Finite Group Databasebeta

← #29714515

Subgroups — #29714515

Order 15616.

1022 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840374C211yesmaximal, characteristic
2C2×#16840070C211yesmaximal, characteristic, complemented
3C2×#16839973C211yesmaximal, characteristic, complemented
4#21448501C211yesmaximal, characteristic
5#21447023C211yesmaximal, characteristic
6#21447016C211yesmaximal, characteristic
7#21446829C211yesmaximal, characteristic
8#21446829C211yesmaximal, characteristic
9#21446774C211yesmaximal, characteristic, complemented
10#21446741C211yesmaximal, characteristic, complemented
11#21446159C211yesmaximal, characteristic, complemented
12#21446159C211yesmaximal, characteristic, complemented
13#21446090C211yesmaximal, characteristic, complemented
14#21446076C211yesmaximal, characteristic, complemented
15C61×#1798C211yesmaximal, characteristic
16C22×Dic244C2211yescharacteristic
17C22×Dic244C2211yescharacteristic
18#16840374—24no—
19#16840374—24no—
20C2×#14519777C2211yescharacteristic
21C2×#14519770—21no—
22C2×#14519770—21no—
23C2×#14519769C2211yescharacteristic
24C2×#14519767—21no—
25C2×#14519767—21no—
26C2×#14519756C2211yescharacteristic
27C2×#14519756C2211yescharacteristic
28C2×#14519756C2211yescharacteristic
29C2×C4×Dic122C2211yescharacteristic
30C2×C4×Dic122C2211yescharacteristic
31C2×C4×Dic122C2211yescharacteristic
32#16840070—24no—
33#16839973—24no—
34C2×#14519711C2211yescharacteristic
35C2×#14519709C2211yescharacteristic
36C2×#14519702C2211yescharacteristic
37C2×#14519702C2211yescharacteristic
38C2×#14519702C2211yescharacteristic
39C2×#14519702C2211yescharacteristic
40C2×#14519702C2211yescharacteristic
41C2×#14519702C2211yescharacteristic
42C2×#14519702C2211yescharacteristic
43C2×#14519702C2211yescharacteristic
44#16839668C2211yescharacteristic
45#16839660C2211yescharacteristic
46#16839657—21no—
47#16839656—21no—
48#16839650—21no—
49#16839643—21no—
50#16839622C2211yescharacteristic
51C4×C4⋊Dic61C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839525C2211yescharacteristic
54#16839525C2211yescharacteristic
55#16839525C2211yescharacteristic
56C2×#117×C61C2211yescharacteristic
57C2×C4⋊C8×C61C2211yescharacteristic
58C2×C4⋊C8×C61C2211yescharacteristic
59C2×C22⋊C8×C61C2211yescharacteristic
60C2×C22⋊C8×C61C2211yescharacteristic
61C61×#336C2211yescharacteristic
62C61×#336C2211yescharacteristic
63C22×Dic122C2311yescharacteristic
64C22×Dic122C2311yescharacteristic
65C22×Dic122C2311yescharacteristic
66C2×Dic244—22no—
67C2×Dic244—22no—
68C2×Dic244—22no—
69C2×Dic244—22no—
70C2×Dic244—41no—
71C2×Dic244—41no—
72#14519777—22no—
73#14519777D414yes—
74#14519770—28no—
75#14519770—28no—
76#14519769—22no—
77#14519769D414yes—
78#14519767—28no—
79#14519767—28no—
80#14519756—22no—
81#14519756—22no—
82#14519756—22no—
83#14519756—22no—
84#14519756—22no—
85#14519756—22no—
86C4×Dic122—22no—
87C4×Dic122—22no—
88C4×Dic122—22no—
89C4×Dic122—22no—
90C4×Dic122—22no—
91C4×Dic122—22no—
92C2×C4⋊Dic61—21no—
93C2×C4⋊Dic61—21no—
94C2×C4⋊Dic61C2311yescharacteristic
95C2×C4⋊Dic61C2311yescharacteristic
96C2×C4⋊Dic61C2311yescharacteristic
97C2×C4⋊Dic61C2311yescharacteristic
98C2×C4⋊Dic61C2311yescharacteristic
99C2×#11758196—21no—
100C2×#11758196—21no—
101C2×#11758196—21no—
102C2×#11758196—21no—
103C2×#11758196—21no—
104C2×#11758196—21no—
105C2×#11758196—21no—
106C2×#11758196—21no—
107C2×#11758196—21no—
108C2×C22⋊Dic61—21no—
109C2×C22⋊Dic61—21no—
110C2×C22⋊Dic61—21no—
111C2×C22⋊Dic61—21no—
112C2×C4×Dic61—21no—
113C2×C4×Dic61—21no—
114C2×C4×Dic61—21no—
115C2×C4×Dic61—21no—
116#14519711—22no—
117#14519709—22no—
118#14519702—22no—
119#14519702—22no—
120#14519702—22no—
121#14519702—22no—
122#14519702—22no—
123#14519702—22no—
124#14519702—41no—
125#14519702—41no—
126#14519684—21no—
127#14519684—21no—
128#14519684—21no—
129#14519684—21no—
130#14519684—21no—
131C23×C4×C61C2311yescharacteristic
132C22×C8×C61C2311yescharacteristic
133C22×C8×C61C2311yescharacteristic
134C22×C8×C61C2311yescharacteristic
135C22×C8×C61C2311yescharacteristic
136#117×C61—22no—
137#117×C61—22no—
138C2×C4⋊C4×C61D411yescharacteristic
139C2×C4⋊C4×C61D411yescharacteristic
140C2×C22⋊C4×C61D411yescharacteristic
141C2×C22⋊C4×C61D411yescharacteristic
142C2×C42×C61C2311yescharacteristic
143C2×C42×C61C2311yescharacteristic
144C4⋊C8×C61—22no—
145C4⋊C8×C61—22no—
146C22⋊C8×C61—41no—
147C22⋊C8×C61—41no—
148C24×C61C2×D411yescharacteristic
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2411yescharacteristic
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—21no—
165C22×Dic61—21no—
166C22×Dic61—21no—
167C22×Dic61—21no—
168C22×Dic61—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122—21no—
181C2×Dic122—21no—
182C2×Dic122—21no—
183C2×Dic122—21no—
184C2×Dic122C2×D412yes—
185C2×Dic122C2×D412yes—
186C2×Dic122#4112yes—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61—21no—
192C2×C8×C61—21no—
193C2×C8×C61—21no—
194C2×C8×C61—21no—
195C2×C8×C61—21no—
196C2×C8×C61—21no—
197C2×C8×C61—21no—
198C2×C8×C61—21no—
199C4⋊C4×C61—22no—
200C4⋊C4×C61—22no—
201C22⋊C4×C61—22no—
202C22⋊C4×C61—22no—
203C42×C61—21no—
204C42×C61—21no—
205C42×C61C2×D412yes—
206C42×C61C2×D412yes—
207C22⋊Dic61—24no—
208C22⋊Dic61—24no—
209C22⋊Dic61—24no—
210C22⋊Dic61—81no—
211C4⋊Dic61—24no—
212C4⋊Dic61—24no—
213C4⋊Dic61—21no—
214C4⋊Dic61—21no—
215C4⋊Dic61—21no—
216C4⋊Dic61—21no—
217C4⋊Dic61—21no—
218C4⋊Dic61—21no—
219C4⋊Dic61—21no—
220C4⋊Dic61C2×D412yes—
221C4⋊Dic61C2×D412yes—
222C4⋊Dic61#4112yes—
223#11758196—24no—
224#11758196—24no—
225#11758196—24no—
226#11758196—24no—
227#11758196—24no—
228#11758196—24no—
229#11758196—24no—
230#11758196—42no—
231#11758196—81no—
232C4×Dic61—24no—
233C4×Dic61—24no—
234C4×Dic61—42no—
235C4×Dic61—81no—
236Dic244—44no—
237Dic244—44no—
238C23×C61—21no—
239C23×C61—21no—
240C23×C61—21no—
241C23×C61—21no—
242C23×C61—21no—
243C23×C61—21no—
244C23×C61—21no—
245C23×C61C22×D411yescharacteristic
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61#12012yes—
264C2×C4×C61#12012yes—
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×C4×C61H(5,𝔽2)11yescharacteristic
270C2×C4×C612−1+411yescharacteristic
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—22no—
274C2×Dic61—22no—
275C2×Dic61—22no—
276C2×Dic61—22no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61—41no—
292C2×Dic61—41no—
293C2×Dic61—41no—
294C2×Dic61—41no—
295Dic122—22no—
296Dic122—22no—
297Dic122—22no—
298Dic122—22no—
299Dic122—22no—
300Dic122—22no—
301Dic122—41no—
302Dic122—41no—
303Dic122—41no—
304Dic122—41no—
305Dic122—41no—
306Dic122—41no—
307C8×C61—41no—
308C8×C61—41no—
309C8×C61—41no—
310C8×C61—41no—
311#24165—611nomaximal, Sylow
312C22×C61—41no—
313C22×C61—41no—
314C22×C61—41no—
315C22×C61—41no—
316C22×C61—41no—
317C22×C61—41no—
318C22×C61—41no—
319C22×C61C2×#12011yescharacteristic
320C22×C61#52511yescharacteristic
321C22×C61#53711yescharacteristic
322C22×C61#57711yescharacteristic
323C22×C61#57811yescharacteristic
324C22×C61#57811yescharacteristic
325C22×C61#57811yescharacteristic
326C4×C61—22no—
327C4×C61—22no—
328C4×C61—41no—
329C4×C61—41no—
330C4×C61—41no—
331C4×C61#54611yescharacteristic
332C4×C61#54711yescharacteristic
333C4×C61#54811yescharacteristic
334C4×C61#55011yescharacteristic
335Dic61—42no—
336Dic61—42no—
337Dic61—42no—
338Dic61—42no—
339Dic61—42no—
340Dic61—42no—
341Dic61—81no—
342Dic61—81no—
343C2×#541—611no—
344C2×#479—611no—
345C2×#462—611no—
346#2323—611no—
347#2036—611no—
348#2035—611no—
349#2001—611no—
350#2001—611no—
351#1989—611no—
352#1981—611no—
353#1878—611no—
354#1878—611no—
355#1865—611no—
356#1863—611no—
357#1798D6111yescharacteristic
358C2×C61—81no—
359C2×C61#240711yescharacteristic
360C2×C61#301111yescharacteristic
361C2×C61#301311yescharacteristic
362C2×C61#312011yescharacteristic
363C2×C61#312711yescharacteristic
364C2×C61#325811yescharacteristic
365C2×C61#326111yescharacteristic
366C22×Q16—611no—
367C22×Q16—611no—
368#541—1224no—
369#541—1224no—
370C2×C4⋊Q8—611no—
371C2×#126—1221no—
372C2×#126—1221no—
373C2×#125—611no—
374C2×#125—1221no—
375C2×#125—1221no—
376C2×C22⋊Q8—611no—
377C2×C22⋊Q8—611no—
378C2×C22⋊Q8—611no—
379C2×C4×Q8—611no—
380C2×C4×Q8—611no—
381C2×C4×Q8—611no—
382C2×#117C2×D6111yescharacteristic
383#479—1224no—
384#462—1224no—
385C2×#107—611no—
386C2×#106—611no—
387C2×C4⋊C8C2×D6111yescharacteristic
388C2×C4⋊C8C2×D6111yescharacteristic
389C2×#103—611no—
390C2×#103—611no—
391C2×#103—611no—
392C2×#103—611no—
393C2×#103—611no—
394C2×#103—611no—
395C2×#103—611no—
396C2×#103—611no—
397C2×C22⋊C8C2×D6111yescharacteristic
398C2×C22⋊C8C2×D6111yescharacteristic
399#391—611no—
400#389—611no—
401#389—1221no—
402#388—1221no—
403#388—1221no—
404#387—1221no—
405#380—611no—
406C4×C4⋊C4—611no—
407#340—611no—
408#340—611no—
409#340—611no—
410#340—611no—
411#336C2×D6111yescharacteristic
412#336C2×D6111yescharacteristic
413C61#2416511yescharacteristic, complemented, Sylow
414C22×Q8—611no—
415C22×Q8—611no—
416C22×Q8—611no—
417C23×C4C22×D6111yescharacteristic
418C2×Q16—1222no—
419C2×Q16—1222no—
420C2×Q16—1222no—
421C2×Q16—1222no—
422C2×Q16—2441no—
423C2×Q16—2441no—
424C22×C8C22×D6111yescharacteristic
425C22×C8C22×D6111yescharacteristic
426C22×C8C22×D6111yescharacteristic
427C22×C8C22×D6111yescharacteristic
428C4⋊Q8—614no—
429C4⋊Q8—1222no—
430#126—1228no—
431#126—1228no—
432#125—614no—
433#125—1228no—
434#125—1228no—
435#125—1222no—
436C22⋊Q8—1222no—
437C22⋊Q8—1222no—
438C22⋊Q8—1222no—
439C22⋊Q8—1222no—
440C22⋊Q8—1222no—
441C22⋊Q8—1222no—
442C4×Q8—1222no—
443C4×Q8—1222no—
444C4×Q8—1222no—
445C4×Q8—1222no—
446C4×Q8—1222no—
447C4×Q8—1222no—
448#117—22no—
449#117—22no—
450C2×C4⋊C4—611no—
451C2×C4⋊C4—611no—
452C2×C4⋊C4—611no—
453C2×C4⋊C4—611no—
454C2×C4⋊C4—611no—
455C2×C4⋊C4—1221no—
456C2×C4⋊C4—1221no—
457C2×C4⋊C4—1221no—
458C2×C4⋊C4—1221no—
459C2×C4⋊C4—1221no—
460C2×C4⋊C4—1221no—
461C2×C4⋊C4—1221no—
462C2×C4⋊C4—1221no—
463C2×C4⋊C4—1221no—
464C2×C4⋊C4—1221no—
465C2×C4⋊C4—1221no—
466C2×C4⋊C4D24411yescharacteristic
467C2×C4⋊C4D24411yescharacteristic
468C2×C22⋊C4—1221no—
469C2×C22⋊C4—1221no—
470C2×C22⋊C4—1221no—
471C2×C22⋊C4—1221no—
472C2×C22⋊C4D24411yescharacteristic
473C2×C22⋊C4D24411yescharacteristic
474C2×C42—1221no—
475C2×C42—1221no—
476C2×C42—1221no—
477C2×C42—1221no—
478C2×C42C22×D6111yescharacteristic
479C2×C42C22×D6111yescharacteristic
480#107—1222no—
481#106—1222no—
482C4⋊C8—22no—
483C4⋊C8—22no—
484#103—1222no—
485#103—1222no—
486#103—1222no—
487#103—1222no—
488#103—1222no—
489#103—1222no—
490#103—2441no—
491#103—2441no—
492C22⋊C8—41no—
493C22⋊C8—41no—
494#95—1221no—
495#95—1221no—
496#95—1221no—
497#95—1221no—
498#95—1221no—
499C24C2×D24411yescharacteristic
500C2×Q8—612no—
501C2×Q8—612no—
502C2×Q8—612no—
503C2×Q8—1221no—
504C2×Q8—1221no—
505C2×Q8—1221no—
506C2×Q8—1221no—
507C2×Q8—1221no—
508C2×Q8—1221no—
509C2×Q8—1221no—
510C2×Q8—1221no—
511C2×Q8—1221no—
512C2×Q8—1221no—
513C2×Q8—1221no—
514C2×Q8—1221no—
515C2×Q8—1221no—
516C2×Q8—1221no—
517C2×Q8—1221no—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—21no—
521C22×C4—21no—
522C22×C4—21no—
523C22×C4—21no—
524C22×C4—1221no—
525C22×C4—1221no—
526C22×C4—1221no—
527C22×C4—1221no—
528C22×C4—1221no—
529C22×C4—1221no—
530C22×C4—1221no—
531C22×C4—1221no—
532C22×C4C2×D24411yescharacteristic
533C22×C4C2×D24411yescharacteristic
534C22×C4C2×D24411yescharacteristic
535C22×C4C2×D24411yescharacteristic
536C22×C4C2×D24411yescharacteristic
537C22×C4C23×D6111yescharacteristic
538Q16—2444no—
539Q16—2444no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—21no—
544C2×C8—21no—
545C2×C8—21no—
546C2×C8—21no—
547C2×C8—21no—
548C2×C8—21no—
549C2×C8—21no—
550C2×C8—21no—
551C2×C8—21no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—612no—
555C4⋊C4—612no—
556C4⋊C4—612no—
557C4⋊C4—1224no—
558C4⋊C4—1224no—
559C4⋊C4—1224no—
560C4⋊C4—1224no—
561C4⋊C4—1224no—
562C4⋊C4—1221no—
563C4⋊C4—1221no—
564C4⋊C4—1221no—
565C4⋊C4—1224no—
566C4⋊C4—1224no—
567C4⋊C4—1224no—
568C4⋊C4—1224no—
569C4⋊C4—1221no—
570C4⋊C4—1221no—
571C4⋊C4—1221no—
572C4⋊C4—1221no—
573C4⋊C4—4881no—
574C4⋊C4—2442no—
575C22⋊C4—22no—
576C22⋊C4—22no—
577C22⋊C4—1224no—
578C22⋊C4—1224no—
579C22⋊C4—1224no—
580C22⋊C4—4881no—
581C42—21no—
582C42—21no—
583C42—1224no—
584C42—1224no—
585C42—4881no—
586C42—2442no—
587C42D4×D6112yes—
588C42D4×D6112yes—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23C22×D24411yescharacteristic
597Q8—1222no—
598Q8—1222no—
599Q8—1222no—
600Q8—1222no—
601Q8—1222no—
602Q8—1222no—
603Q8—2441no—
604Q8—2441no—
605Q8—2441no—
606Q8—2441no—
607Q8—2441no—
608Q8—2441no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—1222no—
627C2×C4—1222no—
628C2×C4—1222no—
629C2×C4—1222no—
630C2×C4—1222no—
631C2×C4—1222no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2441no—
635C2×C4—2441no—
636C2×C4—2441no—
637C2×C4—2441no—
638C2×C4—2441no—
639C2×C4—2441no—
640C2×C4—2441no—
641C2×C4—2441no—
642C2×C4—2441no—
643C2×C4—2441no—
644C2×C4—2441no—
645C2×C4—2441no—
646C2×C4—2441no—
647C2×C4—2441no—
648C2×C4—2441no—
649C2×C4—2441no—
650C2×C4#1451974712yes—
651C2×C4#1451974712yes—
652C2×C4C2×D4×D6111yescharacteristic
653C2×C4C2×D4×D6111yescharacteristic
654C2×C4#41×D6111yescharacteristic
655C2×C4#41×D6111yescharacteristic
656C2×C4#1451982211yescharacteristic
657C2×C4#1451982511yescharacteristic
658C8—41no—
659C8—41no—
660C8—41no—
661C8—41no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22C2×#1451974711yescharacteristic
670C22#1684027411yescharacteristic
671C22#1684034311yescharacteristic
672C22#1684057411yescharacteristic
673C22#1684057911yescharacteristic
674C22#1684057911yescharacteristic
675C22#1684057911yescharacteristic
676C4—22no—
677C4—22no—
678C4—41no—
679C4—41no—
680C4—41no—
681C4—4881no—
682C4—4881no—
683C4—2442no—
684C4—2442no—
685C4—2442no—
686C4—2442no—
687C4—2442no—
688C4—2442no—
689C4#1684040911yescharacteristic
690C4#1684041411yescharacteristic
691C4#1684042211yescharacteristic
692C4#1684043011yescharacteristic
693C2—81no—
694C2#2144916511yescharacteristic
695C2#2145511611yescharacteristic
696C2#2145513711yescharacteristic
697C2#2145603511yescharacteristic
698C2#2145612011yescharacteristic
699C2#2145724711yescharacteristic
700C2#2145727311yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1798C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1798D61
hypocenterC61#24165
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1798

Complements

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

NH ≅ G/Ntype
2. C2×#16840070C2semidirect
3. C2×#16839973C2semidirect
9. #21446774C2semidirect
10. #21446741C2semidirect
11. #21446159C2semidirect
12. #21446159C2semidirect
13. #21446090C2semidirect
14. #21446076C2semidirect
413. C61#24165semidirect

Maximal subgroups

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

subgroupindex
C2×#168403742
C2×#168400702
C2×#168399732
#214485012
#214470232
#214470162
#214468292
#214467742
#214467412
#214461592
#214460902
#214460762
C61×#17982
#2416561