Finite Group Databasebeta

← #29689831

Subgroups — #29689831

Order 15616.

1254 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840349C211yesmaximal, characteristic
2#21448432C211yesmaximal, characteristic, complemented
3#21448414C211yesmaximal, characteristic, complemented
4#21446804C211yesmaximal, characteristic, complemented
5#21446803C211yesmaximal, characteristic, complemented
6#21446656C211yesmaximal, characteristic, complemented
7#21446269C214yesmaximal, complemented
8#21445805C214yesmaximal, complemented
9C61×#1964C211yesmaximal, characteristic, complemented
10#16840349—22no—
11#16840349—22no—
12#16840349—22no—
13#16840349—22no—
14C2×#14519772C412yescomplemented
15C2×#14519763C412yescomplemented
16C2×#14519761C412yescomplemented
17C2×#14519756C2211yescharacteristic
18C2×#14519749C412yescomplemented
19C2×#14519745C2212yes—
20C2×D4×Dic61C2212yes—
21C22×#11758198C2211yescharacteristic
22C2×C22⋊C4×D61C2212yes—
23C2×#14519692C2214yes—
24#16839653—22no—
25#16839651—22no—
26#16839636—22no—
27#16839633—22no—
28#16839622C2211yescharacteristic
29#16839620C2212yes—
30#16839620C2212yes—
31#16839617C2212yes—
32C4×C22⋊Dic61C2211yescharacteristic
33#16839528C2214yes—
34#16839528C2214yes—
35C2×#121×C61C2211yescharacteristic
36C2×#99×C61C2214yes—
37C61×#380C2211yescharacteristic
38C4×C22⋊C4×C61C2211yescharacteristic
39C24×D61C2311yescharacteristic
40C22×Dic122C2×C411yescharacteristic, complemented
41C22×C22⋊D61C2×C412yes—
42C22×C22⋊D61C2×C412yes—
43C23×Dic61C2×C412yescomplemented
44C23×Dic61C2311yescharacteristic
45C22×C4×D61—22no—
46#14519772—24no—
47#14519772—24no—
48#14519763—24no—
49#14519763—24no—
50#14519761—24no—
51#14519761—24no—
52#14519756—21no—
53#14519756—21no—
54#14519756—21no—
55#14519756—21no—
56#14519749—24no—
57#14519749—24no—
58#14519745—24no—
59#14519745—24no—
60D4×Dic61—24no—
61D4×Dic61—24no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61C2×C412yescomplemented
64C2×C4⋊Dic61C2×C411yescharacteristic, complemented
65C2×#11758196—22no—
66C2×#11758196—22no—
67C2×#11758196C2×C412yescomplemented
68C2×C22⋊Dic61C2×C412yescomplemented
69C2×C22⋊Dic61C2312yes—
70C2×C22⋊Dic61C2312yes—
71C2×C22⋊Dic61C2312yes—
72C2×#11758198—22no—
73C2×#11758198—22no—
74C2×#11758198—22no—
75C2×#11758198C2×C412yes—
76C2×#11758198C2×C412yes—
77C2×#11758198C2×C412yes—
78C2×#11758198C2×C412yes—
79C2×#11758198D412yes—
80C2×#11758198D412yes—
81C2×#11758198D412yes—
82C2×#11758198D412yes—
83C22⋊C4×D61—44no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61C2×C412yescomplemented
86#14519692—44no—
87#14519684—21no—
88#14519684—21no—
89#14519684—22no—
90#14519684—22no—
91C22×D4×C61C2×C411yescharacteristic, complemented
92C22×D4×C61C2312yes—
93C23×C4×C61C2311yescharacteristic
94#121×C61—21no—
95#121×C61—21no—
96#121×C61—21no—
97#121×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61C2×C411yescharacteristic, complemented
100C2×C22⋊C4×C61C2×C412yescomplemented
101C2×C22⋊C4×C61C2312yes—
102C2×C22⋊C4×C61C2312yes—
103C2×C42×C61—21no—
104#99×C61—44no—
105#95×C61—21no—
106#95×C61—21no—
107C24×C61C22×C412yes—
108C24×C61C2411yescharacteristic
109C23×D61—22no—
110C23×D61—22no—
111C23×D61—81no—
112C23×D61—81no—
113C23×D61C22×C412yes—
114C23×D61C2×D412yes—
115C23×D61C2×D412yes—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—24no—
119C2×D4×C61—24no—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61—81no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—81no—
126C22×C4×C61C22×C412yes—
127C22×C4×C61C22×C411yescharacteristic
128C22×C4×C61C22×C411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C2×C22⋊D61—24no—
134C2×C22⋊D61—24no—
135C2×C22⋊D61—82no—
136C2×C22⋊D61—82no—
137C22×Dic61—24no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—81no—
141C22×Dic61—82no—
142C22×Dic61C22×C412yes—
143C22×Dic61C22×C412yes—
144C22×Dic61C22×C412yes—
145C22×Dic61C2×D412yes—
146C22×Dic61C2×D412yes—
147C2×C4×D61—44no—
148C2×C4×D61—82no—
149C2×C4×D61—82no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—81no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—81no—
156C22⋊C4×C61—22no—
157C22⋊C4×C61—22no—
158C22⋊C4×C61—22no—
159C22⋊C4×C61—24no—
160C22⋊C4×C61—22no—
161C42×C61—81no—
162C22⋊Dic61—22no—
163C22⋊Dic61—22no—
164C22⋊Dic61—24no—
165C22⋊Dic61—22no—
166C22⋊Dic61—24no—
167C22⋊Dic61—22no—
168#11758198—24no—
169#11758198—24no—
170#11758198—44no—
171#11758198—82no—
172#11758198—82no—
173#11758198C22⋊C414yes—
174#11758198C22⋊C414yes—
175#11758198C22⋊C414yes—
176#11758198C22⋊C414yes—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—24no—
180C4⋊Dic61—81no—
181#11758196—22no—
182#11758196—22no—
183#11758196—82no—
184#11758196—82no—
185C4×Dic61—24no—
186C4×Dic61—81no—
187C23×C61—22no—
188C23×C61—24no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61C23×C411yescharacteristic
194C23×C61C22×D411yescharacteristic
195C23×C61C22×D411yescharacteristic
196C23×C61H(5,𝔽2)12yes—
197C23×C612−1+412yes—
198C22×D61—22no—
199C22×D61—22no—
200C22×D61—24no—
201C22×D61—24no—
202C22×D61—42no—
203C22×D61—42no—
204C22×D61—42no—
205C22×D61—42no—
206C22×D61—42no—
207C22×D61—42no—
208C22×D61—84no—
209C22×D61—84no—
210C22×D61C2×C22⋊C414yes—
211D4×C61—42no—
212D4×C61—42no—
213D4×C61—44no—
214D4×C61—81no—
215D4×C61—81no—
216C2×C4×C61—22no—
217C2×C4×C61—24no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61C2×C22⋊C411yescharacteristic
231C2×C4×C61C2×C22⋊C411yescharacteristic
232C2×C4×C61C2×C22⋊C411yescharacteristic
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C22⋊D61—84no—
235C22⋊D61—84no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—84no—
250C2×Dic61C2×C22⋊C414yes—
251C4×D61—88no—
252Dic122—81no—
253Dic122—81no—
254#21876—611nomaximal, Sylow
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—42no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—84no—
267C22×C61C22×C22⋊C411yescharacteristic
268C22×C61#52012yes—
269C22×C61#53511yescharacteristic
270C22×C61#53611yescharacteristic
271C22×C61#53712yes—
272C2×D61—24no—
273C2×D61—24no—
274C2×D61—24no—
275C2×D61—44no—
276C2×D61—44no—
277C2×D61—84no—
278C2×D61—84no—
279C4×C61—21no—
280C4×C61—21no—
281C4×C61—21no—
282C4×C61—21no—
283C4×C61—42no—
284C4×C61—82no—
285C4×C61—82no—
286Dic61—24no—
287Dic61—42no—
288Dic61—42no—
289Dic61—82no—
290Dic61—82no—
291C2×#538—611no—
292#2315—611no—
293#2312—611no—
294#1996—611no—
295#1996—611no—
296#1964—611no—
297#1964D6111yescharacteristic, complemented
298#1898—614no—
299#1799—614no—
300C2×C61—21no—
301C2×C61—22no—
302C2×C61—21no—
303C2×C61—22no—
304C2×C61—82no—
305C2×C61#231111yescharacteristic
306C2×C61#288311yescharacteristic
307C2×C61#288311yescharacteristic
308D61—44no—
309#538—1222no—
310#538—1222no—
311#538—1222no—
312#538—1222no—
313C2×#126—612no—
314C2×#124—612no—
315C2×#123—612no—
316C2×C22⋊Q8—611no—
317C2×#121—612no—
318C2×#121C2×D6111yescharacteristic
319C2×#120—612no—
320C2×C4×D4—612no—
321C22×C22⋊C4—611no—
322C22×C22⋊C4—612no—
323C2×#99—614no—
324C2×#99C2×D6114yes—
325#388—1222no—
326#388—1222no—
327#385—1222no—
328#385—1222no—
329#380—611no—
330#380—612no—
331#380—612no—
332#380C2×D6111yescharacteristic
333#379—612no—
334C4×C22⋊C4—611no—
335C4×C22⋊C4C2×D6111yescharacteristic
336#342—614no—
337#342—614no—
338C61#2187611yescharacteristic, complemented, Sylow
339C25—611no—
340C22×Q8—611no—
341C22×D4—612no—
342C22×D4—612no—
343C22×D4C4×D6111yescharacteristic, complemented
344C22×D4C22×D6112yes—
345C23×C4—611no—
346C23×C4—612no—
347C23×C4—1222no—
348C23×C4C22×D6111yescharacteristic
349#126—1224no—
350#126—1224no—
351#124—1224no—
352#124—1224no—
353#123—1224no—
354#123—1224no—
355C22⋊Q8—1221no—
356C22⋊Q8—1221no—
357C22⋊Q8—1221no—
358C22⋊Q8—1221no—
359#121—21no—
360#121—21no—
361#121—21no—
362#121—21no—
363#121—1224no—
364#121—1224no—
365#120—1224no—
366#120—1224no—
367C4×D4—1224no—
368C4×D4—1224no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—612no—
371C2×C4⋊C4—612no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—1222no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4C4×D6111yescharacteristic, complemented
377C2×C22⋊C4—612no—
378C2×C22⋊C4—612no—
379C2×C22⋊C4—612no—
380C2×C22⋊C4—612no—
381C2×C22⋊C4—612no—
382C2×C22⋊C4—612no—
383C2×C22⋊C4—612no—
384C2×C22⋊C4—612no—
385C2×C22⋊C4—612no—
386C2×C22⋊C4—612no—
387C2×C22⋊C4—612no—
388C2×C22⋊C4—612no—
389C2×C22⋊C4—1222no—
390C2×C22⋊C4—1222no—
391C2×C22⋊C4—1222no—
392C2×C22⋊C4—2444no—
393C2×C22⋊C4C4×D6112yescomplemented
394C2×C22⋊C4C22×D6112yes—
395C2×C22⋊C4C22×D6112yes—
396C2×C42—21no—
397C2×C42—612no—
398C2×C42—1221no—
399#99—44no—
400#99—2444no—
401#95—21no—
402#95—21no—
403#95—1222no—
404#95—1222no—
405#95—1221no—
406#95—1221no—
407C24—612no—
408C24—612no—
409C24—612no—
410C24—1222no—
411C24—1222no—
412C24—4881no—
413C24—4881no—
414C24C2×C4×D6112yes—
415C24C23×D6111yescharacteristic
416C2×Q8—1221no—
417C2×Q8—1221no—
418C2×Q8—4881no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—81no—
426C2×D4—1224no—
427C2×D4—1224no—
428C2×D4—4882no—
429C2×D4—4882no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—81no—
433C22×C4—612no—
434C22×C4—612no—
435C22×C4—612no—
436C22×C4—612no—
437C22×C4—612no—
438C22×C4—1224no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—4882no—
442C22×C4—4882no—
443C22×C4—4882no—
444C22×C4—4881no—
445C22×C4—2444no—
446C22×C4C2×C4×D6112yes—
447C22×C4C2×C4×D6111yescharacteristic
448C22×C4C2×C4×D6111yescharacteristic
449C22×C4D4×D6111yescharacteristic
450C22×C4D4×D6111yescharacteristic
451C22×C4D4×D6111yescharacteristic
452C22×C4D4×D6111yescharacteristic
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—81no—
456C4⋊C4—1224no—
457C4⋊C4—1222no—
458C4⋊C4—1222no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—4882no—
462C4⋊C4—4882no—
463C4⋊C4—4881no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—24no—
468C22⋊C4—22no—
469C22⋊C4—614no—
470C22⋊C4—614no—
471C22⋊C4—614no—
472C22⋊C4—614no—
473C22⋊C4—1224no—
474C22⋊C4—1224no—
475C22⋊C4—1224no—
476C22⋊C4—1222no—
477C22⋊C4—1222no—
478C22⋊C4—1222no—
479C22⋊C4—1224no—
480C22⋊C4—1222no—
481C22⋊C4—4882no—
482C22⋊C4—4882no—
483C22⋊C4—2444no—
484C42—81no—
485C42—1224no—
486C42—4881no—
487C23—22no—
488C23—24no—
489C23—41no—
490C23—41no—
491C23—42no—
492C23—42no—
493C23—614no—
494C23—1224no—
495C23—1224no—
496C23—1222no—
497C23—1222no—
498C23—4884no—
499C23—4884no—
500C23—2442no—
501C23—2442no—
502C23—2442no—
503C23—2442no—
504C23—2442no—
505C23—2442no—
506C23C22×C4×D6111yescharacteristic
507C23C2×D4×D6111yescharacteristic
508C23C2×D4×D6111yescharacteristic
509C23#1451982212yes—
510C23#1451982412yes—
511Q8—4881no—
512Q8—4881no—
513D4—42no—
514D4—42no—
515D4—44no—
516D4—81no—
517D4—81no—
518D4—4884no—
519D4—4884no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—614no—
535C2×C4—1222no—
536C2×C4—1222no—
537C2×C4—4888no—
538C2×C4—4884no—
539C2×C4—2442no—
540C2×C4—2442no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4C22⋊C4×D6111yescharacteristic
551C2×C4C22⋊C4×D6111yescharacteristic
552C2×C4C22⋊C4×D6111yescharacteristic
553C2×C4C22⋊C4×D6111yescharacteristic
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—84no—
566C22—1224no—
567C22—1224no—
568C22—1224no—
569C22—4884no—
570C22—4884no—
571C22—2444no—
572C22—2444no—
573C22C2×C22⋊C4×D6111yescharacteristic
574C22#1684024112yes—
575C22#1684032511yescharacteristic
576C22#1684033211yescharacteristic
577C22#1684034112yes—
578C4—21no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—42no—
583C4—82no—
584C4—82no—
585C4—1224no—
586C4—4882no—
587C4—4882no—
588C4—2442no—
589C4—2442no—
590C2—21no—
591C2—22no—
592C2—22no—
593C2—21no—
594C2—82no—
595C2—2444no—
596C2#2144840911yescharacteristic
597C2#2145427711yescharacteristic
598C2#2145427811yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#1964C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840349C2
cosocleC24×C61C24
hypercenter#1964D61
hypocenterC61#21876
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1964

Complements

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

NH ≅ G/Ntype
2. #21448432C2semidirect
3. #21448414C2semidirect
4. #21446804C2semidirect
5. #21446803C2semidirect
6. #21446656C2semidirect
7. #21446269C2semidirect
8. #21445805C2semidirect
9. C61×#1964C2semidirect
14. C2×#14519772C4semidirect
15. C2×#14519763C4semidirect
16. C2×#14519761C4semidirect
18. C2×#14519749C4semidirect
40. C22×Dic122C2×C4semidirect
43. C23×Dic61C2×C4semidirect
63. C2×C4⋊Dic61C2×C4semidirect
64. C2×C4⋊Dic61C2×C4semidirect
67. C2×#11758196C2×C4semidirect
68. C2×C22⋊Dic61C2×C4semidirect
85. C2×C4×Dic61C2×C4semidirect
91. C22×D4×C61C2×C4semidirect
99. C2×C4⋊C4×C61C2×C4semidirect
100. C2×C22⋊C4×C61C2×C4semidirect
297. #1964D61semidirect
338. C61#21876semidirect
343. C22×D4C4×D61semidirect
376. C2×C4⋊C4C4×D61semidirect
393. C2×C22⋊C4C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403492
#214484322
#214484142
#214468042
#214468032
#214466562
#214462692
#214458052
C61×#19642
#2187661