Finite Group Databasebeta

← #29730757

Subgroups — #29730757

Order 15616.

1018 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840002C212yesmaximal, complemented
2C2×#16839969C211yesmaximal, characteristic, complemented
3C2×C8×D244C211yesmaximal, characteristic
4#21450773C211yesmaximal, characteristic
5#21450742C211yesmaximal, characteristic
6#21447020C212yesmaximal, complemented
7#21446828C212yesmaximal, complemented
8#21446575C211yesmaximal, characteristic, complemented
9#21446239C212yesmaximal, complemented
10C4×#14519702C211yesmaximal, characteristic, complemented
11C61×#1956C211yesmaximal, characteristic, complemented
12C22×#11758202C2212yes—
13C22×C8×D61C2212yes—
14C2×#14519757C2212yes—
15C2×#14519753C2212yes—
16C2×#14519753C2212yes—
17C2×C4×Dic122C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19#16840002—28no—
20#16839969—24no—
21C8×D244—24no—
22C2×#14519710C2212yes—
23C2×C4⋊#92418C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519701C2212yes—
27C2×#14519689C2212yes—
28#16839697—22no—
29#16839684—22no—
30#16839663—21no—
31#16839663—21no—
32#16839661—21no—
33#16839631—21no—
34#16839619C2212yes—
35C4×C4⋊Dic61C2211yescharacteristic
36#16839525C2211yescharacteristic
37#16839525C2211yescharacteristic
38#16839524C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×#103×C61C2212yes—
41C2×#103×C61C2212yes—
42C2×C4×C8×C61C2211yescharacteristic
43C61×#384C2211yescharacteristic
44C22×Dic122C2311yescharacteristic
45C22×D244D411yescharacteristic
46C22×C4×D61C2312yes—
47C2×#11758202—24no—
48C2×#11758202—24no—
49C2×#11758202—42no—
50C22×#92418C2312yes—
51C2×C8×D61—24no—
52C2×C8×D61—24no—
53C2×C8×D61—42no—
54#14519757—24no—
55#14519757—42no—
56#14519753—24no—
57#14519753—24no—
58#14519753—42no—
59#14519753—42no—
60C4×Dic122—22no—
61C4×Dic122—22no—
62C4×D244—22no—
63C4×D244—22no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61D411yescharacteristic
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×#11758196—21no—
70C2×#11758196—22no—
71C2×#11758196—22no—
72C2×#11758198—22no—
73C2×#11758198—22no—
74C2×#11758198—22no—
75C2×#11758198—22no—
76C2×#11758198D412yes—
77C2×C4×Dic61—21no—
78C2×C4×Dic61—21no—
79#14519710—24no—
80C4⋊#92418D414yes—
81#14519702—22no—
82#14519702D414yescomplemented
83#14519701—24no—
84#14519689—24no—
85#14519684—21no—
86#14519684—22no—
87C22×Q8×C61C2312yes—
88C22×C8×C61C2311yescharacteristic
89C22×C8×C61C2311yescharacteristic
90C4⋊Q8×C61—22no—
91C4⋊Q8×C61—22no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61C2312yes—
97C2×C42×C61C2311yescharacteristic
98#103×C61—24no—
99#103×C61—42no—
100C4×C8×C61—22no—
101C23×D61C2×D412yes—
102C2×Q8×C61—22no—
103C2×Q8×C61—22no—
104C2×Q8×C61—22no—
105C2×Q8×C61—22no—
106C2×Q8×C61—22no—
107C2×Q8×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2411yescharacteristic
113C22×Dic61—21no—
114C22×Dic61—21no—
115C22×Dic61—21no—
116C22×Dic61—21no—
117C22×Dic61C2×D412yes—
118C2×D244—22no—
119C2×D244—22no—
120C2×D244Q1614yescomplemented
121C2×C4×D61—22no—
122C2×C4×D61—22no—
123C2×C4×D61—22no—
124C2×C4×D61—22no—
125C2×C4×D61—22no—
126C2×C4×D61—22no—
127C2×Dic122—21no—
128C2×Dic122—21no—
129C2×Dic122—21no—
130C2×Dic122—21no—
131C2×Dic122—21no—
132C2×Dic122C2×D412yes—
133C2×C8×C61—21no—
134C2×C8×C61—21no—
135C2×C8×C61—21no—
136C2×C8×C61—21no—
137C2×C8×C61—21no—
138C2×C8×C61C2×D412yes—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—42no—
142C4⋊C4×C61—42no—
143C4⋊C4×C61—42no—
144C4⋊C4×C61—81no—
145C42×C61C2×D412yes—
146C42×C61#4112yes—
147#11758202—48no—
148#11758198—24no—
149#11758198—44no—
150#11758198—44no—
151#11758198—44no—
152#11758198—82no—
153C4⋊Dic61—22no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61—44no—
159C4⋊Dic61C2×D412yes—
160C4⋊Dic61#4112yes—
161#11758196—44no—
162#11758196—44no—
163#11758196—42no—
164C4×Dic61—42no—
165C4×Dic61—42no—
166C2×#92418—22no—
167C2×#92418—22no—
168C2×#92418C2×D414yes—
169C8×D61—48no—
170C23×C61C22×D411yescharacteristic
171C22×D61—22no—
172C22×D61—22no—
173C22×D61—22no—
174C22×D61—22no—
175C22×D61—22no—
176C22×D61—22no—
177C22×D61C2×Q1614yes—
178Q8×C61—42no—
179Q8×C61—42no—
180Q8×C61—42no—
181Q8×C61—42no—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61#12112yes—
191C2×C4×C61#12112yes—
192C2×C4×C61C2×Q1612yes—
193C2×C4×C61#13512yes—
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C22×D411yescharacteristic
196C2×C4×C61C2×#4111yescharacteristic
197C2×C4×C61C2×#4111yescharacteristic
198C2×C4×C61H(5,𝔽2)11yescharacteristic
199C2×C4×C61H(5,𝔽2)11yescharacteristic
200C2×Dic61—22no—
201C2×Dic61—22no—
202C2×Dic61—22no—
203C2×Dic61—22no—
204C2×Dic61—22no—
205C2×Dic61—22no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215D244—28no—
216C4×D61—42no—
217C4×D61—42no—
218C4×D61—42no—
219C4×D61—42no—
220Dic122—22no—
221Dic122—22no—
222Dic122—41no—
223Dic122—41no—
224C8×C61—22no—
225C8×C61—41no—
226#92418—24no—
227#25386—611nomaximal, Sylow
228C22×C61C2×#12111yescharacteristic
229C22×C61D4211yescharacteristic
230C22×C61#54611yescharacteristic
231C22×C61C22×Q1611yescharacteristic
232C22×C61C2×#13511yescharacteristic
233C22×C61#57711yescharacteristic
234C22×C61#57811yescharacteristic
235C2×D61—24no—
236C2×D61—24no—
237C2×D61—24no—
238C2×D61—42no—
239C2×D61—42no—
240C2×D61—42no—
241C2×D61—42no—
242C4×C61—22no—
243C4×C61—82no—
244C4×C61—82no—
245C4×C61#46714yescomplemented
246C4×C61#53211yescharacteristic
247C4×C61#54011yescharacteristic
248C4×C61D4211yescharacteristic
249C4×C61#55011yescharacteristic
250Dic61—42no—
251Dic61—42no—
252Dic61—42no—
253Dic61—42no—
254Dic61—81no—
255C2×#467—612no—
256C2×#462—611no—
257C2×C8×D4—611no—
258#2542—611no—
259#2538—611no—
260#2036—612no—
261#2001—612no—
262#1956D6111yescharacteristic, complemented
263#1947—611no—
264#1892—612no—
265C4×#103—611no—
266C2×C61#242811yescharacteristic
267C2×C61C2×#46711yescharacteristic
268C2×C61#306311yescharacteristic
269C2×C61#328411yescharacteristic
270C2×C61D4×Q1611yescharacteristic
271C2×C61#329911yescharacteristic
272C2×C61#330211yescharacteristic
273D61—44no—
274C22×Q16—612no—
275C23×C8—612no—
276C2×C4⋊Q8C2×D6111yescharacteristic
277C2×C22⋊Q8—612no—
278C2×C22⋊Q8—612no—
279C2×C22⋊Q8—612no—
280C2×C4×Q8—611no—
281C2×C4×D4—611no—
282#467—1228no—
283#462—1224no—
284C8×D4—1224no—
285C2×#107—612no—
286C2×C4⋊C8—611no—
287C2×#103—611no—
288C2×#103—612no—
289C2×#103—611no—
290C2×#103C2×D6112yes—
291C2×#103C2×D6112yes—
292C2×C22⋊C8—612no—
293C2×C4×C8C2×D6111yescharacteristic
294#399—1222no—
295#396—1222no—
296#390—1221no—
297#390—1221no—
298#389—1221no—
299#384—1221no—
300#384C2×D6111yescharacteristic
301#380—612no—
302C4×C4⋊C4—611no—
303#340—611no—
304#340—611no—
305#340—612no—
306C61#2538611yescharacteristic, complemented, Sylow
307C22×Q8—611no—
308C22×Q8C22×D6112yes—
309C22×D4—611no—
310C23×C4—612no—
311C2×Q16—1224no—
312C2×Q16—1224no—
313C2×Q16—2442no—
314C22×C8—612no—
315C22×C8—1224no—
316C22×C8—1224no—
317C22×C8—2442no—
318C22×C8C22×D6111yescharacteristic
319C22×C8C22×D6111yescharacteristic
320C4⋊Q8—22no—
321C4⋊Q8—22no—
322C22⋊Q8—1224no—
323C22⋊Q8—1224no—
324C22⋊Q8—1224no—
325C22⋊Q8—2442no—
326C22⋊Q8—2442no—
327C22⋊Q8—2442no—
328C4×Q8—1222no—
329C4×Q8—1222no—
330C4×D4—1222no—
331C4×D4—1222no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—611no—
337C2×C4⋊C4—611no—
338C2×C4⋊C4—611no—
339C2×C4⋊C4—611no—
340C2×C4⋊C4—1222no—
341C2×C4⋊C4—1222no—
342C2×C4⋊C4—1222no—
343C2×C4⋊C4—1221no—
344C2×C4⋊C4C22×D6112yes—
345C2×C22⋊C4—612no—
346C2×C22⋊C4—1222no—
347C2×C22⋊C4—1222no—
348C2×C22⋊C4—1222no—
349C2×C22⋊C4—1222no—
350C2×C42—1221no—
351C2×C42—1221no—
352C2×C42C22×D6111yescharacteristic
353#107—1224no—
354C4⋊C8—614no—
355#103—24no—
356#103—42no—
357#103—614no—
358#103—1222no—
359#103—1224no—
360C22⋊C8—1224no—
361C4×C8—22no—
362#95—1222no—
363#95—1221no—
364C24—612no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—612no—
372C2×Q8—1221no—
373C2×Q8—1221no—
374C2×Q8—1221no—
375C2×Q8—1221no—
376C2×Q8—1221no—
377C2×D4—614no—
378C2×D4—1222no—
379C2×D4—1222no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—612no—
383C22×C4—1221no—
384C22×C4—1221no—
385C22×C4—1221no—
386C22×C4—1222no—
387C22×C4—1222no—
388C22×C4—1222no—
389C22×C4—1222no—
390C22×C4—1222no—
391C22×C4—1222no—
392C22×C4—1221no—
393C22×C4D4×D6111yescharacteristic
394C22×C4D4×D6111yescharacteristic
395C22×C4C23×D6111yescharacteristic
396Q16—2448no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—614no—
403C2×C8—1222no—
404C2×C8—1222no—
405C2×C8—2448no—
406C2×C8D4×D6112yes—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—81no—
413C4⋊C4—612no—
414C4⋊C4—612no—
415C4⋊C4—1221no—
416C4⋊C4—1221no—
417C4⋊C4—1222no—
418C4⋊C4—1221no—
419C4⋊C4—1221no—
420C4⋊C4—2444no—
421C4⋊C4—2444no—
422C4⋊C4—2444no—
423C4⋊C4—2442no—
424C22⋊C4—1224no—
425C22⋊C4—4882no—
426C22⋊C4—2444no—
427C22⋊C4—2444no—
428C22⋊C4—2444no—
429C42—2442no—
430C42—2442no—
431C42#1175822212yes—
432C42D4×D6112yes—
433C23—614no—
434C23—1222no—
435C23—1222no—
436C23—1222no—
437C23—1222no—
438C23—1222no—
439C23—1222no—
440C23C2×D4×D6111yescharacteristic
441Q8—42no—
442Q8—42no—
443Q8—42no—
444Q8—42no—
445Q8—1222no—
446Q8—1222no—
447Q8—2441no—
448Q8—2441no—
449D4—1228no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—1222no—
459C2×C4—1222no—
460C2×C4—1222no—
461C2×C4—1222no—
462C2×C4—1222no—
463C2×C4—1222no—
464C2×C4—2441no—
465C2×C4—2441no—
466C2×C4—2441no—
467C2×C4—2441no—
468C2×C4—2441no—
469C2×C4—2441no—
470C2×C4—2441no—
471C2×C4—2441no—
472C2×C4—2441no—
473C2×C4—2442no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4—2442no—
477C2×C4#1451975012yes—
478C2×C4#1451975012yes—
479C2×C4Q16×D6112yes—
480C2×C4#1451979812yes—
481C2×C4C2×D4×D6111yescharacteristic
482C2×C4C2×D4×D6111yescharacteristic
483C2×C4#41×D6111yescharacteristic
484C2×C4C2×#1175822211yescharacteristic
485C2×C4#1451982211yescharacteristic
486C2×C4#1451982311yescharacteristic
487C8—22no—
488C8—41no—
489C8—1224no—
490C22—1224no—
491C22—1224no—
492C22—1224no—
493C22—2442no—
494C22—2442no—
495C22—2442no—
496C22—2442no—
497C22C2×#1451975011yescharacteristic
498C22#1684039711yescharacteristic
499C22#1684040511yescharacteristic
500C22C2×Q16×D6111yescharacteristic
501C22C2×#1451979811yescharacteristic
502C22#1684056811yescharacteristic
503C22#1684057511yescharacteristic
504C4—22no—
505C4—82no—
506C4—82no—
507C4—4881no—
508C4—2442no—
509C4—2442no—
510C4—2442no—
511C4—2442no—
512C4#1684000214yescomplemented
513C4#1684030911yescharacteristic
514C4#1684036311yescharacteristic
515C4#1684039611yescharacteristic
516C4#1684043011yescharacteristic
517C2—2444no—
518C2#2144940811yescharacteristic
519C2C2×#1684000211yescharacteristic
520C2#2145550811yescharacteristic
521C2#2145754811yescharacteristic
522C2#2145756911yescharacteristic
523C2#2145769411yescharacteristic
524C2#2145773511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1956C2
socleC23×C61C22×D4
prime radicalC22×D244D4
hypercenter#1956D61
hypocenterC61#25386
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
1. C2×#16840002C2semidirect
2. C2×#16839969C2semidirect
6. #21447020C2semidirect
7. #21446828C2semidirect
8. #21446575C2semidirect
9. #21446239C2semidirect
10. C4×#14519702C2semidirect
11. C61×#1956C2semidirect
82. #14519702D4semidirect
120. C2×D244Q16semidirect
245. C4×C61#467semidirect
262. #1956D61semidirect
306. C61#25386semidirect
512. C4#16840002semidirect

Maximal subgroups

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

subgroupindex
C2×#168400022
C2×#168399692
C2×C8×D2442
#214507732
#214507422
#214470202
#214468282
#214465752
#214462392
C4×#145197022
C61×#19562
#2538661