Finite Group Databasebeta

← #29712208

Subgroups — #29712208

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×#16839900C211yesmaximal, characteristic, complemented
2C22×#14519700C211yesmaximal, characteristic, complemented
3#21449184C211yesmaximal, characteristic, complemented
4#21446967C212yesmaximal, complemented
5#21446822C212yesmaximal, complemented
6#21446248C211yesmaximal, characteristic, complemented
7#21446153C212yesmaximal, complemented
8#21446131C212yesmaximal, complemented
9#21446112C211yesmaximal, characteristic, complemented
10#21445909C211yesmaximal, characteristic, complemented
11C2×C61×#553C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C22×#11758200C2212yes—
14C23×#92418C2211yescharacteristic
15C2×#14519756C2212yes—
16#16839900—24no—
17C2×#14519700—24no—
18C2×#14519700C2212yes—
19C2×#14519700C2212yes—
20C2×#14519700C2212yes—
21C2×#14519698C2212yes—
22C2×C22⋊#92418C2211yescharacteristic
23C2×C22⋊#92418C2212yes—
24#16839682—21no—
25#16839674—21no—
26#16839668C2212yes—
27#16839642—21no—
28#16839634—21no—
29#16839622C2211yescharacteristic
30#16839524C2212yes—
31#16839524C2212yes—
32#16839524C2212yes—
33#16839517C2212yes—
34C61×#553—24no—
35C61×#553—24no—
36C2×#126×C61—21no—
37C2×#125×C61C2211yescharacteristic
38C2×#123×C61—21no—
39C2×#121×C61—21no—
40C2×#121×C61C2211yescharacteristic
41C2×#121×C61C2211yescharacteristic
42C2×C4×D4×C61—21no—
43C2×C4×D4×C61C2211yescharacteristic
44C2×C4×D4×C61C2211yescharacteristic
45C2×#117×C61C2211yescharacteristic
46C22×C4⋊C4×C61C2211yescharacteristic
47C22×Dic122—22no—
48C22×Dic122—24no—
49C22×Dic122—22no—
50C22×Dic122—21no—
51C22×Dic122—42no—
52C22×Dic122C2312yes—
53C23×Dic61—21no—
54C2×#11758200—24no—
55C2×#11758200—24no—
56C2×#11758200—42no—
57C22×#92418—22no—
58C22×#92418—42no—
59C22×#92418C2312yes—
60C22×#92418C2312yes—
61#14519756—24no—
62#14519756—42no—
63C2×C4⋊Dic61C2312yes—
64C2×#11758196—22no—
65C2×C22⋊Dic61—22no—
66C2×C22⋊Dic61—21no—
67#14519700—24no—
68#14519700—24no—
69#14519700—24no—
70#14519698—42no—
71C22⋊#92418—22no—
72C22⋊#92418—42no—
73#14519684—21no—
74#14519684—22no—
75#14519684—22no—
76C22×D4×C61—21no—
77C22×D4×C61—21no—
78C22×D4×C61D411yescharacteristic
79C22×D4×C61C2311yescharacteristic
80C23×C4×C61—21no—
81C23×C4×C61—21no—
82C23×C4×C61C2311yescharacteristic
83C23×C4×C61C2311yescharacteristic
84#126×C61—28no—
85#125×C61—22no—
86#125×C61—22no—
87#123×C61—28no—
88#121×C61—28no—
89#121×C61—22no—
90#121×C61—22no—
91#121×C61—22no—
92#121×C61D414yes—
93C4×D4×C61—28no—
94C4×D4×C61—22no—
95C4×D4×C61—22no—
96C4×D4×C61—22no—
97C4×D4×C61—22no—
98#117×C61—22no—
99#117×C61—22no—
100C2×C4⋊C4×C61—22no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61D411yescharacteristic
105C2×C4⋊C4×C61D414yes—
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C22⋊C4×C61—21no—
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61D411yescharacteristic
114C2×C22⋊C4×C61D411yescharacteristic
115C2×C42×C61—21no—
116C2×C42×C61C2311yescharacteristic
117C24×C61—21no—
118C24×C61C2×D411yescharacteristic
119C24×C61C2×D411yescharacteristic
120C2×D4×C61—22no—
121C2×D4×C61—24no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—24no—
125C2×D4×C61—21no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—42no—
129C2×D4×C61—42no—
130C2×D4×C61—82no—
131C2×D4×C61C2×D412yes—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—24no—
138C22×C4×C61—24no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—82no—
144C22×C4×C61—82no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D412yes—
150C22×C4×C61C2×D412yes—
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—44no—
155C22×Dic61—42no—
156C2×Dic122—28no—
157C2×Dic122—22no—
158C2×Dic122—22no—
159C2×Dic122—22no—
160C2×Dic122—22no—
161C2×Dic122—22no—
162C2×Dic122—48no—
163C2×Dic122—44no—
164C2×Dic122—44no—
165C2×Dic122—42no—
166C2×Dic122#4114yes—
167C4⋊C4×C61—24no—
168C4⋊C4×C61—24no—
169C4⋊C4×C61—24no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61C2×D412yes—
175C4⋊C4×C61C2×D412yes—
176C4⋊C4×C61C2×D412yes—
177C22⋊C4×C61—24no—
178C22⋊C4×C61—24no—
179C22⋊C4×C61—24no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—24no—
182C22⋊C4×C61—22no—
183C42×C61—24no—
184C42×C61—21no—
185C42×C61—21no—
186C22⋊Dic61—44no—
187C22⋊Dic61—42no—
188#11758200—48no—
189C4⋊Dic61—22no—
190C4⋊Dic61—22no—
191#11758196—44no—
192C2×#92418—22no—
193C2×#92418—22no—
194C2×#92418—22no—
195C2×#92418—22no—
196C2×#92418—22no—
197C2×#92418—22no—
198C2×#92418—44no—
199C23×C61—22no—
200C23×C61—22no—
201C23×C61—21no—
202C23×C61—22no—
203C23×C61—22no—
204C23×C61—21no—
205C23×C61—21no—
206C23×C61—22no—
207C23×C61—42no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—42no—
211C23×C61#12012yes—
212C23×C61#13512yes—
213C23×C61#13712yes—
214C23×C61C22×D411yescharacteristic
215D4×C61—24no—
216D4×C61—44no—
217D4×C61—48no—
218D4×C61—41no—
219D4×C61—41no—
220D4×C61—84no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—24no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—22no—
228C2×C4×C61—21no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—22no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—42no—
242C2×C4×C61—84no—
243C2×C4×C61—84no—
244C2×C4×C61#12012yes—
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C2×#4112yes—
248C2×C4×C61H(5,𝔽2)12yes—
249C2×Dic61—24no—
250C2×Dic61—48no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256Dic122—24no—
257Dic122—24no—
258Dic122—48no—
259Dic122—42no—
260Dic122—42no—
261#92418—42no—
262#92418—42no—
263#23964—611nomaximal, Sylow
264C22×C61—22no—
265C22×C61—22no—
266C22×C61—24no—
267C22×C61—24no—
268C22×C61—41no—
269C22×C61—42no—
270C22×C61—41no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—41no—
274C22×C61—42no—
275C22×C61—42no—
276C22×C61—82no—
277C22×C61—82no—
278C22×C61#45214yes—
279C22×C61C2×#12011yescharacteristic
280C22×C61#54612yes—
281C22×C61C2×#13511yescharacteristic
282C22×C61C2×#13711yescharacteristic
283C22×C61#57712yes—
284C4×C61—22no—
285C4×C61—41no—
286C4×C61—41no—
287C4×C61—42no—
288C4×C61—41no—
289C4×C61—42no—
290C4×C61—42no—
291C4×C61#52511yescharacteristic
292C4×C61#53511yescharacteristic
293C4×C61#54612yes—
294Dic61—44no—
295Dic61—82no—
296C2×#553D6111yescharacteristic
297C2×#448—611no—
298C22×#103—611no—
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—81no—
309C2×C61—82no—
310C2×C61#240411yescharacteristic
311C2×C61C2×#45211yescharacteristic
312C2×C61#300911yescharacteristic
313C2×C61#330112yes—
314C2×C61#332012yes—
315C23×Q8—611no—
316C22×SD16—612no—
317C23×C8—611no—
318#553—24no—
319#553—24no—
320C2×#126—21no—
321C2×#125C2×D6111yescharacteristic
322C2×#123—21no—
323C2×C22⋊Q8—612no—
324C2×#121—21no—
325C2×#121C2×D6111yescharacteristic
326C2×#121C2×D6111yescharacteristic
327C2×C4×D4—21no—
328C2×C4×D4C2×D6111yescharacteristic
329C2×C4×D4C2×D6111yescharacteristic
330C2×#117C2×D6111yescharacteristic
331C22×C4⋊C4C2×D6111yescharacteristic
332#448—1224no—
333C2×#103—612no—
334C2×#103—612no—
335C2×#103—612no—
336C2×#103—1224no—
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—612no—
352C22×Q8—1224no—
353C22×Q8—1221no—
354C22×Q8—1222no—
355C22×Q8—1222no—
356C22×Q8—2442no—
357C22×D4—21no—
358C22×D4—21no—
359C22×D4C22⋊D6111yescharacteristic
360C22×D4C22×D6111yescharacteristic
361C23×C4—21no—
362C23×C4—21no—
363C23×C4—1221no—
364C23×C4C22×D6111yescharacteristic
365C23×C4C22×D6111yescharacteristic
366C2×SD16—1224no—
367C2×SD16—1224no—
368C2×SD16—2442no—
369C22×C8—612no—
370C22×C8—612no—
371C22×C8—1222no—
372C22×C8—2442no—
373#126—28no—
374#125—22no—
375#125—22no—
376#123—28no—
377C22⋊Q8—1224no—
378C22⋊Q8—2442no—
379#121—28no—
380#121—22no—
381#121—22no—
382#121—22no—
383#121C22⋊D6114yes—
384C4×D4—28no—
385C4×D4—22no—
386C4×D4—22no—
387C4×D4—22no—
388C4×D4—22no—
389#117—22no—
390#117—22no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—612no—
396C2×C4⋊C4—1222no—
397C2×C4⋊C4C22⋊D6111yescharacteristic
398C2×C4⋊C4C22⋊D6114yes—
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C4⋊C4C22×D6111yescharacteristic
401C2×C4⋊C4C22×D6111yescharacteristic
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1222no—
407C2×C22⋊C4—1221no—
408C2×C22⋊C4C22⋊D6111yescharacteristic
409C2×C22⋊C4C22⋊D6111yescharacteristic
410C2×C42—21no—
411C2×C42C22×D6111yescharacteristic
412#103—1224no—
413#103—1224no—
414#103—1224no—
415#102—2442no—
416C22⋊C8—1222no—
417C22⋊C8—2442no—
418#95—1221no—
419#95—1222no—
420#95—1222no—
421C24—21no—
422C24C2×C22⋊D6111yescharacteristic
423C24C2×C22⋊D6111yescharacteristic
424C2×Q8—614no—
425C2×Q8—1228no—
426C2×Q8—1222no—
427C2×Q8—1222no—
428C2×Q8—1222no—
429C2×Q8—1222no—
430C2×Q8—1222no—
431C2×Q8—2444no—
432C2×Q8—2448no—
433C2×Q8—2444no—
434C2×Q8—2442no—
435C2×D4—22no—
436C2×D4—24no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—24no—
440C2×D4—21no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4—82no—
446C2×D4C2×C22⋊D6112yes—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—2444no—
463C22×C4—2442no—
464C22×C4C2×C22⋊D6111yescharacteristic
465C22×C4C2×C22⋊D6111yescharacteristic
466C22×C4C2×C22⋊D6111yescharacteristic
467C22×C4C2×C22⋊D6112yes—
468C22×C4C2×C22⋊D6112yes—
469C22×C4C2×C22⋊D6111yescharacteristic
470C22×C4C23×D6111yescharacteristic
471SD16—2448no—
472C2×C8—1222no—
473C2×C8—1222no—
474C2×C8—1222no—
475C2×C8—1222no—
476C2×C8—1222no—
477C2×C8—1222no—
478C2×C8—2444no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—1222no—
487C4⋊C4—1222no—
488C4⋊C4—2444no—
489C4⋊C4C2×C22⋊D6112yes—
490C4⋊C4C2×C22⋊D6112yes—
491C4⋊C4C2×C22⋊D6112yes—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—22no—
496C22⋊C4—24no—
497C22⋊C4—22no—
498C22⋊C4—2444no—
499C22⋊C4—2442no—
500C42—24no—
501C42—21no—
502C42—21no—
503C23—22no—
504C23—22no—
505C23—21no—
506C23—22no—
507C23—21no—
508C23—21no—
509C23—22no—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23#1451974612yes—
516C23#1451979712yes—
517C23#1451980712yes—
518C23C22×C22⋊D6111yescharacteristic
519Q8—1224no—
520Q8—1224no—
521Q8—2448no—
522Q8—2442no—
523Q8—2442no—
524D4—24no—
525D4—44no—
526D4—48no—
527D4—41no—
528D4—41no—
529D4—84no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—84no—
552C2×C4—84no—
553C2×C4—1224no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4—2448no—
558C2×C4—2442no—
559C2×C4—2442no—
560C2×C4#1451974612yes—
561C2×C4C22×C22⋊D6111yescharacteristic
562C2×C4C22×C22⋊D6111yescharacteristic
563C2×C4#41×D6112yes—
564C2×C4#1451982312yes—
565C8—2442no—
566C8—2442no—
567C22—24no—
568C22—24no—
569C22—22no—
570C22—22no—
571C22—42no—
572C22—42no—
573C22—41no—
574C22—42no—
575C22—41no—
576C22—42no—
577C22—42no—
578C22—41no—
579C22—82no—
580C22—82no—
581C22#1683992414yes—
582C22C2×#1451974611yescharacteristic
583C22#1684040012yes—
584C22C2×#1451979711yescharacteristic
585C22C2×#1451980711yescharacteristic
586C22#1684057312yes—
587C4—22no—
588C4—41no—
589C4—41no—
590C4—42no—
591C4—41no—
592C4—42no—
593C4—42no—
594C4—4882no—
595C4—2444no—
596C4#1684026911yescharacteristic
597C4#1684032611yescharacteristic
598C4#1684040012yes—
599C2—24no—
600C2—81no—
601C2—82no—
602C2#2144913411yescharacteristic
603C2C2×#1683992411yescharacteristic
604C2#2145509611yescharacteristic
605C2#2145771512yes—
606C2#2145798412yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#553C2
socleC23×C61C22×D4
hypercenterC2×#553D61
hypocenterC61#23964
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#553

Complements

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

NH ≅ G/Ntype
1. C2×#16839900C2semidirect
2. C22×#14519700C2semidirect
3. #21449184C2semidirect
4. #21446967C2semidirect
5. #21446822C2semidirect
6. #21446248C2semidirect
7. #21446153C2semidirect
8. #21446131C2semidirect
9. #21446112C2semidirect
10. #21445909C2semidirect
350. C61#23964semidirect

Maximal subgroups

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

subgroupindex
C2×#168399002
C22×#145197002
#214491842
#214469672
#214468222
#214462482
#214461532
#214461312
#214461122
#214459092
C2×C61×#5532
#2396461