Finite Group Databasebeta

← #29684665

Subgroups — #29684665

Order 15616.

1142 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840391C211yesmaximal, characteristic
2#21448436C211yesmaximal, characteristic
3#21448174C211yesmaximal, characteristic
4#21446805C211yesmaximal, characteristic
5#21446804C211yesmaximal, characteristic
6#21446803C211yesmaximal, characteristic
7#21446802C214yesmaximal
8#21445550C214yesmaximal
9C61×#1996C211yesmaximal, characteristic
10#16840391—22no—
11#16840391—22no—
12#16840391—22no—
13#16840391—22no—
14C2×#14519776C412yescomplemented
15C2×#14519775C412yescomplemented
16C2×#14519767C412yescomplemented
17C2×#14519767C412yescomplemented
18C2×#14519756C2211yescharacteristic
19C2×#14519754C2212yes—
20C2×#14519733C2212yes—
21C2×#14519733C2212yes—
22C2×#14519733C2212yes—
23C22×#11758196C2211yescharacteristic
24#16839659—22no—
25#16839657—22no—
26#16839622C2211yescharacteristic
27#16839620C2212yes—
28C4×#11758196—22no—
29#16839613—22no—
30C22⋊C4×Dic61C2212yes—
31C4×C22⋊Dic61C2211yescharacteristic
32#16839529C2214yes—
33#16839528C2214yes—
34#16839528C2214yes—
35C2×C22⋊Q8×C61C2211yescharacteristic
36C61×#380C2211yescharacteristic
37C4×C22⋊C4×C61C2211yescharacteristic
38C61×#342C2214yes—
39C22×Dic122C2×C412yes—
40C22×Dic122C2×C411yescharacteristic
41C23×Dic61C2311yescharacteristic
42C23×Dic61C2311yescharacteristic
43#14519776—24no—
44#14519776—24no—
45#14519775—24no—
46#14519775—24no—
47#14519767—24no—
48#14519767—24no—
49#14519767—24no—
50#14519767—24no—
51#14519756—21no—
52#14519756—21no—
53#14519756—21no—
54#14519756—21no—
55#14519754—24no—
56#14519754—24no—
57#14519733—24no—
58#14519733—24no—
59#14519733—24no—
60#14519733—28no—
61#14519733Q818yescomplemented
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61C2×C412yes—
64C2×C4⋊Dic61C2×C411yescharacteristic
65C2×#11758196—22no—
66C2×#11758196—22no—
67C2×#11758196—22no—
68C2×#11758196C2×C412yes—
69C2×#11758196C2×C412yes—
70C2×#11758196C2×C412yes—
71C2×#11758196C2×C412yes—
72C2×#11758196C2×C412yes—
73C2×#11758196C2×C412yes—
74C2×#11758196C2×C412yes—
75C2×#11758196D412yes—
76C2×#11758196D412yes—
77C2×#11758196D412yes—
78C2×#11758196D412yes—
79C2×C22⋊Dic61C2312yes—
80C2×C22⋊Dic61C2312yes—
81C2×C22⋊Dic61C2312yes—
82C2×C4×Dic61—22no—
83C2×C4×Dic61—22no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—22no—
86C2×C4×Dic61C2×C412yes—
87C2×C4×Dic61C2×C412yes—
88#14519684—21no—
89#14519684—21no—
90#14519684—22no—
91#14519684—22no—
92C22×Q8×C61C2×C411yescharacteristic
93C23×C4×C61C2311yescharacteristic
94C22⋊Q8×C61—21no—
95C22⋊Q8×C61—21no—
96C22⋊Q8×C61—21no—
97C22⋊Q8×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61C2×C411yescharacteristic
100C2×C4⋊C4×C61C2×C412yes—
101C2×C22⋊C4×C61C2312yes—
102C2×C22⋊C4×C61C2312yes—
103C2×C22⋊C4×C61C2312yes—
104C2×C42×C61—21no—
105#95×C61—21no—
106#95×C61—21no—
107C24×C61C2411yescharacteristic
108C2×Q8×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—81no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—81no—
114C22×C4×C61C22×C412yes—
115C22×C4×C61C22×C412yes—
116C22×C4×C61C22×C411yescharacteristic
117C22×C4×C61C22×C411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×Dic61—22no—
123C22×Dic61—22no—
124C22×Dic61—81no—
125C22×Dic61C22×C412yes—
126C22×Dic61C22×C412yes—
127C22×Dic61C22×C412yes—
128C22×Dic61C22×C412yes—
129C22×Dic61C2×D412yes—
130C22×Dic61C2×D412yes—
131C22×Dic61C2×D412yes—
132C22×Dic61C2×D412yes—
133C22×Dic61C2×Q814yes—
134C22×Dic61#4114yes—
135C2×Dic122—21no—
136C2×Dic122—21no—
137C2×Dic122—24no—
138C2×Dic122—82no—
139C2×Dic122—81no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—81no—
145C22⋊C4×C61—22no—
146C22⋊C4×C61—22no—
147C22⋊C4×C61—22no—
148C22⋊C4×C61—24no—
149C22⋊C4×C61C2×Q814yes—
150C42×C61—81no—
151C22⋊Dic61—22no—
152C22⋊Dic61—22no—
153C22⋊Dic61—22no—
154C22⋊Dic61—24no—
155C22⋊Dic61C2×Q814yes—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61—24no—
159C4⋊Dic61—81no—
160#11758196—22no—
161#11758196—22no—
162#11758196—24no—
163#11758196—24no—
164#11758196—24no—
165#11758196—24no—
166#11758196—28no—
167#11758196—44no—
168#11758196—82no—
169#11758196C22⋊C414yes—
170#11758196C22⋊C414yes—
171#11758196C22⋊C414yes—
172#11758196C22⋊C414yes—
173C4×Dic61—24no—
174C4×Dic61—24no—
175C4×Dic61—28no—
176C4×Dic61—44no—
177C4×Dic61—82no—
178C4×Dic61—81no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61C23×C411yescharacteristic
182C23×C61C22×D411yescharacteristic
183C23×C61C22×D411yescharacteristic
184C23×C61C22×Q812yes—
185C23×C61C2×#4112yes—
186Q8×C61—81no—
187Q8×C61—81no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—24no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61C2×C22⋊C411yescharacteristic
205C2×C4×C61C2×C22⋊C411yescharacteristic
206C2×C4×C61C2×C22⋊C411yescharacteristic
207C2×C4×C61C2×C22⋊C411yescharacteristic
208C2×Dic61—22no—
209C2×Dic61—22no—
210C2×Dic61—24no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61C2×C22⋊C414yes—
222C2×Dic61C2×C22⋊C414yes—
223C2×Dic61C4×Q818yes—
224C2×Dic61C22⋊Q814yes—
225C2×Dic61C22⋊Q814yes—
226C2×Dic61#12414yes—
227C2×Dic61C4⋊Q814yes—
228Dic122—84no—
229Dic122—81no—
230Dic122—81no—
231#21334—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—21no—
236C22×C61—21no—
237C22×C61—21no—
238C22×C61—42no—
239C22×C61—42no—
240C22×C61C22×C22⋊C411yescharacteristic
241C22×C61C2×C4×Q812yes—
242C22×C61C2×C22⋊Q812yes—
243C22×C61C2×#12411yescharacteristic
244C22×C61C2×C4⋊Q811yescharacteristic
245C4×C61—21no—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—21no—
249C4×C61—44no—
250C4×C61—42no—
251C4×C61—42no—
252C4×C61—82no—
253Dic61—24no—
254Dic61—44no—
255Dic61—42no—
256Dic61—42no—
257Dic61—82no—
258Dic61#39118yes—
259C2×#544—611no—
260#2316—611no—
261#2274—611no—
262#1996—611no—
263#1996—611no—
264#1996—614no—
265#1996—611no—
266#1996D6111yescharacteristic
267#1740—614no—
268C2×C61—22no—
269C2×C61—21no—
270C2×C61—22no—
271C2×C61—21no—
272C2×C61C2×#39111yescharacteristic
273C2×C61#288311yescharacteristic
274C2×C61#288411yescharacteristic
275#544—1222no—
276#544—1222no—
277#544—1222no—
278#544—1222no—
279C2×C4⋊Q8—612no—
280C2×C4⋊Q8—612no—
281C2×#125—612no—
282C2×#125—612no—
283C2×C22⋊Q8—611no—
284C2×C22⋊Q8—612no—
285C2×C22⋊Q8C2×D6111yescharacteristic
286C2×#117—612no—
287C2×#117—612no—
288C2×#117—612no—
289C22×C4⋊C4—611no—
290#389—1222no—
291#389—1222no—
292#380—611no—
293#380—612no—
294#380C2×D6111yescharacteristic
295C4×C4⋊C4—1222no—
296C4×C4⋊C4—1222no—
297C4×C22⋊C4—611no—
298C4×C22⋊C4—612no—
299C4×C22⋊C4C2×D6111yescharacteristic
300#342—614no—
301#342—614no—
302#342—614no—
303#342C2×D6114yes—
304C61#2133411yescharacteristic, complemented, Sylow
305C22×Q8—612no—
306C22×Q8—611no—
307C22×Q8C4×D6111yescharacteristic
308C23×C4—611no—
309C23×C4—611no—
310C23×C4C22×D6111yescharacteristic
311C4⋊Q8—1224no—
312C4⋊Q8—1224no—
313C4⋊Q8—1224no—
314C4⋊Q8—1224no—
315#125—1224no—
316#125—1224no—
317#125—1224no—
318#125—1224no—
319C22⋊Q8—21no—
320C22⋊Q8—21no—
321C22⋊Q8—21no—
322C22⋊Q8—21no—
323C22⋊Q8—1224no—
324C22⋊Q8—1224no—
325C22⋊Q8—1221no—
326C22⋊Q8—1221no—
327C22⋊Q8—1221no—
328C22⋊Q8—1221no—
329#117—618no—
330#117—1224no—
331#117—1224no—
332#117—1224no—
333#117—1228no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—612no—
336C2×C4⋊C4—612no—
337C2×C4⋊C4—612no—
338C2×C4⋊C4—612no—
339C2×C4⋊C4—612no—
340C2×C4⋊C4—612no—
341C2×C4⋊C4—612no—
342C2×C4⋊C4—612no—
343C2×C4⋊C4—612no—
344C2×C4⋊C4—611no—
345C2×C4⋊C4—612no—
346C2×C4⋊C4—612no—
347C2×C4⋊C4—612no—
348C2×C4⋊C4—1222no—
349C2×C4⋊C4—1222no—
350C2×C4⋊C4—1221no—
351C2×C4⋊C4—1222no—
352C2×C4⋊C4C4×D6111yescharacteristic
353C2×C4⋊C4C4×D6112yes—
354C2×C22⋊C4—612no—
355C2×C22⋊C4—612no—
356C2×C22⋊C4—612no—
357C2×C22⋊C4C22×D6112yes—
358C2×C22⋊C4C22×D6112yes—
359C2×C22⋊C4C22×D6112yes—
360C2×C42—21no—
361C2×C42—612no—
362C2×C42—612no—
363C2×C42—1222no—
364C2×C42—1222no—
365C2×C42—1221no—
366C2×C42—1222no—
367#95—21no—
368#95—21no—
369#95—1222no—
370#95—1222no—
371#95—1221no—
372#95—1221no—
373C24C23×D6111yescharacteristic
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—81no—
377C2×Q8—1224no—
378C2×Q8—1221no—
379C2×Q8—1221no—
380C2×Q8—4881no—
381C2×Q8—4882no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—81no—
385C22×C4—612no—
386C22×C4—612no—
387C22×C4—612no—
388C22×C4—614no—
389C22×C4—614no—
390C22×C4—612no—
391C22×C4—612no—
392C22×C4—612no—
393C22×C4—612no—
394C22×C4—612no—
395C22×C4—1222no—
396C22×C4—1222no—
397C22×C4—4881no—
398C22×C4C2×C4×D6112yes—
399C22×C4C2×C4×D6112yes—
400C22×C4C2×C4×D6111yescharacteristic
401C22×C4C2×C4×D6111yescharacteristic
402C22×C4D4×D6111yescharacteristic
403C22×C4D4×D6111yescharacteristic
404C22×C4D4×D6111yescharacteristic
405C22×C4D4×D6111yescharacteristic
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—81no—
411C4⋊C4—614no—
412C4⋊C4—614no—
413C4⋊C4—614no—
414C4⋊C4—614no—
415C4⋊C4—1224no—
416C4⋊C4—1224no—
417C4⋊C4—1224no—
418C4⋊C4—1224no—
419C4⋊C4—1224no—
420C4⋊C4—1228no—
421C4⋊C4—1222no—
422C4⋊C4—1222no—
423C4⋊C4—1221no—
424C4⋊C4—1221no—
425C4⋊C4—4882no—
426C4⋊C4—4881no—
427C4⋊C4—2444no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—24no—
432C22⋊C4—614no—
433C22⋊C4—1222no—
434C22⋊C4—1222no—
435C22⋊C4—1222no—
436C22⋊C4—1224no—
437C22⋊C4Q8×D6114yes—
438C42—81no—
439C42—1224no—
440C42—1224no—
441C42—1228no—
442C42—4881no—
443C42—4882no—
444C42—2444no—
445C23—41no—
446C23—41no—
447C23C22×C4×D6111yescharacteristic
448C23C2×D4×D6111yescharacteristic
449C23C2×D4×D6111yescharacteristic
450C23C2×Q8×D6112yes—
451C23#41×D6112yes—
452Q8—81no—
453Q8—81no—
454Q8—4881no—
455Q8—4881no—
456Q8—4884no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—614no—
474C2×C4—618no—
475C2×C4—614no—
476C2×C4—614no—
477C2×C4—614no—
478C2×C4—614no—
479C2×C4—614no—
480C2×C4—1222no—
481C2×C4—1222no—
482C2×C4—1224no—
483C2×C4—2442no—
484C2×C4—2442no—
485C2×C4—2442no—
486C2×C4—2442no—
487C2×C4—2442no—
488C2×C4—2442no—
489C2×C4—2442no—
490C2×C4—2442no—
491C2×C4—2442no—
492C2×C4—2442no—
493C2×C4C22⋊C4×D6111yescharacteristic
494C2×C4C22⋊C4×D6111yescharacteristic
495C2×C4C22⋊C4×D6111yescharacteristic
496C2×C4C22⋊C4×D6111yescharacteristic
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—42no—
504C22—42no—
505C22C2×C22⋊C4×D6111yescharacteristic
506C22C4×Q8×D6112yes—
507C22C22⋊Q8×D6112yes—
508C22#124×D6111yescharacteristic
509C22C4⋊Q8×D6111yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—44no—
515C4—42no—
516C4—42no—
517C4—82no—
518C4—618no—
519C4—1224no—
520C4—4882no—
521C4—2444no—
522C4—2442no—
523C4—2442no—
524C2—22no—
525C2—21no—
526C2—22no—
527C2—21no—
528C2#391×D6111yescharacteristic
529C2#2145427711yescharacteristic
530C2#2145428111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#1996C2
socleC22×C61C22×C22⋊C4
prime radical, cosocleC24×C61C24
hypercenter#1996D61
hypocenterC61#21334
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
14. C2×#14519776C4semidirect
15. C2×#14519775C4semidirect
16. C2×#14519767C4semidirect
17. C2×#14519767C4semidirect
61. #14519733Q8semidirect
304. C61#21334semidirect

Maximal subgroups

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

subgroupindex
C2×#168403912
#214484362
#214481742
#214468052
#214468042
#214468032
#214468022
#214455502
C61×#19962
#2133461