Finite Group Databasebeta

← #29736137

Subgroups — #29736137

Order 15616.

1056 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840468C211yesmaximal, characteristic
2C2×#16840172C211yesmaximal, characteristic, complemented
3C2×#16840043C211yesmaximal, characteristic
4#21451027C211yesmaximal, characteristic, complemented
5#21447160C212yesmaximal, complemented
6#21447146C212yesmaximal, complemented
7#21446758C211yesmaximal, characteristic, complemented
8#21446081C212yesmaximal
9#21446067C212yesmaximal
10#21445647C211yesmaximal, characteristic
11C61×#1989C211yesmaximal, characteristic, complemented
12#16840468—24no—
13#16840468—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519776C2211yescharacteristic
16C2×#14519768C2211yescharacteristic
17C2×#14519757—21no—
18C2×#14519755—21no—
19C2×#14519755—21no—
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×#14519739C2211yescharacteristic
23C2×#14519739C2211yescharacteristic
24C2×C4⋊C4×D61—21no—
25#16840172—24no—
26#16840043—24no—
27C2×#14519711C2212yes—
28C2×#14519710C2212yes—
29C2×#14519708C2212yes—
30C2×#14519699C2212yes—
31C2×#14519697C2212yes—
32C2×#14519688C2211yescharacteristic
33#16839683—21no—
34#16839683—21no—
35#16839668—21no—
36#16839661C2212yes—
37#16839661C2212yes—
38#16839632—21no—
39#16839524C2212yes—
40#16839516C2212yes—
41#16839516C2212yes—
42C2×#125×C61C2211yescharacteristic
43C2×C4⋊C8×C61C2211yescharacteristic
44C4×C4⋊C4×C61C2211yescharacteristic
45C61×#340C2212yes—
46C61×#340C2212yes—
47C22×Dic122—21no—
48C22×Dic122—21no—
49C22×Dic122D411yescharacteristic
50C22×D244C2311yescharacteristic
51C22×C4×D61—21no—
52C22×C4×D61—21no—
53C22×C4×D61—21no—
54C22×#92418C2312yes—
55#14519777—22no—
56#14519777—22no—
57#14519776—22no—
58#14519776—22no—
59#14519768—22no—
60#14519768—22no—
61#14519757—28no—
62#14519755—28no—
63#14519755—28no—
64C4×Dic122—22no—
65C4×Dic122—22no—
66C4×D244—22no—
67C4×D244D414yes—
68#14519739—22no—
69#14519739—22no—
70#14519739—22no—
71#14519739—22no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—22no—
75C2×C4⋊Dic61—22no—
76C2×C4⋊Dic61D411yescharacteristic
77C2×C4⋊Dic61C2312yes—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196D411yescharacteristic
82C2×#11758196D411yescharacteristic
83C4⋊C4×D61—28no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519711—24no—
90#14519710—24no—
91#14519708—24no—
92#14519699—24no—
93#14519697Q818yescomplemented
94#14519688—22no—
95#14519684—22no—
96#14519684—22no—
97C22×C8×C61C2312yes—
98#125×C61—22no—
99#125×C61D414yes—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2312yes—
105C2×C42×C61—21no—
106C2×C42×C61C2311yescharacteristic
107C4⋊C8×C61—22no—
108#95×C61—21no—
109C23×D61—21no—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—22no—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2411yescharacteristic
116C22×Dic61—21no—
117C22×Dic61—22no—
118C22×Dic61C2×D411yescharacteristic
119C22×Dic61C2×D411yescharacteristic
120C22×Dic61C2×D411yescharacteristic
121C22×Dic61C2×D411yescharacteristic
122C2×D244—21no—
123C2×D244—22no—
124C2×D244C2×D412yes—
125C2×D244C2×Q814yes—
126C2×C4×D61—24no—
127C2×C4×D61—24no—
128C2×C4×D61—24no—
129C2×C4×D61—42no—
130C2×C4×D61—42no—
131C2×C4×D61—82no—
132C2×C4×D61—82no—
133C2×Dic122—22no—
134C2×Dic122—22no—
135C2×Dic122—24no—
136C2×Dic122—41no—
137C2×Dic122—42no—
138C2×Dic122—82no—
139C2×Dic122—82no—
140C2×C8×C61—22no—
141C2×C8×C61—24no—
142C4⋊C4×C61—24no—
143C4⋊C4×C61—24no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61C2×D412yes—
148C4⋊C4×C61C2×D412yes—
149C4⋊C4×C61C2×Q814yes—
150C42×C61—42no—
151C42×C61C2×D412yes—
152C42×C61C2×D412yes—
153#11758198—24no—
154#11758198—24no—
155#11758198—24no—
156C4⋊Dic61—24no—
157C4⋊Dic61—22no—
158C4⋊Dic61—21no—
159C4⋊Dic61—22no—
160C4⋊Dic61—22no—
161C4⋊Dic61—42no—
162C4⋊Dic61—82no—
163C4⋊Dic61—82no—
164C4⋊Dic61C2×D412yes—
165#11758196—24no—
166#11758196—24no—
167#11758196—22no—
168#11758196—22no—
169C4×Dic61—21no—
170C4×Dic61—21no—
171C4×Dic61—21no—
172C4×Dic61—21no—
173C2×#92418—22no—
174C2×#92418—22no—
175C2×#92418C2×Q814yes—
176C23×C61C22×D411yescharacteristic
177C22×D61—22no—
178C22×D61—24no—
179C22×D61—41no—
180C22×D61—41no—
181C22×D61—42no—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—24no—
188C2×C4×C61—41no—
189C2×C4×C61—42no—
190C2×C4×C61—41no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61#12012yes—
194C2×C4×C61#12012yes—
195C2×C4×C61C22×D411yescharacteristic
196C2×C4×C61C22×D411yescharacteristic
197C2×C4×C61C22×Q812yes—
198C2×C4×C612−1+412yes—
199C2×Dic61—21no—
200C2×Dic61—21no—
201C2×Dic61—21no—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—22no—
205C2×Dic61—21no—
206C2×Dic61—21no—
207C2×Dic61—22no—
208C2×Dic61—22no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61#13712yes—
214D244—22no—
215D244—24no—
216D244C4⋊Q814yes—
217C4×D61—48no—
218C4×D61—84no—
219C4×D61—84no—
220Dic122—44no—
221Dic122—84no—
222Dic122—84no—
223C8×C61—42no—
224#92418—24no—
225#25850—611nomaximal, Sylow
226C22×C61C2×#12011yescharacteristic
227C22×C61D4×Q812yes—
228C22×C61C2×#13711yescharacteristic
229C22×C61#57511yescharacteristic
230C22×C61#57612yes—
231C2×D61—24no—
232C2×D61—44no—
233C2×D61—44no—
234C2×D61—42no—
235C2×D61—42no—
236C4×C61—22no—
237C4×C61—22no—
238C4×C61—44no—
239C4×C61—42no—
240C4×C61—42no—
241C4×C61C2×C4⋊Q811yescharacteristic
242C4×C61#53611yescharacteristic
243C4×C61D4×Q812yes—
244Dic61—22no—
245Dic61—41no—
246Dic61—41no—
247Dic61—42no—
248Dic61—41no—
249Dic61—82no—
250C2×#554—611no—
251C2×#501—611no—
252C2×#474—611no—
253#2567—611no—
254#2059—612no—
255#2057—612no—
256#1989D6111yescharacteristic, complemented
257#1985—611no—
258#1864—612no—
259#1862—612no—
260#1762—611no—
261C2×C61#243411yescharacteristic
262C2×C61#300711yescharacteristic
263C2×C61#300811yescharacteristic
264C2×C61#338512yes—
265C2×C61#338912yes—
266D61—44no—
267#554—1224no—
268#554—1224no—
269C2×C4⋊Q8—611no—
270C2×C4⋊Q8—611no—
271C2×#125—611no—
272C2×#125C2×D6111yescharacteristic
273C2×C22⋊Q8—1221no—
274C2×C22⋊Q8—1221no—
275C2×C22⋊Q8—1221no—
276C2×C4×Q8—611no—
277C2×C4×D4—611no—
278C2×C4×D4—611no—
279C2×C4×D4—611no—
280C22×C4⋊C4—1221no—
281#501—1224no—
282#474—1224no—
283C2×#107—612no—
284C2×#107—612no—
285C2×#106—612no—
286C2×C4⋊C8C2×D6111yescharacteristic
287C2×#102—612no—
288C2×#102—612no—
289C2×#97—611no—
290#395—1221no—
291#395—1221no—
292#391—1221no—
293#389—612no—
294#389—612no—
295#384—1221no—
296C4×C4⋊C4C2×D6111yescharacteristic
297#340—612no—
298#340C2×D6112yes—
299#340C2×D6112yes—
300#336—612no—
301#336—612no—
302C61#2585011yescharacteristic, complemented, Sylow
303C22×Q8—611no—
304C22×Q8—1221no—
305C22×Q8—1221no—
306C22×D4—611no—
307C23×C4—1221no—
308C23×C4—1221no—
309C23×C4—1221no—
310C22×C8—612no—
311C22×C8C22×D6112yes—
312C4⋊Q8—1222no—
313C4⋊Q8—1222no—
314C4⋊Q8—1222no—
315C4⋊Q8—1222no—
316#125—22no—
317#125—1222no—
318#125—1222no—
319#125D24414yes—
320C22⋊Q8—1228no—
321C22⋊Q8—1228no—
322C22⋊Q8—1228no—
323C4×Q8—1222no—
324C4×Q8—1222no—
325C4×D4—614no—
326C4×D4—1222no—
327C4×D4—1222no—
328C4×D4—1222no—
329C4×D4—1222no—
330C4×D4—1222no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—611no—
334C2×C4⋊C4—611no—
335C2×C4⋊C4—611no—
336C2×C4⋊C4—611no—
337C2×C4⋊C4—612no—
338C2×C4⋊C4—1228no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4—1222no—
343C2×C4⋊C4—1222no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4C22×D6111yescharacteristic
346C2×C4⋊C4C22×D6111yescharacteristic
347C2×C4⋊C4C22×D6112yes—
348C2×C22⋊C4—1221no—
349C2×C22⋊C4—1221no—
350C2×C22⋊C4—1221no—
351C2×C42—21no—
352C2×C42—611no—
353C2×C42—611no—
354C2×C42C22×D6111yescharacteristic
355#107—1224no—
356#107—1224no—
357#106—1224no—
358C4⋊C8—22no—
359#102—618no—
360#102—1224no—
361#97—1222no—
362#95—21no—
363#95—1222no—
364#95—1222no—
365C24—1221no—
366C2×Q8—1222no—
367C2×Q8—1224no—
368C2×Q8—1222no—
369C2×Q8—4882no—
370C2×Q8—4882no—
371C2×Q8—2441no—
372C2×Q8—2442no—
373C2×D4—612no—
374C2×D4—614no—
375C2×D4—1221no—
376C2×D4—1222no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—611no—
381C22×C4—611no—
382C22×C4—611no—
383C22×C4—611no—
384C22×C4—1224no—
385C22×C4—1224no—
386C22×C4—1221no—
387C22×C4—1224no—
388C22×C4—1222no—
389C22×C4—4882no—
390C22×C4—4882no—
391C22×C4—2442no—
392C22×C4—2442no—
393C22×C4D4×D6111yescharacteristic
394C22×C4D4×D6111yescharacteristic
395C22×C4C23×D6111yescharacteristic
396C2×C8—22no—
397C2×C8—24no—
398C2×C8—614no—
399C2×C8—1222no—
400C2×C8—1222no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—612no—
407C4⋊C4—1224no—
408C4⋊C4—1224no—
409C4⋊C4—1224no—
410C4⋊C4—1221no—
411C4⋊C4—1222no—
412C4⋊C4—1222no—
413C4⋊C4—1222no—
414C4⋊C4—1222no—
415C4⋊C4—1222no—
416C4⋊C4—4882no—
417C4⋊C4—4882no—
418C4⋊C4—2442no—
419C4⋊C4C2×D24412yes—
420C4⋊C4C2×D24412yes—
421C4⋊C4Q8×D6114yes—
422C22⋊C4—1224no—
423C22⋊C4—1224no—
424C22⋊C4—1224no—
425C42—42no—
426C42—1221no—
427C42—1221no—
428C42—1221no—
429C42—1221no—
430C42C2×D24412yes—
431C42D4×D6112yes—
432C23—1222no—
433C23—1224no—
434C23—2441no—
435C23—2441no—
436C23—2442no—
437C23C2×D4×D6111yescharacteristic
438Q8—4884no—
439Q8—4884no—
440Q8—2444no—
441D4—614no—
442D4—1222no—
443D4—1224no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—612no—
456C2×C4—1221no—
457C2×C4—1221no—
458C2×C4—1221no—
459C2×C4—1221no—
460C2×C4—1221no—
461C2×C4—1222no—
462C2×C4—1221no—
463C2×C4—1221no—
464C2×C4—1222no—
465C2×C4—1222no—
466C2×C4—4884no—
467C2×C4—4884no—
468C2×C4—2448no—
469C2×C4—2442no—
470C2×C4—2442no—
471C2×C4—2442no—
472C2×C4—2442no—
473C2×C4#1451974712yes—
474C2×C4#1451974712yes—
475C2×C4C2×D4×D6111yescharacteristic
476C2×C4C22×D24411yescharacteristic
477C2×C4C2×Q8×D6112yes—
478C2×C4#1451982412yes—
479C8—42no—
480C8—1224no—
481C22—1224no—
482C22—2444no—
483C22—2444no—
484C22—2442no—
485C22—2442no—
486C22C2×#1451974711yescharacteristic
487C22#1684042512yes—
488C22#137×D6111yescharacteristic
489C22#1684055811yescharacteristic
490C22#1684056412yes—
491C4—22no—
492C4—22no—
493C4—44no—
494C4—42no—
495C4—42no—
496C4—1222no—
497C4—4882no—
498C4—2441no—
499C4—2441no—
500C4—2441no—
501C4—2442no—
502C4C4⋊Q8×D6111yescharacteristic
503C4#1684033211yescharacteristic
504C4Q8×D24412yes—
505C2—2444no—
506C2#2144946211yescharacteristic
507C2#2145508011yescharacteristic
508C2#2145509011yescharacteristic
509C2#2145873212yes—
510C2#2145877612yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1989C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1989D61
hypocenterC61#25850
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1989

Complements

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

NH ≅ G/Ntype
2. C2×#16840172C2semidirect
4. #21451027C2semidirect
5. #21447160C2semidirect
6. #21447146C2semidirect
7. #21446758C2semidirect
11. C61×#1989C2semidirect
93. #14519697Q8semidirect
256. #1989D61semidirect
302. C61#25850semidirect

Maximal subgroups

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

subgroupindex
C2×#168404682
C2×#168401722
C2×#168400432
#214510272
#214471602
#214471462
#214467582
#214460812
#214460672
#214456472
C61×#19892
#2585061