Finite Group Databasebeta

← #29734191

Subgroups — #29734191

Order 15616.

1028 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×#16840481C211yesmaximal, characteristic
2C2×#16840044C211yesmaximal, characteristic, complemented
3#21451028C211yesmaximal, characteristic, complemented
4#21447063C212yesmaximal, complemented
5#21447058C212yesmaximal
6#21446759C211yesmaximal, characteristic, complemented
7#21446752C211yesmaximal, characteristic
8#21446085C212yesmaximal, complemented
9#21446078C212yesmaximal
10#21445650C211yesmaximal, characteristic, complemented
11C2×C61×#489C211yesmaximal, characteristic
12#16840481—24no—
13#16840481—24no—
14C2×#14519775C2211yescharacteristic
15C2×#14519770—21no—
16C2×#14519770—21no—
17C2×#14519764C2211yescharacteristic
18C2×#14519754—21no—
19C2×Q8×Dic61C2211yescharacteristic
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×D4×Dic61C2211yescharacteristic
23C2×#14519733—21no—
24#16840044—24no—
25C2×#14519708C2212yes—
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519702C2212yes—
28C2×#14519701C2212yes—
29C2×#14519700C2212yes—
30C2×#14519698C2212yes—
31#16839681—21no—
32#16839681—21no—
33#16839667—21no—
34#16839658C2212yes—
35#16839658C2212yes—
36#16839631—21no—
37C4×C4⋊Dic61C2211yescharacteristic
38#16839525C2212yes—
39#16839516C2212yes—
40#16839516C2212yes—
41C2×C4⋊Q8×C61C2211yescharacteristic
42C2×#124×C61C2211yescharacteristic
43C61×#489—24no—
44C2×#103×C61C2212yes—
45C2×#102×C61C2212yes—
46C2×#97×C61C2211yescharacteristic
47C22×Dic122D411yescharacteristic
48C22×Dic122C2311yescharacteristic
49C23×Dic61—21no—
50C22×#92418C2312yes—
51#14519775—22no—
52#14519775—22no—
53#14519770—28no—
54#14519770—28no—
55#14519764—22no—
56#14519764—22no—
57#14519754—28no—
58Q8×Dic61—22no—
59Q8×Dic61—22no—
60C4×Dic122—22no—
61C4×Dic122—22no—
62C4×Dic122—22no—
63C4×Dic122D414yes—
64D4×Dic61—22no—
65D4×Dic61—22no—
66#14519733—28no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61D411yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2312yes—
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—22no—
75C2×#11758196—22no—
76C2×#11758196D411yescharacteristic
77C2×#11758196D411yescharacteristic
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61C2311yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86#14519708—24no—
87C4⋊#92418—22no—
88#14519702—24no—
89#14519701—24no—
90#14519700—24no—
91#14519698—24no—
92#14519684—21no—
93#14519684—22no—
94#14519684—22no—
95C22×Q8×C61—21no—
96C22×Q8×C61C2311yescharacteristic
97C22×D4×C61C2311yescharacteristic
98C22×C8×C61C2312yes—
99C4⋊Q8×C61—22no—
100C4⋊Q8×C61—22no—
101#124×C61—22no—
102#124×C61D414yes—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61C2312yes—
105C2×C22⋊C4×C61—21no—
106C2×C22⋊C4×C61—21no—
107C2×C42×C61C2311yescharacteristic
108#103×C61—24no—
109#102×C61—24no—
110#97×C61—22no—
111C24×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—42no—
116C2×Q8×C61—82no—
117C2×Q8×C61C2×D412yes—
118C2×D4×C61—21no—
119C2×D4×C61—22no—
120C2×D4×C61—22no—
121C2×D4×C61C2×D412yes—
122C22×C4×C61—21no—
123C22×C4×C61—22no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2411yescharacteristic
127C22×Dic61—24no—
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—22no—
131C22×Dic61—42no—
132C22×Dic61—42no—
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136C22×Dic61C2×D411yescharacteristic
137C2×Dic122—22no—
138C2×Dic122—22no—
139C2×Dic122—22no—
140C2×Dic122—21no—
141C2×Dic122—22no—
142C2×Dic122—41no—
143C2×Dic122C2×D412yes—
144C2×C8×C61—22no—
145C2×C8×C61—22no—
146C2×C8×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—42no—
150C22⋊C4×C61—24no—
151C22⋊C4×C61—24no—
152C42×C61C2×D412yes—
153C42×C61C2×D412yes—
154C22⋊Dic61—24no—
155C22⋊Dic61—24no—
156C22⋊Dic61—24no—
157C4⋊Dic61—24no—
158C4⋊Dic61—22no—
159C4⋊Dic61—21no—
160C4⋊Dic61—22no—
161C4⋊Dic61C2×D412yes—
162C4⋊Dic61#4114yes—
163#11758196—24no—
164#11758196—24no—
165#11758196—24no—
166#11758196—22no—
167#11758196—22no—
168#11758196—82no—
169#11758196—82no—
170C4×Dic61—24no—
171C4×Dic61—24no—
172C4×Dic61—21no—
173C4×Dic61—21no—
174C4×Dic61—21no—
175C4×Dic61—21no—
176C4×Dic61—42no—
177C2×#92418—22no—
178C2×#92418—22no—
179C2×#92418—22no—
180C23×C61—22no—
181C23×C61—42no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—42no—
185C23×C61C22×D411yescharacteristic
186Q8×C61—24no—
187Q8×C61—41no—
188Q8×C61—41no—
189Q8×C61—84no—
190D4×C61—24no—
191D4×C61—41no—
192D4×C61—41no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61#12012yes—
202C2×C4×C61#12012yes—
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C2×#4112yes—
206C2×C4×C612−1+412yes—
207C2×Dic61—21no—
208C2×Dic61—21no—
209C2×Dic61—21no—
210C2×Dic61—22no—
211C2×Dic61—21no—
212C2×Dic61—22no—
213C2×Dic61—21no—
214C2×Dic61—24no—
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—48no—
220C2×Dic61—42no—
221C2×Dic61—41no—
222C2×Dic61—42no—
223C2×Dic61—41no—
224C2×Dic61—42no—
225C2×Dic61#13712yes—
226Dic122—22no—
227Dic122—22no—
228Dic122—44no—
229Dic122—42no—
230C8×C61—42no—
231#92418—42no—
232#25662—611nomaximal, Sylow
233C22×C61—42no—
234C22×C61—42no—
235C22×C61—42no—
236C22×C61—82no—
237C22×C61—82no—
238C22×C61C2×#12011yescharacteristic
239C22×C61#54812yes—
240C22×C61C2×#13711yescharacteristic
241C22×C61#57511yescharacteristic
242C22×C61#57712yes—
243C4×C61—22no—
244C4×C61—22no—
245C4×C61—42no—
246C4×C61—82no—
247C4×C61#53211yescharacteristic
248C4×C61#53611yescharacteristic
249C4×C61#54712yes—
250Dic61—22no—
251Dic61—44no—
252Dic61—42no—
253Dic61—41no—
254Dic61—42no—
255Dic61—41no—
256Dic61—41no—
257C2×#556—611no—
258C2×#489D6111yescharacteristic
259C2×#475—611no—
260#2567—611no—
261#2043—612no—
262#2042—612no—
263#1985—611no—
264#1983—611no—
265#1865—612no—
266#1864—612no—
267#1763—611no—
268C2×C61—82no—
269C2×C61#243211yescharacteristic
270C2×C61#300711yescharacteristic
271C2×C61#301011yescharacteristic
272C2×C61#336912yes—
273C2×C61#337612yes—
274#556—1224no—
275#556—1224no—
276C2×C4⋊Q8—611no—
277C2×C4⋊Q8C2×D6111yescharacteristic
278C2×#126—1221no—
279C2×#126—1221no—
280C2×#124—611no—
281C2×#124C2×D6111yescharacteristic
282C2×C22⋊Q8—1221no—
283C2×C4×Q8—611no—
284C2×C4×Q8—611no—
285C2×C4×Q8—611no—
286C2×C4×D4—611no—
287C2×#117—1221no—
288#489—24no—
289#475—1224no—
290C2×#106—612no—
291C2×C4⋊C8—611no—
292C2×#103—612no—
293C2×#103—612no—
294C2×#103—612no—
295C2×#103C2×D6112yes—
296C2×#102—612no—
297C2×#102C2×D6112yes—
298C2×#97C2×D6111yescharacteristic
299#395—1221no—
300#395—1221no—
301#391—1221no—
302#389—612no—
303#389—612no—
304#384—1221no—
305C4×C4⋊C4—611no—
306#340—612no—
307#336—612no—
308#336—612no—
309C61#2566211yescharacteristic, complemented, Sylow
310C22×Q8—21no—
311C22×Q8—611no—
312C22×Q8—611no—
313C22×Q8C22×D6111yescharacteristic
314C22×D4C22×D6111yescharacteristic
315C23×C4—1221no—
316C22×C8—612no—
317C22×C8C22×D6112yes—
318C4⋊Q8—22no—
319C4⋊Q8—22no—
320C4⋊Q8—1222no—
321C4⋊Q8—1222no—
322#126—1228no—
323#126—1228no—
324#124—22no—
325#124—1222no—
326#124—1222no—
327#124C22⋊D6114yes—
328C22⋊Q8—1228no—
329C4×Q8—614no—
330C4×Q8—1222no—
331C4×Q8—1222no—
332C4×Q8—1222no—
333C4×Q8—1222no—
334C4×Q8—1222no—
335C4×D4—1222no—
336C4×D4—1222no—
337#117—1228no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—611no—
340C2×C4⋊C4—612no—
341C2×C4⋊C4—611no—
342C2×C4⋊C4—611no—
343C2×C4⋊C4—611no—
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×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—1221no—
354C2×C22⋊C4—1221no—
355C2×C22⋊C4—1221no—
356C2×C42—611no—
357C2×C42—611no—
358C2×C42—1221no—
359C2×C42—1221no—
360C2×C42—1221no—
361C2×C42C22×D6111yescharacteristic
362#106—1224no—
363C4⋊C8—1222no—
364#103—24no—
365#103—1224no—
366#103—1224no—
367#103—1224no—
368#102—24no—
369#102—1224no—
370#97—22no—
371#95—1222no—
372#95—1222no—
373#95—1221no—
374C24—21no—
375C2×Q8—21no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—42no—
379C2×Q8—82no—
380C2×Q8—612no—
381C2×Q8—1222no—
382C2×Q8—1221no—
383C2×Q8—1222no—
384C2×Q8—1222no—
385C2×Q8—1222no—
386C2×Q8—2441no—
387C2×Q8C2×C22⋊D6112yes—
388C2×D4—21no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4C2×C22⋊D6112yes—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—611no—
395C22×C4—611no—
396C22×C4—611no—
397C22×C4—611no—
398C22×C4—1224no—
399C22×C4—1222no—
400C22×C4—1221no—
401C22×C4—1221no—
402C22×C4—2442no—
403C22×C4—2442no—
404C22×C4D4×D6111yescharacteristic
405C22×C4D4×D6111yescharacteristic
406C22×C4C23×D6111yescharacteristic
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—1222no—
411C2×C8—1222no—
412C2×C8—1222no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—42no—
416C4⋊C4—612no—
417C4⋊C4—614no—
418C4⋊C4—1224no—
419C4⋊C4—1224no—
420C4⋊C4—1224no—
421C4⋊C4—1221no—
422C4⋊C4—1222no—
423C4⋊C4—1222no—
424C4⋊C4—1222no—
425C4⋊C4—1222no—
426C4⋊C4—1224no—
427C4⋊C4—4882no—
428C4⋊C4—4882no—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—1224no—
432C22⋊C4—1224no—
433C22⋊C4—1224no—
434C42—1224no—
435C42—1221no—
436C42—1221no—
437C42—1221no—
438C42—1221no—
439C42—1224no—
440C42—2442no—
441C42D4×D6112yes—
442C42C2×C22⋊D6112yes—
443C23—22no—
444C23—41no—
445C23—41no—
446C23—42no—
447C23—42no—
448C23C2×D4×D6111yescharacteristic
449Q8—24no—
450Q8—41no—
451Q8—41no—
452Q8—84no—
453Q8—1222no—
454Q8—1222no—
455Q8—2444no—
456Q8—2442no—
457D4—24no—
458D4—41no—
459D4—41no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—612no—
469C2×C4—1221no—
470C2×C4—1221no—
471C2×C4—1221no—
472C2×C4—1222no—
473C2×C4—1221no—
474C2×C4—1221no—
475C2×C4—1222no—
476C2×C4—1221no—
477C2×C4—1221no—
478C2×C4—1222no—
479C2×C4—1222no—
480C2×C4—1224no—
481C2×C4—2448no—
482C2×C4—2441no—
483C2×C4—2442no—
484C2×C4—2441no—
485C2×C4—2442no—
486C2×C4—2442no—
487C2×C4#1451974512yes—
488C2×C4#1451974512yes—
489C2×C4C2×D4×D6111yescharacteristic
490C2×C4C22×C22⋊D6111yescharacteristic
491C2×C4#41×D6112yes—
492C2×C4#1451982412yes—
493C8—42no—
494C8—2442no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—82no—
499C22—82no—
500C22C2×#1451974511yescharacteristic
501C22#1684042012yes—
502C22#137×D6111yescharacteristic
503C22#1684056111yescharacteristic
504C22#1684057212yes—
505C4—22no—
506C4—22no—
507C4—42no—
508C4—82no—
509C4—1222no—
510C4—2441no—
511C4—2441no—
512C4—2444no—
513C4—2441no—
514C4—2442no—
515C4—2442no—
516C4#1684031111yescharacteristic
517C4#1684033211yescharacteristic
518C4#1684041112yes—
519C2—82no—
520C2#2144945011yescharacteristic
521C2#2145507511yescharacteristic
522C2#2145510411yescharacteristic
523C2#2145855612yes—
524C2#2145864512yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#489C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenterC2×#489D61
hypocenterC61#25662
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#489

Complements

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

NH ≅ G/Ntype
2. C2×#16840044C2semidirect
3. #21451028C2semidirect
4. #21447063C2semidirect
6. #21446759C2semidirect
8. #21446085C2semidirect
10. #21445650C2semidirect
309. C61#25662semidirect

Maximal subgroups

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

subgroupindex
C2×#168404812
C2×#168400442
#214510282
#214470632
#214470582
#214467592
#214467522
#214460852
#214460782
#214456502
C2×C61×#4892
#2566261