Finite Group Databasebeta

← #29718365

Subgroups — #29718365

Order 15616.

1130 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840092C211yesmaximal, characteristic, complemented
2#21449466C211yesmaximal, characteristic
3#21448688C211yesmaximal, characteristic, complemented
4#21447189C212yesmaximal, complemented
5#21447059C212yesmaximal, complemented
6#21446750C212yesmaximal, complemented
7#21446449C212yesmaximal, complemented
8#21446242C211yesmaximal, characteristic, complemented
9#21445997C211yesmaximal, characteristic, complemented
10#21445750C211yesmaximal, characteristic
11C22×#102×C61C211yesmaximal, characteristic
12C2×#14519764C2212yes—
13C2×#14519756C2211yescharacteristic
14C2×#14519736—21no—
15#16840092—24no—
16C2×#14519708C2211yescharacteristic, complemented
17C2×#14519708C2211yescharacteristic, complemented
18C2×#14519706C2212yescomplemented
19C2×#14519702C2212yescomplemented
20C2×#14519700C2211yescharacteristic, complemented
21C2×#14519700C2211yescharacteristic, complemented
22C2×#14519698C2212yes—
23C2×#14519698C2212yes—
24C2×C22⋊#92418C2211yescharacteristic
25#16839676—22no—
26#16839665C2212yes—
27#16839661C2212yescomplemented
28#16839648—22no—
29#16839625—21no—
30#16839622C2211yescharacteristic
31#16839618—21no—
32#16839618—21no—
33C4⋊C4×Dic61C2212yescomplemented
34C4×C22⋊Dic61C2211yescharacteristic
35#16839525C2212yescomplemented
36#16839516C2212yescomplemented
37C23×D4×C61C2211yescharacteristic
38C23×C8×C61C2211yescharacteristic
39C22×C4⋊C4×C61C2211yescharacteristic
40C2×#102×C61—24no—
41C2×#102×C61C2212yes—
42C2×#102×C61C2212yes—
43C22×Dic122C2311yescharacteristic
44C22×#92418C2311yescharacteristic
45C22×#92418C2311yescharacteristic
46#14519764—24no—
47#14519764—42no—
48#14519756—22no—
49#14519756—22no—
50#14519736—44no—
51C2×C4⋊Dic61—22no—
52C2×C4⋊Dic61C2311yescharacteristic
53C2×C4⋊Dic61C2312yes—
54C2×#11758196—21no—
55C2×C22⋊Dic61—21no—
56C2×C22⋊Dic61—21no—
57C2×C22⋊Dic61—22no—
58C2×C22⋊Dic61—22no—
59C2×C22⋊Dic61—22no—
60C2×C22⋊Dic61—22no—
61C2×C22⋊Dic61D411yescharacteristic
62C2×C22⋊Dic61D411yescharacteristic
63C2×C4×Dic61—22no—
64C2×C4×Dic61C2312yes—
65#14519708—22no—
66#14519708—22no—
67#14519706—42no—
68#14519702—42no—
69#14519700—22no—
70#14519700—22no—
71#14519698—24no—
72#14519698—42no—
73C22⋊#92418—22no—
74#14519684—21no—
75#14519684—22no—
76#14519684—22no—
77#14519684D412yes—
78C25×C61—21no—
79C22×D4×C61—24no—
80C22×D4×C61—22no—
81C22×D4×C61—21no—
82C22×D4×C61—22no—
83C22×D4×C61—42no—
84C22×D4×C61C2312yes—
85C23×C4×C61—21no—
86C23×C4×C61C2311yescharacteristic
87C22×C8×C61—22no—
88C22×C8×C61—42no—
89C22×C8×C61C2312yes—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61—22no—
92C2×C4⋊C4×C61D414yes—
93C2×C4⋊C4×C61C2311yescharacteristic
94C2×C4⋊C4×C61C2311yescharacteristic
95#102×C61—24no—
96#102×C61—24no—
97C24×C61—22no—
98C24×C61—44no—
99C24×C61—42no—
100C24×C61—42no—
101C24×C61—42no—
102C24×C61—82no—
103C24×C61C2×D411yescharacteristic
104C2×D4×C61—28no—
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61—48no—
111C2×D4×C61—44no—
112C2×D4×C61—44no—
113C2×D4×C61—42no—
114C2×D4×C61#4114yes—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—24no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—42no—
121C22×C4×C61—42no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D412yes—
124C22×C4×C61#4112yes—
125C22×C4×C61C2411yescharacteristic
126C22×Dic61—21no—
127C22×Dic61—22no—
128C22×Dic61—21no—
129C22×Dic61C2×D412yes—
130C22×Dic61C2×D412yes—
131C2×Dic122—21no—
132C2×Dic122—21no—
133C2×Dic122—22no—
134C2×Dic122—22no—
135C2×C8×C61—22no—
136C2×C8×C61—22no—
137C2×C8×C61—22no—
138C2×C8×C61—44no—
139C4⋊C4×C61—24no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61C2×D412yes—
143C4⋊C4×C61C2×D412yes—
144C22⋊Dic61—22no—
145C22⋊Dic61—22no—
146C22⋊Dic61—44no—
147C22⋊Dic61—44no—
148C22⋊Dic61—42no—
149C22⋊Dic61—42no—
150C22⋊Dic61—82no—
151C22⋊Dic61—82no—
152C4⋊Dic61—21no—
153C4⋊Dic61—22no—
154C4⋊Dic61—21no—
155C4⋊Dic61—22no—
156C4⋊Dic61—44no—
157#11758196—42no—
158C4×Dic61—22no—
159C4×Dic61—22no—
160C4×Dic61—44no—
161C2×#92418—21no—
162C2×#92418—21no—
163C2×#92418—22no—
164C2×#92418—22no—
165C23×C61—22no—
166C23×C61—22no—
167C23×C61—24no—
168C23×C61—48no—
169C23×C61—42no—
170C23×C61—42no—
171C23×C61—42no—
172C23×C61—42no—
173C23×C61—42no—
174C23×C61—42no—
175C23×C61—88no—
176C23×C61—88no—
177C23×C61—84no—
178C23×C61—84no—
179C23×C61#12112yes—
180C23×C61#13512yes—
181C23×C61AGL(1,ℤ/8)12yes—
182C23×C61C22×D411yescharacteristic
183D4×C61—24no—
184D4×C61—24no—
185D4×C61—48no—
186D4×C61—42no—
187D4×C61—42no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—24no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—48no—
199C2×C4×C61#12112yes—
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C2×#4112yes—
202C2×C4×C61C2×#4111yescharacteristic
203C2×C4×C61H(5,𝔽2)12yes—
204C2×Dic61—22no—
205C2×Dic61—22no—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—22no—
209C2×Dic61—22no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217Dic122—41no—
218Dic122—41no—
219Dic122—42no—
220C8×C61—42no—
221#92418—41no—
222#92418—41no—
223#24477—611nomaximal, Sylow
224C22×C61—22no—
225C22×C61—24no—
226C22×C61—22no—
227C22×C61—24no—
228C22×C61—44no—
229C22×C61—44no—
230C22×C61—44no—
231C22×C61—816no—
232C22×C61—82no—
233C22×C61—82no—
234C22×C61—82no—
235C22×C61—82no—
236C22×C61#46314yes—
237C22×C61C2×#12111yescharacteristic
238C22×C61#54612yes—
239C22×C61C2×#13511yescharacteristic
240C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
241C22×C61#57712yes—
242C4×C61—22no—
243C4×C61—42no—
244C4×C61—42no—
245C4×C61#52511yescharacteristic
246C4×C61#53511yescharacteristic
247C4×C61#55112yes—
248Dic61—42no—
249Dic61—42no—
250Dic61—81no—
251Dic61—81no—
252Dic61—82no—
253C2×#483—611no—
254C22×#102D6111yescharacteristic
255#2435—611no—
256#2344—611no—
257#2065—612no—
258#2042—612no—
259#1983—612no—
260#1926—612no—
261#1892—611no—
262#1850—611no—
263#1786—611no—
264C2×C61—24no—
265C2×C61—84no—
266C2×C61#241111yescharacteristic
267C2×C61C2×#46311yescharacteristic
268C2×C61#304311yescharacteristic
269C2×C61#329612yes—
270C2×C61#330612yes—
271C23×D4C2×D6111yescharacteristic
272C23×C8C2×D6111yescharacteristic
273C2×#124—612no—
274C2×C22⋊Q8—611no—
275C2×#117—1221no—
276C22×C4⋊C4C2×D6111yescharacteristic
277#483—1224no—
278C2×#106—611no—
279C2×#106—611no—
280C2×C4⋊C8—612no—
281C2×#103—611no—
282C2×#103—611no—
283C2×#103—612no—
284C2×#102—24no—
285C2×#102—612no—
286C2×#102—612no—
287C2×#102C2×D6112yes—
288C2×#102C2×D6112yes—
289C2×C22⋊C8—611no—
290#394—1222no—
291#390—612no—
292#389—612no—
293#387—1222no—
294#381—1221no—
295#380—611no—
296#379—1221no—
297#379—1221no—
298C4×C4⋊C4—612no—
299C4×C22⋊C4—611no—
300#340—612no—
301#336—612no—
302C61#2447711yescharacteristic, complemented, Sylow
303C25—21no—
304C22×Q8—611no—
305C22×D4—24no—
306C22×D4—21no—
307C22×D4—22no—
308C22×D4—22no—
309C22×D4—42no—
310C22×D4C22×D6112yes—
311C23×C4—21no—
312C23×C4C22×D6111yescharacteristic
313C22×C8—22no—
314C22×C8—42no—
315C22×C8—611no—
316C22×C8—611no—
317C22×C8C22×D6112yes—
318#124—1224no—
319#124—2442no—
320C22⋊Q8—1222no—
321C22⋊Q8—1222no—
322#117—2444no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—611no—
326C2×C4⋊C4—612no—
327C2×C4⋊C4—1222no—
328C2×C4⋊C4—1221no—
329C2×C4⋊C4C22⋊D6114yes—
330C2×C4⋊C4C22×D6111yescharacteristic
331C2×C4⋊C4C22×D6111yescharacteristic
332C2×C22⋊C4—611no—
333C2×C22⋊C4—611no—
334C2×C22⋊C4—1222no—
335C2×C22⋊C4—1222no—
336C2×C22⋊C4—1222no—
337C2×C22⋊C4—1222no—
338C2×C22⋊C4—1221no—
339C2×C22⋊C4—1221no—
340C2×C42—612no—
341C2×C42—1222no—
342#106—1222no—
343#106—1222no—
344C4⋊C8—2442no—
345#103—1222no—
346#103—1222no—
347#103—2442no—
348#102—24no—
349#102—24no—
350#102—1224no—
351#102—2442no—
352C22⋊C8—1222no—
353#95—612no—
354#95—1222no—
355#95—1222no—
356#95—1221no—
357C24—22no—
358C24—44no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—82no—
363C24D4×D6111yescharacteristic
364C2×Q8—1221no—
365C2×Q8—1221no—
366C2×Q8—1222no—
367C2×Q8—1222no—
368C2×D4—28no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—48no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—42no—
378C2×D4#1175822214yes—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—612no—
387C22×C4—612no—
388C22×C4—1222no—
389C22×C4—1221no—
390C22×C4—1221no—
391C22×C4D4×D6111yescharacteristic
392C22×C4#1175822412yes—
393C22×C4C2×C22⋊D6112yes—
394C22×C4C23×D6111yescharacteristic
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—44no—
399C2×C8—1221no—
400C2×C8—1221no—
401C2×C8—1222no—
402C2×C8—1222no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—24no—
406C4⋊C4—1221no—
407C4⋊C4—1221no—
408C4⋊C4—1222no—
409C4⋊C4—1222no—
410C4⋊C4—2444no—
411C4⋊C4—2442no—
412C4⋊C4C2×C22⋊D6112yes—
413C4⋊C4C2×C22⋊D6112yes—
414C22⋊C4—1222no—
415C22⋊C4—1222no—
416C22⋊C4—4882no—
417C22⋊C4—4882no—
418C22⋊C4—2444no—
419C22⋊C4—2444no—
420C22⋊C4—2442no—
421C22⋊C4—2442no—
422C42—1222no—
423C42—1222no—
424C42—2444no—
425C23—22no—
426C23—22no—
427C23—24no—
428C23—48no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—88no—
436C23—88no—
437C23—84no—
438C23—84no—
439C23#1451974912yes—
440C23#1451979612yes—
441C23#1451980112yes—
442C23C2×D4×D6111yescharacteristic
443Q8—2441no—
444Q8—2441no—
445Q8—2442no—
446D4—24no—
447D4—24no—
448D4—48no—
449D4—42no—
450D4—42no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—48no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—1222no—
463C2×C4—1222no—
464C2×C4—1222no—
465C2×C4—1222no—
466C2×C4—1222no—
467C2×C4—1222no—
468C2×C4—2441no—
469C2×C4—2441no—
470C2×C4—2442no—
471C2×C4—2442no—
472C2×C4—2442no—
473C2×C4—2442no—
474C2×C4—2442no—
475C2×C4#1451974912yes—
476C2×C4C22×C22⋊D6111yescharacteristic
477C2×C4C2×#1175822411yescharacteristic
478C2×C4C2×#1175822212yes—
479C2×C4#1451982212yes—
480C8—42no—
481C8—2441no—
482C8—2441no—
483C22—24no—
484C22—22no—
485C22—24no—
486C22—22no—
487C22—44no—
488C22—44no—
489C22—44no—
490C22—816no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22#1683997514yes—
496C22C2×#1451974911yescharacteristic
497C22#1684040212yes—
498C22C2×#1451979611yescharacteristic
499C22C2×#1451980111yescharacteristic
500C22#1684056812yes—
501C4—22no—
502C4—42no—
503C4—42no—
504C4—4882no—
505C4—4881no—
506C4—4881no—
507C4—2442no—
508C4—2442no—
509C4#1684027311yescharacteristic
510C4#1684032911yescharacteristic
511C4#1684043812yes—
512C2—24no—
513C2—84no—
514C2#2144920411yescharacteristic
515C2C2×#1683997511yescharacteristic
516C2#2145533711yescharacteristic
517C2#2145765012yes—
518C2#2145777612yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×#102×C61C2
socleC23×C61C22×D4
prime radicalC23×D4×C61C22
hypercenterC22×#102D61
hypocenterC61#24477
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#102

Complements

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

NH ≅ G/Ntype
1. C2×#16840092C2semidirect
3. #21448688C2semidirect
4. #21447189C2semidirect
5. #21447059C2semidirect
6. #21446750C2semidirect
7. #21446449C2semidirect
8. #21446242C2semidirect
9. #21445997C2semidirect
16. C2×#14519708C22semidirect
17. C2×#14519708C22semidirect
18. C2×#14519706C22semidirect
19. C2×#14519702C22semidirect
20. C2×#14519700C22semidirect
21. C2×#14519700C22semidirect
27. #16839661C22semidirect
33. C4⋊C4×Dic61C22semidirect
35. #16839525C22semidirect
36. #16839516C22semidirect
302. C61#24477semidirect

Maximal subgroups

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

subgroupindex
C2×#168400922
#214494662
#214486882
#214471892
#214470592
#214467502
#214464492
#214462422
#214459972
#214457502
C22×#102×C612
#2447761