Finite Group Databasebeta

← #29684526

Subgroups — #29684526

Order 15616.

1194 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840318C211yesmaximal, characteristic
2C2×C4⋊C8×D61C211yesmaximal, characteristic
3C4×#14519735C211yesmaximal, characteristic
4#21446777C211yesmaximal, characteristic, complemented
5#21446776C211yesmaximal, characteristic, complemented
6#21446774C211yesmaximal, characteristic, complemented
7#21446769C212yesmaximal, complemented
8#21446765C212yesmaximal, complemented
9#21445534C212yesmaximal
10#21445534C212yesmaximal
11C61×#1989C211yesmaximal, characteristic, complemented
12C22×C8×D61C2212yes—
13#16840318—24no—
14#16840318C418yescomplemented
15C2×#14519776C2211yescharacteristic
16C2×#14519776C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C2×#14519755—21no—
19C2×#14519755—21no—
20C2×#14519742—21no—
21C2×#14519742—21no—
22C2×#14519735C2211yescharacteristic
23C2×#14519735C2211yescharacteristic
24C2×#14519735C2212yes—
25C2×C42×D61C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519706C2212yes—
28C4⋊C8×D61—24no—
29C4⋊C4×Dic61C2212yes—
30C4×C4⋊Dic61C2211yescharacteristic
31C4×#11758198—21no—
32#16839609—21no—
33#16839607—21no—
34C42×Dic61—21no—
35#16839525C2212yes—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×#125×C61C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C4×C4⋊C4×C61C2211yescharacteristic
44C61×#340C2212yes—
45C61×#340C2212yes—
46C22×Dic122—21no—
47C22×Dic122—21no—
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C22×#92418C2312yes—
52C2×C8×D61—24no—
53C2×C8×D61—28no—
54#14519776—22no—
55#14519776—22no—
56#14519776C2×C414yes—
57#14519776C2×C414yes—
58#14519769—22no—
59#14519769C2×C414yes—
60#14519755—28no—
61#14519755—28no—
62#14519742—28no—
63#14519742—28no—
64#14519735—22no—
65#14519735—22no—
66#14519735—24no—
67#14519735C2×C414yes—
68#14519735C2×C414yes—
69#14519735Q818yescomplemented
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2312yes—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—22no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—22no—
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88C42×D61C2×C414yes—
89C42×D61D414yes—
90C4⋊#92418—22no—
91#14519706—24no—
92#14519684—21no—
93#14519684—22no—
94C22×C8×C61C2312yes—
95#125×C61—22no—
96#125×C61C2×C414yes—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C4⋊C4×C61C2312yes—
102C2×C42×C61—21no—
103C2×C42×C61C2311yescharacteristic
104C4⋊C8×C61—22no—
105#95×C61—21no—
106C23×D61C2×D411yescharacteristic
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61—22no—
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2411yescharacteristic
113C22×Dic61—21no—
114C22×Dic61—21no—
115C22×Dic61—22no—
116C22×Dic61C2×D411yescharacteristic
117C22×Dic61C2×D411yescharacteristic
118C22×Dic61C2×D411yescharacteristic
119C2×C4×D61—24no—
120C2×C4×D61—41no—
121C2×C4×D61—41no—
122C2×C4×D61C22⋊C414yes—
123C2×C4×D61C22⋊C414yes—
124C2×C4×D61C22×C412yes—
125C2×C4×D61C2×D412yes—
126C2×C4×D61C2×Q814yes—
127C2×C4×D61#4114yes—
128C2×Dic122—24no—
129C2×Dic122—24no—
130C2×Dic122—42no—
131C2×Dic122—42no—
132C2×Dic122—44no—
133C2×C8×C61—22no—
134C2×C8×C61—24no—
135C4⋊C4×C61—24no—
136C4⋊C4×C61—24no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61C22×C412yes—
141C4⋊C4×C61C22×C412yes—
142C4⋊C4×C61C2×Q814yes—
143C42×C61—42no—
144C42×C61C22×C412yes—
145C42×C61C2×D412yes—
146#11758198—24no—
147#11758198—24no—
148#11758198—28no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—22no—
152C4⋊Dic61C22×C412yes—
153C4⋊Dic61C22×C412yes—
154C4⋊Dic61C2×Q814yes—
155#11758196—24no—
156#11758196—24no—
157#11758196—24no—
158#11758196—24no—
159C4×Dic61—24no—
160C4×Dic61—24no—
161C4×Dic61—28no—
162C4×Dic61—42no—
163C4×Dic61—44no—
164C4×Dic61C22⋊C414yes—
165C4×Dic61C22⋊C414yes—
166C4×Dic61C22×C412yes—
167C4×Dic61C2×D412yes—
168C2×#92418—22no—
169C2×#92418—24no—
170C8×D61—48no—
171C23×C61C22×D411yescharacteristic
172C22×D61—22no—
173C22×D61—21no—
174C22×D61—21no—
175C22×D61C2×C22⋊C412yes—
176C22×D61C22⋊Q814yes—
177C2×C4×C61—22no—
178C2×C4×C61—21no—
179C2×C4×C61—21no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—24no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61C2×C22⋊C412yes—
189C2×C4×C61C2×C22⋊C412yes—
190C2×C4×C61C23×C411yescharacteristic
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C22×Q812yes—
193C2×C4×C61C2×#4112yes—
194C2×Dic61—21no—
195C2×Dic61—21no—
196C2×Dic61—22no—
197C2×Dic61—22no—
198C2×Dic61—24no—
199C2×Dic61—42no—
200C2×Dic61—42no—
201C2×Dic61—42no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61C2×C22⋊C412yes—
205C2×Dic61C2×C22⋊C412yes—
206C2×Dic61C2×C22⋊C412yes—
207C2×Dic61C22⋊Q814yes—
208C2×Dic61#13514yes—
209C4×D61—28no—
210C4×D61—44no—
211C4×D61C4×Q818yes—
212C4×D61#12414yes—
213C4×D61C4⋊Q814yes—
214Dic122—48no—
215Dic122—48no—
216C8×C61—42no—
217#92418—42no—
218#21311—611nomaximal, Sylow
219C22×C61C22×C22⋊C411yescharacteristic
220C22×C61C2×C22⋊Q812yes—
221C22×C61C2×#13512yes—
222C22×C61#57511yescharacteristic
223C22×C61#57511yescharacteristic
224C2×D61—24no—
225C2×D61—24no—
226C2×D61—22no—
227C2×D61—22no—
228C2×D61#39114yes—
229C4×C61—22no—
230C4×C61—22no—
231C4×C61—44no—
232C4×C61—42no—
233C4×C61—42no—
234C4×C61C2×C4×Q812yes—
235C4×C61C2×#12411yescharacteristic
236C4×C61C2×C4⋊Q811yescharacteristic
237Dic61—22no—
238Dic61—44no—
239Dic61—42no—
240Dic61—42no—
241Dic61#39114yes—
242Dic61#41614yes—
243C2×#533—611no—
244C22×C4⋊C8—611no—
245C4×#117—611no—
246#1989—611no—
247#1989—611no—
248#1989—611no—
249#1989D6111yescharacteristic, complemented
250#1988—612no—
251#1987—612no—
252#1735—612no—
253#1735—612no—
254C2×C61C2×#39111yescharacteristic
255C2×C61C2×#41611yescharacteristic
256C2×C61#289611yescharacteristic
257C2×C61#307612yes—
258C2×C61#307612yes—
259D61—24no—
260C23×C8—612no—
261#533—618no—
262#533—1224no—
263C2×C4⋊Q8—611no—
264C2×C4⋊Q8—611no—
265C2×#125—611no—
266C2×#125C2×D6111yescharacteristic
267C2×C22⋊Q8—1221no—
268C2×C22⋊Q8—1221no—
269C2×C4×Q8—1221no—
270C2×C4×Q8—1221no—
271C2×#117—611no—
272C2×#117—611no—
273C2×#117—612no—
274C22×C42—611no—
275C2×C4⋊C8—612no—
276C2×C4⋊C8—611no—
277C2×C4⋊C8—1224no—
278C2×C4⋊C8C2×D6111yescharacteristic
279C4×C4⋊C4—612no—
280C4×C4⋊C4—611no—
281C4×C4⋊C4C2×D6111yescharacteristic
282C4×C22⋊C4—1221no—
283C4×C22⋊C4—1221no—
284#376—1221no—
285C43—1221no—
286#340—612no—
287#340—612no—
288#340—612no—
289#340—612no—
290#340—612no—
291#340—612no—
292#340C2×D6112yes—
293#340C2×D6112yes—
294C61#2131111yescharacteristic, complemented, Sylow
295C22×Q8—1221no—
296C22×Q8—1221no—
297C23×C4—611no—
298C23×C4—611no—
299C23×C4—611no—
300C22×C8—612no—
301C22×C8—1224no—
302C22×C8—1228no—
303C22×C8C22×D6112yes—
304C4⋊Q8—614no—
305C4⋊Q8—614no—
306C4⋊Q8—1222no—
307C4⋊Q8—1222no—
308#125—22no—
309#125—614no—
310#125—1222no—
311#125C4×D6114yes—
312C22⋊Q8—1228no—
313C22⋊Q8—1228no—
314C4×Q8—1228no—
315C4×Q8—1228no—
316#117—614no—
317#117—614no—
318#117—618no—
319#117—1222no—
320#117—1222no—
321#117—1224no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—611no—
325C2×C4⋊C4—611no—
326C2×C4⋊C4—612no—
327C2×C4⋊C4—1221no—
328C2×C4⋊C4—1221no—
329C2×C4⋊C4—1221no—
330C2×C4⋊C4—1221no—
331C2×C4⋊C4C22×D6111yescharacteristic
332C2×C4⋊C4C22×D6111yescharacteristic
333C2×C4⋊C4C22×D6112yes—
334C2×C22⋊C4—1221no—
335C2×C22⋊C4—1221no—
336C2×C22⋊C4—1222no—
337C2×C42—21no—
338C2×C42—611no—
339C2×C42—611no—
340C2×C42—611no—
341C2×C42—614no—
342C2×C42—614no—
343C2×C42—1221no—
344C2×C42—1221no—
345C2×C42—1222no—
346C2×C42—1222no—
347C2×C42—1221no—
348C2×C42C22×D6111yescharacteristic
349C4⋊C8—22no—
350C4⋊C8—1224no—
351C4⋊C8—1222no—
352#95—21no—
353#95—1222no—
354#95—1221no—
355C24—611no—
356C2×Q8—1224no—
357C2×Q8—1224no—
358C2×Q8—2444no—
359C2×Q8—2442no—
360C2×Q8—2442no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—611no—
365C22×C4—611no—
366C22×C4—612no—
367C22×C4—612no—
368C22×C4—614no—
369C22×C4—614no—
370C22×C4—611no—
371C22×C4—614no—
372C22×C4—614no—
373C22×C4—1224no—
374C22×C4—1222no—
375C22×C4—1221no—
376C22×C4—1221no—
377C22×C4—2441no—
378C22×C4—2441no—
379C22×C4D4×D6111yescharacteristic
380C22×C4D4×D6111yescharacteristic
381C22×C4C23×D6111yescharacteristic
382C2×C8—22no—
383C2×C8—24no—
384C2×C8—1222no—
385C2×C8—1224no—
386C2×C8—2448no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—612no—
393C4⋊C4—612no—
394C4⋊C4—614no—
395C4⋊C4—1224no—
396C4⋊C4—1224no—
397C4⋊C4—1221no—
398C4⋊C4—1221no—
399C4⋊C4—1222no—
400C4⋊C4—1224no—
401C4⋊C4—1224no—
402C4⋊C4C2×C4×D6112yes—
403C4⋊C4C2×C4×D6112yes—
404C4⋊C4Q8×D6114yes—
405C22⋊C4—1228no—
406C22⋊C4—1224no—
407C22⋊C4—1224no—
408C42—42no—
409C42—612no—
410C42—612no—
411C42—614no—
412C42—614no—
413C42—1228no—
414C42—1224no—
415C42—1224no—
416C42—2442no—
417C42—2444no—
418C42C2×C4×D6112yes—
419C42D4×D6112yes—
420C23—612no—
421C23—614no—
422C23—1222no—
423C23—1221no—
424C23—1221no—
425C23C2×D4×D6111yescharacteristic
426Q8—2448no—
427Q8—2448no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—612no—
440C2×C4—612no—
441C2×C4—614no—
442C2×C4—614no—
443C2×C4—618no—
444C2×C4—614no—
445C2×C4—614no—
446C2×C4—612no—
447C2×C4—1228no—
448C2×C4—1221no—
449C2×C4—1221no—
450C2×C4—1222no—
451C2×C4—1222no—
452C2×C4—1224no—
453C2×C4—2444no—
454C2×C4—2442no—
455C2×C4—2442no—
456C2×C4—2441no—
457C2×C4—2442no—
458C2×C4—2441no—
459C2×C4C22⋊C4×D6112yes—
460C2×C4C22⋊C4×D6112yes—
461C2×C4C22×C4×D6111yescharacteristic
462C2×C4C2×D4×D6111yescharacteristic
463C2×C4C2×Q8×D6112yes—
464C2×C4#41×D6112yes—
465C8—42no—
466C8—2442no—
467C22—614no—
468C22—1222no—
469C22—1224no—
470C22—1224no—
471C22—1222no—
472C22C2×C22⋊C4×D6111yescharacteristic
473C22C22⋊Q8×D6112yes—
474C22#135×D6112yes—
475C22#1684055811yescharacteristic
476C22#1684056111yescharacteristic
477C4—22no—
478C4—22no—
479C4—42no—
480C4—44no—
481C4—42no—
482C4—614no—
483C4—614no—
484C4—1222no—
485C4—2442no—
486C4—2444no—
487C4—2442no—
488C4C4×Q8×D6112yes—
489C4#124×D6111yescharacteristic
490C4C4⋊Q8×D6111yescharacteristic
491C2—1224no—
492C2#391×D6111yescharacteristic
493C2#416×D6111yescharacteristic
494C2#2145433511yescharacteristic
495C2#2145560412yes—
496C2#2145560612yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1989C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1989D61
hypocenterC61#21311
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1989

Complements

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

NH ≅ G/Ntype
4. #21446777C2semidirect
5. #21446776C2semidirect
6. #21446774C2semidirect
7. #21446769C2semidirect
8. #21446765C2semidirect
11. C61×#1989C2semidirect
14. #16840318C4semidirect
69. #14519735Q8semidirect
249. #1989D61semidirect
294. C61#21311semidirect

Maximal subgroups

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

subgroupindex
C2×#168403182
C2×C4⋊C8×D612
C4×#145197352
#214467772
#214467762
#214467742
#214467692
#214467652
#214455342
C61×#19892
#2131161