Finite Group Databasebeta

← #29648056

Subgroups — #29648056

Order 15616.

974 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459114C211yesmaximal, characteristic
2#21454449C211yesmaximal, characteristic, complemented
3#432×D61C211yesmaximal, characteristic, complemented
4#21444456C211yesmaximal, characteristic, complemented
5#21444456C211yesmaximal, characteristic, complemented
6#21444454C211yesmaximal, characteristic, complemented
7#21444447C214yesmaximal, complemented
8#21444404C214yesmaximal, complemented
9C61×#1536C211yesmaximal, characteristic, complemented
10#16840319C2211yescharacteristic
11#16840317C411yescharacteristic
12#16840317C411yescharacteristic
13#16840310C411yescharacteristic
14#16840310C411yescharacteristic
15#16840294—21no—
16#16840294—21no—
17#16840293—21no—
18#16840293—21no—
19#16840291C2212yes—
20#16840291C2212yes—
21#16840272—21no—
22#16840272—21no—
23#16840269C412yes—
24#41×Dic61C412yes—
25C4×#11758224—21no—
26C4×#11758224—21no—
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#16839503C2214yescomplemented
43#16839503C2214yescomplemented
44#16839502C2214yescomplemented
45C61×#532C2211yescharacteristic
46C61×#432C2211yescharacteristic
47C61×#432C2211yescharacteristic
48C61×#329C2214yescomplemented
49C2×#11758224—21no—
50C2×#11758224—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#14519770—22no—
59#14519770—22no—
60#14519769C2312yes—
61#14519768—21no—
62#14519768—21no—
63#14519764—21no—
64#14519764—21no—
65#14519759—22no—
66#14519759—22no—
67#14519758—22no—
68#14519757—21no—
69#14519757—21no—
70#14519756—21no—
71#14519756—21no—
72#14519753C2×C412yes—
73#14519753C2×C412yes—
74#14519748—21no—
75#14519748—21no—
76#14519748C2×C412yes—
77#14519748C2×C412yes—
78#14519745—22no—
79Q8×Dic61C2×C412yes—
80Q8×Dic61C2×C412yes—
81C4×Dic122—21no—
82C4×Dic122—21no—
83#14519742—21no—
84#14519742—21no—
85D4×Dic61—22no—
86D4×Dic61—22no—
87D4×Dic61—21no—
88D4×Dic61—21no—
89D4×Dic61C2×C412yes—
90D4×Dic61C2×C412yes—
91C4×C22⋊D61—22no—
92C4×C22⋊D61—22no—
93#14519737—22no—
94#14519737—22no—
95#14519736—22no—
96#14519736C2×C412yes—
97#14519735—21no—
98#14519735—21no—
99#14519733—22no—
100#14519733—22no—
101C2×C4×Dic61—22no—
102C2×C4×Dic61C2×C411yescharacteristic
103C2×C4×Dic61D411yescharacteristic
104C2×C4×Dic61D411yescharacteristic
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—
125C2×#41×C61C2×C412yes—
126C4⋊Q8×C61C2×C411yescharacteristic
127#127×C61C2×C411yescharacteristic
128#124×C61C2312yes—
129#121×C61—21no—
130#121×C61—21no—
131C4×D4×C61—21no—
132C4×D4×C61—21no—
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
141#41×C61—44no—
142C2×Q8×C61C22×C412yes—
143C2×D4×C61—22no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61C22×C412yes—
147C22×C4×C61—22no—
148C22×C4×C61C22×C411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C2×C22⋊D61—22no—
152C2×C22⋊D61—22no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61C22⋊C411yescharacteristic
156C22×Dic61C22⋊C411yescharacteristic
157C22×Dic61C22⋊C411yescharacteristic
158#11758224—44no—
159#11758224—44no—
160C2×C4×D61—42no—
161C2×C4×D61—42no—
162C2×C4×D61—42no—
163C2×C4×D61C22⋊C411yescharacteristic
164C2×C4×D61C22⋊C411yescharacteristic
165C2×C4×D61C22⋊C411yescharacteristic
166C2×C4×D61C22⋊C411yescharacteristic
167C2×C4×D61C22⋊C411yescharacteristic
168C2×C4×D61C22⋊C411yescharacteristic
169C2×C4×D61C22×C411yescharacteristic
170C2×C4×D61C22×C411yescharacteristic
171C2×C4×D61C2×D411yescharacteristic
172C2×C4×D61C2×D411yescharacteristic
173C2×C4×D61C2×D411yescharacteristic
174C2×C4×D61C2×D411yescharacteristic
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×C8×C61—22no—
178C2×C8×C61—22no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—22no—
183C42×C61C22×C411yescharacteristic
184C42×C61C2×D411yescharacteristic
185C42×C61C2×D411yescharacteristic
186C42×C61C2411yescharacteristic
187C22⋊Dic61—22no—
188C22⋊Dic61—22no—
189C22⋊Dic61—22no—
190C22⋊Dic61—22no—
191C22⋊Dic61—22no—
192#11758198—22no—
193#11758198—22no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—22no—
197C4⋊Dic61—22no—
198C4⋊Dic61C22×C412yes—
199C4⋊Dic61C22×C412yes—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204C4×Dic61—22no—
205C4×Dic61—22no—
206C4×Dic61—22no—
207C4×Dic61C22⋊C411yescharacteristic
208C4×Dic61C22⋊C411yescharacteristic
209C4×Dic61C22⋊C411yescharacteristic
210C4×Dic61C22⋊C411yescharacteristic
211C4×Dic61C22⋊C411yescharacteristic
212C4×Dic61C22⋊C411yescharacteristic
213C4×Dic61C22×C411yescharacteristic
214C4×Dic61C22×C411yescharacteristic
215C4×Dic61C2×D411yescharacteristic
216C4×Dic61C2×D411yescharacteristic
217C4×Dic61C2×D411yescharacteristic
218C4×Dic61C2×D411yescharacteristic
219C2×#92418—22no—
220C2×#92418—22no—
221#11758189—44no—
222C8×D61—44no—
223C23×C61—22no—
224C23×C61C2×C22⋊C411yescharacteristic
225C22×D61—22no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61C2×C22⋊C411yescharacteristic
229C22×D61C2×C22⋊C411yescharacteristic
230Q8×C61—42no—
231D4×C61—42no—
232D4×C61—42no—
233D4×C61—42no—
234D4×C61—44no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—44no—
240C2×C4×C61C2×C22⋊C411yescharacteristic
241C2×C4×C61C2×C22⋊C411yescharacteristic
242C2×C4×C61C2×C22⋊C411yescharacteristic
243C2×C4×C61C23×C411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C22⋊D61—44no—
247C22⋊D61—44no—
248C2×Dic61—22no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—44no—
254C2×Dic61—44no—
255C2×Dic61C2×C22⋊C411yescharacteristic
256C2×Dic61C2×C22⋊C411yescharacteristic
257C2×Dic61C2×C22⋊C411yescharacteristic
258C2×Dic61C2×C22⋊C411yescharacteristic
259C2×Dic61C2×C22⋊C411yescharacteristic
260C2×Dic61C2×C22⋊C411yescharacteristic
261C4×D61—22no—
262C4×D61—22no—
263C4×D61—22no—
264C4×D61—22no—
265C4×D61—22no—
266C4×D61—22no—
267C4×D61—41no—
268C4×D61—41no—
269C4×D61—41no—
270C4×D61—41no—
271C4×D61—41no—
272C4×D61—41no—
273Dic122—42no—
274Dic122—42no—
275C8×C61—42no—
276C8×C61—42no—
277#92418—42no—
278#92418—42no—
279#17662—611nomaximal, Sylow
280C22×C61—21no—
281C22×C61—21no—
282C22×C61—21no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61C22×C22⋊C411yescharacteristic
287C2×D61—21no—
288C2×D61—21no—
289C2×D61—21no—
290C2×D61—21no—
291C2×D61—21no—
292C2×D61—44no—
293C2×D61#42112yes—
294C4×C61—21no—
295C4×C61—21no—
296C4×C61—21no—
297C4×C61—21no—
298C4×C61—21no—
299C4×C61—42no—
300C4×C61#41312yes—
301Dic61—21no—
302Dic61—21no—
303Dic61—21no—
304Dic61—21no—
305Dic61—21no—
306Dic61—42no—
307Dic61—42no—
308Dic61#42112yes—
309#3444—611no—
310#2920—611no—
311C2×#432—611no—
312#1537—611no—
313#1537—611no—
314#1536—611no—
315#1536D6111yescharacteristic, complemented
316#1535—614no—
317#1524—614no—
318C2×C61—41no—
319C2×C61—82no—
320C2×C61C2×#41311yescharacteristic
321C2×C61C2×#42111yescharacteristic
322C2×C61C2×#42111yescharacteristic
323D61—22no—
324D61—41no—
325#533—611no—
326#533—611no—
327#533—611no—
328#532—611no—
329#532—611no—
330#532C2×D6111yescharacteristic
331#529—612no—
332#529—612no—
333#529—1221no—
334#529—1221no—
335#529—1221no—
336#529—1221no—
337#525—612no—
338#525—1221no—
339#525—1221no—
340C4×#41—612no—
341C4×#41—1221no—
342C4×#41—1221no—
343C22×C42—611no—
344#432—611no—
345#432—611no—
346#432—612no—
347#432—612no—
348#432—1221no—
349#432—1221no—
350#432C2×D6111yescharacteristic
351#432C2×D6111yescharacteristic
352#431—1221no—
353#431—1221no—
354#423—1221no—
355C2×C4⋊C8—1221no—
356#407—1221no—
357C2×C22⋊C8—1221no—
358C4×M16—612no—
359C2×#97—612no—
360#330—614no—
361#330—614no—
362#329—614no—
363#329C2×D6114yescomplemented
364C61#1766211yescharacteristic, complemented, Sylow
365C2×#41—1221no—
366C2×#41—1221no—
367C2×#41C4×D6112yes—
368C23×C4—611no—
369C23×C4—611no—
370C23×C4—611no—
371C2×M16—1222no—
372C22×C8—1222no—
373C4⋊Q8—611no—
374C4⋊Q8—611no—
375C4⋊Q8C4×D6111yescharacteristic
376#127C4×D6111yescharacteristic
377#126—1222no—
378#126—1222no—
379#125—612no—
380#125—1221no—
381#125—1221no—
382#124—1221no—
383#124—1221no—
384#124C22×D6112yes—
385#123—1222no—
386#123—1222no—
387#123—1222no—
388C22⋊Q8—612no—
389C22⋊Q8—612no—
390C22⋊Q8—1221no—
391C22⋊Q8—1221no—
392C22⋊Q8—1221no—
393C22⋊Q8—1221no—
394#121—21no—
395#121—21no—
396#121—612no—
397#121—612no—
398#121—1221no—
399#121—1221no—
400#120—1222no—
401C4×Q8—612no—
402C4×Q8—612no—
403C4×Q8—1221no—
404C4×Q8—1221no—
405C4×Q8—1221no—
406C4×Q8—1221no—
407C4×D4—21no—
408C4×D4—21no—
409C4×D4—612no—
410C4×D4—612no—
411C4×D4—1221no—
412C4×D4—1221no—
413C4×D4—1222no—
414C4×D4—1222no—
415C4×D4—1222no—
416C4×D4—1222no—
417C4×D4—1222no—
418C4×D4—1222no—
419#117—612no—
420#117—1222no—
421#117—1222no—
422#117—1222no—
423#117—1221no—
424#117—1221no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—611no—
428C2×C42—611no—
429C2×C42—611no—
430C2×C42—611no—
431C2×C42—611no—
432C2×C42—611no—
433C2×C42—611no—
434C2×C42—611no—
435C2×C42—611no—
436C2×C42—1222no—
437C2×C42C22×D6111yescharacteristic
438C4⋊C8—21no—
439C4⋊C8—21no—
440C4⋊C8—1222no—
441C4⋊C8—1222no—
442C4⋊C8—1221no—
443C4⋊C8—1221no—
444C22⋊C8—21no—
445C22⋊C8—21no—
446C22⋊C8—1222no—
447C22⋊C8—1222no—
448C22⋊C8—1221no—
449C22⋊C8—1221no—
450#97—612no—
451#97—612no—
452#97—1222no—
453#97C22×D6112yes—
454#97C22×D6112yes—
455C4×C8—1222no—
456C24—611no—
457#41—44no—
458#41—2444no—
459#41—2444no—
460C2×Q8—1221no—
461C2×Q8—1221no—
462C2×Q8C2×C4×D6112yes—
463C2×D4—22no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—1222no—
467C2×D4—1222no—
468C2×D4C2×C4×D6112yes—
469C22×C4—22no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—2442no—
488C22×C4—2442no—
489C22×C4—2442no—
490C22×C4C2×C4×D6111yescharacteristic
491C22×C4D4×D6111yescharacteristic
492C22×C4D4×D6111yescharacteristic
493M16—2444no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—1222no—
497C2×C8—1222no—
498C2×C8—2444no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—612no—
502C4⋊C4—612no—
503C4⋊C4—1222no—
504C4⋊C4—1222no—
505C4⋊C4—1222no—
506C4⋊C4—1222no—
507C4⋊C4—1222no—
508C4⋊C4—1222no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—1222no—
514C22⋊C4—1222no—
515C22⋊C4—1222no—
516C22⋊C4—1222no—
517C22⋊C4—1222no—
518C22⋊C4—1222no—
519C22⋊C4—1222no—
520C42—611no—
521C42—611no—
522C42—611no—
523C42—611no—
524C42—611no—
525C42—611no—
526C42—611no—
527C42—611no—
528C42—611no—
529C42—611no—
530C42—611no—
531C42—611no—
532C42—1222no—
533C42—1222no—
534C42—1222no—
535C42C2×C4×D6111yescharacteristic
536C42D4×D6111yescharacteristic
537C42D4×D6111yescharacteristic
538C42C23×D6111yescharacteristic
539C23—22no—
540C23—611no—
541C23—611no—
542C23—1222no—
543C23—2441no—
544C23—2441no—
545C23C22⋊C4×D6111yescharacteristic
546Q8—42no—
547Q8—2442no—
548Q8—2442no—
549D4—42no—
550D4—42no—
551D4—42no—
552D4—44no—
553D4—2444no—
554D4—2444no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—44no—
560C2×C4—611no—
561C2×C4—611no—
562C2×C4—611no—
563C2×C4—611no—
564C2×C4—611no—
565C2×C4—611no—
566C2×C4—1222no—
567C2×C4—1222no—
568C2×C4—1222no—
569C2×C4—1222no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2444no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2444no—
585C2×C4C22⋊C4×D6111yescharacteristic
586C2×C4C22⋊C4×D6111yescharacteristic
587C2×C4C22⋊C4×D6111yescharacteristic
588C2×C4C22×C4×D6111yescharacteristic
589C2×C4C2×D4×D6111yescharacteristic
590C2×C4C2×D4×D6111yescharacteristic
591C8—42no—
592C8—42no—
593C8—2442no—
594C8—2442no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—612no—
602C22—1221no—
603C22—1221no—
604C22—1221no—
605C22—1221no—
606C22—1221no—
607C22—2444no—
608C22C2×C22⋊C4×D6111yescharacteristic
609C4—21no—
610C4—21no—
611C4—21no—
612C4—21no—
613C4—21no—
614C4—42no—
615C4—612no—
616C4—1221no—
617C4—1221no—
618C4—1221no—
619C4—1221no—
620C4—1221no—
621C4—2442no—
622C4—2442no—
623C4#1683974312yes—
624C2—41no—
625C2—82no—
626C2—1222no—
627C2—2441no—
628C2C2×#1683974311yescharacteristic
629C2#421×D6111yescharacteristic
630C2#421×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1536C2
socleC22×C61C22×C22⋊C4
prime radical#21459114C2
cosocleC42×C61C24
hypercenter#1536D61
hypocenterC61#17662
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1536

Complements

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

NH ≅ G/Ntype
2. #21454449C2semidirect
3. #432×D61C2semidirect
4. #21444456C2semidirect
5. #21444456C2semidirect
6. #21444454C2semidirect
7. #21444447C2semidirect
8. #21444404C2semidirect
9. C61×#1536C2semidirect
42. #16839503C22semidirect
43. #16839503C22semidirect
44. #16839502C22semidirect
48. C61×#329C22semidirect
315. #1536D61semidirect
363. #329C2×D61semidirect
364. C61#17662semidirect

Maximal subgroups

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

subgroupindex
#214591142
#214544492
#432×D612
#214444562
#214444542
#214444472
#214444042
C61×#15362
#1766261