Finite Group Databasebeta

← #29783107

Subgroups — #29783107

Order 15616.

746 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#21454201C211yesmaximal, characteristic, complemented
2#21447321C211yesmaximal, characteristic, complemented
3#21447211C211yesmaximal, characteristic, complemented
4#21447207C211yesmaximal, characteristic
5#21447136C211yesmaximal, characteristic
6#21446672C211yesmaximal, characteristic
7#21446549C211yesmaximal, characteristic, complemented
8#21446505C211yesmaximal, characteristic
9#21446494C211yesmaximal, characteristic, complemented
10#21446460C211yesmaximal, characteristic, complemented
11#21446397C211yesmaximal, characteristic, complemented
12#21446153C211yesmaximal, characteristic
13#21446066C211yesmaximal, characteristic
14C4×C4⋊#92418C211yesmaximal, characteristic, complemented
15C61×#2293C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×#14519710C2211yescharacteristic
18C2×#14519710C2211yescharacteristic
19C2×#14519708C2211yescharacteristic
20C2×C4⋊#92418C2211yescharacteristic
21C2×C4⋊#92418C2211yescharacteristic
22C2×#14519700C2211yescharacteristic
23C2×#14519700C2211yescharacteristic
24C2×#14519700C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×C4×#92418C2211yescharacteristic
29C2×C4×#92418C2211yescharacteristic
30#16839668C2211yescharacteristic
31#16839668C2211yescharacteristic
32#16839668C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839660C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839646—21no—
37#16839646—21no—
38#16839646—21no—
39#16839630—21no—
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48C2×C4×D4×C61C2211yescharacteristic
49C2×#117×C61—21no—
50C61×#390C2211yescharacteristic
51C61×#390C2211yescharacteristic
52C61×#389C2211yescharacteristic
53C61×#389C2211yescharacteristic
54C61×#388—21no—
55C61×#388—21no—
56C4×C4⋊C4×C61C2211yescharacteristic
57C4×C22⋊C4×C61—21no—
58C43×C61C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C2×#11758200—22no—
61C2×#11758200—22no—
62C2×#11758200—41no—
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×#11758196—21no—
71C2×#11758196—21no—
72C2×#11758196—21no—
73#14519710—22no—
74#14519710—41no—
75#14519708—22no—
76C4⋊#92418—22no—
77C4⋊#92418—22no—
78#14519700—22no—
79#14519700—22no—
80#14519700—41no—
81#14519698—22no—
82#14519698—22no—
83#14519698—41no—
84C4×#92418—41no—
85C4×#92418—41no—
86#14519684—21no—
87#14519684—21no—
88#14519684—21no—
89#14519684—21no—
90#14519684—21no—
91#14519684—21no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95C22×D4×C61C2311yescharacteristic
96C23×C4×C61—21no—
97C4×D4×C61—22no—
98C4×D4×C61—22no—
99#117×C61—44no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C22⋊C4×C61—21no—
106C2×C22⋊C4×C61—21no—
107C2×C22⋊C4×C61—21no—
108C2×C22⋊C4×C61—21no—
109C2×C22⋊C4×C61—21no—
110C2×C42×C61—21no—
111C2×C42×C61D411yescharacteristic
112C2×C42×C61D411yescharacteristic
113C2×C42×C61D411yescharacteristic
114C2×C42×C61D411yescharacteristic
115C2×C42×C61C2311yescharacteristic
116C2×C42×C61C2311yescharacteristic
117C2×C42×C61C2311yescharacteristic
118#95×C61—21no—
119#95×C61—21no—
120#95×C61—21no—
121C24×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61#4112yes—
128C22×C4×C61—24no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—42no—
133C22×C4×C61—42no—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2411yescharacteristic
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122—21no—
148C2×Dic122—21no—
149C2×Dic122—21no—
150C2×Dic122—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—42no—
157C4⋊C4×C61—42no—
158C4⋊C4×C61#4112yes—
159C22⋊C4×C61—24no—
160C22⋊C4×C61—42no—
161C22⋊C4×C61—42no—
162C22⋊C4×C61—42no—
163C22⋊C4×C61—42no—
164C42×C61—22no—
165C42×C61—22no—
166C42×C61—22no—
167C42×C61—22no—
168C42×C61—21no—
169C42×C61—21no—
170C42×C61—21no—
171C42×C61—21no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61—42no—
175#11758200—44no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182#11758196—42no—
183#11758196—42no—
184#11758196—42no—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418#4112yes—
196C2×#92418#4112yes—
197C23×C61—22no—
198C23×C61—41no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61C22×D411yescharacteristic
205D4×C61—22no—
206D4×C61—22no—
207D4×C61—41no—
208D4×C61—41no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—22no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—48no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61#13512yes—
234C2×C4×C61#13512yes—
235C2×C4×C61AGL(1,ℤ/8)12yes—
236C2×C4×C61#13712yes—
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61H(5,𝔽2)11yescharacteristic
242C2×C4×C612−1+411yescharacteristic
243C2×Dic61—41no—
244C2×Dic61—41no—
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—
255Dic122—41no—
256Dic122—41no—
257Dic122—41no—
258Dic122—41no—
259#92418—22no—
260#92418—22no—
261#92418—41no—
262#92418—41no—
263#29433—611nomaximal, Sylow
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—42no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61#52511yescharacteristic
272C22×C61#53211yescharacteristic
273C22×C61#53711yescharacteristic
274C22×C61C2×#13511yescharacteristic
275C22×C61C2×#13511yescharacteristic
276C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C61C2×#13711yescharacteristic
278C4×C61—22no—
279C4×C61—22no—
280C4×C61—22no—
281C4×C61—22no—
282C4×C61—42no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61—81no—
286C4×C61—81no—
287C4×C61#55111yescharacteristic
288C4×C61#55211yescharacteristic
289C4×C61#55311yescharacteristic
290C4×C61#55611yescharacteristic
291Dic61—81no—
292Dic61—81no—
293Dic61—81no—
294Dic61—81no—
295#2863—611no—
296#2293D6111yescharacteristic
297#2088—611no—
298#2068—611no—
299#2067—611no—
300#2055—611no—
301#1968—611no—
302#1942—611no—
303#1934—611no—
304#1933—611no—
305#1928—611no—
306#1919—611no—
307#1877—611no—
308#1862—611no—
309C4×C4⋊C8—611no—
310C2×C61—82no—
311C2×C61#260111yescharacteristic
312C2×C61#310511yescharacteristic
313C2×C61#310711yescharacteristic
314C2×C61#316811yescharacteristic
315C2×C61#318211yescharacteristic
316C2×C61#324711yescharacteristic
317C2×C61#324811yescharacteristic
318C22×SD16—611no—
319C2×C4×D4C2×D6111yescharacteristic
320C2×#117—21no—
321C2×#107—611no—
322C2×#107—611no—
323C2×#106—611no—
324C2×C4⋊C8—611no—
325C2×C4⋊C8—611no—
326C2×#103—611no—
327C2×#103—611no—
328C2×#103—611no—
329C2×#102—611no—
330C2×#102—611no—
331C2×#102—611no—
332C2×C4×C8—611no—
333C2×C4×C8—611no—
334#391—611no—
335#391—611no—
336#391—611no—
337#390C2×D6111yescharacteristic
338#390C2×D6111yescharacteristic
339#389—611no—
340#389—611no—
341#389—611no—
342#389C2×D6111yescharacteristic
343#389C2×D6111yescharacteristic
344#388—21no—
345#388—21no—
346#387—1221no—
347#387—1221no—
348#387—1221no—
349#383—1221no—
350C4×C4⋊C4C2×D6111yescharacteristic
351C4×C22⋊C4—21no—
352C43C2×D6111yescharacteristic
353#340—611no—
354#340—611no—
355#340—611no—
356#340—611no—
357#340—611no—
358#340—611no—
359#336—611no—
360#336—611no—
361C61#2943311yescharacteristic, complemented, Sylow
362C22×Q8—611no—
363C22×D4C22×D6111yescharacteristic
364C23×C4—21no—
365C2×SD16—1222no—
366C2×SD16—1222no—
367C2×SD16—2441no—
368C22×C8—611no—
369C22×C8—611no—
370C22×C8—611no—
371C22×C8—611no—
372C4×D4—22no—
373C4×D4—22no—
374#117—44no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4C22×D6111yescharacteristic
384C2×C4⋊C4C22×D6111yescharacteristic
385C2×C4⋊C4C22×D6111yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C42—21no—
392C2×C42C22⋊D6111yescharacteristic
393C2×C42C22⋊D6111yescharacteristic
394C2×C42C22⋊D6111yescharacteristic
395C2×C42C22⋊D6111yescharacteristic
396C2×C42C22×D6111yescharacteristic
397C2×C42C22×D6111yescharacteristic
398C2×C42C22×D6111yescharacteristic
399#107—1222no—
400#107—2441no—
401#106—1222no—
402C4⋊C8—1222no—
403C4⋊C8—1222no—
404#103—1222no—
405#103—1222no—
406#103—2441no—
407#102—1222no—
408#102—1222no—
409#102—2441no—
410C4×C8—2441no—
411C4×C8—2441no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—1221no—
416#95—1221no—
417#95—1221no—
418#95—1221no—
419#95—1221no—
420#95—1221no—
421#95—1221no—
422#95—1221no—
423#95—1221no—
424C24—21no—
425C2×Q8—1221no—
426C2×Q8—1221no—
427C2×Q8—1221no—
428C2×Q8—1221no—
429C2×Q8—1221no—
430C2×Q8—1221no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4#1175822212yes—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—1221no—
444C22×C4—1221no—
445C22×C4—1221no—
446C22×C4—1221no—
447C22×C4C2×C22⋊D6111yescharacteristic
448C22×C4C2×C22⋊D6111yescharacteristic
449C22×C4C2×C22⋊D6111yescharacteristic
450C22×C4C2×C22⋊D6111yescharacteristic
451C22×C4C2×C22⋊D6111yescharacteristic
452C22×C4C2×C22⋊D6111yescharacteristic
453C22×C4C23×D6111yescharacteristic
454SD16—2444no—
455C2×C8—612no—
456C2×C8—612no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—1221no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—2442no—
481C4⋊C4—2442no—
482C4⋊C4—2442no—
483C4⋊C4#1175822212yes—
484C22⋊C4—24no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—42no—
500C23—22no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23C22×C22⋊D6111yescharacteristic
508Q8—2441no—
509Q8—2441no—
510Q8—2441no—
511Q8—2441no—
512D4—22no—
513D4—22no—
514D4—41no—
515D4—41no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—48no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4#1451979712yes—
553C2×C4#1451979712yes—
554C2×C4#1451980312yes—
555C2×C4#1451980712yes—
556C2×C4#41×D6111yescharacteristic
557C2×C4#41×D6111yescharacteristic
558C2×C4C2×#1175822211yescharacteristic
559C2×C4C2×#1175822211yescharacteristic
560C2×C4#1451982311yescharacteristic
561C2×C4#1451982411yescharacteristic
562C8—1222no—
563C8—1222no—
564C8—2441no—
565C8—2441no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22#1684027311yescharacteristic
574C22#1684030811yescharacteristic
575C22#1684034011yescharacteristic
576C22C2×#1451979711yescharacteristic
577C22C2×#1451979711yescharacteristic
578C22C2×#1451980311yescharacteristic
579C22C2×#1451980711yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—41no—
585C4—41no—
586C4—42no—
587C4—81no—
588C4—81no—
589C4—4881no—
590C4—4881no—
591C4—4881no—
592C4—4881no—
593C4#1684043611yescharacteristic
594C4#1684044511yescharacteristic
595C4#1684045611yescharacteristic
596C4#1684048011yescharacteristic
597C2—82no—
598C2#2145141611yescharacteristic
599C2#2145586611yescharacteristic
600C2#2145589011yescharacteristic
601C2#2145638311yescharacteristic
602C2#2145653311yescharacteristic
603C2#2145713911yescharacteristic
604C2#2145715611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2293C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2293D61
hypocenterC61#29433
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C43 ◁ #2293

Complements

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

NH ≅ G/Ntype
1. #21454201C2semidirect
2. #21447321C2semidirect
3. #21447211C2semidirect
7. #21446549C2semidirect
9. #21446494C2semidirect
10. #21446460C2semidirect
11. #21446397C2semidirect
14. C4×C4⋊#92418C2semidirect
361. C61#29433semidirect

Maximal subgroups

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

subgroupindex
#214542012
#214473212
#214472112
#214472072
#214471362
#214466722
#214465492
#214465052
#214464942
#214464602
#214463972
#214461532
#214460662
C4×C4⋊#924182
C61×#22932
#2943361