222no—
417C22⋊C8—2442no—
418#95—1222no—
419#95—1222no—
420#95—1221no—
421C24—611no—
422C24—1221no—
423C24D4×D6111yescharacteristic
424C2×Q8—22no—
425C2×Q8—28no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—44no—
431C2×Q8—48no—
432C2×Q8—44no—
433C2×Q8—42no—
434C2×Q8#1175822214yes—
435C2×D4—612no—
436C2×D4—1224no—
437C2×D4—1222no—
438C2×D4—1224no—
439C2×D4—1222no—
440C2×D4—1222no—
441C2×D4—1221no—
442C2×D4—1222no—
443C2×D4—1222no—
444C2×D4—4882no—
445C2×D4—2442no—
446C2×D4—2442no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—44no—
454C22×C4—42no—
455C22×C4—82no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—1224no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—4882no—
467C22×C4D4×D6112yes—
468C22×C4D4×D6111yescharacteristic
469C22×C4C2×C22⋊D6112yes—
470C22×C4C23×D6111yescharacteristic
471SD16—2448no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—44no—
476C2×C8—1222no—
477C2×C8—1222no—
478C2×C8—1222no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—24no—
482C4⋊C4—612no—
483C4⋊C4—1224no—
484C4⋊C4—1224no—
485C4⋊C4—1221no—
486C4⋊C4—1222no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C4⋊C4—2444no—
490C4⋊C4C2×C22⋊D6112yes—
491C4⋊C4C2×C22⋊D6112yes—
492C22⋊C4—1224no—
493C22⋊C4—1224no—
494C22⋊C4—1224no—
495C22⋊C4—1222no—
496C22⋊C4—1222no—
497C22⋊C4—1224no—
498C22⋊C4—2444no—
499C22⋊C4—2442no—
500C42—1224no—
501C42—1221no—
502C42—1221no—
503C23—22no—
504C23—22no—
505C23—1222no—
506C23—1221no—
507C23—1221no—
508C23—1221no—
509C23—1222no—
510C23—1222no—
511C23—2441no—
512C23—2441no—
513C23—2442no—
514C23—2442no—
515C23#1451974512yes—
516C23#1451980012yes—
517C23#1451980912yes—
518C23C2×D4×D6111yescharacteristic
519Q8—24no—
520Q8—24no—
521Q8—48no—
522Q8—42no—
523Q8—42no—
524D4—1224no—
525D4—4884no—
526D4—2444no—
527D4—2448no—
528D4—2441no—
529D4—2441no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—48no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—84no—
543C2×C4—1221no—
544C2×C4—1222no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1222no—
550C2×C4—1222no—
551C2×C4—4884no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2442no—
560C2×C4#1451974512yes—
561C2×C4C2×D4×D6111yescharacteristic
562C2×C4C22×C22⋊D6111yescharacteristic
563C2×C4C2×#1175822212yes—
564C2×C4#1451982212yes—
565C8—42no—
566C8—2442no—
567C22—24no—
568C22—24no—
569C22—22no—
570C22—22no—
571C22—4882no—
572C22—4882no—
573C22—2442no—
574C22—2442no—
575C22—2441no—
576C22—2442no—
577C22—2441no—
578C22—2442no—
579C22—2442no—
580C22—2441no—
581C22#1683992314yes—
582C22C2×#1451974511yescharacteristic
583C22#1684040212yes—
584C22C2×#1451980011yescharacteristic
585C22C2×#1451980911yescharacteristic
586C22#1684056812yes—
587C4—22no—
588C4—44no—
589C4—42no—
590C4—42no—
591C4—4882no—
592C4—2441no—
593C4—2441no—
594C4—2442no—
595C4—2441no—
596C4#1684027311yescharacteristic
597C4#1684032911yescharacteristic
598C4#1684040212yes—
599C2—24no—
600C2—4882no—
601C2—4881no—
602C2#2144913811yescharacteristic
603C2C2×#1683992311yescharacteristic
604C2#2145509211yescharacteristic
605C2#2145771712yes—
606C2#2145798612yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×#103×C61C2
socleC23×C61C22×D4
prime radicalC2×#16840460C2
hypercenterC22×#103D61
hypocenterC61#23964
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
2. C2×#16839901C2semidirect
3. #21449178C2semidirect
4. #21446968C2semidirect
5. #21446821C2semidirect
6. #21446247C2semidirect
7. #21446150C2semidirect
8. #21446128C2semidirect
9. #21446111C2semidirect
10. #21445909C2semidirect
11. C22×#103×C61C2semidirect
298. C22×#103D61semidirect
350. C61#23964semidirect

Maximal subgroups

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

subgroupindex
C2×#168404602
C2×#168399012
#214491782
#214469682
#214468212
#214462472
#214461502
#214461282
#214461112
#214459092
C22×#103×C612
#2396461