Finite Group Databasebeta

← #29648113

Subgroups — #29648113

Order 15616.

952 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
1#533×D61C211yesmaximal, characteristic
2#21454449C211yesmaximal, characteristic
3#432×D61C211yesmaximal, characteristic
4#21444457C211yesmaximal, characteristic, complemented
5#21444456C211yesmaximal, characteristic, complemented
6#21444456C211yesmaximal, characteristic, complemented
7#21444432C214yesmaximal, complemented
8#21444419C214yesmaximal, complemented
9C61×#1537C211yesmaximal, characteristic, complemented
10#16840319C2211yescharacteristic
11#16840318C2212yes—
12#16840317C411yescharacteristic
13#16840317C411yescharacteristic
14#16840316—21no—
15#16840316—21no—
16#16840315C412yes—
17#16840314—21no—
18#16840314—21no—
19C4⋊Q8×D61C411yescharacteristic
20C4⋊Q8×D61C411yescharacteristic
21#125×D61C2212yes—
22C22⋊Q8×D61—21no—
23C22⋊Q8×D61—21no—
24C4×Q8×D61—21no—
25C4×Q8×D61—21no—
26#117×D61C412yes—
27C2×C42×D61C2211yescharacteristic
28#16839819C2211yescharacteristic
29#16839819C2211yescharacteristic
30#16839818—21no—
31#16839817C2212yes—
32#16839817C2212yes—
33#16839816—21no—
34#16839814—21no—
35#16839812—21no—
36#16839784—21no—
37C4⋊C8×D61—21no—
38#16839724—21no—
39C22⋊C8×D61—21no—
40#16839714C2212yes—
41#97×D61C2212yes—
42#16839504C2214yescomplemented
43#16839503C2214yescomplemented
44#16839503C2214yescomplemented
45C61×#533C2211yescharacteristic
46C61×#432C2211yescharacteristic
47C61×#432C2211yescharacteristic
48C61×#330C2214yescomplemented
49C2×Q8×D61—21no—
50C2×Q8×D61—21no—
51C22×C4×D61C2×C411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61D411yescharacteristic
54C2×#11758189—22no—
55C2×C8×D61—22no—
56#14519777C2×C411yescharacteristic
57#14519777C2×C411yescharacteristic
58#14519776—21no—
59#14519776—21no—
60#14519776C2×C412yes—
61#14519776C2×C412yes—
62#14519775—21no—
63#14519775—21no—
64#14519769C2312yes—
65#14519768C2×C412yes—
66#14519768C2×C412yes—
67#14519767—22no—
68#14519767—22no—
69#14519757—21no—
70#14519757—21no—
71#14519756—21no—
72#14519756—21no—
73#14519755—22no—
74#14519755—22no—
75#14519754—22no—
76#14519754—22no—
77#14519753—21no—
78#14519753—21no—
79Q8×Dic61—21no—
80Q8×Dic61—21no—
81C4×Dic122—21no—
82C4×Dic122—21no—
83C4×Dic122—22no—
84C4×Dic122—22no—
85#14519742—21no—
86#14519742—21no—
87#14519742—22no—
88#14519742—22no—
89#14519736C2×C412yes—
90#14519735C2×C412yes—
91#14519735C2×C412yes—
92#14519734—22no—
93#14519733—22no—
94C4⋊C4×D61—21no—
95C4⋊C4×D61—21no—
96C4⋊C4×D61—22no—
97C4⋊C4×D61—22no—
98C4⋊C4×D61C2×C412yes—
99C4⋊C4×D61C2×C412yes—
100C22⋊C4×D61—22no—
101C2×C4×Dic61C2×C411yescharacteristic
102C2×C4×Dic61D411yescharacteristic
103C2×C4×Dic61D411yescharacteristic
104C42×D61—22no—
105C42×D61C2×C411yescharacteristic
106C42×D61C2×C411yescharacteristic
107C42×D61D411yescharacteristic
108C42×D61D411yescharacteristic
109C42×D61D411yescharacteristic
110C42×D61D411yescharacteristic
111C42×D61C2311yescharacteristic
112C42×D61C2311yescharacteristic
113C4⋊#92418—21no—
114C4⋊#92418—21no—
115#14519706—22no—
116#14519706—22no—
117C22⋊#92418—21no—
118C22⋊#92418—21no—
119#14519689—22no—
120#14519689—22no—
121#14519688C2312yes—
122#14519688C2312yes—
123#14519687—22no—
124C8×Dic61—22no—
125C4⋊Q8×C61C2×C411yescharacteristic
126C4⋊Q8×C61C2×C411yescharacteristic
127#125×C61C2312yes—
128C22⋊Q8×C61—21no—
129C22⋊Q8×C61—21no—
130C4×Q8×C61—21no—
131C4×Q8×C61—21no—
132#117×C61C2×C412yes—
133C2×C42×C61C2311yescharacteristic
134C4⋊C8×C61—21no—
135C4⋊C8×C61—21no—
136C22⋊C8×C61—21no—
137C22⋊C8×C61—21no—
138#97×C61C2312yes—
139#97×C61C2312yes—
140C23×D61C22⋊C411yescharacteristic
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C22×C4×C61C22×C411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×Dic61C22⋊C411yescharacteristic
147C22×Dic61C22⋊C411yescharacteristic
148C22×Dic61C22⋊C411yescharacteristic
149Q8×D61—44no—
150Q8×D61—44no—
151C2×C4×D61—24no—
152C2×C4×D61—22no—
153C2×C4×D61—22no—
154C2×C4×D61—42no—
155C2×C4×D61—42no—
156C2×C4×D61C22⋊C411yescharacteristic
157C2×C4×D61C22⋊C411yescharacteristic
158C2×C4×D61C22⋊C411yescharacteristic
159C2×C4×D61C22⋊C411yescharacteristic
160C2×C4×D61C22⋊C411yescharacteristic
161C2×C4×D61C22⋊C411yescharacteristic
162C2×C4×D61C22×C411yescharacteristic
163C2×C4×D61C22×C411yescharacteristic
164C2×C4×D61C2×D411yescharacteristic
165C2×C4×D61C2×D411yescharacteristic
166C2×C4×D61C2×D411yescharacteristic
167C2×C4×D61C2×D411yescharacteristic
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—22no—
171C2×Dic122—22no—
172C2×C8×C61—22no—
173C2×C8×C61—22no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61C22×C412yes—
179C4⋊C4×C61C22×C412yes—
180C22⋊C4×C61—22no—
181C42×C61—22no—
182C42×C61C22×C411yescharacteristic
183C42×C61C2×D411yescharacteristic
184C42×C61C2×D411yescharacteristic
185C42×C61C2411yescharacteristic
186C22⋊Dic61—22no—
187#11758198—22no—
188#11758198—22no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—22no—
192C4⋊Dic61—22no—
193C4⋊Dic61C22×C412yes—
194C4⋊Dic61C22×C412yes—
195#11758196—22no—
196#11758196—22no—
197#11758196—22no—
198#11758196—22no—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203C4×Dic61—22no—
204C4×Dic61—22no—
205C4×Dic61—22no—
206C4×Dic61C22⋊C411yescharacteristic
207C4×Dic61C22⋊C411yescharacteristic
208C4×Dic61C22⋊C411yescharacteristic
209C4×Dic61C22⋊C411yescharacteristic
210C4×Dic61C22⋊C411yescharacteristic
211C4×Dic61C22⋊C411yescharacteristic
212C4×Dic61C22×C411yescharacteristic
213C4×Dic61C22×C411yescharacteristic
214C4×Dic61C2×D411yescharacteristic
215C4×Dic61C2×D411yescharacteristic
216C4×Dic61C2×D411yescharacteristic
217C4×Dic61C2×D411yescharacteristic
218C2×#92418—22no—
219C2×#92418—22no—
220#11758189—44no—
221C8×D61—44no—
222C23×C61C2×C22⋊C411yescharacteristic
223C22×D61—22no—
224C22×D61—22no—
225C22×D61—41no—
226C22×D61C2×C22⋊C411yescharacteristic
227C22×D61C2×C22⋊C411yescharacteristic
228Q8×C61—42no—
229Q8×C61—42no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61C2×C22⋊C411yescharacteristic
236C2×C4×C61C2×C22⋊C411yescharacteristic
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61C23×C411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—41no—
246C2×Dic61C2×C22⋊C411yescharacteristic
247C2×Dic61C2×C22⋊C411yescharacteristic
248C2×Dic61C2×C22⋊C411yescharacteristic
249C2×Dic61C2×C22⋊C411yescharacteristic
250C2×Dic61C2×C22⋊C411yescharacteristic
251C2×Dic61C2×C22⋊C411yescharacteristic
252C4×D61—22no—
253C4×D61—22no—
254C4×D61—22no—
255C4×D61—22no—
256C4×D61—22no—
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—41no—
260C4×D61—41no—
261C4×D61—41no—
262C4×D61—41no—
263C4×D61—44no—
264C4×D61—44no—
265C4×D61#10114yes—
266Dic122—44no—
267Dic122—42no—
268Dic122—42no—
269Dic122—44no—
270C8×C61—42no—
271C8×C61—42no—
272#92418—42no—
273#92418—42no—
274#17672—611nomaximal, Sylow
275C22×C61—21no—
276C22×C61—21no—
277C22×C61—21no—
278C22×C61C22×C22⋊C411yescharacteristic
279C2×D61—21no—
280C2×D61—21no—
281C2×D61—21no—
282C2×D61—21no—
283C2×D61—44no—
284C2×D61C2×#10112yes—
285C2×D61#42112yes—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—21no—
289C4×C61—21no—
290C4×C61—21no—
291C4×C61—42no—
292C4×C61—42no—
293C4×C61C2×#10112yes—
294Dic61—21no—
295Dic61—21no—
296Dic61—21no—
297Dic61—21no—
298Dic61—42no—
299Dic61—42no—
300Dic61C2×#10112yes—
301Dic61#42112yes—
302C2×#533—611no—
303#2920—611no—
304C2×#432—611no—
305#1537—611no—
306#1537—611no—
307#1537—611no—
308#1537D6111yescharacteristic, complemented
309#1531—614no—
310#1528—614no—
311C2×C61—41no—
312C2×C61C22×#10111yescharacteristic
313C2×C61C2×#42111yescharacteristic
314C2×C61C2×#42111yescharacteristic
315D61—22no—
316D61—22no—
317#533—611no—
318#533—611no—
319#533—612no—
320#533—612no—
321#533—611no—
322#533—1221no—
323#533—1221no—
324#533—1221no—
325#533—1221no—
326#533C2×D6111yescharacteristic
327C2×C4⋊Q8—611no—
328C2×C4⋊Q8—611no—
329C2×#125—612no—
330C2×C22⋊Q8—1221no—
331C2×C22⋊Q8—1221no—
332C2×C4×Q8—1221no—
333C2×C4×Q8—1221no—
334C2×#117—612no—
335C22×C42—611no—
336#432—611no—
337#432—611no—
338#432—612no—
339#432—612no—
340#432—1221no—
341#432—1221no—
342#432C2×D6111yescharacteristic
343#432C2×D6111yescharacteristic
344#431—1221no—
345#431—1221no—
346#423—1221no—
347C2×C4⋊C8—1221no—
348#407—1221no—
349C2×C22⋊C8—1221no—
350C4×M16—612no—
351C2×#97—612no—
352#330—614no—
353#330—614no—
354#330—614no—
355#330C2×D6114yescomplemented
356C61#1767211yescharacteristic, complemented, Sylow
357C22×Q8—1221no—
358C22×Q8—1221no—
359C23×C4—611no—
360C23×C4—611no—
361C23×C4—611no—
362C2×M16—1222no—
363C22×C8—1222no—
364C4⋊Q8—612no—
365C4⋊Q8—612no—
366C4⋊Q8—611no—
367C4⋊Q8—611no—
368C4⋊Q8—1221no—
369C4⋊Q8—1221no—
370C4⋊Q8—1221no—
371C4⋊Q8—1221no—
372C4⋊Q8C4×D6111yescharacteristic
373C4⋊Q8C4×D6111yescharacteristic
374#125—612no—
375#125—612no—
376#125—612no—
377#125—1222no—
378#125—1222no—
379#125C22×D6112yes—
380C22⋊Q8—21no—
381C22⋊Q8—21no—
382C22⋊Q8—1222no—
383C22⋊Q8—1222no—
384C22⋊Q8—1222no—
385C22⋊Q8—1222no—
386C22⋊Q8—1221no—
387C22⋊Q8—1221no—
388C22⋊Q8—1221no—
389C22⋊Q8—1221no—
390C22⋊Q8—1221no—
391C22⋊Q8—1221no—
392C4×Q8—21no—
393C4×Q8—21no—
394C4×Q8—1222no—
395C4×Q8—1222no—
396C4×Q8—1222no—
397C4×Q8—1222no—
398C4×Q8—1221no—
399C4×Q8—1221no—
400C4×Q8—1221no—
401C4×Q8—1221no—
402C4×Q8—1221no—
403C4×Q8—1221no—
404#117—612no—
405#117—612no—
406#117—612no—
407#117—1222no—
408#117—1222no—
409#117C4×D6112yes—
410C2×C4⋊C4—612no—
411C2×C4⋊C4—612no—
412C2×C4⋊C4—1222no—
413C2×C4⋊C4—1222no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C22⋊C4—1222no—
417C2×C42—611no—
418C2×C42—611no—
419C2×C42—611no—
420C2×C42—611no—
421C2×C42—611no—
422C2×C42—611no—
423C2×C42—611no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—611no—
428C2×C42—1222no—
429C2×C42C22×D6111yescharacteristic
430C4⋊C8—21no—
431C4⋊C8—21no—
432C4⋊C8—1222no—
433C4⋊C8—1222no—
434C4⋊C8—1221no—
435C4⋊C8—1221no—
436C22⋊C8—21no—
437C22⋊C8—21no—
438C22⋊C8—1222no—
439C22⋊C8—1222no—
440C22⋊C8—1221no—
441C22⋊C8—1221no—
442#97—612no—
443#97—612no—
444#97—1222no—
445#97C22×D6112yes—
446#97C22×D6112yes—
447C4×C8—1222no—
448C24—611no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—1222no—
452C2×Q8—1222no—
453C2×Q8—1221no—
454C2×Q8—1221no—
455C2×Q8—2444no—
456C2×Q8—2444no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—1224no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—2442no—
476C22×C4—2442no—
477C22×C4C2×C4×D6111yescharacteristic
478C22×C4D4×D6111yescharacteristic
479C22×C4D4×D6111yescharacteristic
480M16—2444no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—1222no—
484C2×C8—1222no—
485C2×C8—2444no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—612no—
491C4⋊C4—612no—
492C4⋊C4—1222no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C4⋊C4—1222no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4C2×C4×D6112yes—
505C4⋊C4C2×C4×D6112yes—
506C22⋊C4—22no—
507C22⋊C4—1222no—
508C22⋊C4—1222no—
509C22⋊C4—1222no—
510C42—22no—
511C42—611no—
512C42—611no—
513C42—611no—
514C42—611no—
515C42—611no—
516C42—611no—
517C42—611no—
518C42—611no—
519C42—611no—
520C42—611no—
521C42—611no—
522C42—611no—
523C42—1222no—
524C42—1222no—
525C42—1222no—
526C42C2×C4×D6111yescharacteristic
527C42D4×D6111yescharacteristic
528C42D4×D6111yescharacteristic
529C42C23×D6111yescharacteristic
530C23—611no—
531C23—611no—
532C23—1222no—
533C23—1222no—
534C23—2441no—
535C23C22⋊C4×D6111yescharacteristic
536Q8—42no—
537Q8—42no—
538Q8—2442no—
539Q8—2442no—
540Q8—2444no—
541Q8—2444no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—611no—
548C2×C4—611no—
549C2×C4—614no—
550C2×C4—611no—
551C2×C4—611no—
552C2×C4—611no—
553C2×C4—611no—
554C2×C4—1222no—
555C2×C4—1222no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2444no—
570C2×C4—2444no—
571C2×C4—2441no—
572C2×C4C22⋊C4×D6111yescharacteristic
573C2×C4C22⋊C4×D6111yescharacteristic
574C2×C4C22⋊C4×D6111yescharacteristic
575C2×C4C22×C4×D6111yescharacteristic
576C2×C4C2×D4×D6111yescharacteristic
577C2×C4C2×D4×D6111yescharacteristic
578C8—42no—
579C8—42no—
580C8—2442no—
581C8—2442no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—612no—
586C22—612no—
587C22—1221no—
588C22—1221no—
589C22—1221no—
590C22—1221no—
591C22—2444no—
592C22C2×C22⋊C4×D6111yescharacteristic
593C4—21no—
594C4—21no—
595C4—21no—
596C4—21no—
597C4—21no—
598C4—42no—
599C4—42no—
600C4—612no—
601C4—612no—
602C4—1221no—
603C4—1221no—
604C4—1221no—
605C4—1221no—
606C4—2442no—
607C4—2442no—
608C4#101×D6112yes—
609C2—41no—
610C2—1222no—
611C2—1222no—
612C2C2×#101×D6111yescharacteristic
613C2#421×D6111yescharacteristic
614C2#421×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1537C2
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC42×C61C24
hypercenter#1537D61
hypocenterC61#17672
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1537

Complements

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

NH ≅ G/Ntype
4. #21444457C2semidirect
5. #21444456C2semidirect
6. #21444456C2semidirect
7. #21444432C2semidirect
8. #21444419C2semidirect
9. C61×#1537C2semidirect
42. #16839504C22semidirect
43. #16839503C22semidirect
44. #16839503C22semidirect
48. C61×#330C22semidirect
308. #1537D61semidirect
355. #330C2×D61semidirect
356. C61#17672semidirect

Maximal subgroups

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

subgroupindex
#533×D612
#214544492
#432×D612
#214444572
#214444562
#214444322
#214444192
C61×#15372
#1767261