Finite Group Databasebeta

← #29708830

Subgroups — #29708830

Order 15616.

1010 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839761C211yesmaximal, characteristic
2C2×#16839759C211yesmaximal, characteristic
3#21448443C211yesmaximal, characteristic, complemented
4#21448412C211yesmaximal, characteristic
5#21447241C212yesmaximal, complemented
6#21447236C212yesmaximal, complemented
7#21447231C212yesmaximal, complemented
8#21447226C212yesmaximal, complemented
9#21445538C211yesmaximal, characteristic, complemented
10#21445515C211yesmaximal, characteristic, complemented
11C2×C61×#340C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C22×#11758194C2211yescharacteristic
14C2×#14519736C2211yescharacteristic
15C2×#14519736C2211yescharacteristic
16#16839761—24no—
17#16839759—24no—
18C2×#14519702C2211yescharacteristic
19C2×#14519702C2211yescharacteristic
20C2×#14519700C2212yes—
21C2×#14519699C2211yescharacteristic
22C2×#14519699C2211yescharacteristic
23C2×#14519697C2212yes—
24#16839667C2212yes—
25#16839663C2212yes—
26#16839658—21no—
27#16839658—21no—
28#16839641—22no—
29#16839635—22no—
30#16839631C2212yes—
31#16839627C2212yes—
32#16839619—21no—
33#16839619—21no—
34#16839524C2212yes—
35#16839524C2212yes—
36#16839516C2212yes—
37#16839516C2212yes—
38C23×C8×C61C2211yescharacteristic
39C22×C4⋊C4×C61C2211yescharacteristic
40C22×C4⋊C4×C61C2211yescharacteristic
41C61×#340C2212yes—
42C61×#340C2212yes—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222D414yes—
49C22×Dic122C2311yescharacteristic
50C22×D244C2311yescharacteristic
51C22×C22⋊D61—21no—
52C22×C4×D61—21no—
53C2×#11758194—22no—
54C2×#11758194—22no—
55C2×#11758194—22no—
56C22×#92418C2312yes—
57#14519736—22no—
58#14519736—22no—
59#14519736—22no—
60#14519736—22no—
61C2×C4⋊Dic61D411yescharacteristic
62C2×C4⋊Dic61D411yescharacteristic
63C2×C4⋊Dic61C2311yescharacteristic
64C2×C4⋊Dic61C2311yescharacteristic
65C2×#11758196—22no—
66C2×#11758196—22no—
67C2×C22⋊Dic61—21no—
68C2×C22⋊Dic61D412yes—
69C2×#11758198—22no—
70C2×#11758198—22no—
71C2×C4×Dic61—21no—
72C2×C4×Dic61C2312yes—
73#14519702—22no—
74#14519702—22no—
75#14519700—24no—
76#14519699—22no—
77#14519699—22no—
78#14519697—24no—
79#14519684—22no—
80#14519684—22no—
81#14519684—22no—
82#14519684—22no—
83C23×C4×C61—21no—
84C23×C4×C61—21no—
85C23×C4×C61C2311yescharacteristic
86C22×C8×C61—22no—
87C22×C8×C61—22no—
88C22×C8×C61—22no—
89C22×C8×C61C2311yescharacteristic
90C22×C8×C61C2311yescharacteristic
91C2×C4⋊C4×C61—22no—
92C2×C4⋊C4×C61—22no—
93C2×C4⋊C4×C61—22no—
94C2×C4⋊C4×C61—22no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61C2312yes—
98C2×C4⋊C4×C61C2312yes—
99C24×C61C2×D411yescharacteristic
100C23×D61—21no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—42no—
104C22×C4×C61—42no—
105C22×C4×C61—81no—
106C22×C4×C61—81no—
107C22×C4×C61—81no—
108C22×C4×C61—81no—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D412yes—
111C22×C4×C61#4112yes—
112C22×C4×C61#4112yes—
113C22×C4×C61#4112yes—
114C22×C4×C61#4112yes—
115C22×C4×C61#4112yes—
116C22×C4×C61C2411yescharacteristic
117C2×C22⋊D61—24no—
118C2×C22⋊D61—42no—
119C2×C22⋊D61—42no—
120C22×Dic61—21no—
121C22×Dic61—21no—
122C22×Dic61—21no—
123C22×Dic61C2×D412yes—
124C22×Dic61C2×D412yes—
125#11758222—28no—
126#11758222—28no—
127#11758222—44no—
128#11758222—44no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244C2×D412yes—
135C2×C4×D61—24no—
136C2×C4×D61—42no—
137C2×C4×D61—42no—
138C2×Dic122—21no—
139C2×Dic122—21no—
140C2×Dic122—21no—
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122C2×D412yes—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61—21no—
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C2×C8×C61—21no—
150C2×C8×C61—44no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—42no—
156C4⋊C4×C61—42no—
157C22⋊Dic61—24no—
158C22⋊Dic61—42no—
159#11758198—82no—
160#11758198—82no—
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163C4⋊Dic61—21no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61—21no—
167#11758196—82no—
168#11758196—82no—
169C4×Dic61—22no—
170C4×Dic61—22no—
171C4×Dic61—42no—
172#11758194—44no—
173C2×#92418—22no—
174C2×#92418—22no—
175C2×#92418—22no—
176C23×C61—21no—
177C23×C61—21no—
178C23×C61—21no—
179C23×C61—21no—
180C23×C61#12112yes—
181C23×C61#12312yes—
182C23×C61#12412yes—
183C23×C61C22×D411yescharacteristic
184C22×D61—22no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—41no—
190C22×D61—41no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61—84no—
198C2×C4×C61—84no—
199C2×C4×C61#12112yes—
200C2×C4×C61#12314yes—
201C2×C4×C61#12314yes—
202C2×C4×C61#12312yes—
203C2×C4×C61#12412yes—
204C2×C4×C61#12614yes—
205C2×C4×C61#12614yes—
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C2×#4111yescharacteristic
208C2×C4×C61C2×#4111yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C2×C4×C61C2×#4111yescharacteristic
211C2×C4×C61C2×#4111yescharacteristic
212C22⋊D61—48no—
213C2×Dic61—22no—
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—22no—
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—22no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228D244—22no—
229D244—22no—
230D244—41no—
231D244—41no—
232C4×D61—48no—
233Dic122—22no—
234Dic122—22no—
235Dic122—41no—
236Dic122—41no—
237C8×C61—41no—
238C8×C61—41no—
239#92418—42no—
240#23634—611nomaximal, Sylow
241C22×C61—22no—
242C22×C61—22no—
243C22×C61—22no—
244C22×C61—22no—
245C22×C61—22no—
246C22×C61—22no—
247C22×C61#39714yes—
248C22×C61C2×#12111yescharacteristic
249C22×C61C2×#12311yescharacteristic
250C22×C61C2×#12411yescharacteristic
251C22×C61#57611yescharacteristic
252C22×C61#57711yescharacteristic
253C22×C61#57711yescharacteristic
254C22×C61#57811yescharacteristic
255C2×D61—42no—
256C2×D61—42no—
257C2×D61—42no—
258C2×D61—81no—
259C2×D61—81no—
260C2×D61—81no—
261C2×D61—81no—
262C4×C61—82no—
263C4×C61—82no—
264C4×C61#39714yes—
265C4×C61C2×#12311yescharacteristic
266C4×C61C2×#12311yescharacteristic
267C4×C61C2×#12611yescharacteristic
268C4×C61C2×#12611yescharacteristic
269Dic61—42no—
270Dic61—42no—
271Dic61—42no—
272Dic61—81no—
273Dic61—81no—
274C2×#417—611no—
275C2×#416—611no—
276#2317—611no—
277#2311—611no—
278#2074—612no—
279#2073—612no—
280#2072—612no—
281#2071—612no—
282#1736—611no—
283C2×#340D6111yescharacteristic, complemented
284#1729—611no—
285C2×C61—24no—
286C2×C61C2×#39711yescharacteristic
287C2×C61#304511yescharacteristic
288C2×C61#304611yescharacteristic
289C2×C61#309811yescharacteristic
290C2×C61#309911yescharacteristic
291C2×C61#313611yescharacteristic
292C2×C61#313711yescharacteristic
293D61—82no—
294C22×#41—611no—
295C22×M16—611no—
296C23×C8C2×D6111yescharacteristic
297C2×#117—611no—
298C2×#117—611no—
299C22×C4⋊C4C2×D6111yescharacteristic
300C22×C4⋊C4C2×D6111yescharacteristic
301#417—1224no—
302#416—1224no—
303C2×#103—611no—
304C2×#103—611no—
305C2×#103—612no—
306C2×#102—611no—
307C2×#102—611no—
308C2×#102—612no—
309#391—612no—
310#390—612no—
311#389—1221no—
312#389—1221no—
313#386—1222no—
314#385—1222no—
315#384—612no—
316#382—612no—
317#380—1221no—
318#380—1221no—
319#340—612no—
320#340—612no—
321#340C2×D6112yes—
322#340C2×D6112yes—
323#336—612no—
324#336—612no—
325C61#2363411yescharacteristic, complemented, Sylow
326C2×#41—614no—
327C2×#41—1222no—
328C2×#41—1222no—
329C2×#41—1222no—
330C2×#41—1222no—
331C2×#41—1222no—
332C22×Q8—611no—
333C22×D4—611no—
334C22×D4—1221no—
335C23×C4—21no—
336C23×C4—21no—
337C23×C4—1221no—
338C23×C4C22×D6111yescharacteristic
339C2×M16—1222no—
340C2×M16—1222no—
341C2×M16—1222no—
342C22×C8—22no—
343C22×C8—22no—
344C22×C8—22no—
345C22×C8—612no—
346C22×C8C22×D6111yescharacteristic
347C22×C8C22×D6111yescharacteristic
348#117—1222no—
349#117—1222no—
350#117—1222no—
351#117—1222no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—611no—
359C2×C4⋊C4—611no—
360C2×C4⋊C4—611no—
361C2×C4⋊C4—611no—
362C2×C4⋊C4—1222no—
363C2×C4⋊C4—1222no—
364C2×C4⋊C4C22×D6112yes—
365C2×C4⋊C4C22×D6112yes—
366C2×C22⋊C4—612no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—1222no—
369C2×C22⋊C4—1221no—
370C2×C42—612no—
371C2×C42—1221no—
372#103—1224no—
373#103—1222no—
374#103—1222no—
375#102—1224no—
376#102—1222no—
377#102—1222no—
378#95—1222no—
379#95—1222no—
380#95—1222no—
381#95—1222no—
382C24—1221no—
383C24D4×D6111yescharacteristic
384#41—1228no—
385#41—1228no—
386#41—2444no—
387#41—2444no—
388C2×Q8—612no—
389C2×Q8—1221no—
390C2×Q8—1221no—
391C2×Q8—1221no—
392C2×Q8—1221no—
393C2×Q8—1221no—
394C2×D4—612no—
395C2×D4—1224no—
396C2×D4—1221no—
397C2×D4—1221no—
398C2×D4—1221no—
399C2×D4—1221no—
400C2×D4—1221no—
401C2×D4—2442no—
402C2×D4—2442no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4—81no—
411C22×C4—612no—
412C22×C4—612no—
413C22×C4—1221no—
414C22×C4—1224no—
415C22×C4—1221no—
416C22×C4—1221no—
417C22×C4—2442no—
418C22×C4—2442no—
419C22×C4#1175822212yes—
420C22×C4#1175822212yes—
421C22×C4#1175822212yes—
422C22×C4D4×D6112yes—
423C22×C4D4×D6111yescharacteristic
424C22×C4#1175822412yes—
425C22×C4#1175822612yes—
426C22×C4C23×D6111yescharacteristic
427M16—2444no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—44no—
435C2×C8—1222no—
436C2×C8—1222no—
437C2×C8—1222no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—1222no—
445C4⋊C4—1222no—
446C4⋊C4—1221no—
447C4⋊C4—1221no—
448C4⋊C4—1221no—
449C4⋊C4—1221no—
450C4⋊C4—4882no—
451C4⋊C4—4882no—
452C22⋊C4—1224no—
453C22⋊C4—4882no—
454C22⋊C4—4882no—
455C22⋊C4—2442no—
456C42—1222no—
457C42—1222no—
458C42—2442no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—1222no—
464C23—2441no—
465C23—2441no—
466C23—2441no—
467C23—2441no—
468C23—2441no—
469C23—2441no—
470C23#1451975012yes—
471C23#1451976212yes—
472C23#1451976312yes—
473C23C2×D4×D6111yescharacteristic
474Q8—1222no—
475Q8—1222no—
476Q8—2441no—
477Q8—2441no—
478D4—1222no—
479D4—1222no—
480D4—2448no—
481D4—2441no—
482D4—2441no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—84no—
490C2×C4—84no—
491C2×C4—1222no—
492C2×C4—1222no—
493C2×C4—1222no—
494C2×C4—1222no—
495C2×C4—1222no—
496C2×C4—1222no—
497C2×C4—1222no—
498C2×C4—2441no—
499C2×C4—2441no—
500C2×C4—2441no—
501C2×C4—2441no—
502C2×C4—2441no—
503C2×C4—2441no—
504C2×C4—2441no—
505C2×C4—2441no—
506C2×C4—2448no—
507C2×C4#1451975012yes—
508C2×C4#1451975914yes—
509C2×C4#1451976212yes—
510C2×C4#1451976214yes—
511C2×C4#1451976312yes—
512C2×C4#1451977114yes—
513C2×C4#1451977214yes—
514C2×C4C2×D4×D6111yescharacteristic
515C2×C4C2×#1175822411yescharacteristic
516C2×C4C2×#1175822211yescharacteristic
517C2×C4C2×#1175822211yescharacteristic
518C2×C4C2×#1175822211yescharacteristic
519C2×C4C2×#1175822611yescharacteristic
520C8—41no—
521C8—41no—
522C8—2442no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—4881no—
530C22—4881no—
531C22—4881no—
532C22—4881no—
533C22—2442no—
534C22—2442no—
535C22—2442no—
536C22#1683969114yes—
537C22C2×#1451975011yescharacteristic
538C22C2×#1451976211yescharacteristic
539C22C2×#1451976311yescharacteristic
540C22#1684056311yescharacteristic
541C22#1684056811yescharacteristic
542C22#1684056811yescharacteristic
543C22#1684057511yescharacteristic
544C4—82no—
545C4—82no—
546C4—4881no—
547C4—4881no—
548C4—2442no—
549C4—2442no—
550C4—2442no—
551C4#1683969114yes—
552C4C2×#1451975911yescharacteristic
553C4C2×#1451976211yescharacteristic
554C4C2×#1451977111yescharacteristic
555C4C2×#1451977211yescharacteristic
556C2—24no—
557C2—4882no—
558C2C2×#1683969111yescharacteristic
559C2#2145536311yescharacteristic
560C2#2145537411yescharacteristic
561C2#2145580211yescharacteristic
562C2#2145581311yescharacteristic
563C2#2145617111yescharacteristic
564C2#2145618211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#340C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenterC2×#340D61
hypocenterC61#23634
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
3. #21448443C2semidirect
5. #21447241C2semidirect
6. #21447236C2semidirect
7. #21447231C2semidirect
8. #21447226C2semidirect
9. #21445538C2semidirect
10. #21445515C2semidirect
11. C2×C61×#340C2semidirect
283. C2×#340D61semidirect
325. C61#23634semidirect

Maximal subgroups

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

subgroupindex
C2×#168397612
C2×#168397592
#214484432
#214484122
#214472412
#214472362
#214472312
#214472262
#214455382
#214455152
C2×C61×#3402
#2363461