Finite Group Databasebeta

← #29689694

Subgroups — #29689694

Order 15616.

1222 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840316C211yesmaximal, characteristic
2C2×#16839814C211yesmaximal, characteristic
3#384×D61C211yesmaximal, characteristic
4#21446778C212yesmaximal, complemented
5#21446619C212yesmaximal, complemented
6#21446125C212yesmaximal
7#21446123C212yesmaximal
8#21445825C211yesmaximal, characteristic, complemented
9#21445824C211yesmaximal, characteristic, complemented
10#21445821C211yesmaximal, characteristic, complemented
11C61×#1803C211yesmaximal, characteristic, complemented
12#16840316—24no—
13#16840316C418yescomplemented
14C2×#14519776C2211yescharacteristic
15C2×#14519775C2211yescharacteristic
16C2×#14519767—21no—
17C2×#14519757C2211yescharacteristic
18C2×#14519755—21no—
19C2×#14519753C2211yescharacteristic
20C2×C4×Dic122—21no—
21C2×C4×Dic122C2211yescharacteristic
22C2×#14519734—21no—
23C2×C4⋊C4×D61—21no—
24C2×C4⋊C4×D61—21no—
25C2×C4⋊C4×D61C2212yes—
26C2×C42×D61C2211yescharacteristic
27#16839814—24no—
28C2×C4⋊#92418C2211yescharacteristic
29C2×#14519702C2212yes—
30C2×#14519701C2212yes—
31C2×#14519700C2212yes—
32C2×#14519689C2212yes—
33C2×C8×Dic61C2212yes—
34#16839632C2211yescharacteristic
35#16839631—21no—
36#16839631—21no—
37#16839631C2212yes—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C4×Q8×C61C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#103×C61C2212yes—
44C61×#384C2211yescharacteristic
45C61×#340C2212yes—
46C22×Dic122—21no—
47C22×Dic122C2311yescharacteristic
48C22×C4×D61—22no—
49C22×C4×D61D411yescharacteristic
50C22×C4×D61D411yescharacteristic
51C22×C4×D61C2311yescharacteristic
52C22×#92418C2312yes—
53#14519776—22no—
54#14519776C2×C414yescomplemented
55#14519775—22no—
56#14519775C2×C414yescomplemented
57#14519767—28no—
58#14519757—22no—
59#14519757C2×C414yes—
60#14519755—28no—
61#14519753—22no—
62#14519753C2×C414yes—
63C4×Dic122—22no—
64C4×Dic122—28no—
65C4×Dic122C2×C414yescomplemented
66#14519734—28no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2312yes—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—22no—
78C2×#11758196—22no—
79C2×#11758196—22no—
80C4⋊C4×D61—42no—
81C4⋊C4×D61—42no—
82C4⋊C4×D61—44no—
83C4⋊C4×D61—44no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61D411yescharacteristic
88C2×C4×Dic61D411yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90C42×D61C2×C414yes—
91C42×D61D414yes—
92C4⋊#92418—22no—
93#14519702—42no—
94#14519701—42no—
95#14519700—42no—
96#14519689—24no—
97C8×Dic61—24no—
98C22×Q8×C61C2311yescharacteristic
99C22×C8×C61C2312yes—
100C4×Q8×C61—22no—
101C4×Q8×C61C2×C414yescomplemented
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61C2312yes—
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C42×C61—21no—
108C2×C42×C61C2311yescharacteristic
109C4⋊C8×C61—22no—
110#103×C61—42no—
111C23×D61C2×D411yescharacteristic
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61—21no—
117C2×Q8×C61C22×C412yes—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—22no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2411yescharacteristic
124C22×Dic61—21no—
125C22×Dic61—21no—
126C22×Dic61—22no—
127C22×Dic61C2×D411yescharacteristic
128C22×Dic61C2×D411yescharacteristic
129C22×Dic61C2×D411yescharacteristic
130C2×C4×D61—21no—
131C2×C4×D61—21no—
132C2×C4×D61—21no—
133C2×C4×D61—21no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—44no—
137C2×C4×D61—44no—
138C2×C4×D61—44no—
139C2×C4×D61C22⋊C414yes—
140C2×C4×D61C22⋊C414yes—
141C2×C4×D61SD1614yes—
142C2×C4×D61Q1614yes—
143C2×C4×D61C22×C412yes—
144C2×C4×D61C2×D412yes—
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122—21no—
148C2×Dic122—21no—
149C2×Dic122—21no—
150C2×Dic122—24no—
151C2×Dic122—42no—
152C2×Dic122—42no—
153C2×Dic122C22×C412yes—
154C2×C8×C61—22no—
155C2×C8×C61—22no—
156C2×C8×C61—22no—
157C4⋊C4×C61—24no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—42no—
162C4⋊C4×C61—42no—
163C4⋊C4×C61C22×C412yes—
164C42×C61—24no—
165C42×C61C22×C412yes—
166C42×C61C2×D412yes—
167#11758198—24no—
168#11758198—24no—
169C4⋊Dic61—21no—
170C4⋊Dic61—24no—
171C4⋊Dic61—22no—
172C4⋊Dic61—22no—
173C4⋊Dic61—42no—
174C4⋊Dic61—42no—
175C4⋊Dic61C22×C412yes—
176#11758196—24no—
177#11758196—24no—
178#11758196—24no—
179#11758196—24no—
180#11758196—24no—
181#11758196—44no—
182#11758196—44no—
183#11758196—44no—
184C4×Dic61—24no—
185C4×Dic61C22⋊C414yes—
186C4×Dic61C22⋊C414yes—
187C4×Dic61C22×C412yes—
188C4×Dic61C2×D412yes—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418—22no—
192C23×C61C22×D411yescharacteristic
193C22×D61—21no—
194C22×D61—21no—
195C22×D61—21no—
196C22×D61—21no—
197C22×D61C2×C22⋊C412yes—
198C22×D61C2×SD1612yes—
199C22×D61C2×Q1612yes—
200Q8×C61—22no—
201Q8×C61—22no—
202Q8×C61—41no—
203Q8×C61—41no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61C2×C22⋊C412yes—
216C2×C4×C61C2×C22⋊C412yes—
217C2×C4×C61C2×SD1612yes—
218C2×C4×C61C2×Q1612yes—
219C2×C4×C61C23×C411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61H(5,𝔽2)11yescharacteristic
222C2×C4×C61H(5,𝔽2)11yescharacteristic
223C2×C4×C612−1+411yescharacteristic
224C2×C4×C612−1+411yescharacteristic
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61C2×C22⋊C412yes—
237C2×Dic61C2×C22⋊C412yes—
238C2×Dic61C2×C22⋊C412yes—
239C2×Dic61C2×SD1612yes—
240C2×Dic61C2×Q1612yes—
241C2×Dic61AGL(1,ℤ/8)12yes—
242C2×Dic61#13712yes—
243C4×D61—22no—
244C4×D61—22no—
245C4×D61—22no—
246C4×D61—22no—
247C4×D61—28no—
248C4×D61—88no—
249C4×D61#103116yescomplemented
250Dic122—22no—
251Dic122—22no—
252Dic122—41no—
253Dic122—41no—
254Dic122—48no—
255C8×C61—42no—
256#92418—42no—
257#21860—611nomaximal, Sylow
258C22×C61C22×C22⋊C411yescharacteristic
259C22×C61#53511yescharacteristic
260C22×C61#53611yescharacteristic
261C22×C61C22×SD1611yescharacteristic
262C22×C61C22×Q1611yescharacteristic
263C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
264C22×C61C2×#13711yescharacteristic
265C2×D61—22no—
266C2×D61—22no—
267C2×D61—22no—
268C2×D61—22no—
269C2×D61—22no—
270C2×D61—22no—
271C2×D61C2×#10314yes—
272C4×C61—22no—
273C4×C61—42no—
274C4×C61—42no—
275C4×C61—82no—
276C4×C61C2×#10314yes—
277C4×C61#52011yescharacteristic
278C4×C61#52011yescharacteristic
279C4×C61#53711yescharacteristic
280C4×C61#53711yescharacteristic
281Dic61—22no—
282Dic61—42no—
283Dic61—42no—
284Dic61—82no—
285Dic61C2×#10314yes—
286Dic61#41714yes—
287C2×#533—611no—
288C2×#431—611no—
289C2×#384—611no—
290#1990—612no—
291#1956—612no—
292#1872—612no—
293#1872—612no—
294#1803—611no—
295#1803—611no—
296#1803—611no—
297#1803D6111yescharacteristic, complemented
298C2×C61#231111yescharacteristic
299C2×C61C22×#10311yescharacteristic
300C2×C61C2×#41711yescharacteristic
301C2×C61#320811yescharacteristic
302C2×C61#320911yescharacteristic
303C2×C61#322911yescharacteristic
304C2×C61#323011yescharacteristic
305D61—24no—
306#533—618no—
307#533—1224no—
308C2×C4⋊Q8—611no—
309C2×C4⋊Q8—611no—
310C2×#125—1221no—
311C2×C22⋊Q8—611no—
312C2×C22⋊Q8—611no—
313C2×C22⋊Q8—1221no—
314C2×C4×Q8—611no—
315C2×C4×Q8—1221no—
316C2×C4×Q8C2×D6111yescharacteristic
317C2×#117—1221no—
318C22×C4⋊C4—612no—
319C22×C4⋊C4—1221no—
320C22×C4⋊C4—1221no—
321C22×C42—611no—
322#431—1224no—
323C2×C4⋊C8—611no—
324C2×C4⋊C8C2×D6111yescharacteristic
325C2×#103—612no—
326C2×#103—612no—
327C2×#103—612no—
328C2×#103C2×D6112yes—
329C2×C22⋊C8—612no—
330C2×C4×C8—612no—
331#384—612no—
332#384—611no—
333#384—1221no—
334#384—1221no—
335#384C2×D6111yescharacteristic
336#340—612no—
337#340—612no—
338#340—612no—
339#340C2×D6112yes—
340C61#2186011yescharacteristic, complemented, Sylow
341C22×Q8—611no—
342C22×Q8—1221no—
343C22×Q8C22×D6111yescharacteristic
344C23×C4—611no—
345C23×C4—611no—
346C23×C4—611no—
347C23×C4—1222no—
348C22×C8—612no—
349C22×C8C22×D6112yes—
350C4⋊Q8—614no—
351C4⋊Q8—614no—
352C4⋊Q8—1222no—
353C4⋊Q8—1222no—
354#125—1228no—
355C22⋊Q8—614no—
356C22⋊Q8—614no—
357C22⋊Q8—1222no—
358C22⋊Q8—1228no—
359C22⋊Q8—1222no—
360C4×Q8—22no—
361C4×Q8—614no—
362C4×Q8—1228no—
363C4×Q8—1222no—
364C4×Q8C4×D6114yescomplemented
365#117—1228no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—612no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4—1222no—
379C2×C4⋊C4—1222no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—2442no—
383C2×C4⋊C4—2442no—
384C2×C4⋊C4—2444no—
385C2×C4⋊C4—2444no—
386C2×C4⋊C4C22×D6112yes—
387C2×C4⋊C4C22×D6111yescharacteristic
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C42—21no—
391C2×C42—611no—
392C2×C42—611no—
393C2×C42—611no—
394C2×C42—614no—
395C2×C42—614no—
396C2×C42—1221no—
397C2×C42C22×D6111yescharacteristic
398C4⋊C8—22no—
399C4⋊C8—1222no—
400#103—42no—
401#103—2442no—
402#103—2442no—
403#103—2442no—
404C22⋊C8—1224no—
405C4×C8—1224no—
406C24—611no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—612no—
413C2×Q8—1224no—
414C2×Q8—1221no—
415C2×Q8—1221no—
416C2×Q8—1221no—
417C2×Q8—1221no—
418C2×Q8—1221no—
419C2×Q8—2442no—
420C2×Q8—2442no—
421C2×Q8C2×C4×D6112yes—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—614no—
428C22×C4—614no—
429C22×C4—614no—
430C22×C4—611no—
431C22×C4—612no—
432C22×C4—612no—
433C22×C4—614no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1221no—
437C22×C4—1221no—
438C22×C4—1221no—
439C22×C4—1221no—
440C22×C4—1221no—
441C22×C4—1221no—
442C22×C4—1222no—
443C22×C4—2444no—
444C22×C4—2444no—
445C22×C4—2444no—
446C22×C4D4×D6111yescharacteristic
447C22×C4D4×D6111yescharacteristic
448C22×C4C23×D6111yescharacteristic
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—1222no—
453C2×C8—1222no—
454C2×C8—1222no—
455C4⋊C4—21no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—612no—
462C4⋊C4—1224no—
463C4⋊C4—1224no—
464C4⋊C4—1224no—
465C4⋊C4—1224no—
466C4⋊C4—1224no—
467C4⋊C4—1221no—
468C4⋊C4—1224no—
469C4⋊C4—1222no—
470C4⋊C4—1222no—
471C4⋊C4—2442no—
472C4⋊C4—2444no—
473C4⋊C4—2444no—
474C4⋊C4—2444no—
475C4⋊C4—2442no—
476C4⋊C4C2×C4×D6112yes—
477C22⋊C4—1224no—
478C22⋊C4—1224no—
479C42—24no—
480C42—612no—
481C42—612no—
482C42—614no—
483C42—614no—
484C42—1224no—
485C42C2×C4×D6112yes—
486C42D4×D6112yes—
487C23—612no—
488C23—612no—
489C23—612no—
490C23—1221no—
491C23—1221no—
492C23—1221no—
493C23—1221no—
494C23C2×D4×D6111yescharacteristic
495Q8—22no—
496Q8—22no—
497Q8—41no—
498Q8—41no—
499Q8—1222no—
500Q8—1222no—
501Q8—2448no—
502Q8—2441no—
503Q8—2441no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—612no—
516C2×C4—612no—
517C2×C4—612no—
518C2×C4—612no—
519C2×C4—612no—
520C2×C4—612no—
521C2×C4—6116no—
522C2×C4—612no—
523C2×C4—1228no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—4888no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4—2442no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4C22⋊C4×D6112yes—
541C2×C4C22⋊C4×D6112yes—
542C2×C4SD16×D6112yes—
543C2×C4Q16×D6112yes—
544C2×C4C22×C4×D6111yescharacteristic
545C2×C4C2×D4×D6111yescharacteristic
546C2×C4#1451982211yescharacteristic
547C2×C4#1451982311yescharacteristic
548C2×C4#1451982411yescharacteristic
549C2×C4#1451982511yescharacteristic
550C8—42no—
551C8—2442no—
552C22—614no—
553C22—1222no—
554C22—1222no—
555C22—1222no—
556C22—1222no—
557C22—1222no—
558C22—1222no—
559C22C2×C22⋊C4×D6111yescharacteristic
560C22#1684032811yescharacteristic
561C22#1684033511yescharacteristic
562C22C2×SD16×D6111yescharacteristic
563C22C2×Q16×D6111yescharacteristic
564C22AGL(1,ℤ/8)×D6111yescharacteristic
565C22#137×D6111yescharacteristic
566C4—22no—
567C4—42no—
568C4—42no—
569C4—82no—
570C4—614no—
571C4—614no—
572C4—1222no—
573C4—4882no—
574C4—2442no—
575C4—2442no—
576C4#103×D6114yes—
577C4#1684024211yescharacteristic
578C4#1684024311yescharacteristic
579C4#1684034111yescharacteristic
580C4#1684034211yescharacteristic
581C2—1224no—
582C2#2144840811yescharacteristic
583C2C2×#103×D6111yescharacteristic
584C2#417×D6111yescharacteristic
585C2#2145675911yescharacteristic
586C2#2145676611yescharacteristic
587C2#2145696211yescharacteristic
588C2#2145696911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1803C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1803D61
hypocenterC61#21860
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
4. #21446778C2semidirect
5. #21446619C2semidirect
8. #21445825C2semidirect
9. #21445824C2semidirect
10. #21445821C2semidirect
11. C61×#1803C2semidirect
13. #16840316C4semidirect
54. #14519776C2×C4semidirect
56. #14519775C2×C4semidirect
65. C4×Dic122C2×C4semidirect
101. C4×Q8×C61C2×C4semidirect
249. C4×D61#103semidirect
297. #1803D61semidirect
340. C61#21860semidirect
364. C4×Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403162
C2×#168398142
#384×D612
#214467782
#214466192
#214461252
#214461232
#214458252
#214458242
#214458212
C61×#18032
#2186061