Finite Group Databasebeta

← #29716404

Subgroups — #29716404

Order 15616.

1070 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840487C211yesmaximal, characteristic, complemented
2C2×#16840039C211yesmaximal, characteristic
3C2×C4×#11758190C211yesmaximal, characteristic
4#21450742C211yesmaximal, characteristic
5#21447156C212yesmaximal, complemented
6#21446964C212yesmaximal
7#21446755C211yesmaximal, characteristic, complemented
8#21446175C212yesmaximal
9#21446074C212yesmaximal, complemented
10#21445821C211yesmaximal, characteristic, complemented
11C61×#1990C211yesmaximal, characteristic, complemented
12C22×#11758190C2211yescharacteristic
13#16840487—24no—
14#16840487—24no—
15C2×#14519775—21no—
16C2×#14519768C2212yes—
17C2×#14519767—22no—
18C2×#14519757—21no—
19C2×Q8×Dic61—21no—
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×#14519742C2212yes—
23C2×C4×D244C2211yescharacteristic
24#16840039—24no—
25C4×#11758190—24no—
26C2×#14519710C2212yes—
27C2×#14519709C2211yescharacteristic
28C2×C4⋊#92418C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519700C2212yes—
31C2×#14519699C2211yescharacteristic
32C2×#14519697C2212yes—
33#16839684—22no—
34#16839663C2212yes—
35#16839661C2212yes—
36#16839619—21no—
37#16839524C2212yes—
38#16839516C2212yes—
39#16839516C2212yes—
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×C4×C8×C61C2211yescharacteristic
42C61×#384C2211yescharacteristic
43C61×#340C2212yes—
44C61×#340C2212yes—
45C22×Dic122D411yescharacteristic
46C22×Dic122C2311yescharacteristic
47C22×D244C2311yescharacteristic
48C22×C4×D61—21no—
49C2×#11758190—22no—
50C2×#11758190—22no—
51C2×#11758190Q814yescomplemented
52C22×#92418C2312yes—
53#14519775—28no—
54#14519768—24no—
55#14519768—24no—
56#14519767—216no—
57#14519757—44no—
58Q8×Dic61—28no—
59C4×Dic122—22no—
60C4×Dic122—22no—
61C4×Dic122—22no—
62C4×Dic122D414yes—
63#14519742—24no—
64#14519742—24no—
65C4×D244—22no—
66C4×D244—22no—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61—22no—
69C2×C4⋊Dic61D411yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—21no—
73C2×#11758196—22no—
74C2×#11758196—22no—
75C2×#11758196D412yes—
76C2×#11758198—21no—
77C2×#11758198—22no—
78C2×#11758198—22no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—22no—
81C2×C4×Dic61C2312yes—
82#14519710—24no—
83#14519709—22no—
84C4⋊#92418—22no—
85#14519702—22no—
86#14519700—24no—
87#14519699—22no—
88#14519697—24no—
89#14519684—22no—
90C22×Q8×C61—21no—
91C22×C8×C61C2311yescharacteristic
92C22×C8×C61C2311yescharacteristic
93C4⋊Q8×C61—22no—
94C4⋊Q8×C61D414yes—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61C2312yes—
99C2×C4⋊C4×C61C2312yes—
100C2×C42×C61C2311yescharacteristic
101C4×C8×C61—22no—
102C23×D61—21no—
103C2×Q8×C61—24no—
104C2×Q8×C61—42no—
105C2×Q8×C61—42no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2411yescharacteristic
111C22×Dic61—21no—
112C22×Dic61—21no—
113C22×Dic61—21no—
114C22×Dic61C2×D412yes—
115C22×Dic61C2×D412yes—
116C2×D244—21no—
117C2×D244—21no—
118C2×D244—21no—
119C2×D244—21no—
120C2×D244—21no—
121C2×D244C2×Q812yes—
122C2×C4×D61—42no—
123C2×C4×D61—42no—
124C2×C4×D61—42no—
125C2×Dic122—21no—
126C2×Dic122—21no—
127C2×Dic122—21no—
128C2×Dic122—22no—
129C2×Dic122—22no—
130C2×Dic122—22no—
131C2×Dic122C2×D412yes—
132C2×Dic122C2×Q812yes—
133C2×Dic122#4112yes—
134C2×C8×C61—21no—
135C2×C8×C61—21no—
136C2×C8×C61—21no—
137C2×C8×C61—21no—
138C2×C8×C61—21no—
139C2×C8×C61C2×Q812yes—
140C4⋊C4×C61—24no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—42no—
145C4⋊C4×C61—81no—
146C4⋊C4×C61C2×D414yes—
147C42×C61C2×D412yes—
148C42×C61C2×D412yes—
149#11758198—44no—
150#11758198—42no—
151#11758198—82no—
152C4⋊Dic61—28no—
153C4⋊Dic61—22no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—44no—
158C4⋊Dic61C2×D412yes—
159#11758196—28no—
160#11758196—28no—
161#11758196—24no—
162#11758196—24no—
163C4×Dic61—28no—
164C4×Dic61—24no—
165C4×Dic61—22no—
166C4×Dic61—22no—
167C2×#92418—22no—
168C2×#92418—22no—
169C2×#92418—22no—
170#11758190—28no—
171C23×C61C22×D411yescharacteristic
172C22×D61—22no—
173C22×D61—41no—
174C22×D61—41no—
175C22×D61—41no—
176C22×D61—41no—
177C22×D61—41no—
178C22×D61—41no—
179Q8×C61—48no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—24no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61#12012yes—
189C2×C4×C61#12012yes—
190C2×C4×C61#13512yes—
191C2×C4×C61AGL(1,ℤ/8)12yes—
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C22×Q811yescharacteristic
195C2×C4×C61C2×#4111yescharacteristic
196C2×C4×C612−1+411yescharacteristic
197C2×C4×C612−1+411yescharacteristic
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—22no—
201C2×Dic61—22no—
202C2×Dic61—22no—
203C2×Dic61—22no—
204C2×Dic61—22no—
205C2×Dic61—22no—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213D244—22no—
214D244—22no—
215D244—41no—
216D244—41no—
217C4×D61—84no—
218Dic122—22no—
219Dic122—22no—
220Dic122—22no—
221Dic122—44no—
222Dic122C22⋊Q814yescomplemented
223C8×C61—22no—
224C8×C61—41no—
225#92418—42no—
226#24328—611nomaximal, Sylow
227C22×C61C2×#12011yescharacteristic
228C22×C61#54811yescharacteristic
229C22×C61D4×Q811yescharacteristic
230C22×C61C2×#13511yescharacteristic
231C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
232C22×C61#57711yescharacteristic
233C22×C61#57811yescharacteristic
234C2×D61—42no—
235C2×D61—42no—
236C2×D61—42no—
237C2×D61—81no—
238C2×D61—81no—
239C2×D61—81no—
240C2×D61—81no—
241C4×C61—22no—
242C4×C61—44no—
243C4×C61—82no—
244C4×C61#44914yes—
245C4×C61C2×C22⋊Q811yescharacteristic
246C4×C61#53611yescharacteristic
247C4×C61#54811yescharacteristic
248C4×C61D4×Q811yescharacteristic
249Dic61—24no—
250Dic61—42no—
251Dic61—42no—
252Dic61—42no—
253Dic61—81no—
254C2×#557—611no—
255C2×#474—611no—
256C2×C4×SD16—611no—
257#2538—611no—
258#2059—612no—
259#2025—612no—
260#1990D6111yescharacteristic, complemented
261#1984—611no—
262#1880—612no—
263#1863—612no—
264#1803—611no—
265C2×C61#240911yescharacteristic
266C2×C61C2×#44911yescharacteristic
267C2×C61#301311yescharacteristic
268C2×C61#336411yescharacteristic
269C2×C61#336811yescharacteristic
270C2×C61#338111yescharacteristic
271C2×C61#338611yescharacteristic
272D61—82no—
273C22×SD16—611no—
274#557—1224no—
275#557—1224no—
276C2×C4⋊Q8—1221no—
277C2×C4⋊Q8C2×D6111yescharacteristic
278C2×#125—612no—
279C2×#125—1222no—
280C2×C22⋊Q8—1221no—
281C2×C4×Q8—611no—
282C2×C4×Q8—612no—
283C2×C4×Q8—611no—
284C2×C4×Q8—1221no—
285C2×C4×D4—611no—
286#474—1224no—
287C4×SD16—1224no—
288C2×#107—612no—
289C2×#106—611no—
290C2×C4⋊C8—611no—
291C2×#103—611no—
292C2×#103—612no—
293C2×#102—611no—
294C2×#102—612no—
295C2×C4×C8C2×D6111yescharacteristic
296#396—1222no—
297#390—612no—
298#389—612no—
299#384C2×D6111yescharacteristic
300#380—1221no—
301#340—612no—
302#340C2×D6112yes—
303#340C2×D6112yes—
304#336—612no—
305#336—612no—
306C61#2432811yescharacteristic, complemented, Sylow
307C22×Q8—21no—
308C22×Q8—611no—
309C22×Q8—611no—
310C22×D4—611no—
311C23×C4—1221no—
312C2×SD16—614no—
313C2×SD16—1222no—
314C2×SD16—1222no—
315C22×C8—612no—
316C22×C8C22×D6111yescharacteristic
317C22×C8C22×D6111yescharacteristic
318C4⋊Q8—22no—
319C4⋊Q8—1228no—
320C4⋊Q8D24414yes—
321#125—1224no—
322#125—12216no—
323#125—1224no—
324C22⋊Q8—2444no—
325C4×Q8—614no—
326C4×Q8—1224no—
327C4×Q8—1228no—
328C4×Q8—1224no—
329C4×Q8—1222no—
330C4×Q8—1222no—
331C4×Q8—1222no—
332C4×D4—1222no—
333C4×D4—1222no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—611no—
338C2×C4⋊C4—611no—
339C2×C4⋊C4—612no—
340C2×C4⋊C4—611no—
341C2×C4⋊C4—1222no—
342C2×C4⋊C4—1222no—
343C2×C4⋊C4—1222no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4—1222no—
346C2×C4⋊C4C22×D6112yes—
347C2×C4⋊C4C22×D6112yes—
348C2×C22⋊C4—1222no—
349C2×C22⋊C4—1222no—
350C2×C22⋊C4—1221no—
351C2×C42—612no—
352C2×C42—1222no—
353C2×C42—1221no—
354C2×C42C22×D6111yescharacteristic
355#107—1224no—
356#106—1222no—
357C4⋊C8—1222no—
358#103—1224no—
359#103—1222no—
360#102—1222no—
361#102—1224no—
362C4×C8—22no—
363#95—1222no—
364C24—1221no—
365C2×Q8—24no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8—612no—
369C2×Q8—612no—
370C2×Q8—612no—
371C2×Q8—1221no—
372C2×Q8—1221no—
373C2×Q8—1221no—
374C2×Q8—1222no—
375C2×Q8—1222no—
376C2×Q8—1222no—
377C2×D4—612no—
378C2×D4—1221no—
379C2×D4—1221no—
380C2×D4—1221no—
381C2×D4—1221no—
382C2×D4—1221no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—612no—
386C22×C4—612no—
387C22×C4—1221no—
388C22×C4—1221no—
389C22×C4—1221no—
390C22×C4—2442no—
391C22×C4—2442no—
392C22×C4—2442no—
393C22×C4D4×D6111yescharacteristic
394C22×C4D4×D6111yescharacteristic
395C22×C4C23×D6111yescharacteristic
396SD16—1228no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—1222no—
403C2×C8—1222no—
404C2×C8—1222no—
405C2×C8Q8×D6112yes—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—42no—
411C4⋊C4—81no—
412C4⋊C4—612no—
413C4⋊C4—1228no—
414C4⋊C4—1228no—
415C4⋊C4—1228no—
416C4⋊C4—1221no—
417C4⋊C4—1224no—
418C4⋊C4—1222no—
419C4⋊C4—1224no—
420C4⋊C4—1221no—
421C4⋊C4—1221no—
422C4⋊C4—2444no—
423C4⋊C4C2×D24414yes—
424C22⋊C4—4882no—
425C22⋊C4—2444no—
426C22⋊C4—2442no—
427C42—1228no—
428C42—1222no—
429C42—1222no—
430C42—1224no—
431C42C2×D24412yes—
432C42D4×D6112yes—
433C23—1222no—
434C23—2441no—
435C23—2441no—
436C23—2441no—
437C23—2441no—
438C23—2441no—
439C23—2441no—
440C23C2×D4×D6111yescharacteristic
441Q8—48no—
442Q8—614no—
443Q8—1222no—
444Q8—1222no—
445Q8—1222no—
446Q8—2444no—
447D4—1222no—
448D4—1222no—
449D4—2441no—
450D4—2441no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—24no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—1222no—
460C2×C4—1222no—
461C2×C4—1222no—
462C2×C4—1222no—
463C2×C4—1222no—
464C2×C4—1222no—
465C2×C4—1222no—
466C2×C4—1222no—
467C2×C4—1222no—
468C2×C4—1222no—
469C2×C4—4884no—
470C2×C4—2441no—
471C2×C4—2441no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4#1451974712yes—
476C2×C4#1451974712yes—
477C2×C4#1451979812yes—
478C2×C4#1451980112yes—
479C2×C4C2×D4×D6111yescharacteristic
480C2×C4C22×D24411yescharacteristic
481C2×C4C2×Q8×D6111yescharacteristic
482C2×C4#41×D6111yescharacteristic
483C2×C4#1451982411yescharacteristic
484C2×C4#1451982511yescharacteristic
485C8—22no—
486C8—41no—
487C8—2442no—
488C22—4881no—
489C22—4881no—
490C22—4881no—
491C22—4881no—
492C22—2442no—
493C22—2442no—
494C22—2442no—
495C22C2×#1451974711yescharacteristic
496C22#1684042011yescharacteristic
497C22#1684042711yescharacteristic
498C22C2×#1451979811yescharacteristic
499C22C2×#1451980111yescharacteristic
500C22#1684056811yescharacteristic
501C22#1684057511yescharacteristic
502C4—22no—
503C4—44no—
504C4—82no—
505C4—1224no—
506C4—4881no—
507C4—2442no—
508C4—2442no—
509C4—2442no—
510C4#1683990814yes—
511C4C22⋊Q8×D6111yescharacteristic
512C4#1684033511yescharacteristic
513C4#1684042411yescharacteristic
514C4Q8×D24411yescharacteristic
515C2—4882no—
516C2#2144918211yescharacteristic
517C2C2×#1683990811yescharacteristic
518C2#2145513911yescharacteristic
519C2#2145850911yescharacteristic
520C2#2145855311yescharacteristic
521C2#2145869911yescharacteristic
522C2#2145874511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1990C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1990D61
hypocenterC61#24328
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
1. C2×#16840487C2semidirect
5. #21447156C2semidirect
7. #21446755C2semidirect
9. #21446074C2semidirect
10. #21445821C2semidirect
11. C61×#1990C2semidirect
51. C2×#11758190Q8semidirect
222. Dic122C22⋊Q8semidirect
260. #1990D61semidirect
306. C61#24328semidirect

Maximal subgroups

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

subgroupindex
C2×#168404872
C2×#168400392
C2×C4×#117581902
#214507422
#214471562
#214469642
#214467552
#214461752
#214460742
#214458212
C61×#19902
#2432861