Finite Group Databasebeta

← #29714329

Subgroups — #29714329

Order 15616.

1198 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic, complemented
2C2×#16840053C211yesmaximal, characteristic, complemented
3C2×#16839964C211yesmaximal, characteristic, complemented
4C4×#14519752C211yesmaximal, characteristic
5#21447014C212yesmaximal, complemented
6#21446821C212yesmaximal
7#21446767C211yesmaximal, characteristic, complemented
8#21446172C212yesmaximal, complemented
9#21446089C212yesmaximal, complemented
10#21445796C211yesmaximal, characteristic, complemented
11C61×#1981C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13#16840317—24no—
14#16840317—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519768—21no—
17C2×#14519757—21no—
18C2×#14519753C2212yes—
19C2×#14519752C2211yescharacteristic
20C2×Q8×Dic61C2212yes—
21C2×#14519742—21no—
22C2×C4×D244—21no—
23C2×C4×D244C2211yescharacteristic
24C2×#14519735—21no—
25C2×C42×D61C2211yescharacteristic
26#16840053—24no—
27#16839964—24no—
28C2×#14519709C2212yes—
29C2×C4⋊#92418C2211yescharacteristic
30C2×#14519701C2212yes—
31C2×#14519699C2212yes—
32C2×#14519689C2212yes—
33#16839663C2211yescharacteristic
34#16839661C2211yescharacteristic
35#16839654—21no—
36#16839640—21no—
37#16839619C2211yescharacteristic
38C4×#11758198—21no—
39#16839524C2212yes—
40#16839524C2212yes—
41#16839516C2212yes—
42C2×C4⋊Q8×C61C2211yescharacteristic
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×#103×C61C2212yes—
45C2×#103×C61C2212yes—
46C4×C4⋊C4×C61C2211yescharacteristic
47C22×Dic122—21no—
48C22×D244—21no—
49C22×D244C2311yescharacteristic
50C22×C4×D61—21no—
51C22×C4×D61D411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61C2311yescharacteristic
54C2×#11758201—24no—
55C2×#11758201—24no—
56C2×#11758201—42no—
57C22×#92418C2312yes—
58#14519777—22no—
59#14519777D414yescomplemented
60#14519768—28no—
61#14519757—28no—
62#14519753—24no—
63#14519753—24no—
64#14519752—22no—
65#14519752—22no—
66Q8×Dic61—24no—
67Q8×Dic61—24no—
68#14519742—28no—
69C4×D244—22no—
70C4×D244—22no—
71C4×D244—44no—
72#14519735—28no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—22no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2312yes—
78C2×#11758196—22no—
79C2×#11758198—22no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×C4×Dic61—22no—
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61D411yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89C42×D61—22no—
90C42×D61—22no—
91#14519709—24no—
92C4⋊#92418—22no—
93#14519701—24no—
94#14519699—24no—
95#14519689—42no—
96#14519684—21no—
97#14519684—21no—
98C22×Q8×C61C2312yes—
99C22×C8×C61C2312yes—
100C4⋊Q8×C61—22no—
101C4⋊Q8×C61D414yescomplemented
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C42×C61—21no—
107C2×C42×C61C2311yescharacteristic
108C4⋊C8×C61—22no—
109#103×C61—24no—
110#103×C61—42no—
111#95×C61—21no—
112C23×D61—21no—
113C23×D61C2×D411yescharacteristic
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61—22no—
117C2×Q8×C61—22no—
118C2×Q8×C61—22no—
119C2×Q8×C61C2×D414yes—
120C22×C4×C61—21no—
121C22×C4×C61—22no—
122C22×C4×C61—21no—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2411yescharacteristic
126C22×Dic61—22no—
127C22×Dic61—21no—
128C22×Dic61C2×D411yescharacteristic
129C22×Dic61C2×D411yescharacteristic
130C22×Dic61C2×D411yescharacteristic
131C2×D244—21no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244—21no—
135C2×D244—24no—
136C2×D244—21no—
137C2×D244—42no—
138C2×D244—42no—
139C2×D244#4112yes—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—21no—
144C2×C4×D61—21no—
145C2×C4×D61—21no—
146C2×C4×D61—21no—
147C2×C4×D61—21no—
148C2×C4×D61—21no—
149C2×C4×D61—41no—
150C2×C4×D61—41no—
151C2×C4×D61—42no—
152C2×C4×D61—42no—
153C2×C4×D61—42no—
154C2×C4×D61SD1614yescomplemented
155C2×Dic122—24no—
156C2×Dic122—42no—
157C2×Dic122—42no—
158C2×C8×C61—22no—
159C2×C8×C61—22no—
160C2×C8×C61—22no—
161C4⋊C4×C61—24no—
162C4⋊C4×C61—24no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61#4112yes—
167C42×C61—42no—
168C42×C61C2×D412yes—
169C42×C61C2×D412yes—
170#11758201—48no—
171#11758198—28no—
172#11758198—24no—
173#11758198—42no—
174#11758198—42no—
175#11758198—42no—
176#11758198—42no—
177C4⋊Dic61—28no—
178C4⋊Dic61—24no—
179C4⋊Dic61—22no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—42no—
183C4⋊Dic61C2×D414yes—
184#11758196—28no—
185C4×Dic61—28no—
186C4×Dic61—22no—
187C4×Dic61—22no—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418—22no—
193C23×C61C22×D411yescharacteristic
194C22×D61—21no—
195C22×D61—21no—
196C22×D61—21no—
197C22×D61—21no—
198C22×D61—21no—
199C22×D61—21no—
200C22×D61—22no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61C2×SD1612yes—
208Q8×C61—24no—
209Q8×C61—24no—
210Q8×C61—42no—
211Q8×C61—42no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—24no—
215C2×C4×C61—22no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61#12012yes—
224C2×C4×C61#12012yes—
225C2×C4×C61C2×SD1612yes—
226C2×C4×C61#13712yes—
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61H(5,𝔽2)11yescharacteristic
232C2×C4×C612−1+411yescharacteristic
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—24no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61C2×SD1612yes—
248D244—22no—
249D244—22no—
250D244—48no—
251D244—41no—
252D244—41no—
253C4×D61—28no—
254C4×D61—44no—
255C4×D61—41no—
256C4×D61—41no—
257C4×D61—41no—
258C4×D61—41no—
259C4×D61—84no—
260Dic122—48no—
261C8×C61—42no—
262#92418—42no—
263#24151—611nomaximal, Sylow
264C22×C61C2×#12011yescharacteristic
265C22×C61#52511yescharacteristic
266C22×C61#53711yescharacteristic
267C22×C61C22×SD1611yescharacteristic
268C22×C61C2×#13711yescharacteristic
269C22×C61#57511yescharacteristic
270C22×C61#57511yescharacteristic
271C2×D61—22no—
272C2×D61—22no—
273C2×D61—22no—
274C2×D61—41no—
275C2×D61—41no—
276C2×D61—41no—
277C2×D61—41no—
278C2×D61—42no—
279C2×D61—42no—
280C2×D61—42no—
281C2×D61—81no—
282C2×D61—81no—
283C2×D61—81no—
284C2×D61—81no—
285C4×C61—22no—
286C4×C61—42no—
287C4×C61—44no—
288C4×C61—81no—
289C4×C61#44814yes—
290C4×C61#54611yescharacteristic
291C4×C61#54711yescharacteristic
292C4×C61#54811yescharacteristic
293C4×C61#55011yescharacteristic
294Dic61—22no—
295Dic61—41no—
296Dic61—44no—
297Dic61—41no—
298Dic61—81no—
299C2×#533—611no—
300C2×#476—611no—
301C2×#461—611no—
302C4×#121—611no—
303#2035—612no—
304#2000—612no—
305#1987—611no—
306#1981D6111yescharacteristic, complemented
307#1880—612no—
308#1865—612no—
309#1797—611no—
310C2×C61#240711yescharacteristic
311C2×C61C2×#44811yescharacteristic
312C2×C61#300511yescharacteristic
313C2×C61#310111yescharacteristic
314C2×C61#311611yescharacteristic
315C2×C61#324611yescharacteristic
316C2×C61#325211yescharacteristic
317D61—42no—
318D61—82no—
319C22×SD16—612no—
320#533—1224no—
321#533—1224no—
322C2×C4⋊Q8—611no—
323C2×C4⋊Q8C2×D6111yescharacteristic
324C2×#125—1221no—
325C2×C22⋊Q8—612no—
326C2×C22⋊Q8—1221no—
327C2×#121—611no—
328C2×C4×Q8—612no—
329C2×C4×Q8—1221no—
330C2×C4×D4—611no—
331C2×C4×D4—1221no—
332C2×#117—1221no—
333C22×C42—611no—
334#476—1224no—
335#461—1224no—
336C2×#106—612no—
337C2×C4⋊C8—611no—
338C2×C4⋊C8C2×D6111yescharacteristic
339C2×#103—612no—
340C2×#103C2×D6112yes—
341C2×#103C2×D6112yes—
342C2×#102—612no—
343C2×C22⋊C8—612no—
344#390—611no—
345#389—611no—
346#388—1221no—
347#386—1221no—
348#380—611no—
349C4×C4⋊C4C2×D6111yescharacteristic
350C4×C22⋊C4—1221no—
351#340—612no—
352#340—612no—
353#336—612no—
354C61#2415111yescharacteristic, complemented, Sylow
355C22×Q8—1221no—
356C22×Q8C22×D6112yes—
357C22×D4—611no—
358C22×D4—1221no—
359C23×C4—611no—
360C23×C4—611no—
361C23×C4—611no—
362C23×C4—1221no—
363C2×SD16—1224no—
364C2×SD16—1224no—
365C2×SD16—2442no—
366C22×C8—612no—
367C22×C8C22×D6112yes—
368C4⋊Q8—22no—
369C4⋊Q8—614no—
370C4⋊Q8—1222no—
371C4⋊Q8D24414yescomplemented
372#125—1228no—
373C22⋊Q8—1228no—
374C22⋊Q8—1224no—
375C22⋊Q8—1224no—
376#121—1222no—
377#121—1222no—
378C4×Q8—1228no—
379C4×Q8—1224no—
380C4×Q8—1224no—
381C4×D4—1222no—
382C4×D4—1222no—
383C4×D4—2444no—
384#117—1228no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—612no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—1222no—
390C2×C4⋊C4—1222no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4C22×D6111yescharacteristic
394C2×C4⋊C4C22×D6111yescharacteristic
395C2×C22⋊C4—1221no—
396C2×C22⋊C4—1222no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C42—21no—
402C2×C42—611no—
403C2×C42—611no—
404C2×C42—611no—
405C2×C42—1222no—
406C2×C42—1222no—
407C2×C42—1222no—
408C2×C42C22×D6111yescharacteristic
409#106—1224no—
410C4⋊C8—22no—
411C4⋊C8—1222no—
412#103—24no—
413#103—42no—
414#103—1224no—
415#102—1224no—
416C22⋊C8—2442no—
417#95—21no—
418#95—1221no—
419#95—1221no—
420C24—611no—
421C24—1221no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—1224no—
428C2×Q8—2442no—
429C2×Q8—2442no—
430C2×Q8C2×D24414yes—
431C2×D4—612no—
432C2×D4—1224no—
433C2×D4—1221no—
434C2×D4—1221no—
435C2×D4—1221no—
436C2×D4—1221no—
437C2×D4—1221no—
438C2×D4—2442no—
439C2×D4—2442no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—611no—
444C22×C4—614no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1222no—
457C22×C4—1221no—
458C22×C4—2441no—
459C22×C4—2441no—
460C22×C4—2442no—
461C22×C4—2442no—
462C22×C4—2442no—
463C22×C4D4×D6111yescharacteristic
464C22×C4D4×D6111yescharacteristic
465C22×C4C23×D6111yescharacteristic
466SD16—2448no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—1222no—
471C2×C8—1222no—
472C2×C8—1222no—
473C4⋊C4—21no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—614no—
479C4⋊C4—1228no—
480C4⋊C4—1228no—
481C4⋊C4—1222no—
482C4⋊C4—1224no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—2442no—
486C4⋊C4#1175822412yes—
487C22⋊C4—1228no—
488C22⋊C4—1224no—
489C22⋊C4—2442no—
490C22⋊C4—2442no—
491C22⋊C4—2442no—
492C22⋊C4—2442no—
493C42—42no—
494C42—1228no—
495C42—1221no—
496C42—1221no—
497C42—1222no—
498C42—1222no—
499C42C2×D24412yes—
500C42D4×D6112yes—
501C23—612no—
502C23—1221no—
503C23—1221no—
504C23—1221no—
505C23—1221no—
506C23—1221no—
507C23—1221no—
508C23—1222no—
509C23—2441no—
510C23—2441no—
511C23—2441no—
512C23—2441no—
513C23—2441no—
514C23—2441no—
515C23C2×D4×D6111yescharacteristic
516Q8—24no—
517Q8—24no—
518Q8—42no—
519Q8—42no—
520Q8—2448no—
521D4—1222no—
522D4—1222no—
523D4—2448no—
524D4—2441no—
525D4—2441no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—612no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1228no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1224no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—4884no—
549C2×C4—2444no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4#1451974712yes—
560C2×C4#1451974712yes—
561C2×C4SD16×D6112yes—
562C2×C4#1451980912yes—
563C2×C4C2×D4×D6111yescharacteristic
564C2×C4C22×D24411yescharacteristic
565C2×C4#41×D6111yescharacteristic
566C2×C4C2×#1175822411yescharacteristic
567C2×C4#1451982211yescharacteristic
568C2×C4#1451982411yescharacteristic
569C8—42no—
570C8—2442no—
571C22—1222no—
572C22—1222no—
573C22—1222no—
574C22—4881no—
575C22—4881no—
576C22—4881no—
577C22—4881no—
578C22—2442no—
579C22—2442no—
580C22—2442no—
581C22—2441no—
582C22—2441no—
583C22—2441no—
584C22—2441no—
585C22C2×#1451974711yescharacteristic
586C22#1684027211yescharacteristic
587C22#1684034111yescharacteristic
588C22C2×SD16×D6111yescharacteristic
589C22C2×#1451980911yescharacteristic
590C22#1684056111yescharacteristic
591C22#1684056111yescharacteristic
592C4—22no—
593C4—44no—
594C4—42no—
595C4—81no—
596C4—1222no—
597C4—4881no—
598C4—2441no—
599C4—2441no—
600C4—2444no—
601C4#1683990214yes—
602C4#1684040711yescharacteristic
603C4#1684041611yescharacteristic
604C4#1684042011yescharacteristic
605C4#1684043211yescharacteristic
606C2—4882no—
607C2—2442no—
608C2#2144916211yescharacteristic
609C2C2×#1683990211yescharacteristic
610C2#2145506511yescharacteristic
611C2#2145582611yescharacteristic
612C2#2145598011yescharacteristic
613C2#2145713211yescharacteristic
614C2#2145719011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1981C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
hypercenter#1981D61
hypocenterC61#24151
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
1. C2×#16840317C2semidirect
2. C2×#16840053C2semidirect
3. C2×#16839964C2semidirect
5. #21447014C2semidirect
7. #21446767C2semidirect
8. #21446172C2semidirect
9. #21446089C2semidirect
10. #21445796C2semidirect
11. C61×#1981C2semidirect
59. #14519777D4semidirect
101. C4⋊Q8×C61D4semidirect
154. C2×C4×D61SD16semidirect
306. #1981D61semidirect
354. C61#24151semidirect
371. C4⋊Q8D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
C2×#168400532
C2×#168399642
C4×#145197522
#214470142
#214468212
#214467672
#214461722
#214460892
#214457962
C61×#19812
#2415161