Finite Group Databasebeta

← #29713911

Subgroups — #29713911

Order 15616.

1238 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840049C211yesmaximal, characteristic, complemented
2C2×C4×#11758199C211yesmaximal, characteristic, complemented
3#21450732C211yesmaximal, characteristic, complemented
4#21447005C212yesmaximal, complemented
5#21446815C212yesmaximal, complemented
6#21446781C211yesmaximal, characteristic, complemented
7#21446614C211yesmaximal, characteristic, complemented
8#21446141C212yesmaximal, complemented
9#21446018C212yesmaximal, complemented
10#21445819C211yesmaximal, characteristic, complemented
11C2×C61×#555C211yesmaximal, characteristic, complemented
12C22×#11758199C2211yescharacteristic
13C2×#14519751C2212yes—
14C2×#14519748—21no—
15C2×C4×D244C2211yescharacteristic
16#16840049—24no—
17C4×#11758199—24no—
18C2×#14519710C2211yescharacteristic
19C2×#14519708C2212yes—
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519698C2211yescharacteristic
22C2×#14519698C2212yes—
23C2×#14519697C2212yes—
24C2×#14519697C2211yescharacteristic
25C2×C22⋊#92418C2212yes—
26C2×C22⋊#92418C2212yes—
27C2×C4×#92418C2211yescharacteristic
28#16839678—22no—
29#16839632C2211yescharacteristic
30#16839622C2212yes—
31#16839619—21no—
32#16839524C2212yes—
33#16839524C2212yes—
34#16839524C2212yes—
35C61×#555—24no—
36C61×#555—24no—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C2×#124×C61—22no—
39C2×#121×C61—21no—
40C2×#121×C61C2212yes—
41C2×C4×D4×C61C2211yescharacteristic
42C2×C4×D4×C61C2211yescharacteristic
43C2×#117×C61—21no—
44C2×#117×C61C2212yes—
45C22×D244C2311yescharacteristic
46C22×C22⋊D61—22no—
47C22×C4×D61—21no—
48C2×#11758199—22no—
49C2×#11758199—22no—
50C2×#11758199—22no—
51C22×#92418C2311yescharacteristic
52C22×#92418C2312yes—
53C22×#92418C2311yescharacteristic
54#14519751—24no—
55#14519751—42no—
56#14519748—44no—
57C4×D244—22no—
58C4×D244—22no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61C2311yescharacteristic
62C2×C4⋊Dic61C2312yes—
63C2×C22⋊Dic61—22no—
64C2×#11758198—21no—
65C2×#11758198—22no—
66#14519710—22no—
67#14519708—24no—
68C4⋊#92418—22no—
69#14519698—22no—
70#14519698—42no—
71#14519697—24no—
72#14519697—22no—
73C22⋊#92418—24no—
74C22⋊#92418—42no—
75C4×#92418—22no—
76#14519684—22no—
77C22×Q8×C61—21no—
78C22×D4×C61—22no—
79C22×D4×C61D411yescharacteristic
80C22×D4×C61C2311yescharacteristic
81C23×C4×C61—21no—
82C23×C4×C61C2312yes—
83C4⋊Q8×C61—22no—
84C4⋊Q8×C61D414yescomplemented
85#124×C61—216no—
86#121×C61—28no—
87#121×C61—24no—
88#121×C61—24no—
89C4×D4×C61—22no—
90C4×D4×C61—22no—
91C4×D4×C61—22no—
92C4×D4×C61D414yescomplemented
93#117×C61—28no—
94#117×C61—24no—
95#117×C61—24no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61D411yescharacteristic
98C2×C4⋊C4×C61C2312yes—
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C22⋊C4×C61—22no—
101C2×C22⋊C4×C61—22no—
102C2×C22⋊C4×C61—21no—
103C2×C22⋊C4×C61D412yes—
104C2×C42×C61—22no—
105C2×C42×C61C2311yescharacteristic
106C24×C61—21no—
107C24×C61C2×D412yes—
108C23×D61—21no—
109C2×Q8×C61—24no—
110C2×Q8×C61—81no—
111C2×Q8×C61—81no—
112C2×D4×C61—28no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—82no—
122C2×D4×C61—82no—
123C2×D4×C61C2×D412yes—
124C22×C4×C61—21no—
125C22×C4×C61—22no—
126C22×C4×C61—24no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—42no—
134C22×C4×C61—42no—
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2411yescharacteristic
139C2×C22⋊D61—44no—
140C2×C22⋊D61—44no—
141C2×C22⋊D61—82no—
142C22×Dic61—21no—
143C22×Dic61—22no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244#4112yes—
150C2×C4×D61—42no—
151C2×C4×D61—42no—
152C2×C4×D61—42no—
153C4⋊C4×C61—24no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61C2×D412yes—
158C4⋊C4×C61C2×D414yes—
159C22⋊C4×C61—28no—
160C22⋊C4×C61—28no—
161C22⋊C4×C61—24no—
162C22⋊C4×C61—24no—
163C42×C61—28no—
164C42×C61C2×D412yes—
165C42×C61C2×D412yes—
166C22⋊Dic61—44no—
167#11758199—28no—
168#11758198—44no—
169#11758198—42no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—22no—
173C4⋊Dic61—22no—
174C4⋊Dic61—42no—
175C4⋊Dic61—81no—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418—21no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—21no—
182C2×#92418—22no—
183C2×#92418—21no—
184C2×#92418#4112yes—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61—22no—
190C23×C61—22no—
191C23×C61—22no—
192C23×C61—22no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61C22×D411yescharacteristic
200C22×D61—22no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—41no—
207Q8×C61—84no—
208D4×C61—22no—
209D4×C61—22no—
210D4×C61—41no—
211D4×C61—41no—
212D4×C61—44no—
213D4×C61—88no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—24no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—48no—
230C2×C4×C61#12012yes—
231C2×C4×C61#12012yes—
232C2×C4×C61#13512yes—
233C2×C4×C61AGL(1,ℤ/8)12yes—
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61H(5,𝔽2)11yescharacteristic
239C2×C4×C612−1+411yescharacteristic
240C22⋊D61—88no—
241C2×Dic61—41no—
242C2×Dic61—42no—
243C2×Dic61—41no—
244C2×Dic61—42no—
245C2×Dic61—41no—
246C2×Dic61—42no—
247D244—22no—
248D244—22no—
249D244—41no—
250D244—41no—
251C4×D61—84no—
252#92418—22no—
253#92418—41no—
254#92418—42no—
255#24117—611nomaximal, Sylow
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—42no—
260C22×C61—42no—
261C22×C61—42no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61C2×#12011yescharacteristic
271C22×C61#54611yescharacteristic
272C22×C61#54711yescharacteristic
273C22×C61C2×#13511yescharacteristic
274C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C61#57611yescharacteristic
276C22×C61#57711yescharacteristic
277C2×D61—42no—
278C2×D61—42no—
279C2×D61—42no—
280C2×D61—81no—
281C2×D61—81no—
282C2×D61—81no—
283C2×D61—81no—
284C4×C61—22no—
285C4×C61—42no—
286C4×C61—42no—
287C4×C61—44no—
288C4×C61#44914yes—
289C4×C61#52511yescharacteristic
290C4×C61#53711yescharacteristic
291C4×C61#54811yescharacteristic
292C4×C61#55011yescharacteristic
293Dic61—81no—
294Dic61—82no—
295C2×#555D6111yescharacteristic, complemented
296C2×#476—611no—
297C2×C4×D8—611no—
298#2537—611no—
299#2034—612no—
300#1999—612no—
301#1990—611no—
302#1954—611no—
303#1875—612no—
304#1854—612no—
305#1802—611no—
306C2×C61—82no—
307C2×C61—82no—
308C2×C61#240711yescharacteristic
309C2×C61C2×#44911yescharacteristic
310C2×C61#301211yescharacteristic
311C2×C61#330311yescharacteristic
312C2×C61#332511yescharacteristic
313C2×C61#334111yescharacteristic
314C2×C61#335411yescharacteristic
315D61—82no—
316C22×D8—611no—
317#555—24no—
318#555—24no—
319C2×C4⋊Q8C2×D6111yescharacteristic
320C2×#124—22no—
321C2×#121—21no—
322C2×#121—612no—
323C2×#121—1221no—
324C2×#121C2×D6112yes—
325C2×C4×D4—611no—
326C2×C4×D4C2×D6111yescharacteristic
327C2×C4×D4C2×D6111yescharacteristic
328C2×#117—21no—
329C2×#117C2×D6112yes—
330#476—1224no—
331C4×D8—1224no—
332C2×#107—611no—
333C2×#106—612no—
334C2×C4⋊C8—611no—
335C2×#102—611no—
336C2×#102—611no—
337C2×#102—612no—
338C2×#102—612no—
339C2×C22⋊C8—612no—
340C2×C22⋊C8—612no—
341C2×C4×C8—611no—
342#394—1222no—
343#384—611no—
344#380—612no—
345#380—1221no—
346#340—612no—
347#340—612no—
348#340—612no—
349C61#2411711yescharacteristic, complemented, Sylow
350C22×Q8—21no—
351C22×D4—22no—
352C22×D4—611no—
353C22×D4—1222no—
354C22×D4C22⋊D6111yescharacteristic
355C22×D4C22×D6111yescharacteristic
356C23×C4—21no—
357C23×C4—1221no—
358C23×C4C22×D6112yes—
359C2×D8—1222no—
360C2×D8—1222no—
361C2×D8—1222no—
362C22×C8—611no—
363C22×C8—611no—
364C22×C8—612no—
365C4⋊Q8—22no—
366C4⋊Q8C22⋊D6114yescomplemented
367#124—216no—
368#121—28no—
369#121—24no—
370#121—24no—
371#121—1224no—
372#121—2442no—
373#121—2444no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4—22no—
377C4×D4—1222no—
378C4×D4—1222no—
379C4×D4C22⋊D6114yescomplemented
380#117—28no—
381#117—24no—
382#117—24no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—612no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4C22⋊D6111yescharacteristic
389C2×C4⋊C4C22×D6111yescharacteristic
390C2×C4⋊C4C22×D6112yes—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—1222no—
395C2×C22⋊C4—1221no—
396C2×C22⋊C4—1222no—
397C2×C22⋊C4C22⋊D6112yes—
398C2×C42—22no—
399C2×C42C22×D6111yescharacteristic
400#107—1222no—
401#106—1224no—
402C4⋊C8—1222no—
403#102—1222no—
404#102—1222no—
405#102—1224no—
406#102—2442no—
407C22⋊C8—1224no—
408C22⋊C8—2442no—
409C4×C8—1222no—
410#95—1222no—
411C24—21no—
412C24—1221no—
413C24C2×C22⋊D6112yes—
414C2×Q8—24no—
415C2×Q8—81no—
416C2×Q8—81no—
417C2×D4—28no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—82no—
427C2×D4—82no—
428C2×D4—612no—
429C2×D4—1221no—
430C2×D4—1221no—
431C2×D4—1221no—
432C2×D4—1221no—
433C2×D4—1221no—
434C2×D4—4882no—
435C2×D4—2444no—
436C2×D4—2444no—
437C2×D4C2×C22⋊D6112yes—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—1221no—
450C22×C4—1222no—
451C22×C4—2442no—
452C22×C4—2442no—
453C22×C4—2442no—
454C22×C4C2×C22⋊D6112yes—
455C22×C4C2×C22⋊D6111yescharacteristic
456C22×C4C2×C22⋊D6111yescharacteristic
457C22×C4C23×D6111yescharacteristic
458D8—1228no—
459C2×C8—612no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—1222no—
464C2×C8—1222no—
465C2×C8—1221no—
466C2×C8—1222no—
467C2×C8—1221no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—4881no—
477C4⋊C4—2442no—
478C4⋊C4C2×C22⋊D6114yes—
479C4⋊C4C2×C22⋊D6112yes—
480C22⋊C4—28no—
481C22⋊C4—28no—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—2444no—
485C22⋊C4—2444no—
486C22⋊C4—2442no—
487C42—28no—
488C42C2×C22⋊D6112yes—
489C42C2×C22⋊D6112yes—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—1222no—
505C23—2441no—
506C23—2441no—
507C23—2441no—
508C23—2441no—
509C23—2441no—
510C23—2441no—
511C23C22×C22⋊D6111yescharacteristic
512Q8—84no—
513D4—22no—
514D4—22no—
515D4—44no—
516D4—41no—
517D4—41no—
518D4—88no—
519D4—1222no—
520D4—1222no—
521D4—4888no—
522D4—2441no—
523D4—2441no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—48no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—4884no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4#1451974612yes—
548C2×C4#1451974612yes—
549C2×C4#1451979712yes—
550C2×C4#1451980212yes—
551C2×C4C22×C22⋊D6111yescharacteristic
552C2×C4C22×C22⋊D6111yescharacteristic
553C2×C4#41×D6111yescharacteristic
554C2×C4#41×D6111yescharacteristic
555C2×C4#1451982311yescharacteristic
556C2×C4#1451982411yescharacteristic
557C8—1222no—
558C8—2441no—
559C8—2442no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—4881no—
575C22—4881no—
576C22—4881no—
577C22—4881no—
578C22—2442no—
579C22—2442no—
580C22—2442no—
581C22C2×#1451974611yescharacteristic
582C22#1684040011yescharacteristic
583C22#1684041111yescharacteristic
584C22C2×#1451979711yescharacteristic
585C22C2×#1451980211yescharacteristic
586C22#1684056511yescharacteristic
587C22#1684057311yescharacteristic
588C4—22no—
589C4—44no—
590C4—42no—
591C4—42no—
592C4—4882no—
593C4—4881no—
594C4#1683990714yes—
595C4#1684026911yescharacteristic
596C4#1684034011yescharacteristic
597C4#1684041911yescharacteristic
598C4#1684043111yescharacteristic
599C2—82no—
600C2—82no—
601C2—4882no—
602C2#2144915811yescharacteristic
603C2C2×#1683990711yescharacteristic
604C2#2145512311yescharacteristic
605C2#2145774111yescharacteristic
606C2#2145805511yescharacteristic
607C2#2145823711yescharacteristic
608C2#2145840811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#555C2
socleC23×C61C22×D4
hypercenterC2×#555D61
hypocenterC61#24117
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. C2×#16840049C2semidirect
2. C2×C4×#11758199C2semidirect
3. #21450732C2semidirect
4. #21447005C2semidirect
5. #21446815C2semidirect
6. #21446781C2semidirect
7. #21446614C2semidirect
8. #21446141C2semidirect
9. #21446018C2semidirect
10. #21445819C2semidirect
11. C2×C61×#555C2semidirect
84. C4⋊Q8×C61D4semidirect
92. C4×D4×C61D4semidirect
295. C2×#555D61semidirect
349. C61#24117semidirect
366. C4⋊Q8C22⋊D61semidirect
379. C4×D4C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400492
C2×C4×#117581992
#214507322
#214470052
#214468152
#214467812
#214466142
#214461412
#214460182
#214458192
C2×C61×#5552
#2411761