Finite Group Databasebeta

← #29708423

Subgroups — #29708423

Order 15616.

1058 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×#16840370C211yesmaximal, characteristic
2C2×#16839867C211yesmaximal, characteristic, complemented
3C2×C4×Dic244C211yesmaximal, characteristic, complemented
4#21448502C211yesmaximal, characteristic
5#21447160C211yesmaximal, characteristic, complemented
6#21447064C211yesmaximal, characteristic, complemented
7#21447021C211yesmaximal, characteristic
8#21447018C211yesmaximal, characteristic
9#21446969C211yesmaximal, characteristic, complemented
10#21446920C211yesmaximal, characteristic, complemented
11#21446830C211yesmaximal, characteristic
12#21446828C211yesmaximal, characteristic
13#21445652C211yesmaximal, characteristic, complemented
14#21445586C211yesmaximal, characteristic
15C4×#103×C61C211yesmaximal, characteristic, complemented
16C22×Dic244C2211yescharacteristic
17C22×#11758202C2211yescharacteristic
18#16840370—24no—
19#16840370—24no—
20C2×#14519775C2211yescharacteristic
21C2×#14519772—21no—
22C2×#14519771—21no—
23C2×#14519769—21no—
24C2×#14519768C2211yescharacteristic
25C2×#14519767—21no—
26C2×#14519757C2211yescharacteristic
27C2×#14519757C2211yescharacteristic
28C2×#14519753C2211yescharacteristic
29C2×C4×Dic122C2211yescharacteristic
30C2×C4×Dic122C2211yescharacteristic
31C2×#14519734C2211yescharacteristic
32#16839867—24no—
33C4×Dic244—24no—
34C2×#14519711C2211yescharacteristic
35C2×#14519708C2211yescharacteristic
36C2×#14519702C2211yescharacteristic
37C2×#14519702C2211yescharacteristic
38C2×#14519701C2211yescharacteristic
39C2×#14519700C2211yescharacteristic
40C2×#14519700C2211yescharacteristic
41C2×#14519700C2211yescharacteristic
42C2×#14519689C2211yescharacteristic
43C2×#14519689C2211yescharacteristic
44C2×#14519688C2211yescharacteristic
45#16839667C2211yescharacteristic
46#16839661C2211yescharacteristic
47#16839660—21no—
48#16839654—21no—
49#16839653—21no—
50#16839647—21no—
51#16839619C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839516C2211yescharacteristic
55#16839516C2211yescharacteristic
56C2×C4×Q8×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C2×C4×C8×C61C2211yescharacteristic
60C4×C4⋊C4×C61C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C61×#340C2211yescharacteristic
63C22×Dic122C2311yescharacteristic
64C22×Dic122C2311yescharacteristic
65C22×C4×D61C2311yescharacteristic
66C2×Dic244—22no—
67C2×Dic244—22no—
68C2×Dic244—22no—
69C2×#11758202—22no—
70C2×#11758202—22no—
71C2×#11758202—22no—
72C22×#92418C2311yescharacteristic
73C22×#92418C2311yescharacteristic
74#14519775—22no—
75#14519775—22no—
76#14519772—28no—
77#14519771—28no—
78#14519769—28no—
79#14519768—22no—
80#14519768—22no—
81#14519767—28no—
82#14519757—22no—
83#14519757—22no—
84#14519757—22no—
85#14519757—22no—
86#14519753—22no—
87#14519753—22no—
88C4×Dic122—22no—
89C4×Dic122—22no—
90C4×Dic122—22no—
91C4×Dic122D414yes—
92#14519734—22no—
93#14519734—22no—
94C2×C4⋊Dic61—21no—
95C2×C4⋊Dic61—21no—
96C2×C4⋊Dic61—21no—
97C2×C4⋊Dic61—21no—
98C2×C4⋊Dic61C2311yescharacteristic
99C2×C4⋊Dic61C2311yescharacteristic
100C2×#11758196—21no—
101C2×#11758196—21no—
102C2×#11758196—21no—
103C2×#11758196—21no—
104C2×#11758196—21no—
105C2×#11758196—21no—
106C2×#11758196D411yescharacteristic
107C2×#11758196D411yescharacteristic
108C2×#11758198—21no—
109C2×#11758198—21no—
110C2×#11758198—21no—
111C2×#11758198—21no—
112C2×#11758198D411yescharacteristic
113C2×#11758198D411yescharacteristic
114C2×C4×Dic61—21no—
115C2×C4×Dic61—21no—
116C2×C4×Dic61C2311yescharacteristic
117#14519711—22no—
118#14519708—22no—
119#14519702—22no—
120#14519702—22no—
121#14519701—22no—
122#14519700—22no—
123#14519700—22no—
124#14519700—22no—
125#14519689—22no—
126#14519689—22no—
127#14519688—22no—
128#14519684—21no—
129#14519684—21no—
130#14519684—21no—
131#14519684—21no—
132C22×Q8×C61C2311yescharacteristic
133C22×C8×C61C2311yescharacteristic
134C22×C8×C61C2311yescharacteristic
135C4×Q8×C61—22no—
136C4×Q8×C61D414yes—
137C2×C4⋊C4×C61—21no—
138C2×C4⋊C4×C61C2311yescharacteristic
139C2×C4⋊C4×C61C2311yescharacteristic
140C2×C4⋊C4×C61C2311yescharacteristic
141C2×C42×C61—21no—
142C2×C42×C61—21no—
143C2×C42×C61C2311yescharacteristic
144#103×C61—22no—
145#103×C61—22no—
146C4×C8×C61—22no—
147#95×C61—21no—
148C23×D61C2×D411yescharacteristic
149C2×Q8×C61—21no—
150C2×Q8×C61—21no—
151C2×Q8×C61—21no—
152C2×Q8×C61—21no—
153C2×Q8×C61—21no—
154C2×Q8×C61C2×D412yes—
155C22×C4×C61—21no—
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2411yescharacteristic
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—21no—
165C22×Dic61—21no—
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168C22×Dic61C2×D411yescharacteristic
169C2×C4×D61—21no—
170C2×C4×D61—21no—
171C2×C4×D61—21no—
172C2×C4×D61—21no—
173C2×C4×D61—21no—
174C2×C4×D61—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×Dic122—21no—
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#4112yes—
193C4⋊C4×C61—24no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—21no—
198C4⋊C4×C61C2×D412yes—
199C4⋊C4×C61#4112yes—
200C42×C61—24no—
201C42×C61—81no—
202C42×C61C2×D412yes—
203C42×C61C2×D412yes—
204#11758202—28no—
205#11758198—24no—
206#11758198—24no—
207#11758198—24no—
208#11758198—22no—
209#11758198—22no—
210#11758198—81no—
211C4⋊Dic61—24no—
212C4⋊Dic61—24no—
213C4⋊Dic61—24no—
214C4⋊Dic61—21no—
215C4⋊Dic61—21no—
216C4⋊Dic61—21no—
217C4⋊Dic61—81no—
218C4⋊Dic61C2×D412yes—
219#11758196—24no—
220#11758196—24no—
221#11758196—24no—
222#11758196—24no—
223#11758196—24no—
224#11758196—22no—
225#11758196—22no—
226#11758196—42no—
227C4×Dic61—24no—
228C4×Dic61—21no—
229C4×Dic61—21no—
230C4×Dic61—42no—
231C2×#92418—21no—
232C2×#92418—21no—
233C2×#92418—21no—
234C2×#92418—21no—
235C2×#92418—21no—
236C2×#92418#4112yes—
237Dic244—28no—
238C23×C61C22×D411yescharacteristic
239C22×D61—21no—
240C22×D61—21no—
241C22×D61—21no—
242C22×D61—21no—
243C22×D61—21no—
244C22×D61—21no—
245C22×D61—21no—
246Q8×C61—22no—
247Q8×C61—22no—
248Q8×C61—41no—
249Q8×C61—41no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61#12012yes—
267C2×C4×C61#12012yes—
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61C2×#4111yescharacteristic
271C2×C4×C61C2×#4111yescharacteristic
272C2×C4×C61C2×#4111yescharacteristic
273C2×C4×C61C2×#4111yescharacteristic
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—22no—
277C2×Dic61—22no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—22no—
281C2×Dic61—21no—
282C2×Dic61—21no—
283C2×Dic61—21no—
284C2×Dic61—21no—
285C2×Dic61—21no—
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—
295C4×D61—41no—
296C4×D61—41no—
297C4×D61—41no—
298C4×D61—41no—
299Dic122—22no—
300Dic122—22no—
301Dic122—22no—
302Dic122—22no—
303Dic122—41no—
304Dic122—41no—
305Dic122—41no—
306Dic122—41no—
307C8×C61—22no—
308C8×C61—41no—
309#92418—22no—
310#92418—41no—
311#23586—611nomaximal, Sylow
312C22×C61C2×#12011yescharacteristic
313C22×C61#52511yescharacteristic
314C22×C61#53211yescharacteristic
315C22×C61#57711yescharacteristic
316C22×C61#57811yescharacteristic
317C22×C61#57811yescharacteristic
318C22×C61#57811yescharacteristic
319C2×D61—41no—
320C2×D61—41no—
321C2×D61—41no—
322C2×D61—41no—
323C2×D61—41no—
324C2×D61—41no—
325C2×D61—41no—
326C4×C61—22no—
327C4×C61—22no—
328C4×C61—42no—
329C4×C61—42no—
330C4×C61—42no—
331C4×C61—81no—
332C4×C61#52511yescharacteristic
333C4×C61#52511yescharacteristic
334C4×C61#53211yescharacteristic
335C4×C61#53211yescharacteristic
336Dic61—42no—
337Dic61—42no—
338Dic61—41no—
339Dic61—41no—
340Dic61—41no—
341Dic61—42no—
342Dic61—81no—
343C2×#541—611no—
344C2×#441—611no—
345C2×C4×Q16—611no—
346#2323—611no—
347#2059—611no—
348#2043—611no—
349#2036—611no—
350#2035—611no—
351#2026—611no—
352#2016—611no—
353#2001—611no—
354#2001—611no—
355#1763—611no—
356C4×#103D6111yescharacteristic, complemented
357#1750—611no—
358C2×C61#238811yescharacteristic
359C2×C61#301111yescharacteristic
360C2×C61#301311yescharacteristic
361C2×C61#312011yescharacteristic
362C2×C61#312711yescharacteristic
363C2×C61#317711yescharacteristic
364C2×C61#317811yescharacteristic
365D61—81no—
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×Q8C2×D6111yescharacteristic
382C2×#117—611no—
383#441—1224no—
384C4×Q16—1224no—
385C2×#107—611no—
386C2×#106—611no—
387C2×#103—611no—
388C2×#103—611no—
389C2×#103—611no—
390C2×#103—611no—
391C2×#103—611no—
392C2×#103—611no—
393C2×#103C2×D6111yescharacteristic
394C2×#103C2×D6111yescharacteristic
395C2×C22⋊C8—611no—
396C2×C22⋊C8—611no—
397C2×#97—611no—
398C2×C4×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⋊C4C2×D6111yescharacteristic
407#340—611no—
408#340—611no—
409#340C2×D6111yescharacteristic
410#340C2×D6111yescharacteristic
411#336—611no—
412#336—611no—
413C61#2358611yescharacteristic, complemented, Sylow
414C22×Q8—611no—
415C22×Q8—611no—
416C22×Q8C22×D6111yescharacteristic
417C23×C4—611no—
418C2×Q16—1222no—
419C2×Q16—1222no—
420C2×Q16—1222no—
421C2×Q16—1222no—
422C2×Q16—1222no—
423C2×Q16—1222no—
424C22×C8—611no—
425C22×C8—611no—
426C22×C8C22×D6111yescharacteristic
427C22×C8C22×D6111yescharacteristic
428C4⋊Q8—1222no—
429C4⋊Q8—1222no—
430#126—1228no—
431#126—1228no—
432#125—1228no—
433#125—1222no—
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—22no—
443C4×Q8—614no—
444C4×Q8—1222no—
445C4×Q8—1222no—
446C4×Q8—1222no—
447C4×Q8D24414yes—
448#117—1222no—
449#117—1222no—
450C2×C4⋊C4—21no—
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⋊C4C22×D6111yescharacteristic
466C2×C4⋊C4C22×D6111yescharacteristic
467C2×C4⋊C4C22×D6111yescharacteristic
468C2×C22⋊C4—611no—
469C2×C22⋊C4—611no—
470C2×C22⋊C4—1221no—
471C2×C22⋊C4—1221no—
472C2×C22⋊C4—1221no—
473C2×C22⋊C4—1221no—
474C2×C42—21no—
475C2×C42—21no—
476C2×C42—611no—
477C2×C42—1221no—
478C2×C42—1221no—
479C2×C42C22×D6111yescharacteristic
480#107—1222no—
481#106—1222no—
482#103—22no—
483#103—22no—
484#103—1222no—
485#103—1222no—
486#103—1222no—
487#103—1222no—
488#103—1222no—
489#103—1222no—
490C22⋊C8—1222no—
491C22⋊C8—1222no—
492#97—1222no—
493C4×C8—22no—
494#95—21no—
495#95—1221no—
496#95—1221no—
497#95—1221no—
498#95—1221no—
499C24—611no—
500C2×Q8—21no—
501C2×Q8—21no—
502C2×Q8—21no—
503C2×Q8—21no—
504C2×Q8—21no—
505C2×Q8—612no—
506C2×Q8—612no—
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×Q8C2×D24412yes—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—21no—
521C22×C4—21no—
522C22×C4—611no—
523C22×C4—611no—
524C22×C4—611no—
525C22×C4—1221no—
526C22×C4—1221no—
527C22×C4—1221no—
528C22×C4—1221no—
529C22×C4—1221no—
530C22×C4—1221no—
531C22×C4—1221no—
532C22×C4—1221no—
533C22×C4—1221no—
534C22×C4—1221no—
535C22×C4D4×D6111yescharacteristic
536C22×C4D4×D6111yescharacteristic
537C22×C4C23×D6111yescharacteristic
538Q16—1228no—
539Q16—1228no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—21no—
544C2×C8—21no—
545C2×C8—612no—
546C2×C8—1221no—
547C2×C8—1221no—
548C2×C8—1221no—
549C2×C8—1221no—
550C2×C8—1221no—
551C2×C8#1175822212yes—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4—24no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—612no—
558C4⋊C4—1224no—
559C4⋊C4—1224no—
560C4⋊C4—1224no—
561C4⋊C4—1224no—
562C4⋊C4—1224no—
563C4⋊C4—1224no—
564C4⋊C4—1224no—
565C4⋊C4—1221no—
566C4⋊C4—1222no—
567C4⋊C4—1222no—
568C4⋊C4—1224no—
569C4⋊C4—1221no—
570C4⋊C4—1221no—
571C4⋊C4—4881no—
572C4⋊C4—2442no—
573C4⋊C4C2×D24412yes—
574C4⋊C4#1175822212yes—
575C22⋊C4—1224no—
576C22⋊C4—1224no—
577C22⋊C4—1224no—
578C22⋊C4—1222no—
579C22⋊C4—1222no—
580C22⋊C4—4881no—
581C42—24no—
582C42—81no—
583C42—1221no—
584C42—1221no—
585C42—1224no—
586C42—2442no—
587C42C2×D24412yes—
588C42D4×D6112yes—
589C23—1221no—
590C23—1221no—
591C23—1221no—
592C23—1221no—
593C23—1221no—
594C23—1221no—
595C23—1221no—
596C23C2×D4×D6111yescharacteristic
597Q8—22no—
598Q8—22no—
599Q8—41no—
600Q8—41no—
601Q8—1222no—
602Q8—1222no—
603Q8—1222no—
604Q8—1222no—
605Q8—2441no—
606Q8—2441no—
607Q8—2441no—
608Q8—2441no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—1222no—
626C2×C4—1221no—
627C2×C4—1221no—
628C2×C4—1222no—
629C2×C4—1221no—
630C2×C4—1222no—
631C2×C4—1221no—
632C2×C4—1221no—
633C2×C4—1221no—
634C2×C4—1221no—
635C2×C4—1221no—
636C2×C4—1221no—
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×C4C22×D24411yescharacteristic
654C2×C4#41×D6111yescharacteristic
655C2×C4#41×D6111yescharacteristic
656C2×C4C2×#1175822211yescharacteristic
657C2×C4C2×#1175822211yescharacteristic
658C8—22no—
659C8—41no—
660C8—1222no—
661C8—2441no—
662C22—2441no—
663C22—2441no—
664C22—2441no—
665C22—2441no—
666C22—2441no—
667C22—2441no—
668C22—2441no—
669C22C2×#1451974711yescharacteristic
670C22#1684027111yescharacteristic
671C22#1684030911yescharacteristic
672C22#1684056811yescharacteristic
673C22#1684057511yescharacteristic
674C22#1684057611yescharacteristic
675C22#1684057611yescharacteristic
676C4—22no—
677C4—22no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—81no—
682C4—4881no—
683C4—2441no—
684C4—2441no—
685C4—2442no—
686C4—2441no—
687C4—2442no—
688C4—2442no—
689C4#1684027111yescharacteristic
690C4#1684027411yescharacteristic
691C4#1684030911yescharacteristic
692C4#1684031211yescharacteristic
693C2—4881no—
694C2#2144903511yescharacteristic
695C2#2145511711yescharacteristic
696C2#2145513911yescharacteristic
697C2#2145602911yescharacteristic
698C2#2145611611yescharacteristic
699C2#2145647411yescharacteristic
700C2#2145648311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×#103×C61C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenterC4×#103D61
hypocenterC61#23586
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
2. C2×#16839867C2semidirect
3. C2×C4×Dic244C2semidirect
5. #21447160C2semidirect
6. #21447064C2semidirect
9. #21446969C2semidirect
10. #21446920C2semidirect
13. #21445652C2semidirect
15. C4×#103×C61C2semidirect
356. C4×#103D61semidirect
413. C61#23586semidirect

Maximal subgroups

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

subgroupindex
C2×#168403702
C2×#168398672
C2×C4×Dic2442
#214485022
#214471602
#214470642
#214470212
#214470182
#214469692
#214469202
#214468302
#214468282
#214456522
#214455862
C4×#103×C612
#2358661