Finite Group Databasebeta

← #29736971

Subgroups — #29736971

Order 15616.

1098 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840469C211yesmaximal, characteristic, complemented
2C2×#16840121C211yesmaximal, characteristic, complemented
3#21451308C211yesmaximal, characteristic, complemented
4#21446968C212yesmaximal, complemented
5#21446963C212yesmaximal, complemented
6#21446770C211yesmaximal, characteristic, complemented
7#21446758C211yesmaximal, characteristic, complemented
8#21446169C212yesmaximal
9#21446155C212yesmaximal, complemented
10#21445644C211yesmaximal, characteristic, complemented
11C2×C61×#463C211yesmaximal, characteristic
12#16840469—24no—
13#16840469—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519768C2212yes—
16C2×#14519756—21no—
17C2×#14519754—22no—
18C2×C4×Dic122C2211yescharacteristic
19C2×D4×Dic61C2212yes—
20C22×C4⋊Dic61—21no—
21#16840121—24no—
22C2×#14519701C2212yes—
23C2×#14519698C2212yes—
24C2×#14519698C2212yes—
25C2×#14519688C2211yescharacteristic
26#16839687—21no—
27#16839674—21no—
28#16839669C2212yes—
29#16839665C2212yes—
30#16839656—22no—
31C4×C4⋊Dic61C2211yescharacteristic
32#16839525C2212yes—
33#16839525C2212yes—
34#16839524C2212yes—
35#16839516C2212yes—
36#16839516C2212yes—
37C22×SD16×C61C2211yescharacteristic
38C22×D8×C61C2211yescharacteristic
39C2×#124×C61C2211yescharacteristic
40C2×C4×D4×C61C2211yescharacteristic
41C61×#463—24no—
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#103×C61C2211yescharacteristic
44C2×#102×C61C2211yescharacteristic
45C22×Dic122—21no—
46C22×Dic122D411yescharacteristic
47C23×Dic61—22no—
48C22×#92418C2312yes—
49#14519777—22no—
50#14519777D414yes—
51#14519768—24no—
52#14519768—24no—
53#14519756—28no—
54#14519754—216no—
55C4×Dic122—22no—
56C4×Dic122—22no—
57D4×Dic61—24no—
58D4×Dic61—24no—
59C2×C4⋊Dic61—28no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61—22no—
62C2×C4⋊Dic61D411yescharacteristic
63C2×C4⋊Dic61C2312yes—
64C2×C4⋊Dic61C2312yes—
65C2×#11758196—22no—
66C2×#11758196D412yes—
67C2×C22⋊Dic61—22no—
68C2×C22⋊Dic61—22no—
69C2×C22⋊Dic61—22no—
70C2×C4×Dic61—21no—
71C2×C4×Dic61C2312yes—
72#14519701—24no—
73#14519698—24no—
74#14519698—24no—
75#14519688—22no—
76#14519684—21no—
77#14519684—22no—
78#14519684—22no—
79C22×Q8×C61C2311yescharacteristic
80C22×D4×C61C2311yescharacteristic
81C22×D4×C61C2311yescharacteristic
82C23×C4×C61—21no—
83C2×SD16×C61—22no—
84C2×SD16×C61—24no—
85C2×D8×C61—22no—
86C2×D8×C61—24no—
87C22×C8×C61C2311yescharacteristic
88C22×C8×C61C2311yescharacteristic
89#124×C61—22no—
90#124×C61—22no—
91C4×D4×C61—22no—
92C4×D4×C61D414yes—
93C2×C4⋊C4×C61C2311yescharacteristic
94C2×C22⋊C4×C61—21no—
95C2×C22⋊C4×C61—21no—
96C2×C22⋊C4×C61—21no—
97C2×C42×C61C2311yescharacteristic
98C4⋊C8×C61—22no—
99#103×C61—22no—
100#102×C61—22no—
101C24×C61—21no—
102C24×C61—21no—
103C2×Q8×C61—21no—
104C2×Q8×C61—21no—
105C2×Q8×C61—22no—
106C2×Q8×C61—22no—
107C2×D4×C61—21no—
108C2×D4×C61—21no—
109C2×D4×C61—21no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61C2×D412yes—
114C2×D4×C61#4114yes—
115C22×C4×C61—21no—
116C22×C4×C61—24no—
117C22×C4×C61—21no—
118C22×C4×C61—44no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2411yescharacteristic
122C22×Dic61—22no—
123C22×Dic61—28no—
124C22×Dic61—22no—
125C22×Dic61—82no—
126C22×Dic61—82no—
127C22×Dic61C2×D412yes—
128C22×Dic61C2×D412yes—
129C2×Dic122—22no—
130C2×Dic122—24no—
131C2×Dic122—42no—
132C2×Dic122—81no—
133C2×Dic122—81no—
134SD16×C61—44no—
135D8×C61—44no—
136C2×C8×C61—21no—
137C2×C8×C61—21no—
138C2×C8×C61—22no—
139C2×C8×C61—22no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61C2×D412yes—
142C22⋊C4×C61—24no—
143C22⋊C4×C61—42no—
144C22⋊C4×C61—42no—
145C42×C61C2×D412yes—
146C42×C61C2×D412yes—
147C22⋊Dic61—28no—
148C22⋊Dic61—44no—
149C22⋊Dic61—82no—
150C4⋊Dic61—28no—
151C4⋊Dic61—22no—
152C4⋊Dic61—24no—
153C4⋊Dic61—22no—
154C4⋊Dic61—22no—
155C4⋊Dic61—22no—
156C4⋊Dic61C2×D414yes—
157#11758196—28no—
158#11758196—24no—
159C4×Dic61—22no—
160C4×Dic61—22no—
161C4×Dic61—81no—
162C2×#92418—22no—
163C2×#92418—22no—
164C2×#92418—22no—
165C23×C61—22no—
166C23×C61—24no—
167C23×C61—42no—
168C23×C61—41no—
169C23×C61—41no—
170C23×C61—41no—
171C23×C61—42no—
172C23×C61—42no—
173C23×C61—41no—
174C23×C61—41no—
175C23×C61C22×D411yescharacteristic
176Q8×C61—41no—
177Q8×C61—41no—
178Q8×C61—42no—
179D4×C61—22no—
180D4×C61—24no—
181D4×C61—41no—
182D4×C61—41no—
183D4×C61—42no—
184D4×C61#12414yes—
185C2×C4×C61—22no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—22no—
189C2×C4×C61—48no—
190C2×C4×C61—42no—
191C2×C4×C61—41no—
192C2×C4×C61—42no—
193C2×C4×C61#12012yes—
194C2×C4×C61#12012yes—
195C2×C4×C61C22×D411yescharacteristic
196C2×C4×C61C22×D411yescharacteristic
197C2×C4×C61C2×#4112yes—
198C2×C4×C612−1+412yes—
199C2×Dic61—22no—
200C2×Dic61—22no—
201C2×Dic61—24no—
202C2×Dic61—22no—
203C2×Dic61—22no—
204C2×Dic61—22no—
205C2×Dic61—22no—
206C2×Dic61—42no—
207C2×Dic61—42no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61—88no—
212Dic122—44no—
213Dic122—84no—
214C8×C61—41no—
215C8×C61—41no—
216#92418—42no—
217#25940—611nomaximal, Sylow
218C22×C61—24no—
219C22×C61—44no—
220C22×C61—44no—
221C22×C61—42no—
222C22×C61—42no—
223C22×C61—82no—
224C22×C61—82no—
225C22×C61—81no—
226C22×C61—81no—
227C22×C61—81no—
228C22×C61C2×#12011yescharacteristic
229C22×C61#54812yes—
230C22×C61#57512yes—
231C22×C61#57611yescharacteristic
232C22×C61#57711yescharacteristic
233C4×C61—22no—
234C4×C61—22no—
235C4×C61—42no—
236C4×C61—81no—
237C4×C61C2×#12411yescharacteristic
238C4×C61#53611yescharacteristic
239C4×C61#54812yes—
240Dic61—42no—
241Dic61—44no—
242Dic61—42no—
243Dic61—82no—
244C2×#554—611no—
245C2×#489—611no—
246C2×#463D6111yescharacteristic
247#2592—611no—
248#2026—612no—
249#2025—612no—
250#1988—611no—
251#1985—611no—
252#1879—612no—
253#1877—612no—
254#1762—611no—
255C2×C61—44no—
256C2×C61—161no—
257C2×C61#243611yescharacteristic
258C2×C61#300511yescharacteristic
259C2×C61#301211yescharacteristic
260C2×C61#337012yes—
261C2×C61#337612yes—
262C22×SD16C2×D6111yescharacteristic
263C22×D8C2×D6111yescharacteristic
264#554—1224no—
265#554—1224no—
266C2×C4⋊Q8—611no—
267C2×#125—612no—
268C2×#124C2×D6111yescharacteristic
269C2×C22⋊Q8—1222no—
270C2×C22⋊Q8—1221no—
271C2×C4×Q8—611no—
272C2×C4×D4—612no—
273C2×C4×D4C2×D6111yescharacteristic
274C22×C4⋊C4—1221no—
275#489—1224no—
276#463—24no—
277C2×C4⋊C8C2×D6111yescharacteristic
278C2×#103—612no—
279C2×#103C2×D6111yescharacteristic
280C2×#102—612no—
281C2×#102—612no—
282C2×#102C2×D6111yescharacteristic
283C2×#97—611no—
284#397—1221no—
285#393—1221no—
286#391—612no—
287#390—612no—
288#388—1222no—
289C4×C4⋊C4—611no—
290#340—612no—
291#340—612no—
292#340—612no—
293#336—612no—
294#336—612no—
295C61#2594011yescharacteristic, complemented, Sylow
296C22×Q8—611no—
297C22×Q8—1221no—
298C22×Q8C22×D6111yescharacteristic
299C22×D4C22×D6111yescharacteristic
300C22×D4C22×D6111yescharacteristic
301C23×C4—21no—
302C23×C4—1222no—
303C2×SD16—24no—
304C2×SD16—22no—
305C2×D8—24no—
306C2×D8—22no—
307C22×C8—612no—
308C22×C8C22×D6111yescharacteristic
309C22×C8C22×D6111yescharacteristic
310C4⋊Q8—614no—
311C4⋊Q8—1222no—
312#125—1224no—
313#125—1224no—
314#124—22no—
315#124—22no—
316C22⋊Q8—12216no—
317C22⋊Q8—1228no—
318C4×Q8—1222no—
319C4×Q8—1222no—
320C4×D4—22no—
321C4×D4—1224no—
322C4×D4—1224no—
323C4×D4C22⋊D6114yes—
324C2×C4⋊C4—612no—
325C2×C4⋊C4—611no—
326C2×C4⋊C4—612no—
327C2×C4⋊C4—612no—
328C2×C4⋊C4—1222no—
329C2×C4⋊C4—1222no—
330C2×C4⋊C4—1228no—
331C2×C4⋊C4—1221no—
332C2×C4⋊C4C22×D6111yescharacteristic
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—1222no—
337C2×C22⋊C4—1222no—
338C2×C22⋊C4—1222no—
339C2×C42—612no—
340C2×C42—1221no—
341C2×C42C22×D6111yescharacteristic
342C4⋊C8—22no—
343#103—22no—
344#103—1224no—
345#102—22no—
346#102—1224no—
347#102—1224no—
348#97—1222no—
349#95—1222no—
350#95—1222no—
351#95—1221no—
352C24—21no—
353C24—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—1224no—
359C2×Q8—1222no—
360C2×Q8—4881no—
361C2×Q8—4881no—
362C2×Q8—2442no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4#1175822414yes—
370C2×D4C2×C22⋊D6112yes—
371C22×C4—21no—
372C22×C4—24no—
373C22×C4—21no—
374C22×C4—44no—
375C22×C4—612no—
376C22×C4—612no—
377C22×C4—1228no—
378C22×C4—1222no—
379C22×C4—1222no—
380C22×C4—4882no—
381C22×C4—4882no—
382C22×C4D4×D6111yescharacteristic
383C22×C4D4×D6111yescharacteristic
384C22×C4C23×D6111yescharacteristic
385SD16—44no—
386D8—44no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—1222no—
392C2×C8—1222no—
393C2×C8—1222no—
394C4⋊C4—21no—
395C4⋊C4—614no—
396C4⋊C4—1228no—
397C4⋊C4—1228no—
398C4⋊C4—1222no—
399C4⋊C4—1222no—
400C4⋊C4—1224no—
401C4⋊C4—1224no—
402C4⋊C4—1222no—
403C4⋊C4—1222no—
404C4⋊C4C2×C22⋊D6112yes—
405C22⋊C4—24no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—1228no—
409C22⋊C4—4882no—
410C22⋊C4—2444no—
411C42—1222no—
412C42—1222no—
413C42—4881no—
414C42D4×D6112yes—
415C42C2×C22⋊D6112yes—
416C23—22no—
417C23—24no—
418C23—42no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—42no—
423C23—41no—
424C23—41no—
425C23—42no—
426C23C2×D4×D6111yescharacteristic
427Q8—41no—
428Q8—41no—
429Q8—42no—
430Q8—4884no—
431Q8—2444no—
432D4—22no—
433D4—24no—
434D4—41no—
435D4—41no—
436D4—42no—
437D4#1451976414yes—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—48no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—1222no—
447C2×C4—1222no—
448C2×C4—1224no—
449C2×C4—1222no—
450C2×C4—1222no—
451C2×C4—1222no—
452C2×C4—1222no—
453C2×C4—4888no—
454C2×C4—2442no—
455C2×C4—2442no—
456C2×C4—2442no—
457C2×C4—2442no—
458C2×C4—2442no—
459C2×C4#1451974512yes—
460C2×C4#1451974512yes—
461C2×C4C2×D4×D6111yescharacteristic
462C2×C4C22×C22⋊D6111yescharacteristic
463C2×C4C2×#1175822412yes—
464C2×C4#1451982412yes—
465C8—41no—
466C8—41no—
467C8—2442no—
468C22—24no—
469C22—42no—
470C22—44no—
471C22—42no—
472C22—44no—
473C22—81no—
474C22—81no—
475C22—82no—
476C22—81no—
477C22—82no—
478C22C2×#1451974511yescharacteristic
479C22#1684041812yes—
480C22#1684055812yes—
481C22#1684056411yescharacteristic
482C22#1684056911yescharacteristic
483C4—22no—
484C4—22no—
485C4—42no—
486C4—81no—
487C4—4882no—
488C4—2442no—
489C4—2444no—
490C4—2442no—
491C4C2×#1451976411yescharacteristic
492C4#1684033311yescharacteristic
493C4#1684041812yes—
494C2—44no—
495C2—161no—
496C2#2144947811yescharacteristic
497C2#2145505811yescharacteristic
498C2#2145511911yescharacteristic
499C2#2145856912yes—
500C2#2145864312yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#463C2
socleC23×C61C22×D4
prime radicalC22×D8×C61C22
hypercenterC2×#463D61
hypocenterC61#25940
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#463

Complements

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

NH ≅ G/Ntype
1. C2×#16840469C2semidirect
2. C2×#16840121C2semidirect
3. #21451308C2semidirect
4. #21446968C2semidirect
5. #21446963C2semidirect
6. #21446770C2semidirect
7. #21446758C2semidirect
9. #21446155C2semidirect
10. #21445644C2semidirect
295. C61#25940semidirect

Maximal subgroups

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

subgroupindex
C2×#168404692
C2×#168401212
#214513082
#214469682
#214469632
#214467702
#214467582
#214461692
#214461552
#214456442
C2×C61×#4632
#2594061