Finite Group Databasebeta

← #29752723

Subgroups — #29752723

Order 15616.

728 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450243C211yesmaximal, characteristic
2#21448828C211yesmaximal, characteristic
3#21447230C211yesmaximal, characteristic, complemented
4#21447196C211yesmaximal, characteristic, complemented
5#21447012C211yesmaximal, characteristic
6#21446579C211yesmaximal, characteristic, complemented
7#21446546C211yesmaximal, characteristic, complemented
8#21446422C211yesmaximal, characteristic
9#21446406C211yesmaximal, characteristic, complemented
10#21446400C211yesmaximal, characteristic, complemented
11#21446085C211yesmaximal, characteristic, complemented
12#21446011C211yesmaximal, characteristic
13#21445910C211yesmaximal, characteristic
14#103×Dic61C211yesmaximal, characteristic, complemented
15C61×#2035C211yesmaximal, characteristic
16C2×#14519770—21no—
17C2×Q8×Dic61C2211yescharacteristic
18C22×C4⋊Dic61C2211yescharacteristic
19C2×#14519708C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×C22⋊#92418C2211yescharacteristic
26C2×C8×Dic61C2211yescharacteristic
27#16839704—21no—
28#16839690—21no—
29#16839669C2211yescharacteristic
30#16839661—21no—
31#16839661C2211yescharacteristic
32#16839661C2211yescharacteristic
33#16839658C2211yescharacteristic
34#16839648—21no—
35#16839644—21no—
36#16839637—21no—
37#16839631C2211yescharacteristic
38#16839622C2211yescharacteristic
39C4⋊C4×Dic61C2211yescharacteristic
40C4⋊C4×Dic61C2211yescharacteristic
41C22⋊C4×Dic61—21no—
42C4×C22⋊Dic61C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×C22⋊Q8×C61C2211yescharacteristic
53C2×#106×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×C22⋊C8×C61C2211yescharacteristic
57C61×#380C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C23×Dic61—21no—
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62#14519770—44no—
63Q8×Dic61—22no—
64Q8×Dic61—41no—
65C2×C4⋊Dic61—22no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—41no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61D411yescharacteristic
80C2×C22⋊Dic61D411yescharacteristic
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61C2311yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86#14519708—22no—
87#14519708—22no—
88#14519708—41no—
89#14519706—22no—
90#14519701—22no—
91#14519701—22no—
92C22⋊#92418—22no—
93C8×Dic61—41no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684D411yescharacteristic
103#14519684D411yescharacteristic
104C22×Q8×C61C2311yescharacteristic
105C23×C4×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C22⋊Q8×C61—22no—
109C22⋊Q8×C61—22no—
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C22⋊C4×C61—21no—
115C2×C22⋊C4×C61—21no—
116#106×C61—22no—
117#103×C61—22no—
118#103×C61—41no—
119C22⋊C8×C61—41no—
120#95×C61—21no—
121C24×C61C2×D411yescharacteristic
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61#4112yes—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—42no—
145C22×Dic61—42no—
146C22×Dic61—81no—
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C4⋊C4×C61—24no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61#4112yes—
164C22⋊C4×C61—24no—
165C22⋊C4×C61—42no—
166C22⋊Dic61—41no—
167C22⋊Dic61—42no—
168C22⋊Dic61SD1614yes—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—42no—
177C4⋊Dic61—42no—
178C4⋊Dic61—42no—
179C4⋊Dic61—81no—
180C4⋊Dic61#4112yes—
181#11758196—42no—
182#11758196—42no—
183#11758196—42no—
184C4×Dic61—21no—
185C4×Dic61—21no—
186C4×Dic61—21no—
187C4×Dic61—21no—
188C4×Dic61—42no—
189C4×Dic61—42no—
190C4×Dic61—81no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418#4112yes—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—21no—
200C23×C61—21no—
201C23×C61—21no—
202C23×C61—21no—
203C23×C61C2×SD1612yes—
204C23×C61C22×D411yescharacteristic
205Q8×C61—22no—
206Q8×C61—22no—
207Q8×C61—41no—
208Q8×C61—41no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
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—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61H(5,𝔽2)11yescharacteristic
233C2×C4×C612−1+411yescharacteristic
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—22no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—84no—
257C2×Dic61C2×SD1612yes—
258C2×Dic61C2×SD1612yes—
259C8×C61—41no—
260C8×C61—41no—
261#92418—22no—
262#92418—41no—
263#27034—611nomaximal, Sylow
264C22×C61—22no—
265C22×C61—22no—
266C22×C61—22no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61#52511yescharacteristic
272C22×C61#54611yescharacteristic
273C22×C61#54711yescharacteristic
274C22×C61C22×SD1611yescharacteristic
275C22×C61#57511yescharacteristic
276C22×C61#57711yescharacteristic
277C22×C61#57811yescharacteristic
278C4×C61—42no—
279C4×C61—41no—
280C4×C61—42no—
281C4×C61—81no—
282C4×C61—81no—
283C4×C61#52911yescharacteristic
284C4×C61#53911yescharacteristic
285C4×C61#55311yescharacteristic
286C4×C61#55611yescharacteristic
287Dic61—22no—
288Dic61—22no—
289Dic61—41no—
290Dic61—41no—
291Dic61—42no—
292Dic61—81no—
293Dic61—81no—
294Dic61—81no—
295#2496—611no—
296#2360—611no—
297#2072—611no—
298#2066—611no—
299#2035—611no—
300#2035D6111yescharacteristic
301#1948—611no—
302#1941—611no—
303#1923—611no—
304#1920—611no—
305#1919—611no—
306#1865—611no—
307#1853—611no—
308#1824—611no—
309C4×#103—611no—
310C2×C61—42no—
311C2×C61#246611yescharacteristic
312C2×C61#310311yescharacteristic
313C2×C61#312011yescharacteristic
314C2×C61#329811yescharacteristic
315C2×C61#331111yescharacteristic
316C2×C61#333511yescharacteristic
317C2×C61#335211yescharacteristic
318C2×#126—1221no—
319C2×C22⋊Q8C2×D6111yescharacteristic
320C2×C4×Q8—611no—
321C22×C4⋊C4—611no—
322C2×#106—611no—
323C2×#106—611no—
324C2×#106—611no—
325C2×#106C2×D6111yescharacteristic
326C2×C4⋊C8—611no—
327C2×#103—611no—
328C2×#103—611no—
329C2×#103C2×D6111yescharacteristic
330C2×#103C2×D6111yescharacteristic
331C2×C22⋊C8—611no—
332C2×C22⋊C8C2×D6111yescharacteristic
333C2×C4×C8—611no—
334#400—1221no—
335#397—1221no—
336#391—611no—
337#389—611no—
338#389—611no—
339#389—611no—
340#389—1221no—
341#387—1221no—
342#387—1221no—
343#385—1221no—
344#384—611no—
345#380—611no—
346#380C2×D6111yescharacteristic
347C4×C4⋊C4—611no—
348C4×C4⋊C4—611no—
349C4×C22⋊C4—611no—
350C4×C22⋊C4—1221no—
351#340—611no—
352#340—611no—
353#340—611no—
354#340—611no—
355#340—611no—
356#340—611no—
357#340—611no—
358#340C2×D6111yescharacteristic
359#336—611no—
360#336—611no—
361C61#2703411yescharacteristic, complemented, Sylow
362C22×Q8C22×D6111yescharacteristic
363C23×C4—1221no—
364C23×C4C22×D6111yescharacteristic
365C22×C8—611no—
366C22×C8—611no—
367C22×C8C22×D6111yescharacteristic
368C22×C8C22×D6111yescharacteristic
369#126—2444no—
370C22⋊Q8—22no—
371C22⋊Q8—22no—
372C4×Q8—1222no—
373C4×Q8—2441no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—1222no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—2441no—
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C4⋊C4C22×D6111yescharacteristic
390C2×C4⋊C4C22×D6111yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—611no—
394C2×C22⋊C4—611no—
395C2×C22⋊C4—1221no—
396C2×C42—611no—
397C2×C42—611no—
398C2×C42—1221no—
399C2×C42—1221no—
400C2×C42—1221no—
401#106—22no—
402#106—1222no—
403#106—1222no—
404#106—2441no—
405C4⋊C8—1222no—
406#103—22no—
407#103—41no—
408#103—1222no—
409#103—1222no—
410C22⋊C8—41no—
411C22⋊C8—1222no—
412C4×C8—2441no—
413#95—21no—
414#95—611no—
415#95—611no—
416#95—1221no—
417#95—1221no—
418#95—1221no—
419#95—1221no—
420#95—1221no—
421#95—1221no—
422#95—1221no—
423#95—1221no—
424C24D4×D6111yescharacteristic
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8#1175822412yes—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—1221no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—4881no—
450C22×C4—2442no—
451C22×C4—2442no—
452C22×C4D4×D6111yescharacteristic
453C22×C4C23×D6111yescharacteristic
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—612no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C4⋊C4—24no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—612no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—4881no—
481C4⋊C4—2442no—
482C4⋊C4—2442no—
483C4⋊C4—2442no—
484C4⋊C4—2442no—
485C4⋊C4—2442no—
486C4⋊C4—2442no—
487C4⋊C4#1175822412yes—
488C22⋊C4—24no—
489C22⋊C4—42no—
490C22⋊C4—614no—
491C22⋊C4—2441no—
492C22⋊C4—2442no—
493C42—1221no—
494C42—1221no—
495C42—1221no—
496C42—1221no—
497C42—4881no—
498C42—2442no—
499C42—2442no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23SD16×D6112yes—
507C23C2×D4×D6111yescharacteristic
508Q8—22no—
509Q8—22no—
510Q8—41no—
511Q8—41no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—612no—
532C2×C4—612no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1222no—
543C2×C4—1221no—
544C2×C4—4884no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4#41×D6111yescharacteristic
557C2×C4#41×D6111yescharacteristic
558C2×C4C2×#1175822411yescharacteristic
559C2×C4C2×#1175822411yescharacteristic
560C2×C4#1451982211yescharacteristic
561C2×C4#1451982411yescharacteristic
562C8—41no—
563C8—41no—
564C8—1222no—
565C8—2441no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22#1684026811yescharacteristic
574C22#1684040111yescharacteristic
575C22#1684041011yescharacteristic
576C22C2×SD16×D6111yescharacteristic
577C22#1684056111yescharacteristic
578C22#1684057111yescharacteristic
579C22#1684057611yescharacteristic
580C4—41no—
581C4—42no—
582C4—42no—
583C4—81no—
584C4—81no—
585C4—1222no—
586C4—1222no—
587C4—4881no—
588C4—4881no—
589C4—4881no—
590C4—2441no—
591C4—2442no—
592C4—2441no—
593C4#1684028911yescharacteristic
594C4#1684035711yescharacteristic
595C4#1684045911yescharacteristic
596C4#1684048111yescharacteristic
597C2—42no—
598C2#2144984811yescharacteristic
599C2#2145584511yescharacteristic
600C2#2145603411yescharacteristic
601C2#2145767911yescharacteristic
602C2#2145785011yescharacteristic
603C2#2145816911yescharacteristic
604C2#2145838711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2035C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#2035D61
hypocenterC61#27034
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2035

Complements

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

NH ≅ G/Ntype
3. #21447230C2semidirect
4. #21447196C2semidirect
6. #21446579C2semidirect
7. #21446546C2semidirect
9. #21446406C2semidirect
10. #21446400C2semidirect
11. #21446085C2semidirect
14. #103×Dic61C2semidirect
361. C61#27034semidirect

Maximal subgroups

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

subgroupindex
#214502432
#214488282
#214472302
#214471962
#214470122
#214465792
#214465462
#214464222
#214464062
#214464002
#214460852
#214460112
#214459102
#103×Dic612
C61×#20352
#2703461