Finite Group Databasebeta

← #29696804

Subgroups — #29696804

Order 15616.

1074 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840484C211yesmaximal, characteristic
2C2×#16839886C211yesmaximal, characteristic, complemented
3#21448771C211yesmaximal, characteristic, complemented
4#21446749C211yesmaximal, characteristic
5#21446748C211yesmaximal, characteristic
6#21446740C211yesmaximal, characteristic, complemented
7#21446628C211yesmaximal, characteristic
8#21446627C211yesmaximal, characteristic
9#21446619C211yesmaximal, characteristic, complemented
10#21445833C211yesmaximal, characteristic, complemented
11#21445818C211yesmaximal, characteristic
12C4×#14519701C211yesmaximal, characteristic, complemented
13#21445638C211yesmaximal, characteristic, complemented
14#21445633C211yesmaximal, characteristic
15C61×#1805C211yesmaximal, characteristic, complemented
16#16840484—24no—
17#16840484C418yescomplemented
18C2×#14519775C2211yescharacteristic
19C2×#14519772—21no—
20C2×#14519771—21no—
21C2×#14519766C2211yescharacteristic
22C2×#14519765C2211yescharacteristic
23C2×#14519755—21no—
24C2×C4×Dic122C2211yescharacteristic
25C2×#14519742C2211yescharacteristic
26C2×#14519739C2211yescharacteristic
27C2×#14519734—21no—
28#16839886—24no—
29C2×#14519706C2211yescharacteristic
30C2×#14519706C2211yescharacteristic
31C2×#14519702C2211yescharacteristic
32C2×#14519702C2211yescharacteristic
33C2×#14519701C2211yescharacteristic
34C2×#14519701C2211yescharacteristic
35C2×#14519700C2211yescharacteristic
36C2×#14519700C2211yescharacteristic
37C2×#14519699C2211yescharacteristic
38C2×#14519699C2211yescharacteristic
39C2×#14519697C2211yescharacteristic
40C2×#14519697C2211yescharacteristic
41C2×C4×#92418C2211yescharacteristic
42C2×C8×Dic61C2211yescharacteristic
43C2×C8×Dic61C2211yescharacteristic
44#16839666—21no—
45#16839647—21no—
46#16839644—21no—
47#16839631C2211yescharacteristic
48#16839627C2211yescharacteristic
49C4⋊C4×Dic61C2211yescharacteristic
50C4⋊C4×Dic61C2211yescharacteristic
51C4×C4⋊Dic61C2211yescharacteristic
52#16839613—21no—
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55C2×C4×Q8×C61C2211yescharacteristic
56C2×C4⋊C8×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C61×#382C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C22×Dic122D411yescharacteristic
63C22×Dic122C2311yescharacteristic
64C22×D244C2311yescharacteristic
65C22×C4×D61—21no—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519775—22no—
69#14519775C2×C414yes—
70#14519772—28no—
71#14519771—28no—
72#14519766—22no—
73#14519766C2×C414yes—
74#14519765—22no—
75#14519765C2×C414yes—
76#14519755—28no—
77C4×Dic122C2×C414yes—
78C4×Dic122D414yes—
79#14519742—22no—
80#14519742C2×C414yes—
81#14519739—22no—
82#14519739C2×C414yes—
83#14519734—28no—
84C2×C4⋊Dic61D411yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×#11758196D411yescharacteristic
93C2×#11758196D411yescharacteristic
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×#11758198—21no—
98C2×#11758198—21no—
99C2×C4×Dic61—21no—
100C2×C4×Dic61—21no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61—21no—
103C2×C4×Dic61C2311yescharacteristic
104C2×C4×Dic61C2311yescharacteristic
105#14519706—22no—
106#14519706—22no—
107#14519702—22no—
108#14519702—41no—
109#14519701—22no—
110#14519701—22no—
111#14519700—22no—
112#14519700—41no—
113#14519699—22no—
114#14519699—41no—
115#14519697—22no—
116#14519697—41no—
117C4×#92418—22no—
118C8×Dic61—22no—
119C8×Dic61—22no—
120#14519684—21no—
121#14519684—21no—
122#14519684—21no—
123#14519684—21no—
124#14519684—21no—
125C22×Q8×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127C22×C8×C61C2311yescharacteristic
128C4×Q8×C61—22no—
129C4×Q8×C61C2×C414yes—
130C2×C4⋊C4×C61—21no—
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C4⋊C4×C61C2311yescharacteristic
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C42×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—22no—
137#103×C61—41no—
138#103×C61—41no—
139#95×C61—21no—
140#95×C61—21no—
141C23×D61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61C22×C412yes—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2411yescharacteristic
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244C22×C412yes—
168C2×C4×D61—24no—
169C2×C4×D61—42no—
170C2×C4×D61—42no—
171C2×Dic122—22no—
172C2×Dic122—22no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122C22⋊C414yes—
179C2×Dic122C22×C412yes—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C4⋊C4×C61—24no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61C22×C412yes—
193C42×C61—24no—
194C42×C61C22×C412yes—
195C42×C61C2×D412yes—
196#11758198—24no—
197#11758198—24no—
198#11758198—24no—
199#11758198—24no—
200#11758198—24no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61C22⋊C414yes—
205C4⋊Dic61#4112yes—
206#11758196—24no—
207#11758196—24no—
208#11758196—24no—
209#11758196—42no—
210#11758196—81no—
211#11758196C22⋊C414yes—
212#11758196C22⋊C414yes—
213C4×Dic61—24no—
214C4×Dic61—42no—
215C4×Dic61—81no—
216C4×Dic61—81no—
217C4×Dic61C22×C412yes—
218C4×Dic61C22×C412yes—
219C4×Dic61C2×D412yes—
220C4×Dic61C2×D412yes—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418#4112yes—
227C23×C61C22×D411yescharacteristic
228C22×D61—22no—
229C22×D61—41no—
230C22×D61—41no—
231C22×D61—41no—
232C22×D61—41no—
233C22×D61—41no—
234C22×D61—41no—
235Q8×C61—22no—
236Q8×C61—22no—
237Q8×C61—41no—
238Q8×C61—41no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61C2×C22⋊C412yes—
256C2×C4×C61C2×C22⋊C412yes—
257C2×C4×C61C23×C411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61H(5,𝔽2)11yescharacteristic
262C2×C4×C612−1+411yescharacteristic
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61C2×C22⋊C412yes—
279C2×Dic61C2×C22⋊C412yes—
280C2×Dic61C2×C22⋊C412yes—
281C2×Dic61C2×C22⋊C412yes—
282C2×Dic61#13512yes—
283C2×Dic61#13512yes—
284D244—22no—
285D244—22no—
286D244—41no—
287D244—41no—
288C4×D61—48no—
289Dic122—28no—
290Dic122—22no—
291Dic122—22no—
292Dic122—41no—
293Dic122—41no—
294C8×C61—41no—
295C8×C61—41no—
296#92418—22no—
297#92418—41no—
298#22541—611nomaximal, Sylow
299C22×C61C22×C22⋊C411yescharacteristic
300C22×C61#54811yescharacteristic
301C22×C61#55011yescharacteristic
302C22×C61C2×#13511yescharacteristic
303C22×C61C2×#13511yescharacteristic
304C22×C61#57711yescharacteristic
305C22×C61#57811yescharacteristic
306C2×D61—42no—
307C2×D61—42no—
308C2×D61—42no—
309C2×D61—81no—
310C2×D61—81no—
311C2×D61—81no—
312C2×D61—81no—
313C4×C61—22no—
314C4×C61—22no—
315C4×C61—42no—
316C4×C61—42no—
317C4×C61—81no—
318C4×C61—81no—
319C4×C61C4×#4111yescharacteristic
320C4×C61#52011yescharacteristic
321C4×C61#54811yescharacteristic
322C4×C61#55011yescharacteristic
323Dic61—22no—
324Dic61—22no—
325Dic61—22no—
326Dic61—42no—
327Dic61—42no—
328Dic61—81no—
329Dic61#41614yes—
330C2×#556—611no—
331C2×C8×Q8—611no—
332#2355—611no—
333#1983—611no—
334#1983—611no—
335#1981—611no—
336#1958—611no—
337#1958—611no—
338#1956—611no—
339#1805—611no—
340#1805D6111yescharacteristic, complemented
341#1802—611no—
342C4×#103—611no—
343C4×#103—611no—
344C4×#102—611no—
345C2×C61#234111yescharacteristic
346C2×C61C2×#41611yescharacteristic
347C2×C61#289811yescharacteristic
348C2×C61#336011yescharacteristic
349C2×C61#336411yescharacteristic
350C2×C61#339311yescharacteristic
351C2×C61#339411yescharacteristic
352D61—82no—
353#556—618no—
354#556—1224no—
355C2×C4⋊Q8—611no—
356C2×#126—1221no—
357C2×#126—1221no—
358C2×#124—611no—
359C2×#124—611no—
360C2×C22⋊Q8—1221no—
361C2×C4×Q8—611no—
362C2×C4×Q8—611no—
363C2×C4×Q8C2×D6111yescharacteristic
364C2×C4×D4—611no—
365C2×#117—1221no—
366C8×Q8—1224no—
367C2×C4⋊C8—611no—
368C2×C4⋊C8—611no—
369C2×C4⋊C8C2×D6111yescharacteristic
370C2×#103—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×#103—611no—
376C2×#103C2×D6111yescharacteristic
377C2×#103C2×D6111yescharacteristic
378C2×#102—611no—
379C2×#102—611no—
380C2×#102—611no—
381C2×#102—611no—
382C2×C4×C8—611no—
383C2×C4×C8—611no—
384C2×C4×C8—611no—
385#391—1221no—
386#387—1221no—
387#387—1221no—
388#384—611no—
389#382—611no—
390#382C2×D6111yescharacteristic
391C4×C4⋊C4—611no—
392C4×C4⋊C4—611no—
393C4×C4⋊C4—611no—
394C4×C4⋊C4—1221no—
395#340—611no—
396#340—611no—
397#340C2×D6111yescharacteristic
398#340C2×D6111yescharacteristic
399C61#2254111yescharacteristic, complemented, Sylow
400C22×Q8—611no—
401C22×Q8—611no—
402C22×Q8C22×D6111yescharacteristic
403C22×D4—611no—
404C23×C4—1221no—
405C22×C8—611no—
406C22×C8—611no—
407C22×C8C22×D6111yescharacteristic
408C22×C8C22×D6111yescharacteristic
409C4⋊Q8—614no—
410C4⋊Q8—1222no—
411#126—1228no—
412#126—1228no—
413#124—614no—
414#124—614no—
415#124—1222no—
416#124—1222no—
417C22⋊Q8—1228no—
418C4×Q8—22no—
419C4×Q8—614no—
420C4×Q8—614no—
421C4×Q8—614no—
422C4×Q8—1222no—
423C4×Q8C4×D6114yes—
424C4×D4—614no—
425C4×D4—1222no—
426#117—1228no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4C22×D6111yescharacteristic
439C2×C4⋊C4C22×D6111yescharacteristic
440C2×C4⋊C4C22×D6111yescharacteristic
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C42—21no—
447C2×C42—611no—
448C2×C42—611no—
449C2×C42—1221no—
450C2×C42—1221no—
451C2×C42—1221no—
452C2×C42—1221no—
453C2×C42C22×D6111yescharacteristic
454C4⋊C8—22no—
455C4⋊C8—1222no—
456C4⋊C8—1222no—
457#103—41no—
458#103—41no—
459#103—1222no—
460#103—1222no—
461#103—1222no—
462#103—1222no—
463#103—2441no—
464#103—2441no—
465#102—1222no—
466#102—1222no—
467#102—2441no—
468#102—2441no—
469C4×C8—1222no—
470C4×C8—1222no—
471C4×C8—1222no—
472#95—21no—
473#95—21no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479C24—1221no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—614no—
486C2×Q8—612no—
487C2×Q8—1222no—
488C2×Q8—1222no—
489C2×Q8—1221no—
490C2×Q8—1221no—
491C2×Q8—1221no—
492C2×Q8—1221no—
493C2×Q8—1221no—
494C2×Q8C2×C4×D6112yes—
495C2×D4—612no—
496C2×D4—1221no—
497C2×D4—1221no—
498C2×D4—1221no—
499C2×D4—1221no—
500C2×D4—1221no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—611no—
506C22×C4—611no—
507C22×C4—611no—
508C22×C4—611no—
509C22×C4—1224no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—2442no—
514C22×C4—2442no—
515C22×C4D4×D6111yescharacteristic
516C22×C4D4×D6111yescharacteristic
517C22×C4C23×D6111yescharacteristic
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—612no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—1221no—
530C4⋊C4—21no—
531C4⋊C4—24no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—614no—
537C4⋊C4—614no—
538C4⋊C4—614no—
539C4⋊C4—612no—
540C4⋊C4—1224no—
541C4⋊C4—1224no—
542C4⋊C4—1224no—
543C4⋊C4—1221no—
544C4⋊C4—1221no—
545C4⋊C4—1221no—
546C4⋊C4—4881no—
547C4⋊C4—2442no—
548C4⋊C4C2×C4×D6112yes—
549C22⋊C4—1224no—
550C22⋊C4—1224no—
551C22⋊C4—1224no—
552C22⋊C4—1224no—
553C22⋊C4—1224no—
554C42—24no—
555C42—612no—
556C42—612no—
557C42—612no—
558C42—612no—
559C42—1224no—
560C42—4881no—
561C42—4881no—
562C42—2442no—
563C42C2×C4×D6112yes—
564C42D4×D6112yes—
565C23—1222no—
566C23—2441no—
567C23—2441no—
568C23—2441no—
569C23—2441no—
570C23—2441no—
571C23—2441no—
572C23C2×D4×D6111yescharacteristic
573Q8—22no—
574Q8—22no—
575Q8—41no—
576Q8—41no—
577Q8—1228no—
578Q8—1222no—
579Q8—1222no—
580Q8—2441no—
581Q8—2441no—
582D4—1222no—
583D4—1222no—
584D4—2441no—
585D4—2441no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—612no—
603C2×C4—612no—
604C2×C4—612no—
605C2×C4—612no—
606C2×C4—612no—
607C2×C4—612no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1222no—
615C2×C4—1222no—
616C2×C4—2448no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4C22⋊C4×D6112yes—
625C2×C4C22⋊C4×D6112yes—
626C2×C4C22×C4×D6111yescharacteristic
627C2×C4C2×D4×D6111yescharacteristic
628C2×C4#41×D6111yescharacteristic
629C2×C4#41×D6111yescharacteristic
630C2×C4#1451982211yescharacteristic
631C2×C4#1451982511yescharacteristic
632C8—41no—
633C8—41no—
634C8—1222no—
635C8—2441no—
636C22—4881no—
637C22—4881no—
638C22—4881no—
639C22—4881no—
640C22—2442no—
641C22—2442no—
642C22—2442no—
643C22C2×C22⋊C4×D6111yescharacteristic
644C22#1684042211yescharacteristic
645C22#1684043011yescharacteristic
646C22#135×D6111yescharacteristic
647C22#135×D6111yescharacteristic
648C22#1684057111yescharacteristic
649C22#1684057811yescharacteristic
650C4—22no—
651C4—22no—
652C4—42no—
653C4—42no—
654C4—81no—
655C4—81no—
656C4—614no—
657C4—1222no—
658C4—1222no—
659C4—1222no—
660C4—4881no—
661C4—2442no—
662C4—2442no—
663C4#1684023011yescharacteristic
664C4#1684024311yescharacteristic
665C4#1684042211yescharacteristic
666C4#1684043011yescharacteristic
667C2—4882no—
668C2#2144865611yescharacteristic
669C2#416×D6111yescharacteristic
670C2#2145435111yescharacteristic
671C2#2145847111yescharacteristic
672C2#2145850711yescharacteristic
673C2#2145880211yescharacteristic
674C2#2145881011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1805C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1805D61
hypocenterC61#22541
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
2. C2×#16839886C2semidirect
3. #21448771C2semidirect
6. #21446740C2semidirect
9. #21446619C2semidirect
10. #21445833C2semidirect
12. C4×#14519701C2semidirect
13. #21445638C2semidirect
15. C61×#1805C2semidirect
17. #16840484C4semidirect
340. #1805D61semidirect
399. C61#22541semidirect

Maximal subgroups

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

subgroupindex
C2×#168404842
C2×#168398862
#214487712
#214467492
#214467482
#214467402
#214466282
#214466272
#214466192
#214458332
#214458182
C4×#145197012
#214456382
#214456332
C61×#18052
#2254161