Finite Group Databasebeta

← #29685252

Subgroups — #29685252

Order 15616.

1186 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840314C211yesmaximal, characteristic
2C2×M16×Dic61C211yesmaximal, characteristic
3#21448221C211yesmaximal, characteristic
4#21446778C212yesmaximal, complemented
5#21446619C212yesmaximal, complemented
6#21446253C211yesmaximal, characteristic
7#21446252C211yesmaximal, characteristic
8#21446250C211yesmaximal, characteristic
9#21445651C212yesmaximal, complemented
10#21445648C212yesmaximal, complemented
11C61×#1894C211yesmaximal, characteristic
12C22×#11758194C2211yescharacteristic
13#16840314—24no—
14#16840314C418yescomplemented
15C2×#14519776C2211yescharacteristic
16C2×#14519775C2211yescharacteristic
17C2×#14519767—21no—
18C2×#14519756C2211yescharacteristic
19C2×#14519754—21no—
20C2×Q8×Dic61C2211yescharacteristic
21C2×#14519742—21no—
22C2×#14519742C2211yescharacteristic
23C2×#14519733—21no—
24C2×#14519733—21no—
25C22×C4×Dic61C2211yescharacteristic
26C2×#14519702C2212yes—
27C2×#14519701C2212yes—
28C2×#14519700C2212yes—
29M16×Dic61—24no—
30C2×#14519687C2212yes—
31C2×C8×Dic61C2212yes—
32#16839631C2212yes—
33#16839626—21no—
34#16839622C2211yescharacteristic
35#16839620—21no—
36C4⋊C4×Dic61C2212yes—
37#16839613—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C22×M16×C61C2211yescharacteristic
42C2×C22⋊Q8×C61C2211yescharacteristic
43C2×#103×C61C2212yes—
44C61×#380C2211yescharacteristic
45C61×#340C2212yes—
46C22×Dic122—21no—
47C22×Dic122C2311yescharacteristic
48C23×Dic61D411yescharacteristic
49C23×Dic61D411yescharacteristic
50C2×#11758194—22no—
51C2×#11758194—24no—
52C22×#92418C2312yes—
53#14519776—22no—
54#14519776C2×C414yes—
55#14519775—22no—
56#14519775C2×C414yes—
57#14519767—28no—
58#14519756—22no—
59#14519756C2×C414yes—
60#14519754—28no—
61Q8×Dic61—22no—
62Q8×Dic61C2×C414yes—
63#14519742—28no—
64#14519742—22no—
65#14519742C2×C414yes—
66#14519733—28no—
67#14519733—44no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61C2312yes—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—22no—
77C2×#11758196—22no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61C2×C414yes—
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D414yes—
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519702—24no—
90#14519701—24no—
91#14519700—24no—
92#14519687—24no—
93C8×Dic61—24no—
94#14519684—21no—
95#14519684—22no—
96C22×Q8×C61C2311yescharacteristic
97C23×C4×C61C2311yescharacteristic
98C2×M16×C61—22no—
99C2×M16×C61—24no—
100C22×C8×C61C2312yes—
101C22⋊Q8×C61—22no—
102C22⋊Q8×C61C2×C414yes—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61C2312yes—
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C22⋊C4×C61—21no—
107C2×C22⋊C4×C61—21no—
108#103×C61—24no—
109#95×C61—21no—
110C24×C61C2×D411yescharacteristic
111C2×Q8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61C22×C412yes—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61C22×C412yes—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2411yescharacteristic
124C22×Dic61—22no—
125C22×Dic61—22no—
126C22×Dic61—24no—
127C22×Dic61—21no—
128C22×Dic61—21no—
129C22×Dic61—22no—
130C22×Dic61C22⋊C414yes—
131C22×Dic61C22⋊C414yes—
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136C2×Dic122—22no—
137C2×Dic122—22no—
138C2×Dic122—21no—
139C2×Dic122—24no—
140C2×Dic122—42no—
141C2×Dic122—42no—
142C2×Dic122C22×C412yes—
143M16×C61—28no—
144C2×C8×C61—22no—
145C2×C8×C61—22no—
146C2×C8×C61#4114yes—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61C22×C412yes—
152C22⋊C4×C61—24no—
153C22⋊C4×C61—42no—
154C22⋊Dic61—24no—
155C22⋊Dic61—42no—
156C4⋊Dic61—21no—
157C4⋊Dic61—24no—
158C4⋊Dic61—22no—
159C4⋊Dic61—22no—
160C4⋊Dic61C22×C412yes—
161#11758196—24no—
162#11758196—24no—
163#11758196—24no—
164#11758196—24no—
165#11758196—24no—
166#11758196—44no—
167#11758196—44no—
168C4×Dic61—24no—
169C4×Dic61—24no—
170C4×Dic61—44no—
171C4×Dic61C22⋊C414yes—
172C4×Dic61C22⋊C414yes—
173C4×Dic61C22×C412yes—
174C4×Dic61C22×C412yes—
175C4×Dic61C2×D412yes—
176C4×Dic61C2×D412yes—
177#11758194—28no—
178C2×#92418—22no—
179C2×#92418—22no—
180C2×#92418#4114yes—
181C23×C61—22no—
182C23×C61—21no—
183C23×C61—22no—
184C23×C61—21no—
185C23×C61C2×C22⋊C412yes—
186C23×C61C22×D411yescharacteristic
187Q8×C61—24no—
188Q8×C61—41no—
189Q8×C61—41no—
190C2×C4×C61—22no—
191C2×C4×C61—24no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—42no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61C2×C22⋊C412yes—
203C2×C4×C61C23×C411yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C2×#4112yes—
206C2×C4×C61C2×#4112yes—
207C2×Dic61—21no—
208C2×Dic61—22no—
209C2×Dic61—21no—
210C2×Dic61—28no—
211C2×Dic61—28no—
212C2×Dic61—22no—
213C2×Dic61—22no—
214C2×Dic61—42no—
215C2×Dic61—41no—
216C2×Dic61—41no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61C2×C22⋊C412yes—
221C2×Dic61C2×C22⋊C412yes—
222C2×Dic61C2×C22⋊C412yes—
223C2×Dic61C2×C22⋊C412yes—
224C2×Dic61AGL(1,ℤ/8)12yes—
225C2×Dic61#13712yes—
226C2×Dic61#13714yes—
227Dic122—24no—
228Dic122—41no—
229Dic122—41no—
230Dic122—48no—
231C8×C61—24no—
232#92418—24no—
233#21409—611nomaximal, Sylow
234C22×C61—22no—
235C22×C61—22no—
236C22×C61—22no—
237C22×C61—42no—
238C22×C61—42no—
239C22×C61C22×C22⋊C411yescharacteristic
240C22×C61#53212yes—
241C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
242C22×C61C2×#13712yes—
243C22×C61C2×#13711yescharacteristic
244C4×C61—22no—
245C4×C61—42no—
246C4×C61—42no—
247C4×C61—82no—
248C4×C61C4×#4112yes—
249C4×C61#52511yescharacteristic
250C4×C61#52511yescharacteristic
251Dic61—22no—
252Dic61—22no—
253Dic61—42no—
254Dic61—42no—
255Dic61—82no—
256Dic61#41714yes—
257Dic61#41914yes—
258C2×#533—611no—
259C2×C4×M16—611no—
260#2285—611no—
261#1990—612no—
262#1956—612no—
263#1894—611no—
264#1894—611no—
265#1894—611no—
266#1894D6111yescharacteristic
267#1763—612no—
268#1763—612no—
269C2×C61—42no—
270C2×C61#229711yescharacteristic
271C2×C61C2×#41911yescharacteristic
272C2×C61C2×#41711yescharacteristic
273C2×C61#318212yes—
274C2×C61#318512yes—
275C22×M16—611no—
276C22×M16C2×D6111yescharacteristic
277#533—618no—
278#533—1224no—
279C2×C4⋊Q8—611no—
280C2×C4⋊Q8—611no—
281C2×#125—1221no—
282C2×C22⋊Q8—611no—
283C2×C22⋊Q8—1221no—
284C2×C22⋊Q8C2×D6111yescharacteristic
285C2×C4×Q8—611no—
286C2×C4×Q8—611no—
287C2×C4×Q8—1221no—
288C2×#117—1221no—
289C2×#117—1221no—
290C22×C42—611no—
291C2×#103—612no—
292C2×#103—612no—
293C2×#103—612no—
294C2×#103C2×D6112yes—
295C4×M16—1224no—
296C2×#97—612no—
297C2×C4×C8—612no—
298#384—612no—
299#382—1221no—
300#380—611no—
301#380—1221no—
302#380C2×D6111yescharacteristic
303C4×C4⋊C4—612no—
304C4×C4⋊C4—1221no—
305#340—612no—
306#340—612no—
307#340—612no—
308#340C2×D6112yes—
309C61#2140911yescharacteristic, complemented, Sylow
310C22×Q8—611no—
311C22×Q8—1221no—
312C22×Q8C22×D6111yescharacteristic
313C23×C4—611no—
314C23×C4—611no—
315C23×C4C22×D6111yescharacteristic
316C2×M16—22no—
317C2×M16—24no—
318C2×M16—1224no—
319C2×M16—1222no—
320C22×C8—612no—
321C22×C8C22×D6112yes—
322C4⋊Q8—614no—
323C4⋊Q8—614no—
324C4⋊Q8—1222no—
325C4⋊Q8—1222no—
326#125—1228no—
327C22⋊Q8—22no—
328C22⋊Q8—614no—
329C22⋊Q8—1228no—
330C22⋊Q8—1222no—
331C22⋊Q8C4×D6114yes—
332C4×Q8—614no—
333C4×Q8—614no—
334C4×Q8—1222no—
335C4×Q8—1228no—
336C4×Q8—1222no—
337#117—1228no—
338#117—2444no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—611no—
341C2×C4⋊C4—612no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4—1221no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4—1221no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4—1222no—
349C2×C4⋊C4—1222no—
350C2×C4⋊C4C22×D6112yes—
351C2×C4⋊C4C22×D6111yescharacteristic
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—1221no—
355C2×C22⋊C4—1221no—
356C2×C42—614no—
357C2×C42—611no—
358C2×C42—611no—
359C2×C42—614no—
360C2×C42—611no—
361C2×C42—611no—
362C2×C42—1221no—
363C2×C42—1221no—
364C2×C42—1222no—
365#103—24no—
366#103—1224no—
367#103—1224no—
368#103—1224no—
369#97—1224no—
370C4×C8—1224no—
371#95—21no—
372#95—1222no—
373#95—1221no—
374C24D4×D6111yescharacteristic
375C2×Q8—21no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—612no—
379C2×Q8—1224no—
380C2×Q8—1221no—
381C2×Q8—1222no—
382C2×Q8—1222no—
383C2×Q8—2442no—
384C2×Q8—2442no—
385C2×Q8C2×C4×D6112yes—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—611no—
392C22×C4—611no—
393C22×C4—611no—
394C22×C4—611no—
395C22×C4—614no—
396C22×C4—614no—
397C22×C4—1222no—
398C22×C4—1224no—
399C22×C4—1222no—
400C22×C4—1221no—
401C22×C4—1221no—
402C22×C4—1222no—
403C22×C4C2×C4×D6112yes—
404C22×C4D4×D6112yes—
405C22×C4D4×D6111yescharacteristic
406C22×C4C23×D6111yescharacteristic
407M16—28no—
408M16—1228no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—614no—
412C2×C8—1222no—
413C2×C8—1222no—
414C2×C8#1175822414yes—
415C4⋊C4—21no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—612no—
420C4⋊C4—1224no—
421C4⋊C4—1224no—
422C4⋊C4—1224no—
423C4⋊C4—1224no—
424C4⋊C4—1224no—
425C4⋊C4—1221no—
426C4⋊C4—1224no—
427C4⋊C4—1222no—
428C4⋊C4—1222no—
429C4⋊C4—2444no—
430C4⋊C4—2444no—
431C4⋊C4C2×C4×D6112yes—
432C22⋊C4—24no—
433C22⋊C4—42no—
434C22⋊C4—1224no—
435C22⋊C4—2442no—
436C42—612no—
437C42—612no—
438C42—612no—
439C42—612no—
440C42—614no—
441C42—614no—
442C42—1224no—
443C42—1224no—
444C42—2444no—
445C23—21no—
446C23—22no—
447C23—21no—
448C23—22no—
449C23C22⋊C4×D6112yes—
450C23C2×D4×D6111yescharacteristic
451Q8—24no—
452Q8—41no—
453Q8—41no—
454Q8—1224no—
455Q8—2448no—
456Q8—2441no—
457Q8—2441no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—612no—
471C2×C4—612no—
472C2×C4—612no—
473C2×C4—614no—
474C2×C4—612no—
475C2×C4—612no—
476C2×C4—612no—
477C2×C4—1228no—
478C2×C4—1228no—
479C2×C4—1221no—
480C2×C4—1222no—
481C2×C4—1221no—
482C2×C4—1222no—
483C2×C4—1222no—
484C2×C4—2442no—
485C2×C4—2442no—
486C2×C4—2442no—
487C2×C4—2441no—
488C2×C4—2441no—
489C2×C4—2442no—
490C2×C4C22⋊C4×D6112yes—
491C2×C4C22×C4×D6111yescharacteristic
492C2×C4C2×D4×D6111yescharacteristic
493C2×C4#41×D6112yes—
494C2×C4C2×#1175822412yes—
495C8—24no—
496C8—1224no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—42no—
501C22—42no—
502C22C2×C22⋊C4×D6111yescharacteristic
503C22#1684030712yes—
504C22AGL(1,ℤ/8)×D6111yescharacteristic
505C22#137×D6112yes—
506C22#137×D6111yescharacteristic
507C4—22no—
508C4—42no—
509C4—42no—
510C4—82no—
511C4—614no—
512C4—614no—
513C4—1222no—
514C4—1222no—
515C4—4882no—
516C4—2442no—
517C4—2442no—
518C4C4×#1175822412yes—
519C4#1684026811yescharacteristic
520C4#1684027011yescharacteristic
521C2—42no—
522C2#2144830411yescharacteristic
523C2#419×D6111yescharacteristic
524C2#417×D6111yescharacteristic
525C2#2145653212yes—
526C2#2145655912yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1894C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenter#1894D61
hypocenterC61#21409
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1894

Complements

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

NH ≅ G/Ntype
4. #21446778C2semidirect
5. #21446619C2semidirect
9. #21445651C2semidirect
10. #21445648C2semidirect
14. #16840314C4semidirect
309. C61#21409semidirect

Maximal subgroups

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

subgroupindex
C2×#168403142
C2×M16×Dic612
#214482212
#214467782
#214466192
#214462532
#214462522
#214462502
#214456512
#214456482
C61×#18942
#2140961