Finite Group Databasebeta

← #30617899

Subgroups — #30617899

Order 15616.

1074 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#431C211yesmaximal, characteristic
2#21462268C211yesmaximal, characteristic
3#21462000C212yesmaximal, complemented
4#21462000C212yesmaximal, complemented
5#21461999C214yesmaximal, complemented
6#21461724C214yesmaximal
7#533×D61C211yesmaximal, characteristic
8C2×#14519856—21no—
9#16840781—21no—
10#16840780—21no—
11#16840779—21no—
12#16840778—21no—
13D61⋊(C4⋊C8)C2212yes—
14D61⋊(C22⋊C8)C2212yes—
15C4×D61⋊C8C2212yes—
16#16840708C2212yes—
17#16840706—21no—
18#16840704—21no—
19#16840704C2212yes—
20C4×C4⋊#6923—21no—
21C4×C4⋊#6923C2212yes—
22#16840635C2218yes—
23#16840635C2218yes—
24#16840319C411yescharacteristic, complemented
25#16840318C412yescomplemented
26#16840317C411yescharacteristic, complemented
27#16840317C411yescharacteristic, complemented
28#16840316—21no—
29#16840316—21no—
30#16840315C412yescomplemented
31#16840314—21no—
32#16840314—21no—
33C4⋊Q8×D61C2211yescharacteristic
34C4⋊Q8×D61C2211yescharacteristic
35#125×D61C2212yes—
36C22⋊Q8×D61—21no—
37C22⋊Q8×D61—21no—
38C4×Q8×D61—21no—
39C4×Q8×D61—21no—
40#117×D61C2212yes—
41C2×C42×D61C2211yescharacteristic
42C61×#533C411yescharacteristic, complemented
43C2×D61⋊C8C2314yes—
44#14519856—44no—
45C2×C4⋊#6923—22no—
46C2×C4⋊#6923—22no—
47C2×C4⋊#6923C2314yes—
48C2×C22⋊#6923—22no—
49C2×C4×#6923—22no—
50#14519844—22no—
51C4⋊#92420—22no—
52#14519834—22no—
53C22⋊#92420—22no—
54C4×#92420—22no—
55C4×#92420—22no—
56(C2×#6923)⋊C4—22no—
57(C2×#6923)⋊C4—22no—
58C2×Q8×D61—21no—
59C2×Q8×D61—21no—
60C22×C4×D61D411yescharacteristic
61C22×C4×D61D411yescharacteristic
62C22×C4×D61C2311yescharacteristic
63#14519777C2×C411yescharacteristic
64#14519777C2×C411yescharacteristic
65#14519776—21no—
66#14519776—21no—
67#14519776C2×C412yes—
68#14519776C2×C412yes—
69#14519775—21no—
70#14519775—21no—
71#14519769C2×C412yes—
72#14519768C2×C412yes—
73#14519768C2×C412yes—
74#14519767—22no—
75#14519767—22no—
76#14519757—21no—
77#14519757—21no—
78#14519756—21no—
79#14519756—21no—
80#14519755—22no—
81#14519755—22no—
82#14519754—22no—
83#14519754—22no—
84#14519753—21no—
85#14519753—21no—
86Q8×Dic61—21no—
87Q8×Dic61—21no—
88C4×Dic122—21no—
89C4×Dic122—21no—
90C4×Dic122—22no—
91C4×Dic122—22no—
92#14519742—21no—
93#14519742—21no—
94#14519742—22no—
95#14519742—22no—
96#14519736C2×C412yes—
97#14519735C2×C412yes—
98#14519735C2×C412yes—
99#14519734—22no—
100#14519733—22no—
101C4⋊C4×D61—22no—
102C4⋊C4×D61—22no—
103C4⋊C4×D61—21no—
104C4⋊C4×D61—21no—
105C4⋊C4×D61C2312yes—
106C4⋊C4×D61C2312yes—
107C22⋊C4×D61—22no—
108C2×C4×Dic61C2×C411yescharacteristic
109C2×C4×Dic61D411yescharacteristic
110C2×C4×Dic61D411yescharacteristic
111C42×D61—22no—
112C42×D61C2×C411yescharacteristic
113C42×D61C2×C411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61C2311yescharacteristic
119C42×D61C2311yescharacteristic
120C4⋊Q8×C61C2×C411yescharacteristic
121C4⋊Q8×C61C2×C411yescharacteristic
122#125×C61C2×C412yes—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—21no—
125C4×Q8×C61—21no—
126C4×Q8×C61—21no—
127#117×C61C2×C412yes—
128C2×C42×C61C2×C411yescharacteristic
129C23×D61C2×D411yescharacteristic
130C22×#6923—24no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C22×C4×C61C22⋊C411yescharacteristic
134C22×C4×C61C22⋊C411yescharacteristic
135C22×C4×C61C22×C411yescharacteristic
136C22×Dic61C22⋊C411yescharacteristic
137C22×Dic61C22⋊C411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139Q8×D61—44no—
140Q8×D61—42no—
141Q8×D61—42no—
142C2×C4×D61—24no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—24no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61C22⋊C411yescharacteristic
149C2×C4×D61C22⋊C411yescharacteristic
150C2×C4×D61C22⋊C411yescharacteristic
151C2×C4×D61C22⋊C411yescharacteristic
152C2×C4×D61C22×C411yescharacteristic
153C2×C4×D61C2×D411yescharacteristic
154C2×C4×D61C2×D411yescharacteristic
155C2×C4×D61C2×D411yescharacteristic
156C2×C4×D61C2×D411yescharacteristic
157C2×C4×D61C2×D411yescharacteristic
158C2×C4×D61C2×D411yescharacteristic
159C2×C4×D61C2411yescharacteristic
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—22no—
163C2×Dic122—22no—
164C22⋊#6923—44no—
165C2×#92420—24no—
166C4⋊#6923—28no—
167C4⋊#6923—44no—
168C4⋊#6923—44no—
169C4×#6923—44no—
170D61⋊C8—28no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61C22×C412yes—
176C4⋊C4×C61C22×C412yes—
177C22⋊C4×C61—22no—
178C42×C61—22no—
179C42×C61C22⋊C411yescharacteristic
180C42×C61C22⋊C411yescharacteristic
181C42×C61C22×C411yescharacteristic
182C42×C61C22×C411yescharacteristic
183C22⋊Dic61—22no—
184#11758198—22no—
185#11758198—22no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—22no—
189C4⋊Dic61—22no—
190C4⋊Dic61C22×C412yes—
191C4⋊Dic61C22×C412yes—
192#11758196—22no—
193#11758196—22no—
194#11758196—22no—
195#11758196—22no—
196#11758196—22no—
197#11758196—22no—
198#11758196—22no—
199#11758196—22no—
200C4×Dic61—22no—
201C4×Dic61—22no—
202C4×Dic61—22no—
203C4×Dic61C22⋊C411yescharacteristic
204C4×Dic61C22⋊C411yescharacteristic
205C4×Dic61C22⋊C411yescharacteristic
206C4×Dic61C22⋊C411yescharacteristic
207C4×Dic61C22⋊C411yescharacteristic
208C4×Dic61C22⋊C411yescharacteristic
209C4×Dic61C22×C411yescharacteristic
210C4×Dic61C22×C411yescharacteristic
211C4×Dic61C2×D411yescharacteristic
212C4×Dic61C2×D411yescharacteristic
213C4×Dic61C2×D411yescharacteristic
214C4×Dic61C2×D411yescharacteristic
215C23×C61C2×C22⋊C411yescharacteristic
216C22×D61—22no—
217C22×D61—22no—
218C22×D61—21no—
219C22×D61—21no—
220C22×D61C2×C22⋊C411yescharacteristic
221C22×D61C22×D411yescharacteristic
222C2×#6923—44no—
223C2×#6923—44no—
224C2×#6923—44no—
225Q8×C61—42no—
226Q8×C61—42no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61C2×C22⋊C411yescharacteristic
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C2×C4×C61C2×C22⋊C411yescharacteristic
235C2×C4×C61C2×C22⋊C411yescharacteristic
236C2×C4×C61C2×C22⋊C411yescharacteristic
237C2×C4×C61C23×C411yescharacteristic
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61C2×C22⋊C411yescharacteristic
244C2×Dic61C2×C22⋊C411yescharacteristic
245C2×Dic61C2×C22⋊C411yescharacteristic
246C2×Dic61C2×C22⋊C411yescharacteristic
247C2×Dic61C2×C22⋊C411yescharacteristic
248C2×Dic61C22×D411yescharacteristic
249C4×D61—22no—
250C4×D61—22no—
251C4×D61—21no—
252C4×D61—21no—
253C4×D61—22no—
254C4×D61—22no—
255C4×D61—22no—
256C4×D61—22no—
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—44no—
260C4×D61—42no—
261C4×D61—42no—
262C4×D61#13514yes—
263C4×D61AGL(1,ℤ/8)12yes—
264C4×D61#13712yes—
265C4×D612−1+414yes—
266Dic122—42no—
267Dic122—42no—
268Dic122—44no—
269Dic122—44no—
270#92420—44no—
271#21835—611nomaximal, Sylow
272C22×C61—21no—
273C22×C61—21no—
274C22×C61—21no—
275C22×C61C22×C22⋊C411yescharacteristic
276C2×D61—21no—
277C2×D61—21no—
278C2×D61—21no—
279C2×D61—42no—
280C2×D61—42no—
281C2×D61#53612yes—
282C2×D61C2×#13512yes—
283C2×D61C2×AGL(1,ℤ/8)11yescharacteristic
284C2×D61C2×#13711yescharacteristic
285#6923—84no—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—21no—
289C4×C61—42no—
290C4×C61—42no—
291C4×C61#41612yes—
292C4×C61#41712yes—
293C4×C61#51912yes—
294Dic61—21no—
295Dic61—21no—
296Dic61—21no—
297Dic61—42no—
298Dic61—42no—
299Dic61#41612yes—
300Dic61#41712yes—
301Dic61#53612yes—
302C2×#533—611no—
303C2×#431—611no—
304#2285—611no—
305#1990—612no—
306#1990—612no—
307#1989—614no—
308#1825—614no—
309C2×C61—41no—
310C2×C61#231011yescharacteristic
311C2×C61C2×#41611yescharacteristic
312C2×C61C2×#41711yescharacteristic
313D61—41no—
314D61#322812yes—
315D61#323112yes—
316#533—611no—
317#533—611no—
318#533—612no—
319#533—612no—
320#533—611no—
321#533—1221no—
322#533—1221no—
323#533—1221no—
324#533—1221no—
325#533#692311yescharacteristic, complemented
326C2×C4⋊Q8—611no—
327C2×C4⋊Q8—611no—
328C2×#125—612no—
329C2×C22⋊Q8—1221no—
330C2×C22⋊Q8—1221no—
331C2×C4×Q8—1221no—
332C2×C4×Q8—1221no—
333C2×#117—612no—
334C2×#117—1221no—
335C22×C42—611no—
336#431—1221no—
337#431—1221no—
338#431—1221no—
339#431—1221no—
340C2×C4⋊C8—612no—
341C2×C22⋊C8—612no—
342C2×C4×C8—612no—
343#384—612no—
344#382—1221no—
345#380—612no—
346#380—1221no—
347C4×C4⋊C4—612no—
348C4×C4⋊C4—1221no—
349#340—618no—
350#340—618no—
351C61#2183511yescharacteristic, complemented, Sylow
352C22×Q8—1221no—
353C22×Q8—1221no—
354C23×C4—611no—
355C23×C4—611no—
356C23×C4—611no—
357C22×C8—614no—
358C4⋊Q8—612no—
359C4⋊Q8—612no—
360C4⋊Q8—611no—
361C4⋊Q8—611no—
362C4⋊Q8—1221no—
363C4⋊Q8—1221no—
364C4⋊Q8—1221no—
365C4⋊Q8—1221no—
366C4⋊Q8C2×#692311yescharacteristic
367C4⋊Q8C2×#692311yescharacteristic
368#125—612no—
369#125—612no—
370#125—612no—
371#125—1222no—
372#125—1222no—
373#125C2×#692312yes—
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376C22⋊Q8—1222no—
377C22⋊Q8—1222no—
378C22⋊Q8—1222no—
379C22⋊Q8—1222no—
380C22⋊Q8—1221no—
381C22⋊Q8—1221no—
382C22⋊Q8—1221no—
383C22⋊Q8—1221no—
384C22⋊Q8—1221no—
385C22⋊Q8—1221no—
386C4×Q8—21no—
387C4×Q8—21no—
388C4×Q8—1222no—
389C4×Q8—1222no—
390C4×Q8—1222no—
391C4×Q8—1222no—
392C4×Q8—1221no—
393C4×Q8—1221no—
394C4×Q8—1221no—
395C4×Q8—1221no—
396C4×Q8—1221no—
397C4×Q8—1221no—
398#117—612no—
399#117—612no—
400#117—612no—
401#117—1222no—
402#117—1222no—
403#117—2444no—
404#117C2×#692312yes—
405C2×C4⋊C4—612no—
406C2×C4⋊C4—612no—
407C2×C4⋊C4—614no—
408C2×C4⋊C4—1222no—
409C2×C4⋊C4—1222no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1222no—
413C2×C4⋊C4—1222no—
414C2×C22⋊C4—1222no—
415C2×C22⋊C4—1222no—
416C2×C42—611no—
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—1222no—
428C2×C42—1222no—
429C2×C42C2×#692311yescharacteristic
430C4⋊C8—1222no—
431C4⋊C8—1222no—
432C22⋊C8—1222no—
433C22⋊C8—1222no—
434C4×C8—1222no—
435C4×C8—1222no—
436#95—1222no—
437#95—1222no—
438C24—611no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—1222no—
442C2×Q8—1222no—
443C2×Q8—1221no—
444C2×Q8—1221no—
445C2×Q8—2444no—
446C2×Q8—2442no—
447C2×Q8—2442no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—1222no—
464C22×C4—1224no—
465C22×C4—1222no—
466C22×C4—1224no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1224no—
470C22×C4C22⋊#692311yescharacteristic
471C22×C4C22⋊#692311yescharacteristic
472C22×C4C22×#692311yescharacteristic
473C2×C8—1224no—
474C2×C8—1228no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—612no—
480C4⋊C4—612no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C4⋊C4—1222no—
485C4⋊C4—1222no—
486C4⋊C4—1222no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C4⋊C4—1228no—
490C4⋊C4—1222no—
491C4⋊C4—1222no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—2444no—
495C4⋊C4—2444no—
496C4⋊C4C22×#692312yes—
497C4⋊C4C22×#692312yes—
498C22⋊C4—22no—
499C22⋊C4—1222no—
500C22⋊C4—1222no—
501C22⋊C4—1222no—
502C22⋊C4—2444no—
503C42—22no—
504C42—611no—
505C42—611no—
506C42—611no—
507C42—611no—
508C42—611no—
509C42—611no—
510C42—611no—
511C42—611no—
512C42—611no—
513C42—611no—
514C42—611no—
515C42—611no—
516C42—1222no—
517C42—1222no—
518C42—1222no—
519C42—2444no—
520C42C22⋊#692311yescharacteristic
521C42C22⋊#692311yescharacteristic
522C42C22×#692311yescharacteristic
523C42C22×#692311yescharacteristic
524C23—611no—
525C23—611no—
526C23—1221no—
527C23—1222no—
528C23—1222no—
529C23—1221no—
530C23C2×C22⋊#692311yescharacteristic
531Q8—42no—
532Q8—42no—
533Q8—2444no—
534Q8—2444no—
535Q8—2442no—
536Q8—2442no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—611no—
543C2×C4—611no—
544C2×C4—611no—
545C2×C4—614no—
546C2×C4—612no—
547C2×C4—612no—
548C2×C4—614no—
549C2×C4—611no—
550C2×C4—611no—
551C2×C4—611no—
552C2×C4—1222no—
553C2×C4—1222no—
554C2×C4—1222no—
555C2×C4—1221no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—1221no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1222no—
567C2×C4—2444no—
568C2×C4—2442no—
569C2×C4—2442no—
570C2×C4—2444no—
571C2×C4—2444no—
572C2×C4—2444no—
573C2×C4C2×C22⋊#692311yescharacteristic
574C2×C4C2×C22⋊#692311yescharacteristic
575C2×C4C2×C22⋊#692311yescharacteristic
576C2×C4C2×C22⋊#692311yescharacteristic
577C2×C4C2×C22⋊#692311yescharacteristic
578C2×C4C23×#692311yescharacteristic
579C8—2444no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—612no—
584C22—612no—
585C22—611no—
586C22—611no—
587C22—1221no—
588C22—1221no—
589C22—1221no—
590C22—2442no—
591C22—2442no—
592C22C22×C22⋊#692311yescharacteristic
593C4—21no—
594C4—21no—
595C4—21no—
596C4—42no—
597C4—42no—
598C4—612no—
599C4—612no—
600C4—612no—
601C4—1221no—
602C4—1221no—
603C4—1221no—
604C4—4884no—
605C4—2442no—
606C4—2442no—
607C4#1684075212yes—
608C4#1684075312yes—
609C4#1684083012yes—
610C2—41no—
611C2—612no—
612C2—612no—
613C2—2441no—
614C2#2146229311yescharacteristic
615C2C2×#1684075211yescharacteristic
616C2C2×#1684075311yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#533C4
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenter#533#6923
hypocenterC61#21835
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
3. #21462000C2semidirect
4. #21462000C2semidirect
5. #21461999C2semidirect
24. #16840319C4semidirect
25. #16840318C4semidirect
26. #16840317C4semidirect
27. #16840317C4semidirect
30. #16840315C4semidirect
42. C61×#533C4semidirect
325. #533#6923semidirect
351. C61#21835semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4312
#214622682
#214620002
#214619992
#214617242
#533×D612
#2183561