Finite Group Databasebeta

← #29733375

Subgroups — #29733375

Order 15616.

1126 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840452C211yesmaximal, characteristic
2C2×#16840131C211yesmaximal, characteristic, complemented
3C2×#16839973C211yesmaximal, characteristic, complemented
4#21450772C211yesmaximal, characteristic
5#21446829C212yesmaximal, complemented
6#21446829C212yesmaximal, complemented
7#21446774C211yesmaximal, characteristic, complemented
8#21446129C212yesmaximal, complemented
9#21446124C212yesmaximal, complemented
10#21445650C211yesmaximal, characteristic, complemented
11C61×#1955C211yesmaximal, characteristic
12C22×Dic244C2212yes—
13#16840452—24no—
14#16840452—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519770—21no—
17C2×#14519769C2211yescharacteristic
18C2×#14519767—21no—
19C2×#14519758—21no—
20C2×#14519756C2211yescharacteristic
21C2×#14519756C2211yescharacteristic
22C2×#14519756C2212yes—
23C2×#14519756C2212yes—
24C2×C4×Dic122C2211yescharacteristic
25C2×#14519736C2211yescharacteristic
26C2×#14519733—21no—
27C22×C4⋊Dic61C2211yescharacteristic
28#16840131—24no—
29#16839973—24no—
30C2×#14519702C2212yes—
31C2×#14519702C2212yes—
32C2×#14519702C2212yes—
33C2×#14519702C2212yes—
34#16839698—21no—
35#16839698—21no—
36#16839665—21no—
37#16839631—21no—
38C4×C4⋊Dic61C2211yescharacteristic
39#16839525C2212yes—
40#16839525C2212yes—
41#16839525C2212yes—
42C2×C4×D4×C61C2211yescharacteristic
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×C22⋊C8×C61C2212yes—
45C2×C22⋊C8×C61C2212yes—
46C2×#97×C61C2211yescharacteristic
47C22×Dic122C2312yes—
48C22×Dic122C2311yescharacteristic
49C23×Dic61—21no—
50C2×Dic244—24no—
51C2×Dic244—42no—
52C2×Dic244—42no—
53#14519777—22no—
54#14519777—22no—
55#14519770—28no—
56#14519769—22no—
57#14519769D414yes—
58#14519767—28no—
59#14519758—28no—
60#14519756—22no—
61#14519756—22no—
62#14519756—22no—
63#14519756—22no—
64#14519756—24no—
65#14519756—24no—
66#14519756—42no—
67#14519756—42no—
68C4×Dic122—22no—
69C4×Dic122D414yes—
70#14519736—22no—
71#14519736—22no—
72#14519733—28no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—22no—
75C2×C4⋊Dic61—22no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2312yes—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—22no—
88C2×#11758196—22no—
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61—21no—
91C2×C22⋊Dic61—21no—
92C2×C22⋊Dic61—22no—
93C2×C22⋊Dic61—22no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97#14519702—24no—
98#14519702—24no—
99#14519702—24no—
100#14519702—24no—
101#14519684—21no—
102C22×D4×C61D411yescharacteristic
103C23×C4×C61C2311yescharacteristic
104C23×C4×C61C2311yescharacteristic
105C22×C8×C61C2312yes—
106C22×C8×C61C2312yes—
107C4×D4×C61—22no—
108C4×D4×C61—22no—
109C2×C4⋊C4×C61D411yescharacteristic
110C2×C22⋊C4×C61D411yescharacteristic
111C2×C22⋊C4×C61D411yescharacteristic
112C2×C42×C61C2311yescharacteristic
113C4⋊C8×C61—22no—
114C22⋊C8×C61—42no—
115C22⋊C8×C61—42no—
116#97×C61—22no—
117C24×C61C2×D411yescharacteristic
118C24×C61C2×D411yescharacteristic
119C2×D4×C61—22no—
120C2×D4×C61—22no—
121C2×D4×C61—41no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2411yescharacteristic
135C22×Dic61—24no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—22no—
140C22×Dic61—21no—
141C22×Dic61—22no—
142C22×Dic61—42no—
143C22×Dic61—42no—
144C2×Dic122—22no—
145C2×Dic122—22no—
146C2×Dic122—21no—
147C2×Dic122—22no—
148C2×Dic122—22no—
149C2×Dic122—22no—
150C2×Dic122—22no—
151C2×Dic122—22no—
152C2×Dic122—22no—
153C2×Dic122C2×D412yes—
154C2×C8×C61—22no—
155C2×C8×C61—22no—
156C2×C8×C61—22no—
157C2×C8×C61—22no—
158C2×C8×C61—22no—
159C2×C8×C61—22no—
160C4⋊C4×C61—22no—
161C22⋊C4×C61—22no—
162C22⋊C4×C61—22no—
163C42×C61C2×D412yes—
164C42×C61C2×D412yes—
165C22⋊Dic61—24no—
166C22⋊Dic61—24no—
167C22⋊Dic61—24no—
168C22⋊Dic61—44no—
169C22⋊Dic61—44no—
170C4⋊Dic61—24no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—22no—
175C4⋊Dic61—42no—
176C4⋊Dic61—42no—
177C4⋊Dic61C2×D412yes—
178C4⋊Dic61C2×D412yes—
179C4⋊Dic61C2×D412yes—
180C4⋊Dic61#4114yes—
181#11758196—24no—
182#11758196—24no—
183#11758196—24no—
184#11758196—24no—
185#11758196—24no—
186#11758196—44no—
187#11758196—44no—
188C4×Dic61—24no—
189C4×Dic61—24no—
190C4×Dic61—42no—
191Dic244—48no—
192C23×C61—21no—
193C23×C61—22no—
194C23×C61—21no—
195C23×C61—22no—
196C23×C61—21no—
197C23×C61—22no—
198C23×C61—21no—
199C23×C61—21no—
200C23×C61—22no—
201C23×C61#13712yes—
202C23×C61C22×D411yescharacteristic
203D4×C61—44no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—22no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61#12012yes—
218C2×C4×C61#12012yes—
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C2×#4112yes—
222C2×C4×C61H(5,𝔽2)12yes—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—24no—
228C2×Dic61—48no—
229C2×Dic61—42no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—42no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240Dic122—22no—
241Dic122—22no—
242Dic122—42no—
243Dic122—42no—
244Dic122—42no—
245Dic122—42no—
246Dic122—42no—
247C8×C61—42no—
248C8×C61—42no—
249#25584—611nomaximal, Sylow
250C22×C61—22no—
251C22×C61—24no—
252C22×C61—41no—
253C22×C61—41no—
254C22×C61—42no—
255C22×C61—41no—
256C22×C61—41no—
257C22×C61—41no—
258C22×C61—42no—
259C22×C61—42no—
260C22×C61C2×#12011yescharacteristic
261C22×C61#54612yes—
262C22×C61C2×#13711yescharacteristic
263C22×C61#57511yescharacteristic
264C22×C61#57812yes—
265C4×C61—22no—
266C4×C61—22no—
267C4×C61—41no—
268C4×C61—41no—
269C4×C61#53211yescharacteristic
270C4×C61#53511yescharacteristic
271C4×C61#55012yes—
272Dic61—42no—
273Dic61—42no—
274Dic61—42no—
275Dic61—42no—
276Dic61—44no—
277Dic61—82no—
278C2×#552—611no—
279C2×#491—611no—
280C2×#462—611no—
281#2542—611no—
282#2001—612no—
283#2001—612no—
284#1989—611no—
285#1955D6111yescharacteristic
286#1873—612no—
287#1872—612no—
288#1763—611no—
289C2×C61—42no—
290C2×C61—81no—
291C2×C61#243111yescharacteristic
292C2×C61#300711yescharacteristic
293C2×C61#301111yescharacteristic
294C2×C61#331212yes—
295C2×C61#331512yes—
296C22×Q16—612no—
297#552—1224no—
298#552—1224no—
299C2×C4⋊Q8—611no—
300C2×#126—1221no—
301C2×#125—611no—
302C2×#125—1221no—
303C2×#123—1221no—
304C2×C22⋊Q8—611no—
305C2×C22⋊Q8—611no—
306C2×C22⋊Q8—612no—
307C2×C22⋊Q8—612no—
308C2×C4×Q8—611no—
309C2×C4×D4C2×D6111yescharacteristic
310C2×#117—611no—
311C2×#117—1221no—
312C22×C4⋊C4—611no—
313#491—1224no—
314#462—1224no—
315C2×C4⋊C8C2×D6111yescharacteristic
316C2×#103—612no—
317C2×#103—612no—
318C2×#103—612no—
319C2×#103—612no—
320C2×C22⋊C8C2×D6112yes—
321C2×C22⋊C8C2×D6112yes—
322C2×#97C2×D6111yescharacteristic
323#399—1221no—
324#399—1221no—
325#390—1221no—
326#384—1221no—
327C4×C4⋊C4—611no—
328#340—612no—
329#340—612no—
330#340—612no—
331C61#2558411yescharacteristic, complemented, Sylow
332C22×Q8—611no—
333C22×Q8—612no—
334C22×D4D24411yescharacteristic
335C23×C4—1221no—
336C23×C4C22×D6111yescharacteristic
337C23×C4C22×D6111yescharacteristic
338C2×Q16—1224no—
339C2×Q16—2442no—
340C2×Q16—2442no—
341C22×C8C22×D6112yes—
342C22×C8C22×D6112yes—
343C4⋊Q8—1222no—
344C4⋊Q8—1222no—
345#126—1228no—
346#125—614no—
347#125—1228no—
348#125—1222no—
349#123—1228no—
350C22⋊Q8—1222no—
351C22⋊Q8—1222no—
352C22⋊Q8—1222no—
353C22⋊Q8—1222no—
354C22⋊Q8—1224no—
355C22⋊Q8—1224no—
356C22⋊Q8—2442no—
357C22⋊Q8—2442no—
358C4×Q8—614no—
359C4×Q8—1222no—
360C4×D4—22no—
361C4×D4—22no—
362#117—1228no—
363#117—1222no—
364#117—1222no—
365C2×C4⋊C4—611no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—612no—
369C2×C4⋊C4—1222no—
370C2×C4⋊C4—1222no—
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—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4D24411yescharacteristic
382C2×C22⋊C4—1221no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1222no—
386C2×C22⋊C4—1222no—
387C2×C22⋊C4D24411yescharacteristic
388C2×C22⋊C4D24411yescharacteristic
389C2×C42—1221no—
390C2×C42—1221no—
391C2×C42—1221no—
392C2×C42C22×D6111yescharacteristic
393C4⋊C8—22no—
394#103—1224no—
395#103—1224no—
396#103—1224no—
397#103—1224no—
398C22⋊C8—42no—
399C22⋊C8—42no—
400#97—22no—
401#95—1221no—
402C24C2×D24411yescharacteristic
403C24C2×D24411yescharacteristic
404C2×Q8—612no—
405C2×Q8—1221no—
406C2×Q8—1222no—
407C2×Q8—1222no—
408C2×Q8—1222no—
409C2×Q8—1222no—
410C2×Q8—1222no—
411C2×Q8—1222no—
412C2×Q8—1222no—
413C2×Q8—1222no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—41no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—1224no—
426C22×C4—1222no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4—1221no—
430C22×C4—1221no—
431C22×C4—1222no—
432C22×C4—2442no—
433C22×C4—2442no—
434C22×C4C2×D24411yescharacteristic
435C22×C4C2×D24411yescharacteristic
436C22×C4C2×D24411yescharacteristic
437C22×C4C2×D24411yescharacteristic
438C22×C4C23×D6111yescharacteristic
439Q16—2448no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C4⋊C4—22no—
447C4⋊C4—612no—
448C4⋊C4—612no—
449C4⋊C4—612no—
450C4⋊C4—614no—
451C4⋊C4—1224no—
452C4⋊C4—1224no—
453C4⋊C4—1224no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—1222no—
458C4⋊C4—1224no—
459C4⋊C4—1224no—
460C4⋊C4—1224no—
461C4⋊C4—2444no—
462C4⋊C4—2444no—
463C4⋊C4—2442no—
464C4⋊C4—2442no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—1224no—
468C22⋊C4—1224no—
469C22⋊C4—1224no—
470C22⋊C4—2444no—
471C22⋊C4—2444no—
472C42—1224no—
473C42—1224no—
474C42—2442no—
475C42D4×D6112yes—
476C42D4×D6112yes—
477C23—21no—
478C23—21no—
479C23—22no—
480C23—22no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—22no—
485C23—22no—
486C23#1451981012yes—
487C23C22×D24411yescharacteristic
488Q8—1222no—
489Q8—1222no—
490Q8—2442no—
491Q8—2442no—
492Q8—2442no—
493Q8—2442no—
494Q8—2442no—
495D4—44no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—1222no—
512C2×C4—1222no—
513C2×C4—1224no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2448no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4—2442no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4#1451974712yes—
527C2×C4#1451974712yes—
528C2×C4C2×D4×D6111yescharacteristic
529C2×C4C2×D4×D6111yescharacteristic
530C2×C4#41×D6112yes—
531C2×C4#1451982212yes—
532C8—42no—
533C8—42no—
534C22—24no—
535C22—22no—
536C22—41no—
537C22—41no—
538C22—42no—
539C22—41no—
540C22—42no—
541C22—41no—
542C22—41no—
543C22—42no—
544C22C2×#1451974711yescharacteristic
545C22#1684040712yes—
546C22C2×#1451981011yescharacteristic
547C22#1684056211yescharacteristic
548C22#1684057912yes—
549C4—22no—
550C4—22no—
551C4—41no—
552C4—41no—
553C4—4882no—
554C4—2442no—
555C4—2442no—
556C4—2442no—
557C4—2442no—
558C4—2444no—
559C4#1684031111yescharacteristic
560C4#1684032511yescharacteristic
561C4#1684043012yes—
562C2—42no—
563C2—81no—
564C2#2144943711yescharacteristic
565C2#2145508111yescharacteristic
566C2#2145511611yescharacteristic
567C2#2145787212yes—
568C2#2145791712yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1955C2
socleC23×C61C22×D4
prime radicalC22×D4×C61D4
hypercenter#1955D61
hypocenterC61#25584
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1955

Complements

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

NH ≅ G/Ntype
2. C2×#16840131C2semidirect
3. C2×#16839973C2semidirect
5. #21446829C2semidirect
6. #21446829C2semidirect
7. #21446774C2semidirect
8. #21446129C2semidirect
9. #21446124C2semidirect
10. #21445650C2semidirect
331. C61#25584semidirect

Maximal subgroups

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

subgroupindex
C2×#168404522
C2×#168401312
C2×#168399732
#214507722
#214468292
#214467742
#214461292
#214461242
#214456502
C61×#19552
#2558461