Finite Group Databasebeta

← #29733350

Subgroups — #29733350

Order 15616.

1190 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840132C211yesmaximal, characteristic, complemented
2C2×#16839967C211yesmaximal, characteristic, complemented
3#21450772C211yesmaximal, characteristic, complemented
4#21446822C212yesmaximal, complemented
5#21446822C212yesmaximal, complemented
6#21446777C211yesmaximal, characteristic, complemented
7#21446618C211yesmaximal, characteristic, complemented
8#21446131C212yesmaximal, complemented
9#21446126C212yesmaximal, complemented
10#21445644C211yesmaximal, characteristic, complemented
11C2×C61×#553C211yesmaximal, characteristic
12C22×#11758200C2212yes—
13C2×#14519777C2211yescharacteristic
14C2×#14519756C2212yes—
15C2×#14519756C2212yes—
16#16840132—24no—
17#16839967—24no—
18C2×C4⋊#92418C2211yescharacteristic
19C2×#14519700C2212yes—
20C2×#14519700C2212yes—
21C2×#14519700C2212yes—
22C2×#14519698C2212yes—
23C2×C22⋊#92418C2212yes—
24C2×C22⋊#92418C2212yes—
25C2×#14519688C2211yescharacteristic
26#16839698—21no—
27#16839698—21no—
28#16839665—21no—
29#16839631—21no—
30C4×C4⋊Dic61C2211yescharacteristic
31#16839524C2212yes—
32#16839524C2212yes—
33#16839524C2212yes—
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×Dic122C2312yes—
48C2×#11758200—24no—
49C2×#11758200—42no—
50C2×#11758200—42no—
51C22×#92418C2312yes—
52C22×#92418C2312yes—
53#14519777—22no—
54#14519777—22no—
55#14519756—24no—
56#14519756—24no—
57#14519756—42no—
58#14519756—42no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61C2312yes—
62C2×#11758196—22no—
63C2×#11758196—22no—
64C2×C22⋊Dic61—22no—
65C2×C22⋊Dic61—22no—
66C2×C4×Dic61—21no—
67C4⋊#92418—22no—
68#14519700—24no—
69#14519700—24no—
70#14519700—24no—
71#14519698—24no—
72C22⋊#92418—42no—
73C22⋊#92418—42no—
74#14519688—22no—
75#14519684—21no—
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×C61D414yes—
87#123×C61—28no—
88#121×C61—28no—
89#121×C61—22no—
90#121×C61—22no—
91#121×C61—22no—
92#121×C61—22no—
93C4×D4×C61—28no—
94C4×D4×C61—22no—
95C4×D4×C61—22no—
96C4×D4×C61—22no—
97C4×D4×C61D414yes—
98#117×C61—22no—
99#117×C61—22no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—22no—
104C2×C4⋊C4×C61—22no—
105C2×C4⋊C4×C61D411yescharacteristic
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—24no—
121C2×D4×C61—22no—
122C2×D4×C61—24no—
123C2×D4×C61—22no—
124C2×D4×C61—21no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—41no—
128C2×D4×C61—42no—
129C2×D4×C61—42no—
130C2×D4×C61—82no—
131C2×D4×C61C2×D412yes—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—24no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—22no—
144C22×C4×C61—24no—
145C22×C4×C61—82no—
146C22×C4×C61—82no—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—22no—
153C22×Dic61—22no—
154C2×Dic122—22no—
155C2×Dic122—22no—
156C2×Dic122—22no—
157C2×Dic122—22no—
158C2×Dic122—22no—
159C2×Dic122—22no—
160C4⋊C4×C61—24no—
161C4⋊C4×C61—24no—
162C4⋊C4×C61—24no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61C2×D412yes—
168C4⋊C4×C61C2×D412yes—
169C4⋊C4×C61C2×D412yes—
170C22⋊C4×C61—24no—
171C22⋊C4×C61—24no—
172C22⋊C4×C61—24no—
173C22⋊C4×C61—24no—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—22no—
176C42×C61—24no—
177C42×C61C2×D412yes—
178C42×C61C2×D412yes—
179C22⋊Dic61—44no—
180C22⋊Dic61—44no—
181#11758200—48no—
182C4⋊Dic61—22no—
183C4⋊Dic61—42no—
184C4⋊Dic61—42no—
185C4⋊Dic61#4114yes—
186#11758196—44no—
187#11758196—44no—
188C4×Dic61—42no—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418—22no—
193C2×#92418—22no—
194C2×#92418—22no—
195C23×C61—21no—
196C23×C61—22no—
197C23×C61—21no—
198C23×C61—22no—
199C23×C61—21no—
200C23×C61—22no—
201C23×C61—22no—
202C23×C61—21no—
203C23×C61—21no—
204C23×C61—22no—
205C23×C61—42no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—42no—
209C23×C61#13712yes—
210C23×C61C22×D411yescharacteristic
211D4×C61—22no—
212D4×C61—22no—
213D4×C61—44no—
214D4×C61—48no—
215D4×C61—42no—
216D4×C61—84no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—22no—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—21no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—84no—
238C2×C4×C61—84no—
239C2×C4×C61#12012yes—
240C2×C4×C61#12012yes—
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C2×#4112yes—
244C2×C4×C61H(5,𝔽2)12yes—
245C2×Dic61—24no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251Dic122—42no—
252Dic122—42no—
253Dic122—42no—
254Dic122—42no—
255#92418—42no—
256#92418—42no—
257#25582—611nomaximal, Sylow
258C22×C61—22no—
259C22×C61—24no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61—42no—
263C22×C61—41no—
264C22×C61—42no—
265C22×C61—41no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—41no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—82no—
272C22×C61—82no—
273C22×C61C2×#12011yescharacteristic
274C22×C61#54612yes—
275C22×C61C2×#13711yescharacteristic
276C22×C61#57511yescharacteristic
277C22×C61#57712yes—
278C4×C61—22no—
279C4×C61—22no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—42no—
283C4×C61—42no—
284C4×C61—42no—
285C4×C61#53211yescharacteristic
286C4×C61#53511yescharacteristic
287C4×C61#55012yes—
288Dic61—44no—
289Dic61—82no—
290C2×#553D6111yescharacteristic
291C2×#491—611no—
292C2×#461—611no—
293#2542—611no—
294#2000—612no—
295#2000—612no—
296#1989—611no—
297#1955—611no—
298#1873—612no—
299#1872—612no—
300#1762—611no—
301C2×C61—42no—
302C2×C61—81no—
303C2×C61—82no—
304C2×C61#243111yescharacteristic
305C2×C61#300711yescharacteristic
306C2×C61#300911yescharacteristic
307C2×C61#330712yes—
308C2×C61#332012yes—
309C22×SD16—612no—
310#553—24no—
311#553—24no—
312C2×C4⋊Q8—611no—
313C2×#126—21no—
314C2×#125C2×D6111yescharacteristic
315C2×#123—21no—
316C2×C22⋊Q8—612no—
317C2×C22⋊Q8—612no—
318C2×#121—21no—
319C2×#121C2×D6111yescharacteristic
320C2×#121C2×D6111yescharacteristic
321C2×C4×D4—21no—
322C2×C4×D4C2×D6111yescharacteristic
323C2×C4×D4C2×D6111yescharacteristic
324C2×#117C2×D6111yescharacteristic
325C22×C4⋊C4C2×D6111yescharacteristic
326#491—1224no—
327#461—1224no—
328C2×C4⋊C8—611no—
329C2×#103—612no—
330C2×#103—612no—
331C2×#103—612no—
332C2×#102—612no—
333C2×C22⋊C8—612no—
334C2×C22⋊C8—612no—
335C2×#97—611no—
336#399—1221no—
337#399—1221no—
338#390—1221no—
339#384—1221no—
340C4×C4⋊C4—611no—
341#340—612no—
342#340—612no—
343#340—612no—
344C61#2558211yescharacteristic, complemented, Sylow
345C22×Q8—612no—
346C22×D4—21no—
347C22×D4—21no—
348C22×D4C22⋊D6111yescharacteristic
349C22×D4C22×D6111yescharacteristic
350C23×C4—21no—
351C23×C4—21no—
352C23×C4C22×D6111yescharacteristic
353C23×C4C22×D6111yescharacteristic
354C2×SD16—1224no—
355C2×SD16—2442no—
356C2×SD16—2442no—
357C22×C8—612no—
358C22×C8—612no—
359C4⋊Q8—1222no—
360C4⋊Q8—1222no—
361#126—28no—
362#125—22no—
363#125C22⋊D6114yes—
364#123—28no—
365C22⋊Q8—1224no—
366C22⋊Q8—1224no—
367C22⋊Q8—2442no—
368C22⋊Q8—2442no—
369#121—28no—
370#121—22no—
371#121—22no—
372#121—22no—
373#121—22no—
374C4×D4—28no—
375C4×D4—22no—
376C4×D4—22no—
377C4×D4—22no—
378C4×D4C22⋊D6114yes—
379#117—22no—
380#117—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—612no—
387C2×C4⋊C4—1222no—
388C2×C4⋊C4—1222no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4C22⋊D6111yescharacteristic
392C2×C4⋊C4C22×D6111yescharacteristic
393C2×C4⋊C4C22×D6111yescharacteristic
394C2×C4⋊C4C22×D6111yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—1222no—
400C2×C22⋊C4—1222no—
401C2×C22⋊C4C22⋊D6111yescharacteristic
402C2×C22⋊C4C22⋊D6111yescharacteristic
403C2×C42—21no—
404C2×C42—1221no—
405C2×C42C22×D6111yescharacteristic
406C4⋊C8—1222no—
407#103—1224no—
408#103—1224no—
409#103—1224no—
410#102—1224no—
411C22⋊C8—2442no—
412C22⋊C8—2442no—
413#97—1222no—
414#95—1221no—
415C24—21no—
416C24C2×C22⋊D6111yescharacteristic
417C24C2×C22⋊D6111yescharacteristic
418C2×Q8—1222no—
419C2×Q8—1222no—
420C2×Q8—1222no—
421C2×Q8—1222no—
422C2×Q8—1222no—
423C2×Q8—1222no—
424C2×D4—24no—
425C2×D4—22no—
426C2×D4—24no—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—41no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—82no—
435C2×D4C2×C22⋊D6112yes—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4C2×C22⋊D6111yescharacteristic
454C22×C4C2×C22⋊D6111yescharacteristic
455C22×C4C2×C22⋊D6111yescharacteristic
456C22×C4C2×C22⋊D6111yescharacteristic
457C22×C4C23×D6111yescharacteristic
458SD16—2448no—
459C2×C8—1222no—
460C2×C8—1222no—
461C2×C8—1222no—
462C2×C8—1222no—
463C2×C8—1222no—
464C2×C8—1222no—
465C4⋊C4—24no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—614no—
473C4⋊C4—1222no—
474C4⋊C4—2444no—
475C4⋊C4—2444no—
476C4⋊C4—2442no—
477C4⋊C4—2442no—
478C4⋊C4C2×C22⋊D6112yes—
479C4⋊C4C2×C22⋊D6112yes—
480C4⋊C4C2×C22⋊D6112yes—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—2444no—
488C22⋊C4—2444no—
489C42—24no—
490C42—2442no—
491C42C2×C22⋊D6112yes—
492C42C2×C22⋊D6112yes—
493C23—21no—
494C23—21no—
495C23—22no—
496C23—22no—
497C23—21no—
498C23—22no—
499C23—21no—
500C23—21no—
501C23—22no—
502C23—22no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23#1451980712yes—
508C23C22×C22⋊D6111yescharacteristic
509Q8—2442no—
510Q8—2442no—
511Q8—2442no—
512Q8—2442no—
513D4—22no—
514D4—22no—
515D4—44no—
516D4—48no—
517D4—42no—
518D4—84no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—84no—
540C2×C4—84no—
541C2×C4—1224no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4#1451974612yes—
548C2×C4#1451974612yes—
549C2×C4C22×C22⋊D6111yescharacteristic
550C2×C4C22×C22⋊D6111yescharacteristic
551C2×C4#41×D6112yes—
552C2×C4#1451982312yes—
553C8—2442no—
554C8—2442no—
555C22—24no—
556C22—22no—
557C22—42no—
558C22—41no—
559C22—41no—
560C22—42no—
561C22—42no—
562C22—41no—
563C22—42no—
564C22—41no—
565C22—42no—
566C22—42no—
567C22—41no—
568C22—82no—
569C22—82no—
570C22C2×#1451974611yescharacteristic
571C22#1684040012yes—
572C22C2×#1451980711yescharacteristic
573C22#1684056011yescharacteristic
574C22#1684057312yes—
575C4—22no—
576C4—22no—
577C4—41no—
578C4—41no—
579C4—42no—
580C4—42no—
581C4—42no—
582C4—4882no—
583C4—2444no—
584C4#1684030811yescharacteristic
585C4#1684032611yescharacteristic
586C4#1684043112yes—
587C2—42no—
588C2—81no—
589C2—82no—
590C2#2144943411yescharacteristic
591C2#2145507911yescharacteristic
592C2#2145509611yescharacteristic
593C2#2145778912yes—
594C2#2145798412yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#553C2
socleC23×C61C22×D4
hypercenterC2×#553D61
hypocenterC61#25582
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×#16840132C2semidirect
2. C2×#16839967C2semidirect
3. #21450772C2semidirect
4. #21446822C2semidirect
5. #21446822C2semidirect
6. #21446777C2semidirect
7. #21446618C2semidirect
8. #21446131C2semidirect
9. #21446126C2semidirect
10. #21445644C2semidirect
344. C61#25582semidirect

Maximal subgroups

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

subgroupindex
C2×#168401322
C2×#168399672
#214507722
#214468222
#214467772
#214466182
#214461312
#214461262
#214456442
C2×C61×#5532
#2558261