Finite Group Databasebeta

← #29752763

Subgroups — #29752763

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#21450250C211yesmaximal, characteristic
2#21448829C211yesmaximal, characteristic
3#21447227C211yesmaximal, characteristic, complemented
4#21447021C211yesmaximal, characteristic
5#21447021C211yesmaximal, characteristic
6#21446578C211yesmaximal, characteristic, complemented
7#21446550C211yesmaximal, characteristic, complemented
8#21446424C211yesmaximal, characteristic
9#21446410C211yesmaximal, characteristic, complemented
10#21446408C211yesmaximal, characteristic, complemented
11#21446071C211yesmaximal, characteristic, complemented
12#21446005C211yesmaximal, characteristic
13#21445908C211yesmaximal, characteristic
14#103×Dic61C211yesmaximal, characteristic, complemented
15C61×#2068C211yesmaximal, characteristic, complemented
16C2×#14519753C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C2×C4⋊C4×D61C2211yescharacteristic
19C2×#14519711C2211yescharacteristic
20C2×#14519711C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519689C2211yescharacteristic
25C2×#14519689C2211yescharacteristic
26C2×C8×Dic61C2211yescharacteristic
27#16839704—21no—
28#16839684—21no—
29#16839659—21no—
30#16839658C2211yescharacteristic
31#16839658C2211yescharacteristic
32#16839654—21no—
33#16839648—21no—
34#16839644—21no—
35#16839636—21no—
36#16839632C2211yescharacteristic
37#16839619C2211yescharacteristic
38#16839619C2211yescharacteristic
39C4⋊C4×Dic61C2211yescharacteristic
40C4×C4⋊Dic61C2211yescharacteristic
41C4×#11758198C2211yescharacteristic
42#16839609—21no—
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839517C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#107×C61C2211yescharacteristic
53C2×C4⋊C8×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C61×#391C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×C4×D61—21no—
60C22×C4×D61C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63#14519753—22no—
64#14519753—22no—
65Q8×Dic61—22no—
66Q8×Dic61—41no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C4⋊C4×D61—22no—
79C4⋊C4×D61—41no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198D411yescharacteristic
84C2×#11758198D411yescharacteristic
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61C2311yescharacteristic
89#14519711—22no—
90#14519711—22no—
91#14519710—22no—
92#14519701—22no—
93#14519701—22no—
94#14519689—22no—
95#14519689—41no—
96C8×Dic61—41no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684D411yescharacteristic
104#14519684D411yescharacteristic
105C22×Q8×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C42×C61C2311yescharacteristic
113#107×C61—41no—
114C4⋊C8×C61—22no—
115#103×C61—22no—
116#103×C61—41no—
117#95×C61—21no—
118#95×C61—21no—
119#95×C61—21no—
120C23×D61C2×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×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2411yescharacteristic
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141C2×C4×D61—21no—
142C2×C4×D61—21no—
143C2×C4×D61—21no—
144C2×C4×D61—21no—
145C2×C4×D61—21no—
146C2×C4×D61—21no—
147C2×C4×D61—42no—
148C2×C4×D61—42no—
149C2×C4×D61—81no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61#4112yes—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—42no—
163C42×C61—21no—
164C42×C61—21no—
165#11758198—24no—
166#11758198—41no—
167#11758198—81no—
168#11758198—81no—
169#11758198Q1614yes—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—42no—
177C4⋊Dic61—81no—
178C4⋊Dic61—81no—
179C4⋊Dic61#4112yes—
180C4⋊Dic61#4112yes—
181#11758196—24no—
182#11758196—42no—
183#11758196—81no—
184#11758196—81no—
185C4×Dic61—21no—
186C4×Dic61—21no—
187C4×Dic61—42no—
188C4×Dic61—81no—
189C4×Dic61—81no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C23×C61C22×D411yescharacteristic
197C22×D61—21no—
198C22×D61—21no—
199C22×D61—21no—
200C22×D61—21no—
201C22×D61—21no—
202C22×D61—21no—
203C22×D61C2×Q1612yes—
204Q8×C61—22no—
205Q8×C61—22no—
206Q8×C61—41no—
207Q8×C61—41no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
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×C61C2×Q1612yes—
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61H(5,𝔽2)11yescharacteristic
231C2×C4×C612−1+411yescharacteristic
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—22no—
236C2×Dic61—21no—
237C2×Dic61—22no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—41no—
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×Dic61C2×Q1612yes—
253C4×D61—41no—
254C4×D61—41no—
255C4×D61—41no—
256C4×D61—41no—
257C4×D61—84no—
258C8×C61—22no—
259C8×C61—41no—
260#92418—41no—
261#92418—41no—
262#27036—611nomaximal, Sylow
263C22×C61#52511yescharacteristic
264C22×C61#54611yescharacteristic
265C22×C61#54711yescharacteristic
266C22×C61C22×Q1611yescharacteristic
267C22×C61#57511yescharacteristic
268C22×C61#57611yescharacteristic
269C22×C61#57711yescharacteristic
270C2×D61—22no—
271C2×D61—22no—
272C2×D61—22no—
273C2×D61—41no—
274C2×D61—41no—
275C2×D61—41no—
276C2×D61—41no—
277C4×C61—22no—
278C4×C61—41no—
279C4×C61—42no—
280C4×C61—81no—
281C4×C61—81no—
282C4×C61—81no—
283C4×C61#52911yescharacteristic
284C4×C61#53911yescharacteristic
285C4×C61#55311yescharacteristic
286C4×C61#55611yescharacteristic
287Dic61—22no—
288Dic61—42no—
289Dic61—41no—
290Dic61—42no—
291Dic61—41no—
292Dic61—81no—
293Dic61—81no—
294#2497—611no—
295#2360—611no—
296#2072—611no—
297#2068D6111yescharacteristic, complemented
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#246611yescharacteristic
310C2×C61#310411yescharacteristic
311C2×C61#311911yescharacteristic
312C2×C61#329911yescharacteristic
313C2×C61#331911yescharacteristic
314C2×C61#333811yescharacteristic
315C2×C61#335111yescharacteristic
316D61—42no—
317C2×C22⋊Q8—611no—
318C2×C4×Q8—611no—
319C22×C4⋊C4—611no—
320C2×#107—611no—
321C2×#107—611no—
322C2×#107—611no—
323C2×#107C2×D6111yescharacteristic
324C2×C4⋊C8C2×D6111yescharacteristic
325C2×#103—611no—
326C2×#103—611no—
327C2×#103C2×D6111yescharacteristic
328C2×#103C2×D6111yescharacteristic
329C2×C22⋊C8—611no—
330C2×C22⋊C8—611no—
331C2×C4×C8—611no—
332#400—1221no—
333#396—1221no—
334#391C2×D6111yescharacteristic
335#389—611no—
336#389—611no—
337#389—1221no—
338#389C2×D6111yescharacteristic
339#388—1221no—
340#387—1221no—
341#387—1221no—
342#385—1221no—
343#384—611no—
344#380—611no—
345#380—611no—
346C4×C4⋊C4—611no—
347C4×C4⋊C4—611no—
348C4×C22⋊C4—611no—
349C4×C22⋊C4—1221no—
350#340—611no—
351#340—611no—
352#340—611no—
353#340—611no—
354#340—611no—
355#340—611no—
356#340—611no—
357#340C2×D6111yescharacteristic
358#336—611no—
359#336—611no—
360C61#2703611yescharacteristic, complemented, Sylow
361C22×Q8C22×D6111yescharacteristic
362C23×C4—611no—
363C23×C4—1221no—
364C22×C8—611no—
365C22×C8—611no—
366C22×C8C22×D6111yescharacteristic
367C22×C8C22×D6111yescharacteristic
368C22⋊Q8—1222no—
369C22⋊Q8—1222no—
370C4×Q8—1222no—
371C4×Q8—2441no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—2441no—
386C2×C4⋊C4C22×D6111yescharacteristic
387C2×C4⋊C4C22×D6111yescharacteristic
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C22⋊C4—611no—
390C2×C22⋊C4—611no—
391C2×C22⋊C4—1221no—
392C2×C22⋊C4—1221no—
393C2×C22⋊C4—1221no—
394C2×C42—611no—
395C2×C42—1221no—
396C2×C42—1221no—
397C2×C42—1221no—
398C2×C42C22×D6111yescharacteristic
399#107—41no—
400#107—1222no—
401#107—1222no—
402#107—1222no—
403C4⋊C8—22no—
404#103—22no—
405#103—41no—
406#103—1222no—
407#103—1222no—
408C22⋊C8—1222no—
409C22⋊C8—2441no—
410C4×C8—2441no—
411#95—21no—
412#95—21no—
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—
422C24—611no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8#1175822612yes—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—611no—
434C22×C4—611no—
435C22×C4—611no—
436C22×C4—1221no—
437C22×C4—1221no—
438C22×C4—1221no—
439C22×C4—1221no—
440C22×C4—1221no—
441C22×C4—1221no—
442C22×C4—1221no—
443C22×C4—1221no—
444C22×C4—1221no—
445C22×C4—1221no—
446C22×C4—4881no—
447C22×C4—2442no—
448C22×C4—2442no—
449C22×C4D4×D6111yescharacteristic
450C22×C4D4×D6111yescharacteristic
451C22×C4C23×D6111yescharacteristic
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8#1175822412yes—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—42no—
471C4⋊C4—612no—
472C4⋊C4—612no—
473C4⋊C4—1224no—
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—4881no—
484C4⋊C4—2442no—
485C4⋊C4—2442no—
486C22⋊C4—614no—
487C22⋊C4—1224no—
488C22⋊C4—4881no—
489C22⋊C4—4881no—
490C22⋊C4—2441no—
491C42—21no—
492C42—21no—
493C42—1221no—
494C42—1221no—
495C42—4881no—
496C42—4881no—
497C42—2442no—
498C23—612no—
499C23—1221no—
500C23—1221no—
501C23—1221no—
502C23—1221no—
503C23—1221no—
504C23—1221no—
505C23C2×D4×D6111yescharacteristic
506Q8—22no—
507Q8—22no—
508Q8—41no—
509Q8—41no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
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—612no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1222no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1222no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—4884no—
539C2×C4—2441no—
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—2441no—
553C2×C4Q16×D6112yes—
554C2×C4#41×D6111yescharacteristic
555C2×C4#41×D6111yescharacteristic
556C2×C4C2×#1175822411yescharacteristic
557C2×C4C2×#1175822611yescharacteristic
558C2×C4#1451982211yescharacteristic
559C2×C4#1451982411yescharacteristic
560C8—22no—
561C8—41no—
562C8—2441no—
563C8—2441no—
564C22—1222no—
565C22—1222no—
566C22—1222no—
567C22—2441no—
568C22—2441no—
569C22—2441no—
570C22—2441no—
571C22#1684027511yescharacteristic
572C22#1684040911yescharacteristic
573C22#1684041211yescharacteristic
574C22C2×Q16×D6111yescharacteristic
575C22#1684055811yescharacteristic
576C22#1684056411yescharacteristic
577C22#1684057111yescharacteristic
578C4—22no—
579C4—42no—
580C4—41no—
581C4—81no—
582C4—81no—
583C4—81no—
584C4—1222no—
585C4—4881no—
586C4—4881no—
587C4—2441no—
588C4—2442no—
589C4—2441no—
590C4—2442no—
591C4#1684029811yescharacteristic
592C4#1684036111yescharacteristic
593C4#1684045511yescharacteristic
594C4#1684047911yescharacteristic
595C2—2442no—
596C2#2144985811yescharacteristic
597C2#2145586011yescharacteristic
598C2#2145602511yescharacteristic
599C2#2145769811yescharacteristic
600C2#2145798211yescharacteristic
601C2#2145820811yescharacteristic
602C2#2145837411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2068C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2068D61
hypocenterC61#27036
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2068

Complements

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

NH ≅ G/Ntype
3. #21447227C2semidirect
6. #21446578C2semidirect
7. #21446550C2semidirect
9. #21446410C2semidirect
10. #21446408C2semidirect
11. #21446071C2semidirect
14. #103×Dic61C2semidirect
15. C61×#2068C2semidirect
297. #2068D61semidirect
360. C61#27036semidirect

Maximal subgroups

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

subgroupindex
#214502502
#214488292
#214472272
#214470212
#214465782
#214465502
#214464242
#214464102
#214464082
#214460712
#214460052
#214459082
#103×Dic612
C61×#20682
#2703661