Finite Group Databasebeta

← #29752754

Subgroups — #29752754

Order 15616.

708 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450252C211yesmaximal, characteristic
2#21447227C211yesmaximal, characteristic, complemented
3#21447214C211yesmaximal, characteristic, complemented
4#21447019C211yesmaximal, characteristic
5#21447019C211yesmaximal, characteristic
6#21446577C211yesmaximal, characteristic, complemented
7#21446549C211yesmaximal, characteristic, complemented
8#21446427C211yesmaximal, characteristic
9#21446411C211yesmaximal, characteristic, complemented
10#21446408C211yesmaximal, characteristic, complemented
11#21446073C211yesmaximal, characteristic, complemented
12#21446007C211yesmaximal, characteristic
13#21445909C211yesmaximal, characteristic
14C4×#14519701C211yesmaximal, characteristic, complemented
15C61×#2360C211yesmaximal, characteristic
16C2×#14519710C2211yescharacteristic
17C2×#14519710C2211yescharacteristic
18C2×#14519710C2211yescharacteristic
19C2×#14519710C2211yescharacteristic
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519701C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×C22⋊#92418C2211yescharacteristic
26C2×C22⋊#92418C2211yescharacteristic
27C2×C4×#92418C2211yescharacteristic
28#16839704—21no—
29#16839685—21no—
30#16839660C2211yescharacteristic
31#16839660C2211yescharacteristic
32#16839655—21no—
33#16839649—21no—
34#16839632C2211yescharacteristic
35#16839622C2211yescharacteristic
36#16839622C2211yescharacteristic
37C4×C4⋊Dic61C2211yescharacteristic
38#16839524C2211yescharacteristic
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48C2×C22⋊Q8×C61C2211yescharacteristic
49C2×C4×Q8×C61C2211yescharacteristic
50C22×C4⋊C4×C61C2211yescharacteristic
51C61×#391C2211yescharacteristic
52C61×#389—21no—
53C61×#389C2211yescharacteristic
54C61×#387—21no—
55C61×#385—21no—
56C4×C4⋊C4×C61C2211yescharacteristic
57C4×C22⋊C4×C61—21no—
58C4×C22⋊C4×C61C2211yescharacteristic
59C22×#92418C2311yescharacteristic
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61C2311yescharacteristic
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×#11758196—21no—
70C2×#11758196—21no—
71C2×C22⋊Dic61—21no—
72C2×C22⋊Dic61—21no—
73C2×C4×Dic61—21no—
74#14519710—22no—
75#14519710—22no—
76#14519710—22no—
77#14519710—41no—
78C4⋊#92418—22no—
79#14519701—22no—
80#14519701—22no—
81#14519701—22no—
82#14519701—41no—
83C22⋊#92418—22no—
84C22⋊#92418—41no—
85C4×#92418—41no—
86#14519684—21no—
87#14519684—21no—
88#14519684—21no—
89#14519684—21no—
90#14519684—21no—
91C22×Q8×C61C2311yescharacteristic
92C23×C4×C61—21no—
93C23×C4×C61C2311yescharacteristic
94C22⋊Q8×C61—22no—
95C22⋊Q8×C61—22no—
96C4×Q8×C61—22no—
97C4×Q8×C61—41no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—22no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—41no—
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C22⋊C4×C61—21no—
108C2×C22⋊C4×C61D411yescharacteristic
109C2×C22⋊C4×C61D411yescharacteristic
110C2×C42×C61—21no—
111C2×C42×C61—21no—
112C2×C42×C61C2311yescharacteristic
113C2×C42×C61C2311yescharacteristic
114#95×C61—21no—
115#95×C61—21no—
116#95×C61—21no—
117#95×C61—21no—
118#95×C61D411yescharacteristic
119#95×C61D411yescharacteristic
120C24×C61C2×D411yescharacteristic
121C2×Q8×C61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61#4112yes—
127C22×C4×C61—21no—
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—42no—
138C22×C4×C61—42no—
139C22×C4×C61—81no—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2411yescharacteristic
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C4⋊C4×C61—24no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—42no—
156C4⋊C4×C61—42no—
157C4⋊C4×C61—81no—
158C4⋊C4×C61#4112yes—
159C4⋊C4×C61#4112yes—
160C22⋊C4×C61—24no—
161C22⋊C4×C61—41no—
162C22⋊C4×C61Q1614yes—
163C42×C61—21no—
164C42×C61—21no—
165C42×C61—21no—
166C42×C61—21no—
167C42×C61—42no—
168C42×C61—81no—
169C22⋊Dic61—81no—
170C22⋊Dic61—81no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—81no—
180C4⋊Dic61—81no—
181#11758196—42no—
182#11758196—81no—
183C4×Dic61—81no—
184C2×#92418—21no—
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—
196C23×C61—21no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—21no—
200C23×C61—21no—
201C23×C61—21no—
202C23×C61C2×Q1612yes—
203C23×C61C22×D411yescharacteristic
204Q8×C61—22no—
205Q8×C61—22no—
206Q8×C61—41no—
207Q8×C61—41no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—22no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—41no—
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—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—84no—
238C2×C4×C61C2×Q1612yes—
239C2×C4×C61C2×Q1612yes—
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61H(5,𝔽2)11yescharacteristic
245C2×C4×C612−1+411yescharacteristic
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—41no—
257C2×Dic61—41no—
258#92418—22no—
259#92418—41no—
260#92418—41no—
261#92418—41no—
262#27036—611nomaximal, Sylow
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—22no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61#52511yescharacteristic
271C22×C61#54611yescharacteristic
272C22×C61#54711yescharacteristic
273C22×C61C22×Q1611yescharacteristic
274C22×C61#57511yescharacteristic
275C22×C61#57611yescharacteristic
276C22×C61#57711yescharacteristic
277C4×C61—22no—
278C4×C61—22no—
279C4×C61—41no—
280C4×C61—42no—
281C4×C61—41no—
282C4×C61—42no—
283C4×C61—41no—
284C4×C61—42no—
285C4×C61—81no—
286C4×C61#52911yescharacteristic
287C4×C61#53911yescharacteristic
288C4×C61#55311yescharacteristic
289C4×C61#55611yescharacteristic
290Dic61—81no—
291Dic61—81no—
292Dic61—81no—
293Dic61—81no—
294#2497—611no—
295#2360D6111yescharacteristic
296#2072—611no—
297#2068—611no—
298#2036—611no—
299#2036—611no—
300#1947—611no—
301#1942—611no—
302#1923—611no—
303#1921—611no—
304#1921—611no—
305#1863—611no—
306#1852—611no—
307#1824—611no—
308C4×#103—611no—
309C2×C61—42no—
310C2×C61#246611yescharacteristic
311C2×C61#310411yescharacteristic
312C2×C61#311911yescharacteristic
313C2×C61#329911yescharacteristic
314C2×C61#331911yescharacteristic
315C2×C61#333811yescharacteristic
316C2×C61#335111yescharacteristic
317C2×C22⋊Q8C2×D6111yescharacteristic
318C2×C4×Q8C2×D6111yescharacteristic
319C22×C4⋊C4C2×D6111yescharacteristic
320C2×#107—611no—
321C2×#107—611no—
322C2×#107—611no—
323C2×#107—611no—
324C2×C4⋊C8—611no—
325C2×#103—611no—
326C2×#103—611no—
327C2×#103—611no—
328C2×#103—611no—
329C2×C22⋊C8—611no—
330C2×C22⋊C8—611no—
331C2×C4×C8—611no—
332#400—1221no—
333#396—1221no—
334#391C2×D6111yescharacteristic
335#389—21no—
336#389—611no—
337#389—611no—
338#389C2×D6111yescharacteristic
339#388—1221no—
340#387—21no—
341#387—1221no—
342#385—21no—
343#384—611no—
344#380—611no—
345#380—611no—
346C4×C4⋊C4—611no—
347C4×C4⋊C4C2×D6111yescharacteristic
348C4×C22⋊C4—21no—
349C4×C22⋊C4C2×D6111yescharacteristic
350#340—611no—
351#340—611no—
352#340—611no—
353#340—611no—
354#340—611no—
355#340—611no—
356#340—611no—
357#340—611no—
358#336—611no—
359#336—611no—
360C61#2703611yescharacteristic, complemented, Sylow
361C22×Q8C22×D6111yescharacteristic
362C23×C4—21no—
363C23×C4C22×D6111yescharacteristic
364C22×C8—611no—
365C22×C8—611no—
366C22×C8—611no—
367C22×C8—611no—
368C22⋊Q8—22no—
369C22⋊Q8—22no—
370C4×Q8—22no—
371C4×Q8—41no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—41no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4C22×D6111yescharacteristic
387C2×C4⋊C4C22×D6111yescharacteristic
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1221no—
391C2×C22⋊C4—1221no—
392C2×C22⋊C4C22⋊D6111yescharacteristic
393C2×C22⋊C4C22⋊D6111yescharacteristic
394C2×C42—21no—
395C2×C42—21no—
396C2×C42—1221no—
397C2×C42C22×D6111yescharacteristic
398C2×C42C22×D6111yescharacteristic
399#107—1222no—
400#107—1222no—
401#107—1222no—
402#107—2441no—
403C4⋊C8—1222no—
404#103—1222no—
405#103—1222no—
406#103—1222no—
407#103—2441no—
408C22⋊C8—1222no—
409C22⋊C8—2441no—
410C4×C8—2441no—
411#95—21no—
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#95C22⋊D6111yescharacteristic
421#95C22⋊D6111yescharacteristic
422C24C2×C22⋊D6111yescharacteristic
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8#1175822212yes—
429C22×C4—21no—
430C22×C4—21no—
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—42no—
440C22×C4—42no—
441C22×C4—81no—
442C22×C4—1221no—
443C22×C4—1221no—
444C22×C4—1221no—
445C22×C4—1221no—
446C22×C4C2×C22⋊D6111yescharacteristic
447C22×C4C2×C22⋊D6111yescharacteristic
448C22×C4C2×C22⋊D6111yescharacteristic
449C22×C4C2×C22⋊D6111yescharacteristic
450C22×C4C2×C22⋊D6111yescharacteristic
451C22×C4C23×D6111yescharacteristic
452C2×C8—612no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—1221no—
464C4⋊C4—24no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—81no—
472C4⋊C4—1221no—
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—4881no—
482C4⋊C4—4881no—
483C4⋊C4—2442no—
484C4⋊C4#1175822212yes—
485C4⋊C4#1175822412yes—
486C22⋊C4—24no—
487C22⋊C4—41no—
488C22⋊C4—4881no—
489C22⋊C4—4881no—
490C22⋊C4#1175820214yes—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—42no—
496C42—81no—
497C42—4881no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23C2×#1175820212yes—
505C23C22×C22⋊D6111yescharacteristic
506Q8—22no—
507Q8—22no—
508Q8—41no—
509Q8—41no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—84no—
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×C4C2×#1175820212yes—
553C2×C4C2×#1175820212yes—
554C2×C4#41×D6111yescharacteristic
555C2×C4C2×#1175822411yescharacteristic
556C2×C4C2×#1175822211yescharacteristic
557C2×C4C2×#1175822211yescharacteristic
558C2×C4#1451982211yescharacteristic
559C2×C4#1451982411yescharacteristic
560C8—1222no—
561C8—2441no—
562C8—2441no—
563C8—2441no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22#1684027311yescharacteristic
572C22#1684040611yescharacteristic
573C22#1684041111yescharacteristic
574C22C22×#1175820211yescharacteristic
575C22#1684056011yescharacteristic
576C22#1684056511yescharacteristic
577C22#1684057011yescharacteristic
578C4—22no—
579C4—22no—
580C4—41no—
581C4—42no—
582C4—41no—
583C4—42no—
584C4—42no—
585C4—41no—
586C4—81no—
587C4—4881no—
588C4—4881no—
589C4—4881no—
590C4—4881no—
591C4#1684029211yescharacteristic
592C4#1684035411yescharacteristic
593C4#1684046211yescharacteristic
594C4#1684048211yescharacteristic
595C2—42no—
596C2#2144984911yescharacteristic
597C2#2145585811yescharacteristic
598C2#2145601811yescharacteristic
599C2#2145769511yescharacteristic
600C2#2145797511yescharacteristic
601C2#2145820711yescharacteristic
602C2#2145837911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2360C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#2360D61
hypocenterC61#27036
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
2. #21447227C2semidirect
3. #21447214C2semidirect
6. #21446577C2semidirect
7. #21446549C2semidirect
9. #21446411C2semidirect
10. #21446408C2semidirect
11. #21446073C2semidirect
14. C4×#14519701C2semidirect
360. C61#27036semidirect

Maximal subgroups

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

subgroupindex
#214502522
#214472272
#214472142
#214470192
#214465772
#214465492
#214464272
#214464112
#214464082
#214460732
#214460072
#214459092
C4×#145197012
C61×#23602
#2703661