Finite Group Databasebeta

← #29696487

Subgroups — #29696487

Order 15616.

1182 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840486C211yesmaximal, characteristic
2C2×#16839839C211yesmaximal, characteristic
3C4⋊C4×D244C211yesmaximal, characteristic
4#21446781C211yesmaximal, characteristic, complemented
5#21446251C212yesmaximal, complemented
6#21446125C212yesmaximal, complemented
7#21445913C212yesmaximal, complemented
8#21445825C211yesmaximal, characteristic, complemented
9C4×#14519700C211yesmaximal, characteristic, complemented
10#21445537C212yesmaximal, complemented
11C61×#1988C211yesmaximal, characteristic, complemented
12C22×#11758189C2212yes—
13#16840486—24no—
14#16840486C418yescomplemented
15C2×#14519772—22no—
16C2×#14519766—21no—
17C2×#14519757C2212yes—
18C2×Q8×Dic61—21no—
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×D244—21no—
21C2×C4×D244C2211yescharacteristic
22C2×#14519735C2212yes—
23C2×C4⋊C4×D61C2212yes—
24#16839839—24no—
25C2×#14519702C2212yes—
26C2×#14519700C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×#14519689C2212yes—
29C2×C4×#92418C2211yescharacteristic
30#16839660—21no—
31#16839636—22no—
32#16839632C2211yescharacteristic
33#16839619C2212yes—
34#16839525C2212yes—
35#16839524C2211yescharacteristic
36#16839524C2211yescharacteristic
37#16839524C2212yes—
38#16839524C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×C4⋊C8×C61C2211yescharacteristic
41C4×C4⋊C4×C61C2211yescharacteristic
42C61×#340C2212yes—
43C61×#340C2212yes—
44C22×Dic122C2311yescharacteristic
45C22×D244D411yescharacteristic
46C22×C4×D61—22no—
47C22×C4×D61C2312yes—
48C2×#11758189—24no—
49C2×#11758189—24no—
50C2×#11758189—42no—
51C22×#92418C2311yescharacteristic
52C22×#92418C2311yescharacteristic
53#14519772—216no—
54#14519766—28no—
55#14519757—24no—
56#14519757C2×C418yescomplemented
57Q8×Dic61—28no—
58C4×Dic122—22no—
59C4×Dic122C2×C414yescomplemented
60C4×D244—44no—
61C4×D244C2×C414yes—
62C4×D244D414yes—
63#14519735—24no—
64#14519735C2×C418yescomplemented
65C2×C4⋊Dic61—22no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61D411yescharacteristic
69C2×C4⋊Dic61C2312yes—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758196—22no—
73C4⋊C4×D61—24no—
74C4⋊C4×D61—42no—
75C2×#11758198—22no—
76C2×#11758198—22no—
77C2×#11758198—22no—
78C2×#11758198D412yes—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—22no—
81#14519702—42no—
82#14519700—22no—
83#14519700—22no—
84#14519689—24no—
85C4×#92418—22no—
86#14519684—22no—
87C22×Q8×C61—21no—
88C22×C8×C61C2312yes—
89C4⋊Q8×C61—22no—
90C4⋊Q8×C61C2×C414yescomplemented
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61C2311yescharacteristic
93C2×C4⋊C4×C61C2312yes—
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C42×C61—21no—
96C2×C42×C61C2311yescharacteristic
97C4⋊C8×C61—22no—
98#95×C61—21no—
99C23×D61C2×D412yes—
100C2×Q8×C61—24no—
101C2×Q8×C61—42no—
102C2×Q8×C61—42no—
103C22×C4×C61—21no—
104C22×C4×C61—22no—
105C22×C4×C61—21no—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2411yescharacteristic
109C22×Dic61—21no—
110C22×Dic61—21no—
111C22×Dic61—22no—
112C22×Dic61C2×D412yes—
113C2×D244—22no—
114C2×D244—22no—
115C2×D244C22⋊C414yes—
116C2×C4×D61—22no—
117C2×C4×D61—22no—
118C2×C4×D61—22no—
119C2×C4×D61—22no—
120C2×C4×D61—44no—
121C2×C4×D61—44no—
122C2×C4×D61—82no—
123C2×C4×D61C22×C414yes—
124C2×C4×D61C2×D414yes—
125C2×Dic122—21no—
126C2×Dic122—21no—
127C2×Dic122—21no—
128C2×Dic122—21no—
129C2×Dic122—21no—
130C2×Dic122C22×C412yes—
131C2×C8×C61—22no—
132C2×C8×C61—22no—
133C2×C8×C61—22no—
134C4⋊C4×C61—24no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61C22×C414yes—
140C4⋊C4×C61#4112yes—
141C42×C61—42no—
142C42×C61C22×C412yes—
143C42×C61C2×D412yes—
144#11758198—28no—
145#11758198—28no—
146#11758198—44no—
147#11758198C22⋊C418yes—
148C4⋊Dic61—21no—
149C4⋊Dic61—28no—
150C4⋊Dic61—22no—
151C4⋊Dic61—22no—
152C4⋊Dic61—42no—
153C4⋊Dic61—81no—
154C4⋊Dic61C22⋊C414yes—
155C4⋊Dic61C22×C412yes—
156#11758196—24no—
157#11758196—44no—
158C4×Dic61—24no—
159C4×Dic61—28no—
160C2×#92418—21no—
161C2×#92418—21no—
162C2×#92418—21no—
163C2×#92418—21no—
164C2×#92418—21no—
165C2×#92418#4112yes—
166#11758189—48no—
167C23×C61C22×D411yescharacteristic
168C22×D61—22no—
169C22×D61—22no—
170C22×D61—22no—
171C22×D61—22no—
172C22×D61—22no—
173C22×D61—22no—
174C22×D61C2×C22⋊C414yes—
175Q8×C61—48no—
176C2×C4×C61—21no—
177C2×C4×C61—21no—
178C2×C4×C61—24no—
179C2×C4×C61—22no—
180C2×C4×C61—41no—
181C2×C4×C61—42no—
182C2×C4×C61—42no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61C2×C22⋊C412yes—
188C2×C4×C61C2×C22⋊C412yes—
189C2×C4×C61AGL(1,ℤ/8)12yes—
190C2×C4×C61#13712yes—
191C2×C4×C61C23×C411yescharacteristic
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C2×#4111yescharacteristic
194C2×C4×C61C2×#4111yescharacteristic
195C2×C4×C61H(5,𝔽2)11yescharacteristic
196C2×C4×C612−1+411yescharacteristic
197C2×Dic61—22no—
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—22no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—42no—
206C2×Dic61—42no—
207C2×Dic61—42no—
208C2×Dic61C2×C22⋊C414yes—
209D244—28no—
210C4×D61—24no—
211C4×D61—24no—
212C4×D61—24no—
213C4×D61—24no—
214C4×D61—88no—
215Dic122—22no—
216Dic122—22no—
217Dic122—41no—
218Dic122—41no—
219C8×C61—42no—
220#92418—22no—
221#92418—41no—
222#22504—611nomaximal, Sylow
223C22×C61C22×C22⋊C411yescharacteristic
224C22×C61#54611yescharacteristic
225C22×C61#54711yescharacteristic
226C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
227C22×C61C2×#13711yescharacteristic
228C22×C61#57711yescharacteristic
229C22×C61#57811yescharacteristic
230C2×D61—24no—
231C2×D61—24no—
232C2×D61—24no—
233C2×D61—42no—
234C2×D61—42no—
235C2×D61—42no—
236C2×D61—42no—
237C4×C61—22no—
238C4×C61—44no—
239C4×C61—42no—
240C4×C61—81no—
241C4×C61#41714yes—
242C4×C61C4×#4111yescharacteristic
243C4×C61#52011yescharacteristic
244C4×C61#54611yescharacteristic
245C4×C61#54711yescharacteristic
246Dic61—24no—
247Dic61—42no—
248Dic61—42no—
249Dic61—82no—
250C2×#556—611no—
251C2×#436—611no—
252D4×C4⋊C4—611no—
253#1990—611no—
254#1988D6111yescharacteristic, complemented
255#1894—612no—
256#1872—612no—
257#1825—612no—
258#1803—611no—
259C4×#103—611no—
260#1736—612no—
261C2×C61#233911yescharacteristic
262C2×C61C2×#41711yescharacteristic
263C2×C61#289811yescharacteristic
264C2×C61#332411yescharacteristic
265C2×C61#332811yescharacteristic
266C2×C61#335311yescharacteristic
267C2×C61#335611yescharacteristic
268D61—44no—
269C22×M16—612no—
270#556—618no—
271#556—1224no—
272C2×C4⋊Q8C2×D6111yescharacteristic
273C2×#126—1222no—
274C2×#124—1221no—
275C2×C22⋊Q8—612no—
276C2×C4×Q8—611no—
277C2×C4×Q8—1221no—
278C2×C4×D4—611no—
279C2×C4×D4—1221no—
280C2×#117—612no—
281C22×C4⋊C4—612no—
282#436—1224no—
283C2×C4⋊C8C2×D6111yescharacteristic
284C2×#103—611no—
285C2×#103—611no—
286C2×#103—612no—
287C2×C22⋊C8—612no—
288C2×C4×C8—611no—
289#389—1221no—
290#385—1222no—
291#384—611no—
292#380—612no—
293C4×C4⋊C4C2×D6111yescharacteristic
294#340—612no—
295#340—611no—
296#340—612no—
297#340—612no—
298#340—611no—
299#340C2×D6112yes—
300#340C2×D6112yes—
301C61#2250411yescharacteristic, complemented, Sylow
302C22×Q8—21no—
303C22×Q8—611no—
304C22×D4—611no—
305C23×C4—612no—
306C23×C4—1222no—
307C2×M16—1224no—
308C2×M16—1224no—
309C2×M16—2442no—
310C22×C8—611no—
311C22×C8—611no—
312C22×C8C22×D6112yes—
313C4⋊Q8—22no—
314C4⋊Q8C4×D6114yescomplemented
315#126—12216no—
316#124—1228no—
317C22⋊Q8—618no—
318C22⋊Q8—1224no—
319C4×Q8—614no—
320C4×Q8—1228no—
321C4×Q8—1222no—
322C4×D4—614no—
323C4×D4—614no—
324C4×D4—2444no—
325#117—618no—
326#117—1224no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—611no—
329C2×C4⋊C4—611no—
330C2×C4⋊C4—612no—
331C2×C4⋊C4—1222no—
332C2×C4⋊C4—1221no—
333C2×C4⋊C4—1224no—
334C2×C4⋊C4—1221no—
335C2×C4⋊C4—1222no—
336C2×C4⋊C4—1221no—
337C2×C4⋊C4—2442no—
338C2×C4⋊C4C22×D6111yescharacteristic
339C2×C4⋊C4C22×D6112yes—
340C2×C4⋊C4C22×D6111yescharacteristic
341C2×C22⋊C4—612no—
342C2×C22⋊C4—1222no—
343C2×C22⋊C4—1222no—
344C2×C22⋊C4—1222no—
345C2×C42—21no—
346C2×C42—1222no—
347C2×C42—1221no—
348C2×C42C22×D6111yescharacteristic
349C4⋊C8—22no—
350#103—1222no—
351#103—1222no—
352#103—2442no—
353C22⋊C8—1224no—
354C4×C8—1222no—
355#95—21no—
356#95—1222no—
357C24—612no—
358C2×Q8—24no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×Q8—612no—
362C2×Q8—1221no—
363C2×Q8—1221no—
364C2×Q8—1221no—
365C2×Q8—1221no—
366C2×Q8—1221no—
367C2×D4—614no—
368C2×D4—1222no—
369C2×D4—1222no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—612no—
374C22×C4—614no—
375C22×C4—614no—
376C22×C4—1222no—
377C22×C4—1222no—
378C22×C4—1222no—
379C22×C4—1222no—
380C22×C4—1221no—
381C22×C4—1221no—
382C22×C4—1222no—
383C22×C4—4882no—
384C22×C4—2444no—
385C22×C4—2444no—
386C22×C4D4×D6111yescharacteristic
387C22×C4D4×D6111yescharacteristic
388C22×C4C23×D6111yescharacteristic
389M16—2448no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—612no—
394C2×C8—1221no—
395C2×C8—1221no—
396C2×C8—1221no—
397C2×C8—1221no—
398C2×C8—1221no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—24no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—614no—
405C4⋊C4—612no—
406C4⋊C4—1228no—
407C4⋊C4—1221no—
408C4⋊C4—1224no—
409C4⋊C4—1222no—
410C4⋊C4—1222no—
411C4⋊C4—4881no—
412C4⋊C4—2444no—
413C4⋊C4—2442no—
414C4⋊C4C2×C4×D6114yes—
415C4⋊C4#1175822612yes—
416C22⋊C4—618no—
417C22⋊C4—1228no—
418C22⋊C4—1228no—
419C22⋊C4—2444no—
420C42—42no—
421C42—1228no—
422C42—1224no—
423C42C2×C4×D6112yes—
424C42D4×D6112yes—
425C23—614no—
426C23—1222no—
427C23—1222no—
428C23—1222no—
429C23—1222no—
430C23—1222no—
431C23—1222no—
432C23C2×D4×D6111yescharacteristic
433Q8—48no—
434Q8—1222no—
435Q8—1222no—
436Q8—2441no—
437Q8—2441no—
438D4—1228no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—614no—
451C2×C4—1222no—
452C2×C4—1222no—
453C2×C4—1224no—
454C2×C4—1224no—
455C2×C4—1224no—
456C2×C4—1224no—
457C2×C4—1222no—
458C2×C4—1222no—
459C2×C4—4888no—
460C2×C4—2442no—
461C2×C4—2442no—
462C2×C4—2442no—
463C2×C4—2441no—
464C2×C4—2441no—
465C2×C4—2441no—
466C2×C4—2441no—
467C2×C4C22⋊C4×D6112yes—
468C2×C4C22⋊C4×D6112yes—
469C2×C4#1451980112yes—
470C2×C4#1451980612yes—
471C2×C4C22×C4×D6111yescharacteristic
472C2×C4C2×D4×D6111yescharacteristic
473C2×C4#41×D6111yescharacteristic
474C2×C4C2×#1175822611yescharacteristic
475C2×C4#1451982311yescharacteristic
476C2×C4#1451982511yescharacteristic
477C8—42no—
478C8—1222no—
479C8—2441no—
480C22—1224no—
481C22—1224no—
482C22—1224no—
483C22—2442no—
484C22—2442no—
485C22—2442no—
486C22—2442no—
487C22C2×C22⋊C4×D6111yescharacteristic
488C22#1684040311yescharacteristic
489C22#1684041411yescharacteristic
490C22C2×#1451980111yescharacteristic
491C22C2×#1451980611yescharacteristic
492C22#1684057111yescharacteristic
493C22#1684057811yescharacteristic
494C4—22no—
495C4—42no—
496C4—44no—
497C4—81no—
498C4—1224no—
499C4—4882no—
500C4—2442no—
501C4—2442no—
502C4#1683976014yes—
503C4C4×#1175822611yescharacteristic
504C4#1684024411yescharacteristic
505C4#1684040511yescharacteristic
506C4#1684041511yescharacteristic
507C2—2444no—
508C2#2144864811yescharacteristic
509C2C2×#1683976011yescharacteristic
510C2#2145435111yescharacteristic
511C2#2145804711yescharacteristic
512C2#2145809911yescharacteristic
513C2#2145840011yescharacteristic
514C2#2145843111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1988C2
socleC23×C61C22×D4
prime radicalC22×D244D4
cosocleC22×C4×C61C24
hypercenter#1988D61
hypocenterC61#22504
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
4. #21446781C2semidirect
5. #21446251C2semidirect
6. #21446125C2semidirect
7. #21445913C2semidirect
8. #21445825C2semidirect
9. C4×#14519700C2semidirect
10. #21445537C2semidirect
11. C61×#1988C2semidirect
14. #16840486C4semidirect
56. #14519757C2×C4semidirect
59. C4×Dic122C2×C4semidirect
64. #14519735C2×C4semidirect
90. C4⋊Q8×C61C2×C4semidirect
254. #1988D61semidirect
301. C61#22504semidirect
314. C4⋊Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168404862
C2×#168398392
C4⋊C4×D2442
#214467812
#214462512
#214461252
#214459132
#214458252
C4×#145197002
#214455372
C61×#19882
#2250461