Finite Group Databasebeta

← #29685253

Subgroups — #29685253

Order 15616.

1186 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×#16840316C211yesmaximal, characteristic
2C2×#16839714C211yesmaximal, characteristic
3#21448223C211yesmaximal, characteristic
4#21446778C212yesmaximal, complemented
5#21446619C212yesmaximal, complemented
6#21446251C212yesmaximal
7#21446249C212yesmaximal
8#21445652C211yesmaximal, characteristic, complemented
9#21445650C211yesmaximal, characteristic, complemented
10#21445649C211yesmaximal, characteristic, complemented
11C61×#1763C211yesmaximal, characteristic, complemented
12C22×#11758189C2212yes—
13#16840316—24no—
14#16840316C418yescomplemented
15C2×#14519776C2211yescharacteristic
16C2×#14519775C2211yescharacteristic
17C2×#14519767—21no—
18C2×#14519757C2211yescharacteristic
19C2×#14519755—21no—
20C2×#14519753C2211yescharacteristic
21C2×C4×Dic122—21no—
22C2×C4×Dic122C2211yescharacteristic
23C2×#14519734—21no—
24C2×#14519734—21no—
25C2×C42×D61C2211yescharacteristic
26C2×#14519702C2212yes—
27C2×#14519701C2212yes—
28C2×#14519700C2212yes—
29#16839714—24no—
30C2×#14519688C2211yescharacteristic
31C2×C8×Dic61C2212yes—
32#16839631C2212yes—
33#16839626—21no—
34#16839621—21no—
35#16839619C2212yes—
36C4×#11758196—21no—
37C4×C4⋊Dic61C2211yescharacteristic
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C4×Q8×C61C2211yescharacteristic
42C2×#103×C61C2212yes—
43C2×#97×C61C2211yescharacteristic
44C4×C4⋊C4×C61C2211yescharacteristic
45C61×#340C2212yes—
46C22×Dic122—21no—
47C22×Dic122C2311yescharacteristic
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C2×#11758189—24no—
52C2×#11758189—24no—
53C2×#11758189—24no—
54C22×#92418C2312yes—
55#14519776—22no—
56#14519776C2×C414yes—
57#14519775—22no—
58#14519775C2×C414yes—
59#14519767—28no—
60#14519757—22no—
61#14519757C2×C414yes—
62#14519755—28no—
63#14519753—22no—
64#14519753C2×C414yes—
65C4×Dic122—22no—
66C4×Dic122—28no—
67C4×Dic122C2×C414yes—
68#14519734—28no—
69#14519734—44no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2312yes—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—22no—
79C2×#11758196—22no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—22no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88C42×D61C2×C414yes—
89C42×D61D414yes—
90#14519702—24no—
91#14519701—24no—
92#14519700—24no—
93#14519688—22no—
94C8×Dic61—24no—
95#14519684—21no—
96#14519684—22no—
97C22×Q8×C61C2311yescharacteristic
98C22×C8×C61C2312yes—
99C4×Q8×C61—22no—
100C4×Q8×C61C2×C414yes—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61C2312yes—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C42×C61—21no—
105C2×C42×C61—21no—
106C2×C42×C61C2311yescharacteristic
107#103×C61—24no—
108#97×C61—22no—
109#95×C61—21no—
110C23×D61C2×D411yescharacteristic
111C2×Q8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61C22×C412yes—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2411yescharacteristic
121C22×Dic61—21no—
122C22×Dic61—21no—
123C22×Dic61—22no—
124C22×Dic61C2×D411yescharacteristic
125C22×Dic61C2×D411yescharacteristic
126C22×Dic61C2×D411yescharacteristic
127C2×C4×D61—22no—
128C2×C4×D61—22no—
129C2×C4×D61—22no—
130C2×C4×D61—22no—
131C2×C4×D61—24no—
132C2×C4×D61C22⋊C414yes—
133C2×C4×D61C22⋊C414yes—
134C2×C4×D61C22×C412yes—
135C2×C4×D61C2×D412yes—
136C2×Dic122—22no—
137C2×Dic122—22no—
138C2×Dic122—21no—
139C2×Dic122—24no—
140C2×Dic122—42no—
141C2×Dic122—42no—
142C2×Dic122C22×C412yes—
143C2×C8×C61—22no—
144C2×C8×C61—22no—
145C2×C8×C61#4114yes—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61C22×C412yes—
151C42×C61—24no—
152C42×C61—42no—
153C42×C61C22×C412yes—
154C42×C61C2×D412yes—
155#11758198—24no—
156#11758198—24no—
157#11758198—44no—
158C4⋊Dic61—21no—
159C4⋊Dic61—24no—
160C4⋊Dic61—22no—
161C4⋊Dic61—22no—
162C4⋊Dic61C22×C412yes—
163#11758196—24no—
164#11758196—24no—
165#11758196—24no—
166#11758196—24no—
167#11758196—24no—
168#11758196—44no—
169#11758196—44no—
170C4×Dic61—24no—
171C4×Dic61—42no—
172C4×Dic61C22⋊C414yes—
173C4×Dic61C22⋊C414yes—
174C4×Dic61C22×C412yes—
175C4×Dic61C2×D412yes—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418#4114yes—
179#11758189—216no—
180C23×C61C22×D411yescharacteristic
181C22×D61—22no—
182C22×D61—21no—
183C22×D61—22no—
184C22×D61—21no—
185C22×D61C2×C22⋊C412yes—
186Q8×C61—22no—
187Q8×C61—22no—
188Q8×C61—42no—
189C2×C4×C61—21no—
190C2×C4×C61—22no—
191C2×C4×C61—21no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—42no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61C2×C22⋊C412yes—
201C2×C4×C61C2×C22⋊C412yes—
202C2×C4×C61C23×C411yescharacteristic
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61C2×#4112yes—
205C2×C4×C61C2×#4112yes—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—22no—
209C2×Dic61—42no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61C2×C22⋊C412yes—
216C2×Dic61C2×C22⋊C412yes—
217C2×Dic61C2×C22⋊C412yes—
218C2×Dic61AGL(1,ℤ/8)12yes—
219C2×Dic61#13712yes—
220C2×Dic61#13714yes—
221C4×D61—22no—
222C4×D61—22no—
223C4×D61—24no—
224C4×D61—28no—
225C4×D61—28no—
226Dic122—22no—
227Dic122—22no—
228Dic122—42no—
229Dic122—48no—
230C8×C61—24no—
231#92418—24no—
232#21409—611nomaximal, Sylow
233C22×C61C22×C22⋊C411yescharacteristic
234C22×C61#53212yes—
235C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
236C22×C61C2×#13712yes—
237C22×C61C2×#13711yescharacteristic
238C2×D61—22no—
239C2×D61—22no—
240C2×D61—22no—
241C2×D61—42no—
242C2×D61—42no—
243C4×C61—22no—
244C4×C61—22no—
245C4×C61—42no—
246C4×C61—42no—
247C4×C61—82no—
248C4×C61C4×#4111yescharacteristic
249C4×C61C4×#4111yescharacteristic
250C4×C61#52512yes—
251Dic61—22no—
252Dic61—42no—
253Dic61—42no—
254Dic61—82no—
255Dic61#41714yes—
256Dic61#41914yes—
257C2×#533—611no—
258C2×C4×M16—611no—
259#2285—611no—
260#1990—612no—
261#1956—612no—
262#1894—612no—
263#1894—612no—
264#1763—611no—
265#1763—611no—
266#1763—611no—
267#1763D6111yescharacteristic, complemented
268C2×C61#229711yescharacteristic
269C2×C61C2×#41911yescharacteristic
270C2×C61C2×#41711yescharacteristic
271C2×C61#318212yes—
272C2×C61#318512yes—
273D61—42no—
274C22×M16—612no—
275#533—618no—
276#533—1224no—
277C2×C4⋊Q8—611no—
278C2×C4⋊Q8—611no—
279C2×#125—1221no—
280C2×C22⋊Q8—611no—
281C2×C22⋊Q8—611no—
282C2×C22⋊Q8—1221no—
283C2×C4×Q8—611no—
284C2×C4×Q8—1221no—
285C2×C4×Q8C2×D6111yescharacteristic
286C2×#117—1221no—
287C2×#117—1221no—
288C22×C42—611no—
289C2×#103—612no—
290C2×#103—612no—
291C2×#103—612no—
292C2×#103C2×D6112yes—
293C4×M16—1224no—
294C2×#97—611no—
295C2×#97C2×D6111yescharacteristic
296C2×C4×C8—612no—
297#384—612no—
298#382—1221no—
299#380—612no—
300#380—1221no—
301C4×C4⋊C4—611no—
302C4×C4⋊C4—1221no—
303C4×C4⋊C4C2×D6111yescharacteristic
304#340—612no—
305#340—612no—
306#340—612no—
307#340C2×D6112yes—
308C61#2140911yescharacteristic, complemented, Sylow
309C22×Q8—611no—
310C22×Q8—1221no—
311C22×Q8C22×D6111yescharacteristic
312C23×C4—611no—
313C23×C4—611no—
314C23×C4—611no—
315C2×M16—1224no—
316C2×M16—1224no—
317C2×M16—1224no—
318C22×C8—612no—
319C22×C8C22×D6112yes—
320C4⋊Q8—614no—
321C4⋊Q8—614no—
322C4⋊Q8—1222no—
323C4⋊Q8—1222no—
324#125—1228no—
325C22⋊Q8—614no—
326C22⋊Q8—614no—
327C22⋊Q8—1222no—
328C22⋊Q8—1228no—
329C22⋊Q8—1222no—
330C4×Q8—22no—
331C4×Q8—614no—
332C4×Q8—1228no—
333C4×Q8—1222no—
334C4×Q8C4×D6114yes—
335#117—1228no—
336#117—2444no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—611no—
339C2×C4⋊C4—612no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4—1221no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4—1221no—
346C2×C4⋊C4—1222no—
347C2×C4⋊C4—1222no—
348C2×C4⋊C4C22×D6112yes—
349C2×C4⋊C4C22×D6111yescharacteristic
350C2×C22⋊C4—1221no—
351C2×C22⋊C4—1221no—
352C2×C22⋊C4—1222no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—611no—
356C2×C42—611no—
357C2×C42—611no—
358C2×C42—614no—
359C2×C42—614no—
360C2×C42—1221no—
361C2×C42—1221no—
362C2×C42C22×D6111yescharacteristic
363#103—24no—
364#103—1224no—
365#103—1224no—
366#103—1224no—
367#97—22no—
368#97—1222no—
369C4×C8—1224no—
370#95—21no—
371#95—1222no—
372#95—1221no—
373C24—611no—
374C2×Q8—21no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—612no—
378C2×Q8—1224no—
379C2×Q8—1221no—
380C2×Q8—1222no—
381C2×Q8—1222no—
382C2×Q8—2442no—
383C2×Q8—2442no—
384C2×Q8C2×C4×D6112yes—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—611no—
389C22×C4—611no—
390C22×C4—611no—
391C22×C4—612no—
392C22×C4—612no—
393C22×C4—614no—
394C22×C4—614no—
395C22×C4—1222no—
396C22×C4—1224no—
397C22×C4—1222no—
398C22×C4—1222no—
399C22×C4—1222no—
400C22×C4—1221no—
401C22×C4—1221no—
402C22×C4—1222no—
403C22×C4D4×D6111yescharacteristic
404C22×C4D4×D6111yescharacteristic
405C22×C4C23×D6111yescharacteristic
406M16—12216no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—614no—
410C2×C8—1222no—
411C2×C8—1222no—
412C2×C8#1175822214yes—
413C4⋊C4—21no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—612no—
418C4⋊C4—1224no—
419C4⋊C4—1224no—
420C4⋊C4—1224no—
421C4⋊C4—1224no—
422C4⋊C4—1224no—
423C4⋊C4—1221no—
424C4⋊C4—1224no—
425C4⋊C4—1222no—
426C4⋊C4—1222no—
427C4⋊C4—2444no—
428C4⋊C4—2444no—
429C4⋊C4C2×C4×D6112yes—
430C22⋊C4—1224no—
431C22⋊C4—1224no—
432C22⋊C4—2444no—
433C42—24no—
434C42—42no—
435C42—612no—
436C42—612no—
437C42—614no—
438C42—614no—
439C42—1224no—
440C42—2442no—
441C42C2×C4×D6112yes—
442C42D4×D6112yes—
443C23—612no—
444C23—1221no—
445C23—1222no—
446C23—1221no—
447C23—1222no—
448C23C2×D4×D6111yescharacteristic
449Q8—22no—
450Q8—22no—
451Q8—42no—
452Q8—1222no—
453Q8—1222no—
454Q8—2448no—
455Q8—2442no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—612no—
468C2×C4—612no—
469C2×C4—612no—
470C2×C4—614no—
471C2×C4—612no—
472C2×C4—612no—
473C2×C4—1228no—
474C2×C4—1228no—
475C2×C4—1222no—
476C2×C4—1222no—
477C2×C4—1222no—
478C2×C4—1224no—
479C2×C4—1222no—
480C2×C4—1222no—
481C2×C4—2442no—
482C2×C4—2442no—
483C2×C4—2442no—
484C2×C4—2441no—
485C2×C4—2441no—
486C2×C4—2442no—
487C2×C4C22⋊C4×D6112yes—
488C2×C4C22⋊C4×D6112yes—
489C2×C4C22×C4×D6111yescharacteristic
490C2×C4C2×D4×D6111yescharacteristic
491C2×C4#41×D6112yes—
492C2×C4C2×#1175822212yes—
493C8—24no—
494C8—1224no—
495C22—1222no—
496C22—1222no—
497C22—1222no—
498C22—2442no—
499C22—2442no—
500C22C2×C22⋊C4×D6111yescharacteristic
501C22#1684030912yes—
502C22AGL(1,ℤ/8)×D6111yescharacteristic
503C22#137×D6112yes—
504C22#137×D6111yescharacteristic
505C4—22no—
506C4—22no—
507C4—42no—
508C4—42no—
509C4—82no—
510C4—614no—
511C4—614no—
512C4—1222no—
513C4—4882no—
514C4—2442no—
515C4—2442no—
516C4#1684023011yescharacteristic
517C4C4×#1175822211yescharacteristic
518C4#1684027112yes—
519C2—2442no—
520C2#2144830511yescharacteristic
521C2#419×D6111yescharacteristic
522C2#417×D6111yescharacteristic
523C2#2145653412yes—
524C2#2145656112yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1763C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1763D61
hypocenterC61#21409
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
4. #21446778C2semidirect
5. #21446619C2semidirect
8. #21445652C2semidirect
9. #21445650C2semidirect
10. #21445649C2semidirect
11. C61×#1763C2semidirect
14. #16840316C4semidirect
267. #1763D61semidirect
308. C61#21409semidirect

Maximal subgroups

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

subgroupindex
C2×#168403162
C2×#168397142
#214482232
#214467782
#214466192
#214462512
#214462492
#214456522
#214456502
#214456492
C61×#17632
#2140961