Finite Group Databasebeta

← #29730055

Subgroups — #29730055

Order 15616.

776 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839969C211yesmaximal, characteristic
2#21451011C211yesmaximal, characteristic
3#21448879C211yesmaximal, characteristic
4#21447240C211yesmaximal, characteristic
5#21447230C211yesmaximal, characteristic
6#21447140C211yesmaximal, characteristic
7#21447140C211yesmaximal, characteristic
8#21446777C211yesmaximal, characteristic
9#21446542C211yesmaximal, characteristic
10#21446158C211yesmaximal, characteristic
11#21446158C211yesmaximal, characteristic
12#21445832C211yesmaximal, characteristic
13#21445822C211yesmaximal, characteristic
14#21445650C211yesmaximal, characteristic
15C2×C61×#491C211yesmaximal, characteristic
16C22×#11758202C2211yescharacteristic
17C22×#11758202C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×C4×Dic122C2211yescharacteristic
21#16839969—24no—
22C2×C4⋊#92418C2211yescharacteristic
23C2×#14519706C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29#16839695—21no—
30#16839695—21no—
31#16839669C2211yescharacteristic
32#16839669C2211yescharacteristic
33#16839667C2211yescharacteristic
34#16839667C2211yescharacteristic
35#16839661—21no—
36#16839648—21no—
37#16839648—21no—
38#16839643—21no—
39#16839643—21no—
40#16839631C2211yescharacteristic
41#16839627C2211yescharacteristic
42C4⋊C4×Dic61—21no—
43C4×C4⋊Dic61C2211yescharacteristic
44C4×C4⋊Dic61C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×C4⋊Q8×C61C2211yescharacteristic
54C2×#125×C61C2211yescharacteristic
55C61×#491—24no—
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C2×#97×C61C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C2×#11758202—22no—
63C2×#11758202—22no—
64C2×#11758202—22no—
65C2×#11758202—22no—
66C2×#11758202—41no—
67C2×#11758202—41no—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70Q8×Dic61—22no—
71Q8×Dic61—22no—
72Q8×Dic61—41no—
73Q8×Dic61—41no—
74C4×Dic122—22no—
75C4×Dic122—22no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61D411yescharacteristic
79C2×C4⋊Dic61D411yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61C2311yescharacteristic
91C2×C4×Dic61C2311yescharacteristic
92C4⋊#92418—22no—
93#14519706—41no—
94#14519706—41no—
95#14519702—22no—
96#14519702—22no—
97#14519701—22no—
98#14519701—22no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684D411yescharacteristic
111#14519684D411yescharacteristic
112C22×Q8×C61C2311yescharacteristic
113C22×Q8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C4⋊Q8×C61—22no—
117C4⋊Q8×C61D414yes—
118#125×C61—22no—
119#125×C61—22no—
120C2×C4⋊C4×C61—21no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61—21no—
123C2×C4⋊C4×C61—21no—
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C42×C61C2311yescharacteristic
127#103×C61—22no—
128#103×C61—22no—
129#103×C61—22no—
130#103×C61—22no—
131#97×C61—22no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61C2×D412yes—
143C2×Q8×C61C2×D412yes—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122#4112yes—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C4⋊C4×C61—24no—
172C4⋊C4×C61—24no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—42no—
178C4⋊C4×C61—42no—
179C42×C61C2×D412yes—
180C42×C61#4112yes—
181#11758202—44no—
182#11758202—44no—
183C4⋊Dic61—21no—
184C4⋊Dic61—22no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—41no—
190C4⋊Dic61—42no—
191C4⋊Dic61—42no—
192C4⋊Dic61#4112yes—
193#11758196—42no—
194#11758196—42no—
195#11758196—42no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200C4×Dic61—42no—
201C4×Dic61—42no—
202C4×Dic61—42no—
203C4×Dic61—42no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C23×C61C22×D411yescharacteristic
211Q8×C61—22no—
212Q8×C61—22no—
213Q8×C61—22no—
214Q8×C61—22no—
215Q8×C61—41no—
216Q8×C61—41no—
217Q8×C61—41no—
218Q8×C61—41no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
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#12112yes—
236C2×C4×C61#12112yes—
237C2×C4×C61C22×D411yescharacteristic
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—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—22no—
249C2×Dic61—22no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61#13712yes—
267Dic122—22no—
268Dic122—22no—
269Dic122—41no—
270Dic122—41no—
271C8×C61—41no—
272C8×C61—41no—
273#92418—41no—
274#92418—41no—
275#25333—611nomaximal, Sylow
276C22×C61C2×#12111yescharacteristic
277C22×C61#54611yescharacteristic
278C22×C61#54811yescharacteristic
279C22×C61C2×#13711yescharacteristic
280C22×C61#57511yescharacteristic
281C22×C61#57811yescharacteristic
282C22×C61#57811yescharacteristic
283C4×C61—22no—
284C4×C61—22no—
285C4×C61—42no—
286C4×C61—42no—
287C4×C61—81no—
288C4×C61—81no—
289C4×C61#52911yescharacteristic
290C4×C61#54211yescharacteristic
291C4×C61#54611yescharacteristic
292C4×C61#54711yescharacteristic
293Dic61—22no—
294Dic61—42no—
295Dic61—41no—
296Dic61—41no—
297Dic61—41no—
298Dic61—42no—
299Dic61—81no—
300Dic61—81no—
301C2×#491D6111yescharacteristic
302C2×#462—611no—
303#2565—611no—
304#2365—611no—
305#2074—611no—
306#2072—611no—
307#2056—611no—
308#2056—611no—
309#1989—611no—
310#1940—611no—
311#1878—611no—
312#1878—611no—
313#1805—611no—
314#1803—611no—
315#1763—611no—
316C2×C61#242711yescharacteristic
317C2×C61#303711yescharacteristic
318C2×C61#304411yescharacteristic
319C2×C61#331211yescharacteristic
320C2×C61#331511yescharacteristic
321C2×C61#337411yescharacteristic
322C2×C61#337711yescharacteristic
323C22×Q16—611no—
324C22×Q16—611no—
325C2×C4⋊Q8C2×D6111yescharacteristic
326C2×#125C2×D6111yescharacteristic
327C2×C4×Q8—611no—
328C2×C4×Q8—611no—
329C2×C4×Q8—611no—
330#491—24no—
331#462—1224no—
332C2×C4⋊C8—611no—
333C2×C4⋊C8—611no—
334C2×C4⋊C8—611no—
335C2×#103—611no—
336C2×#103—611no—
337C2×#103—611no—
338C2×#103—611no—
339C2×#103C2×D6111yescharacteristic
340C2×#103C2×D6111yescharacteristic
341C2×#103C2×D6111yescharacteristic
342C2×#103C2×D6111yescharacteristic
343C2×#97C2×D6111yescharacteristic
344#398—1221no—
345#398—1221no—
346#391—611no—
347#391—611no—
348#391—611no—
349#391—611no—
350#389—1221no—
351#387—1221no—
352#387—1221no—
353#387—1221no—
354#387—1221no—
355#384—611no—
356#382—611no—
357C4×C4⋊C4—611no—
358C4×C4⋊C4—611no—
359C4×C4⋊C4—1221no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#336—611no—
367#336—611no—
368C61#2533311yescharacteristic, complemented, Sylow
369C22×Q8—611no—
370C22×Q8C22×D6111yescharacteristic
371C22×Q8C22×D6111yescharacteristic
372C2×Q16—1222no—
373C2×Q16—1222no—
374C2×Q16—1222no—
375C2×Q16—1222no—
376C2×Q16—2441no—
377C2×Q16—2441no—
378C22×C8—611no—
379C22×C8—611no—
380C22×C8C22×D6111yescharacteristic
381C22×C8C22×D6111yescharacteristic
382C4⋊Q8—22no—
383C4⋊Q8C22⋊D6114yes—
384#125—22no—
385#125—22no—
386C4×Q8—1222no—
387C4×Q8—1222no—
388C4×Q8—1222no—
389C4×Q8—1222no—
390C4×Q8—2441no—
391C4×Q8—2441no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4C22×D6111yescharacteristic
407C2×C4⋊C4C22×D6111yescharacteristic
408C2×C42—611no—
409C2×C42—611no—
410C2×C42—1221no—
411C2×C42—1221no—
412C2×C42—1221no—
413C2×C42—1221no—
414C2×C42C22×D6111yescharacteristic
415C4⋊C8—1222no—
416C4⋊C8—2441no—
417C4⋊C8—2441no—
418#103—22no—
419#103—22no—
420#103—22no—
421#103—22no—
422#103—1222no—
423#103—1222no—
424#103—1222no—
425#103—1222no—
426#97—22no—
427#95—611no—
428#95—611no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438#95—1221no—
439#95—1221no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—612no—
451C2×Q8—1221no—
452C2×Q8—1221no—
453C2×Q8—1221no—
454C2×Q8—1221no—
455C2×Q8—1221no—
456C2×Q8C2×C22⋊D6112yes—
457C2×Q8C2×C22⋊D6112yes—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4D4×D6111yescharacteristic
471C22×C4D4×D6111yescharacteristic
472C22×C4C23×D6111yescharacteristic
473Q16—2444no—
474Q16—2444no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—612no—
496C4⋊C4—1222no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—2441no—
503C4⋊C4—2442no—
504C4⋊C4—2442no—
505C4⋊C4—2442no—
506C4⋊C4—2442no—
507C4⋊C4—2442no—
508C42—1221no—
509C42—1221no—
510C42—1221no—
511C42—1221no—
512C42—2442no—
513C42—2442no—
514C42—2442no—
515C42—2442no—
516C42#1175822412yes—
517C42C2×C22⋊D6112yes—
518C23C2×D4×D6111yescharacteristic
519Q8—22no—
520Q8—22no—
521Q8—22no—
522Q8—22no—
523Q8—41no—
524Q8—41no—
525Q8—41no—
526Q8—41no—
527Q8—1222no—
528Q8—1222no—
529Q8—2441no—
530Q8—2441no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—612no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4#1451974912yes—
572C2×C4#1451974912yes—
573C2×C4C22×C22⋊D6111yescharacteristic
574C2×C4#41×D6111yescharacteristic
575C2×C4#41×D6111yescharacteristic
576C2×C4C2×#1175822411yescharacteristic
577C2×C4#1451982311yescharacteristic
578C2×C4#1451982411yescharacteristic
579C8—41no—
580C8—41no—
581C8—2441no—
582C8—2441no—
583C22C2×#1451974911yescharacteristic
584C22#1684040511yescharacteristic
585C22#1684042011yescharacteristic
586C22#137×D6111yescharacteristic
587C22#1684056111yescharacteristic
588C22#1684057811yescharacteristic
589C22#1684057811yescharacteristic
590C4—22no—
591C4—22no—
592C4—42no—
593C4—42no—
594C4—81no—
595C4—81no—
596C4—1222no—
597C4—4881no—
598C4—4881no—
599C4—2442no—
600C4—2441no—
601C4—2441no—
602C4—2441no—
603C4—2442no—
604C4#1684029111yescharacteristic
605C4#1684038111yescharacteristic
606C4#1684040011yescharacteristic
607C4#1684041111yescharacteristic
608C2#2144939911yescharacteristic
609C2#2145528811yescharacteristic
610C2#2145535111yescharacteristic
611C2#2145786911yescharacteristic
612C2#2145791411yescharacteristic
613C2#2145861911yescharacteristic
614C2#2145865611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#491C2
socle, prime radicalC23×C61C22×D4
hypercenterC2×#491D61
hypocenterC61#25333
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#491

Complements

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

NH ≅ G/Ntype
368. C61#25333semidirect

Maximal subgroups

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

subgroupindex
C2×#168399692
#214510112
#214488792
#214472402
#214472302
#214471402
#214467772
#214465422
#214461582
#214458322
#214458222
#214456502
C2×C61×#4912
#2533361