Finite Group Databasebeta

← #29736562

Subgroups — #29736562

Order 15616.

1322 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840121C211yesmaximal, characteristic, complemented
2C2×#16840057C211yesmaximal, characteristic, complemented
3#21450811C211yesmaximal, characteristic, complemented
4#21447019C212yesmaximal, complemented
5#21446998C212yesmaximal, complemented
6#21446752C211yesmaximal, characteristic, complemented
7#21446618C211yesmaximal, characteristic, complemented
8#21445759C212yesmaximal, complemented
9#21445754C212yesmaximal, complemented
10#21445649C211yesmaximal, characteristic, complemented
11C2×C61×#548C211yesmaximal, characteristic
12C2×#14519777C2211yescharacteristic
13C2×#14519764—21no—
14#16840121—24no—
15#16840057—24no—
16C2×#14519710C2212yes—
17C2×C4⋊#92418C2211yescharacteristic
18C2×#14519701C2212yes—
19C2×#14519701C2212yes—
20C2×#14519701C2212yes—
21C2×#14519701C2212yes—
22C2×#14519698C2212yes—
23C2×#14519698C2212yes—
24C2×C22⋊#92418C2212yes—
25C2×C22⋊#92418C2212yes—
26C2×#14519688C2211yescharacteristic
27#16839687—21no—
28#16839687—21no—
29#16839667—21no—
30#16839622C2212yes—
31#16839622C2212yes—
32C4×C4⋊Dic61C2211yescharacteristic
33#16839524C2212yes—
34C22×#41×C61C2211yescharacteristic
35C23×Q8×C61C2211yescharacteristic
36C61×#548—24no—
37C61×#548—24no—
38C2×#124×C61—21no—
39C2×#124×C61C2211yescharacteristic
40C2×C22⋊Q8×C61—21no—
41C2×C22⋊Q8×C61C2211yescharacteristic
42C2×#121×C61—21no—
43C2×#121×C61C2211yescharacteristic
44C2×C4×Q8×C61C2211yescharacteristic
45C2×C4×D4×C61—21no—
46C2×C4×D4×C61C2211yescharacteristic
47C22×Dic122—21no—
48C22×#92418C2312yes—
49C22×#92418C2312yes—
50#14519777—22no—
51#14519777—22no—
52#14519764—44no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61C2312yes—
55C2×C4⋊Dic61C2312yes—
56C2×C22⋊Dic61—22no—
57C2×C22⋊Dic61—22no—
58C2×C4×Dic61—21no—
59#14519710—24no—
60C4⋊#92418—22no—
61#14519701—24no—
62#14519701—24no—
63#14519701—24no—
64#14519701—42no—
65#14519698—24no—
66#14519698—42no—
67C22⋊#92418—42no—
68C22⋊#92418—42no—
69#14519688—22no—
70#14519684—21no—
71#14519684—22no—
72#14519684—22no—
73C2×#41×C61—22no—
74C2×#41×C61—22no—
75C2×#41×C61—24no—
76C2×#41×C61—42no—
77C22×Q8×C61—22no—
78C22×Q8×C61—22no—
79C22×Q8×C61—21no—
80C22×Q8×C61—24no—
81C22×Q8×C61—42no—
82C22×Q8×C61C2311yescharacteristic
83C22×Q8×C61C2311yescharacteristic
84C22×D4×C61—21no—
85C22×D4×C61—21no—
86C22×D4×C61D411yescharacteristic
87C22×D4×C61C2311yescharacteristic
88C23×C4×C61—21no—
89C23×C4×C61—21no—
90C23×C4×C61C2311yescharacteristic
91C23×C4×C61C2311yescharacteristic
92#124×C61—28no—
93#124×C61—22no—
94#124×C61D414yes—
95C22⋊Q8×C61—28no—
96C22⋊Q8×C61—22no—
97C22⋊Q8×C61—22no—
98#121×C61—28no—
99#121×C61—22no—
100#121×C61—22no—
101C4×Q8×C61—22no—
102C4×Q8×C61D414yes—
103C4×D4×C61—28no—
104C4×D4×C61—22no—
105C4×D4×C61—22no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61D411yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61D411yescharacteristic
115C2×C22⋊C4×C61D411yescharacteristic
116C2×C42×C61—21no—
117C2×C42×C61C2311yescharacteristic
118C24×C61—21no—
119C24×C61C2×D411yescharacteristic
120C24×C61C2×D411yescharacteristic
121#41×C61—48no—
122#41×C61—44no—
123#41×C61—44no—
124C2×Q8×C61—24no—
125C2×Q8×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61—22no—
128C2×Q8×C61—22no—
129C2×Q8×C61—22no—
130C2×Q8×C61—44no—
131C2×Q8×C61—48no—
132C2×Q8×C61—44no—
133C2×Q8×C61—44no—
134C2×Q8×C61C2×D412yes—
135C2×Q8×C61C2×D412yes—
136C2×Q8×C61#4114yes—
137C2×D4×C61—24no—
138C2×D4×C61—24no—
139C2×D4×C61—22no—
140C2×D4×C61—22no—
141C2×D4×C61—21no—
142C2×D4×C61—22no—
143C2×D4×C61—22no—
144C2×D4×C61—41no—
145C2×D4×C61—44no—
146C2×D4×C61—42no—
147C2×D4×C61—42no—
148C2×D4×C61C2×D412yes—
149C22×C4×C61—21no—
150C22×C4×C61—24no—
151C22×C4×C61—24no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—21no—
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—22no—
159C22×C4×C61—22no—
160C22×C4×C61—22no—
161C22×C4×C61—22no—
162C22×C4×C61—44no—
163C22×C4×C61—44no—
164C22×C4×C61C2×D411yescharacteristic
165C22×C4×C61C2×D411yescharacteristic
166C22×C4×C61C2×D411yescharacteristic
167C22×C4×C61C2×D411yescharacteristic
168C22×C4×C61C2411yescharacteristic
169C22×Dic61—22no—
170C22×Dic61—22no—
171C2×Dic122—42no—
172C2×Dic122—42no—
173C2×Dic122—42no—
174C4⋊C4×C61—24no—
175C4⋊C4×C61—24no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61C2×D412yes—
179C22⋊C4×C61—24no—
180C22⋊C4×C61—24no—
181C22⋊C4×C61—24no—
182C22⋊C4×C61—24no—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—22no—
185C42×C61—24no—
186C42×C61C2×D412yes—
187C42×C61C2×D412yes—
188C22⋊Dic61—44no—
189C22⋊Dic61—44no—
190C4⋊Dic61—22no—
191C4⋊Dic61—22no—
192C4⋊Dic61—22no—
193C4⋊Dic61—22no—
194C4⋊Dic61—42no—
195C4×Dic61—42no—
196C2×#92418—22no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418—22no—
201C2×#92418—22no—
202C23×C61—21no—
203C23×C61—22no—
204C23×C61—21no—
205C23×C61—22no—
206C23×C61—21no—
207C23×C61—22no—
208C23×C61—22no—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61—22no—
212C23×C61—42no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—42no—
216C23×C61AGL(1,ℤ/8)12yes—
217C23×C61C22×D411yescharacteristic
218Q8×C61—22no—
219Q8×C61—24no—
220Q8×C61—24no—
221Q8×C61—48no—
222Q8×C61—41no—
223Q8×C61—41no—
224Q8×C61#12414yescomplemented
225D4×C61—24no—
226D4×C61—44no—
227D4×C61—48no—
228D4×C61—48no—
229D4×C61—41no—
230D4×C61—41no—
231C2×C4×C61—22no—
232C2×C4×C61—24no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—48no—
243C2×C4×C61—48no—
244C2×C4×C61—42no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—42no—
252C2×C4×C61—42no—
253C2×C4×C61#12012yes—
254C2×C4×C61#12012yes—
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C2×#4112yes—
258C2×C4×C61H(5,𝔽2)12yes—
259C2×Dic61—42no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265Dic122—84no—
266#92418—42no—
267#92418—42no—
268#25896—611nomaximal, Sylow
269C22×C61—22no—
270C22×C61—24no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—42no—
274C22×C61—42no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—42no—
278C22×C61—41no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61—82no—
283C22×C61—82no—
284C22×C61C2×#12011yescharacteristic
285C22×C61#54612yes—
286C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
287C22×C61#57511yescharacteristic
288C22×C61#57812yes—
289C4×C61—24no—
290C4×C61—22no—
291C4×C61—22no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—42no—
295C4×C61—42no—
296C4×C61C2×#12411yescharacteristic
297C4×C61#53511yescharacteristic
298C4×C61#54612yes—
299Dic61—82no—
300Dic61—82no—
301C2×#548D6111yescharacteristic
302C2×#489—611no—
303C2×#477—611no—
304#2547—611no—
305#2036—612no—
306#2033—612no—
307#1983—611no—
308#1955—611no—
309#1788—612no—
310#1787—612no—
311#1763—611no—
312C2×C61—42no—
313C2×C61—81no—
314C2×C61—82no—
315C2×C61#243511yescharacteristic
316C2×C61#300611yescharacteristic
317C2×C61#301111yescharacteristic
318C2×C61#330812yes—
319C2×C61#331312yes—
320C22×#41C2×D6111yescharacteristic
321C23×Q8C2×D6111yescharacteristic
322#548—24no—
323#548—24no—
324C2×C4⋊Q8—611no—
325C2×#124—21no—
326C2×#124—1221no—
327C2×#124C2×D6111yescharacteristic
328C2×C22⋊Q8—21no—
329C2×C22⋊Q8C2×D6111yescharacteristic
330C2×#121—21no—
331C2×#121C2×D6111yescharacteristic
332C2×C4×Q8C2×D6111yescharacteristic
333C2×C4×D4—21no—
334C2×C4×D4C2×D6111yescharacteristic
335#489—1224no—
336#477—1224no—
337C2×#107—612no—
338C2×C4⋊C8—611no—
339C2×#103—612no—
340C2×#103—612no—
341C2×#103—612no—
342C2×#103—612no—
343C2×#102—612no—
344C2×#102—612no—
345C2×C22⋊C8—612no—
346C2×C22⋊C8—612no—
347C2×#97—611no—
348#397—1221no—
349#397—1221no—
350#391—1221no—
351#380—612no—
352#380—612no—
353C4×C4⋊C4—611no—
354#340—612no—
355C61#2589611yescharacteristic, complemented, Sylow
356C2×#41—22no—
357C2×#41—22no—
358C2×#41—24no—
359C2×#41—42no—
360C22×Q8—22no—
361C22×Q8—22no—
362C22×Q8—21no—
363C22×Q8—24no—
364C22×Q8—42no—
365C22×Q8—1221no—
366C22×Q8C22×D6111yescharacteristic
367C22×Q8C22×D6111yescharacteristic
368C22×D4—21no—
369C22×D4—21no—
370C22×D4C22⋊D6111yescharacteristic
371C22×D4C22×D6111yescharacteristic
372C23×C4—21no—
373C23×C4—21no—
374C23×C4C22×D6111yescharacteristic
375C23×C4C22×D6111yescharacteristic
376C22×C8—612no—
377C22×C8—612no—
378C4⋊Q8—1222no—
379C4⋊Q8—1222no—
380#124—28no—
381#124—22no—
382#124—2444no—
383#124C22⋊D6114yes—
384C22⋊Q8—28no—
385C22⋊Q8—22no—
386C22⋊Q8—22no—
387#121—28no—
388#121—22no—
389#121—22no—
390C4×Q8—22no—
391C4×Q8C22⋊D6114yes—
392C4×D4—28no—
393C4×D4—22no—
394C4×D4—22no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—612no—
398C2×C4⋊C4—612no—
399C2×C4⋊C4—1221no—
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—1222no—
408C2×C22⋊C4C22⋊D6111yescharacteristic
409C2×C22⋊C4C22⋊D6111yescharacteristic
410C2×C42—21no—
411C2×C42—1221no—
412C2×C42C22×D6111yescharacteristic
413#107—1224no—
414C4⋊C8—1222no—
415#103—1224no—
416#103—1224no—
417#103—1224no—
418#103—2442no—
419#102—1224no—
420#102—2442no—
421C22⋊C8—2442no—
422C22⋊C8—2442no—
423#97—1222no—
424#95—1221no—
425#95—1222no—
426#95—1222no—
427C24—21no—
428C24C2×C22⋊D6111yescharacteristic
429C24C2×C22⋊D6111yescharacteristic
430#41—48no—
431#41—44no—
432#41—44no—
433C2×Q8—24no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—44no—
440C2×Q8—48no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8—2442no—
444C2×Q8—2442no—
445C2×Q8—2442no—
446C2×Q8#1175822414yes—
447C2×Q8C2×C22⋊D6112yes—
448C2×Q8C2×C22⋊D6112yes—
449C2×D4—24no—
450C2×D4—24no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—41no—
457C2×D4—44no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4C2×C22⋊D6112yes—
461C22×C4—21no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4C2×C22⋊D6111yescharacteristic
479C22×C4C2×C22⋊D6111yescharacteristic
480C22×C4C2×C22⋊D6111yescharacteristic
481C22×C4C2×C22⋊D6111yescharacteristic
482C22×C4C23×D6111yescharacteristic
483C2×C8—1222no—
484C2×C8—1222no—
485C2×C8—1222no—
486C2×C8—1222no—
487C2×C8—1222no—
488C2×C8—1222no—
489C4⋊C4—24no—
490C4⋊C4—21no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—2442no—
498C4⋊C4C2×C22⋊D6112yes—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—2444no—
506C22⋊C4—2444no—
507C42—24no—
508C42—2442no—
509C42C2×C22⋊D6112yes—
510C42C2×C22⋊D6112yes—
511C23—21no—
512C23—21no—
513C23—22no—
514C23—22no—
515C23—21no—
516C23—22no—
517C23—21no—
518C23—21no—
519C23—22no—
520C23—22no—
521C23—41no—
522C23—41no—
523C23—42no—
524C23—42no—
525C23#1451980212yes—
526C23C22×C22⋊D6111yescharacteristic
527Q8—22no—
528Q8—24no—
529Q8—24no—
530Q8—48no—
531Q8—41no—
532Q8—41no—
533Q8—4884no—
534Q8#1451976414yescomplemented
535D4—24no—
536D4—44no—
537D4—48no—
538D4—48no—
539D4—41no—
540D4—41no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—48no—
553C2×C4—48no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—2442no—
564C2×C4—2442no—
565C2×C4—2442no—
566C2×C4—2442no—
567C2×C4—2442no—
568C2×C4—2442no—
569C2×C4#1451974612yes—
570C2×C4#1451974612yes—
571C2×C4C22×C22⋊D6111yescharacteristic
572C2×C4C22×C22⋊D6111yescharacteristic
573C2×C4C2×#1175822412yes—
574C2×C4#1451982212yes—
575C8—2442no—
576C8—2442no—
577C22—24no—
578C22—22no—
579C22—41no—
580C22—42no—
581C22—41no—
582C22—41no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—41no—
587C22—42no—
588C22—42no—
589C22—41no—
590C22—82no—
591C22—82no—
592C22C2×#1451974611yescharacteristic
593C22#1684040612yes—
594C22C2×#1451980211yescharacteristic
595C22#1684056011yescharacteristic
596C22#1684057712yes—
597C4—22no—
598C4—22no—
599C4—24no—
600C4—42no—
601C4—41no—
602C4—41no—
603C4—42no—
604C4—4882no—
605C4—4882no—
606C4C2×#1451976411yescharacteristic
607C4#1684032911yescharacteristic
608C4#1684040612yes—
609C2—42no—
610C2—81no—
611C2—82no—
612C2#2144946611yescharacteristic
613C2#2145507111yescharacteristic
614C2#2145511411yescharacteristic
615C2#2145781012yes—
616C2#2145788512yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#548C2
socleC23×C61C22×D4
hypercenterC2×#548D61
hypocenterC61#25896
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#548

Complements

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

NH ≅ G/Ntype
1. C2×#16840121C2semidirect
2. C2×#16840057C2semidirect
3. #21450811C2semidirect
4. #21447019C2semidirect
5. #21446998C2semidirect
6. #21446752C2semidirect
7. #21446618C2semidirect
8. #21445759C2semidirect
9. #21445754C2semidirect
10. #21445649C2semidirect
224. Q8×C61#124semidirect
355. C61#25896semidirect
534. Q8#14519764semidirect

Maximal subgroups

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

subgroupindex
C2×#168401212
C2×#168400572
#214508112
#214470192
#214469982
#214467522
#214466182
#214457592
#214457542
#214456492
C2×C61×#5482
#2589661