Finite Group Databasebeta

← #29708765

Subgroups — #29708765

Order 15616.

1038 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839755C212yesmaximal
2#21448306C211yesmaximal, characteristic
3#21448224C211yesmaximal, characteristic, complemented
4#21447241C212yesmaximal, complemented
5#21447235C212yesmaximal, complemented
6#21447229C211yesmaximal, characteristic, complemented
7#21447228C211yesmaximal, characteristic, complemented
8#21447222C211yesmaximal, characteristic, complemented
9#21445534C212yesmaximal, complemented
10#336×D61C211yesmaximal, characteristic, complemented
11C61×#2071C211yesmaximal, characteristic, complemented
12C22×#11758224C2211yescharacteristic
13C22×C8×D61C2212yes—
14C2×#14519735C2212yes—
15C2×#14519735C2211yescharacteristic
16C2×C42×D61C2211yescharacteristic
17#16839755—28no—
18C2×#14519702C2212yes—
19C2×#14519700C2212yes—
20C2×#14519698C2212yes—
21#16839668C2211yescharacteristic
22#16839667C2211yescharacteristic
23#16839665C2211yescharacteristic
24#16839638—21no—
25#16839638—21no—
26#16839632C2211yescharacteristic
27#16839627C2212yes—
28#16839619—21no—
29#16839619—21no—
30C4⋊C4×Dic61—21no—
31C4⋊C4×Dic61—21no—
32C22⋊C4×Dic61—21no—
33C22⋊C4×Dic61—21no—
34#16839525C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37#16839517C2211yescharacteristic
38#16839516C2212yes—
39C2×#102×C61C2212yes—
40C61×#390C2211yescharacteristic
41C61×#384C2211yescharacteristic
42C61×#340C2212yes—
43C61×#336C2211yescharacteristic
44C2×#11758224—24no—
45C2×#11758224—24no—
46C2×#11758224—22no—
47C2×#11758224D414yes—
48C22×Dic122C2311yescharacteristic
49C22×C22⋊D61—21no—
50C23×Dic61—21no—
51C22×C4×D61D411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61C2311yescharacteristic
54C22×#92418C2312yes—
55C2×C8×D61—24no—
56C2×C8×D61—24no—
57C2×C8×D61—24no—
58#14519735—24no—
59#14519735—24no—
60#14519735—24no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61C2312yes—
64C2×C4⋊Dic61C2311yescharacteristic
65C2×C22⋊Dic61—21no—
66C2×C22⋊Dic61—21no—
67C2×#11758198—22no—
68C2×#11758198—21no—
69C2×C4×Dic61—21no—
70C2×C4×Dic61—22no—
71C2×C4×Dic61D411yescharacteristic
72C2×C4×Dic61D411yescharacteristic
73C2×C4×Dic61C2311yescharacteristic
74C42×D61—24no—
75#14519702—24no—
76#14519700—24no—
77#14519698—24no—
78#14519684—21no—
79#14519684—21no—
80#14519684—21no—
81#14519684—21no—
82#14519684—22no—
83#14519684—22no—
84C22×D4×C61C2311yescharacteristic
85C22×C8×C61C2312yes—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61C2311yescharacteristic
89C2×C4⋊C4×C61C2312yes—
90C2×C22⋊C4×C61—21no—
91C2×C22⋊C4×C61—21no—
92C2×C42×C61C2311yescharacteristic
93#102×C61—24no—
94C24×C61—21no—
95C23×D61C2×D411yescharacteristic
96C2×D4×C61—22no—
97C2×D4×C61—22no—
98C2×D4×C61—21no—
99C2×D4×C61C2×D412yes—
100C22×C4×C61—21no—
101C22×C4×C61—22no—
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2411yescharacteristic
105C2×C22⋊D61—24no—
106C2×C22⋊D61—42no—
107C2×C22⋊D61—42no—
108C22×Dic61—21no—
109C22×Dic61—24no—
110C22×Dic61—21no—
111C22×Dic61—22no—
112C22×Dic61—42no—
113C22×Dic61—42no—
114C22×Dic61C2×D411yescharacteristic
115C22×Dic61C2×D411yescharacteristic
116C22×Dic61C2×D411yescharacteristic
117#11758224—216no—
118#11758224—48no—
119C2×C4×D61—22no—
120C2×C4×D61—22no—
121C2×C4×D61—22no—
122C2×C4×D61—22no—
123C2×C4×D61—41no—
124C2×C4×D61—41no—
125C2×C4×D61C2×D412yes—
126C2×C4×D61#4114yes—
127C2×C4×D61#4114yes—
128C2×C4×D61#4112yes—
129C2×Dic122—22no—
130C2×Dic122—22no—
131C2×Dic122—21no—
132C2×Dic122C2×D412yes—
133C2×C8×C61—22no—
134C2×C8×C61—22no—
135C2×C8×C61—22no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—42no—
140C4⋊C4×C61—42no—
141C22⋊C4×C61—42no—
142C22⋊C4×C61—42no—
143C42×C61—22no—
144C22⋊Dic61—42no—
145C22⋊Dic61—42no—
146#11758198—44no—
147#11758198—42no—
148C4⋊Dic61—22no—
149C4⋊Dic61—22no—
150C4⋊Dic61—22no—
151C4⋊Dic61—42no—
152C4⋊Dic61—42no—
153C4×Dic61—22no—
154C4×Dic61—22no—
155C4×Dic61—22no—
156C4×Dic61—44no—
157C4×Dic61—42no—
158C2×#92418—22no—
159C2×#92418—22no—
160C2×#92418—22no—
161C8×D61—48no—
162C23×C61—22no—
163C23×C61—41no—
164C23×C61—41no—
165C23×C61—41no—
166C23×C61—41no—
167C23×C61—41no—
168C23×C61—41no—
169C23×C61C22×D411yescharacteristic
170C22×D61—21no—
171C22×D61—21no—
172C22×D61—21no—
173C22×D61—21no—
174C22×D61#12112yes—
175C22×D61#12312yes—
176C22×D61#12412yes—
177D4×C61—24no—
178D4×C61—42no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—42no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—42no—
188C2×C4×C61—42no—
189C2×C4×C61#13512yes—
190C2×C4×C61#13512yes—
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C2×#4112yes—
193C2×C4×C61C2×#4112yes—
194C2×C4×C61C2×#4111yescharacteristic
195C22⋊D61—48no—
196C2×Dic61—22no—
197C2×Dic61—21no—
198C2×Dic61—21no—
199C2×Dic61—21no—
200C2×Dic61—21no—
201C2×Dic61—21no—
202C2×Dic61—21no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—48no—
206C2×Dic61—42no—
207C2×Dic61—41no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61#12112yes—
213C2×Dic61#12312yes—
214C2×Dic61#12412yes—
215C4×D61—44no—
216C4×D61—44no—
217C4×D61#12318yes—
218C4×D61#12618yes—
219Dic122—24no—
220Dic122—42no—
221C8×C61—42no—
222#92418—42no—
223#23627—611nomaximal, Sylow
224C22×C61—42no—
225C22×C61—42no—
226C22×C61—42no—
227C22×C61—81no—
228C22×C61—81no—
229C22×C61—81no—
230C22×C61—81no—
231C22×C61C2×#12111yescharacteristic
232C22×C61C2×#12311yescharacteristic
233C22×C61C2×#12411yescharacteristic
234C22×C61C2×#13511yescharacteristic
235C22×C61C2×#13511yescharacteristic
236C22×C61#57511yescharacteristic
237C22×C61#57511yescharacteristic
238C2×D61—22no—
239C2×D61—22no—
240C2×D61—22no—
241C2×D61—22no—
242C2×D61—22no—
243C2×D61—22no—
244C2×D61#39714yes—
245C4×C61—22no—
246C4×C61—22no—
247C4×C61—81no—
248C4×C61—82no—
249C4×C61C2×#12312yes—
250C4×C61C2×#12612yes—
251Dic61—41no—
252Dic61—41no—
253Dic61—42no—
254Dic61—81no—
255Dic61—82no—
256Dic61#39714yes—
257C2×#416—612no—
258#2297—611no—
259#2285—611no—
260#2074—612no—
261#2073—612no—
262#2072—611no—
263#2072—611no—
264#2071—611no—
265#2071D6111yescharacteristic, complemented
266#1735—612no—
267C2×#336—611no—
268C2×C61—82no—
269C2×C61C2×#39711yescharacteristic
270C2×C61#303511yescharacteristic
271C2×C61#303511yescharacteristic
272C2×C61#309111yescharacteristic
273C2×C61#309111yescharacteristic
274C2×C61#313111yescharacteristic
275C2×C61#313511yescharacteristic
276D61—24no—
277C22×#41—611no—
278C23×C8—612no—
279C2×#117—612no—
280C2×#117—611no—
281C22×C42—611no—
282#416—1228no—
283C2×#103—612no—
284C2×#103—612no—
285C2×#102—612no—
286C2×#102C2×D6112yes—
287#391—611no—
288#391—611no—
289#390—611no—
290#390C2×D6111yescharacteristic
291#386—1221no—
292#386—1221no—
293#384—611no—
294#384C2×D6111yescharacteristic
295#382—612no—
296#380—1221no—
297#380—1221no—
298C4×C4⋊C4—1221no—
299C4×C4⋊C4—1221no—
300C4×C22⋊C4—1221no—
301C4×C22⋊C4—1221no—
302#340—612no—
303#340—612no—
304#340—612no—
305#340C2×D6112yes—
306#336—612no—
307#336—611no—
308#336C2×D6111yescharacteristic
309C61#2362711yescharacteristic, complemented, Sylow
310C2×#41—614no—
311C2×#41—1222no—
312C2×#41—1224no—
313C2×#41—1224no—
314C22×Q8—611no—
315C22×D4—1221no—
316C22×D4C22×D6111yescharacteristic
317C23×C4—611no—
318C23×C4—611no—
319C23×C4—611no—
320C23×C4—1221no—
321C22×C8—612no—
322C22×C8—1224no—
323C22×C8—1224no—
324C22×C8—1224no—
325C22×C8C22×D6112yes—
326#117—1224no—
327#117—1224no—
328#117—1224no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—611no—
332C2×C4⋊C4—612no—
333C2×C4⋊C4—1221no—
334C2×C4⋊C4—1221no—
335C2×C4⋊C4C22×D6111yescharacteristic
336C2×C4⋊C4C22×D6112yes—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—1222no—
340C2×C22⋊C4—1221no—
341C2×C22⋊C4—1221no—
342C2×C22⋊C4—1221no—
343C2×C42—611no—
344C2×C42—611no—
345C2×C42—611no—
346C2×C42—1224no—
347C2×C42—1221no—
348C2×C42—1222no—
349C2×C42C22×D6111yescharacteristic
350#103—1224no—
351#103—1224no—
352#102—24no—
353#102—1224no—
354#95—1221no—
355#95—1221no—
356#95—1222no—
357#95—1222no—
358#95—1221no—
359#95—1221no—
360C24—21no—
361C24—611no—
362#41—12216no—
363#41—2448no—
364C2×Q8—612no—
365C2×Q8—1221no—
366C2×Q8—1222no—
367C2×Q8—1222no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—21no—
371C2×D4—1224no—
372C2×D4—2442no—
373C2×D4—2442no—
374C2×D4D4×D6112yes—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—611no—
378C22×C4—612no—
379C22×C4—614no—
380C22×C4—612no—
381C22×C4—614no—
382C22×C4—611no—
383C22×C4—611no—
384C22×C4—1222no—
385C22×C4—1222no—
386C22×C4—1224no—
387C22×C4—1221no—
388C22×C4—1222no—
389C22×C4—1222no—
390C22×C4—1221no—
391C22×C4—1222no—
392C22×C4—2441no—
393C22×C4—2441no—
394C22×C4—2442no—
395C22×C4—2442no—
396C22×C4D4×D6111yescharacteristic
397C22×C4D4×D6111yescharacteristic
398C22×C4C23×D6111yescharacteristic
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—1222no—
403C2×C8—1222no—
404C2×C8—1222no—
405C2×C8—2448no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—1222no—
412C4⋊C4—1222no—
413C4⋊C4—1222no—
414C4⋊C4—2442no—
415C4⋊C4—2442no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—2444no—
419C22⋊C4—2442no—
420C22⋊C4—2442no—
421C22⋊C4—2442no—
422C42—22no—
423C42—1222no—
424C42—1222no—
425C42—1222no—
426C42—2444no—
427C42—2442no—
428C23—22no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—612no—
436C23—612no—
437C23—612no—
438C23—1221no—
439C23—1221no—
440C23—1221no—
441C23—1221no—
442C23C2×D4×D6111yescharacteristic
443Q8—1224no—
444Q8—2442no—
445D4—24no—
446D4—42no—
447D4—2448no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—612no—
459C2×C4—612no—
460C2×C4—612no—
461C2×C4—618no—
462C2×C4—618no—
463C2×C4—1222no—
464C2×C4—1221no—
465C2×C4—1221no—
466C2×C4—1221no—
467C2×C4—1221no—
468C2×C4—1221no—
469C2×C4—1221no—
470C2×C4—2444no—
471C2×C4—2444no—
472C2×C4—2448no—
473C2×C4—2442no—
474C2×C4—2441no—
475C2×C4—2442no—
476C2×C4—2441no—
477C2×C4—2441no—
478C2×C4—2442no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4#1451979612yes—
482C2×C4#1451979812yes—
483C2×C4C2×D4×D6111yescharacteristic
484C2×C4#41×D6112yes—
485C2×C4#41×D6112yes—
486C2×C4#41×D6111yescharacteristic
487C8—42no—
488C8—2442no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22—614no—
497C22—1222no—
498C22—1222no—
499C22—1222no—
500C22—1222no—
501C22—1222no—
502C22—1222no—
503C22#121×D6111yescharacteristic
504C22#123×D6111yescharacteristic
505C22#124×D6111yescharacteristic
506C22C2×#1451979611yescharacteristic
507C22C2×#1451979811yescharacteristic
508C22#1684055811yescharacteristic
509C22#1684056111yescharacteristic
510C4—22no—
511C4—22no—
512C4—81no—
513C4—82no—
514C4—614no—
515C4—4881no—
516C4—4882no—
517C4—2442no—
518C4—2441no—
519C4—2441no—
520C4#123×D6112yes—
521C4#126×D6112yes—
522C2—82no—
523C2—1224no—
524C2#397×D6111yescharacteristic
525C2#2145526711yescharacteristic
526C2#2145527511yescharacteristic
527C2#2145572711yescharacteristic
528C2#2145572911yescharacteristic
529C2#2145613811yescharacteristic
530C2#2145616111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2071C2
socleC23×C61C22×D4
prime radicalC22×#11758224C22
hypercenter#2071D61
hypocenterC61#23627
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2071

Complements

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

NH ≅ G/Ntype
3. #21448224C2semidirect
4. #21447241C2semidirect
5. #21447235C2semidirect
6. #21447229C2semidirect
7. #21447228C2semidirect
8. #21447222C2semidirect
9. #21445534C2semidirect
10. #336×D61C2semidirect
11. C61×#2071C2semidirect
265. #2071D61semidirect
309. C61#23627semidirect

Maximal subgroups

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

subgroupindex
C2×#168397552
#214483062
#214482242
#214472412
#214472352
#214472292
#214472282
#214472222
#214455342
#336×D612
C61×#20712
#2362761