Finite Group Databasebeta

← #29734182

Subgroups — #29734182

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×#16840121C211yesmaximal, characteristic
2C2×#16840047C211yesmaximal, characteristic, complemented
3#21451024C211yesmaximal, characteristic, complemented
4#21447062C212yesmaximal, complemented
5#21447055C212yesmaximal
6#21446760C211yesmaximal, characteristic, complemented
7#21446752C211yesmaximal, characteristic
8#21446087C212yesmaximal, complemented
9#21446080C212yesmaximal
10#21445649C211yesmaximal, characteristic, complemented
11C2×C61×#556C211yesmaximal, characteristic
12C2×#14519777C2211yescharacteristic
13#16840121—24no—
14#16840047—24no—
15C2×#14519708C2212yes—
16C2×C4⋊#92418C2211yescharacteristic
17C2×#14519701C2212yes—
18C2×#14519701C2212yes—
19C2×#14519701C2212yes—
20C2×#14519701C2212yes—
21C2×#14519698C2212yes—
22C2×#14519698C2212yes—
23C2×#14519688C2211yescharacteristic
24#16839682—21no—
25#16839682—21no—
26#16839669—21no—
27#16839660C2212yes—
28#16839660C2212yes—
29#16839631—21no—
30C4×C4⋊Dic61C2211yescharacteristic
31#16839524C2212yes—
32#16839517C2212yes—
33#16839517C2212yes—
34C61×#556—24no—
35C61×#556—24no—
36C2×C4⋊Q8×C61C2211yescharacteristic
37C2×#126×C61—21no—
38C2×#126×C61—21no—
39C2×#124×C61C2211yescharacteristic
40C2×#124×C61C2211yescharacteristic
41C2×C22⋊Q8×C61—21no—
42C2×C4×Q8×C61C2211yescharacteristic
43C2×C4×Q8×C61C2211yescharacteristic
44C2×C4×Q8×C61C2211yescharacteristic
45C2×C4×D4×C61C2211yescharacteristic
46C2×#117×C61—21no—
47C22×Dic122—21no—
48C22×#92418C2312yes—
49C22×#92418C2312yes—
50#14519777—22no—
51#14519777—22no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61C2312yes—
54C2×C4⋊Dic61C2312yes—
55C2×#11758196—22no—
56C2×#11758196—22no—
57C2×C4×Dic61—21no—
58#14519708—24no—
59C4⋊#92418—22no—
60#14519701—24no—
61#14519701—24no—
62#14519701—24no—
63#14519701—24no—
64#14519698—24no—
65#14519698—24no—
66#14519688—22no—
67#14519684—21no—
68#14519684—22no—
69#14519684—22no—
70C22×Q8×C61D411yescharacteristic
71C22×Q8×C61C2311yescharacteristic
72C22×Q8×C61C2311yescharacteristic
73C22×D4×C61C2311yescharacteristic
74C23×C4×C61—21no—
75C4⋊Q8×C61—22no—
76C4⋊Q8×C61—22no—
77#126×C61—28no—
78#126×C61—28no—
79#124×C61—22no—
80#124×C61—22no—
81#124×C61—22no—
82#124×C61D414yes—
83C22⋊Q8×C61—28no—
84C4×Q8×C61—22no—
85C4×Q8×C61—22no—
86C4×Q8×C61—22no—
87C4×Q8×C61—22no—
88C4×Q8×C61—22no—
89C4×Q8×C61D414yes—
90C4×D4×C61—22no—
91C4×D4×C61—22no—
92#117×C61—28no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61D411yescharacteristic
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C22⋊C4×C61—21no—
102C2×C22⋊C4×C61—21no—
103C2×C22⋊C4×C61—21no—
104C2×C22⋊C4×C61—21no—
105C2×C22⋊C4×C61—21no—
106C2×C42×C61—21no—
107C2×C42×C61—21no—
108C2×C42×C61C2311yescharacteristic
109C2×C42×C61C2311yescharacteristic
110C2×C42×C61C2311yescharacteristic
111C24×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61—22no—
117C2×Q8×C61—22no—
118C2×Q8×C61—22no—
119C2×Q8×C61—22no—
120C2×Q8×C61—41no—
121C2×Q8×C61C2×D412yes—
122C2×Q8×C61C2×D412yes—
123C2×D4×C61—21no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61C2×D412yes—
127C22×C4×C61—24no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—42no—
132C22×C4×C61—42no—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C22×Dic61—22no—
141C22×Dic61—22no—
142C2×Dic122—42no—
143C2×Dic122—82no—
144C4⋊C4×C61—24no—
145C4⋊C4×C61—24no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61C2×D412yes—
153C22⋊C4×C61—24no—
154C22⋊C4×C61—24no—
155C22⋊C4×C61—24no—
156C22⋊C4×C61—24no—
157C22⋊C4×C61—24no—
158C42×C61—24no—
159C42×C61—24no—
160C42×C61—21no—
161C42×C61—21no—
162C42×C61—21no—
163C42×C61—21no—
164C42×C61C2×D412yes—
165C42×C61C2×D412yes—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
169C4⋊Dic61—42no—
170C4⋊Dic61#4114yes—
171#11758196—82no—
172#11758196—82no—
173C4×Dic61—42no—
174C2×#92418—22no—
175C2×#92418—22no—
176C2×#92418—22no—
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—22no—
187Q8×C61—22no—
188Q8×C61—24no—
189Q8×C61—44no—
190Q8×C61—41no—
191Q8×C61—41no—
192Q8×C61—42no—
193D4×C61—24no—
194D4×C61—41no—
195D4×C61—41no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—22no—
201C2×C4×C61—21no—
202C2×C4×C61—22no—
203C2×C4×C61—21no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—48no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61#12012yes—
217C2×C4×C61#12012yes—
218C2×C4×C61#13712yes—
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C2×#4112yes—
222C2×C4×C612−1+412yes—
223C2×Dic61—24no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229Dic122—84no—
230#92418—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—22no—
246C4×C61—42no—
247C4×C61—42no—
248C4×C61—41no—
249C4×C61—42no—
250C4×C61—41no—
251C4×C61—41no—
252C4×C61#53211yescharacteristic
253C4×C61#53611yescharacteristic
254C4×C61#54712yes—
255Dic61—44no—
256Dic61—82no—
257C2×#556D6111yescharacteristic
258C2×#489—611no—
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—24no—
275#556—24no—
276C2×C4⋊Q8—611no—
277C2×C4⋊Q8C2×D6111yescharacteristic
278C2×#126—21no—
279C2×#126—21no—
280C2×#124C2×D6111yescharacteristic
281C2×#124C2×D6111yescharacteristic
282C2×C22⋊Q8—21no—
283C2×C4×Q8C2×D6111yescharacteristic
284C2×C4×Q8C2×D6111yescharacteristic
285C2×C4×Q8C2×D6111yescharacteristic
286C2×C4×D4C2×D6111yescharacteristic
287C2×#117—21no—
288#489—1224no—
289#475—1224no—
290C2×#106—612no—
291C2×C4⋊C8—611no—
292C2×#103—612no—
293C2×#103—612no—
294C2×#103—612no—
295C2×#103—612no—
296C2×#102—612no—
297C2×#102—612no—
298C2×#97—611no—
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—1221no—
311C22×Q8C22⋊D6111yescharacteristic
312C22×Q8C22×D6111yescharacteristic
313C22×Q8C22×D6111yescharacteristic
314C22×D4C22×D6111yescharacteristic
315C23×C4—21no—
316C22×C8—612no—
317C22×C8—612no—
318C4⋊Q8—22no—
319C4⋊Q8—22no—
320C4⋊Q8—1222no—
321C4⋊Q8—1222no—
322#126—28no—
323#126—28no—
324#124—22no—
325#124—22no—
326#124—22no—
327#124C22⋊D6114yes—
328C22⋊Q8—28no—
329C4×Q8—22no—
330C4×Q8—22no—
331C4×Q8—22no—
332C4×Q8—22no—
333C4×Q8—22no—
334C4×Q8C22⋊D6114yes—
335C4×D4—22no—
336C4×D4—22no—
337#117—28no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—612no—
343C2×C4⋊C4—612no—
344C2×C4⋊C4—1222no—
345C2×C4⋊C4—1222no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4C22⋊D6111yescharacteristic
348C2×C4⋊C4C22⋊D6111yescharacteristic
349C2×C4⋊C4C22⋊D6111yescharacteristic
350C2×C4⋊C4C22×D6111yescharacteristic
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C42—21no—
357C2×C42—21no—
358C2×C42—1221no—
359C2×C42C22×D6111yescharacteristic
360C2×C42C22×D6111yescharacteristic
361C2×C42C22×D6111yescharacteristic
362#106—1224no—
363C4⋊C8—1222no—
364#103—1224no—
365#103—1224no—
366#103—1224no—
367#103—1224no—
368#102—1224no—
369#102—1224no—
370#97—1222no—
371#95—1221no—
372#95—1222no—
373#95—1222no—
374C24—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—41no—
384C2×Q8—4882no—
385C2×Q8—2442no—
386C2×Q8C2×C22⋊D6112yes—
387C2×Q8C2×C22⋊D6112yes—
388C2×D4—21no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4C2×C22⋊D6112yes—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—1222no—
399C22×C4—1222no—
400C22×C4C2×C22⋊D6111yescharacteristic
401C22×C4C2×C22⋊D6111yescharacteristic
402C22×C4C2×C22⋊D6111yescharacteristic
403C22×C4C2×C22⋊D6111yescharacteristic
404C22×C4C2×C22⋊D6111yescharacteristic
405C22×C4C2×C22⋊D6111yescharacteristic
406C22×C4C23×D6111yescharacteristic
407C2×C8—1222no—
408C2×C8—1222no—
409C2×C8—1222no—
410C2×C8—1222no—
411C2×C8—1222no—
412C2×C8—1222no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—21no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—614no—
422C4⋊C4—1222no—
423C4⋊C4—1222no—
424C4⋊C4—1222no—
425C4⋊C4—4882no—
426C4⋊C4—4882no—
427C4⋊C4—2442no—
428C4⋊C4C2×C22⋊D6112yes—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C42—24no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—24no—
440C42—2442no—
441C42C2×C22⋊D6112yes—
442C42C2×C22⋊D6112yes—
443C23—22no—
444C23—41no—
445C23—41no—
446C23—42no—
447C23—42no—
448C23C22×C22⋊D6111yescharacteristic
449Q8—22no—
450Q8—22no—
451Q8—24no—
452Q8—44no—
453Q8—41no—
454Q8—41no—
455Q8—42no—
456Q8—4884no—
457D4—24no—
458D4—41no—
459D4—41no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—48no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—1224no—
481C2×C4—2442no—
482C2×C4—2442no—
483C2×C4—2442no—
484C2×C4—2442no—
485C2×C4—2442no—
486C2×C4#1451974612yes—
487C2×C4#1451974612yes—
488C2×C4#1451980712yes—
489C2×C4C22×C22⋊D6111yescharacteristic
490C2×C4C22×C22⋊D6111yescharacteristic
491C2×C4#41×D6112yes—
492C2×C4#1451982412yes—
493C8—2442no—
494C8—2442no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—82no—
499C22—82no—
500C22C2×#1451974611yescharacteristic
501C22#1684041912yes—
502C22C2×#1451980711yescharacteristic
503C22#1684056011yescharacteristic
504C22#1684057012yes—
505C4—22no—
506C4—22no—
507C4—22no—
508C4—41no—
509C4—41no—
510C4—42no—
511C4—42no—
512C4—41no—
513C4—42no—
514C4—4882no—
515C4—2444no—
516C4#1684030811yescharacteristic
517C4#1684033311yescharacteristic
518C4#1684041112yes—
519C2—82no—
520C2#2144944311yescharacteristic
521C2#2145507911yescharacteristic
522C2#2145510511yescharacteristic
523C2#2145855512yes—
524C2#2145864012yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#556C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenterC2×#556D61
hypocenterC61#25662
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
2. C2×#16840047C2semidirect
3. #21451024C2semidirect
4. #21447062C2semidirect
6. #21446760C2semidirect
8. #21446087C2semidirect
10. #21445649C2semidirect
309. C61#25662semidirect

Maximal subgroups

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

subgroupindex
C2×#168401212
C2×#168400472
#214510242
#214470622
#214470552
#214467602
#214467522
#214460872
#214460802
#214456492
C2×C61×#5562
#2566261