Finite Group Databasebeta

← #29679699

Subgroups — #29679699

Order 15616.

1330 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519785C211yesmaximal, characteristic
2C22×#14519712C211yesmaximal, characteristic, complemented
3C2×#16839798C211yesmaximal, characteristic, complemented
4#21454339C211yesmaximal, characteristic
5#21454336C211yesmaximal, characteristic
6#21445882C214yesmaximal, complemented
7#21445877C214yesmaximal, complemented
8#21445557C211yesmaximal, characteristic, complemented
9C61×#1742C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C2×#14519785—24no—
12C2×#14519785—21no—
13C2×#14519785—21no—
14C2×#14519785C418yescomplemented
15C2×#14519785C2212yes—
16C2×#14519785C2212yes—
17C22×#11758194C2211yescharacteristic
18C22×#11758189—21no—
19C23×#92418C2211yescharacteristic
20C22×C8×D61—21no—
21C2×#14519736C2211yescharacteristic
22C2×#14519712—24no—
23C2×#14519712C2212yescomplemented
24C2×#14519712C2212yescomplemented
25#16839798—21no—
26#16839798—21no—
27#16839798—21no—
28#16839798—21no—
29C2×#14519710C2212yescomplemented
30C2×#14519708C2212yescomplemented
31#16839772—21no—
32#16839770—21no—
33#16839768—21no—
34#16839766—21no—
35#16839764C2214yes—
36#16839761C2214yes—
37#16839759—22no—
38#16839756—22no—
39#16839530C2214yescomplemented
40C22×M16×C61C2211yescharacteristic
41C22×M16×C61C2211yescharacteristic
42C2×#117×C61C2211yescharacteristic
43C61×#343C2214yescomplemented
44C2×#11758222—22no—
45C2×#11758222—21no—
46C2×#11758222—21no—
47C2×#11758222—22no—
48C2×#11758222C2×C414yes—
49C2×#11758222D412yes—
50C2×#11758222Q814yes—
51C2×#11758222C2312yes—
52C22×Dic122D411yescharacteristic
53C22×D244D411yescharacteristic
54C22×C22⋊D61—21no—
55C22×C4×D61—21no—
56#14519785—28no—
57#14519785—28no—
58#14519785—24no—
59#14519785—24no—
60#14519785C2×C418yes—
61#14519785C2×C418yes—
62C2×#11758194—21no—
63C2×#11758194—22no—
64C2×#11758194—22no—
65C2×#11758194C2312yes—
66C2×#11758189—28no—
67C2×#11758189—22no—
68C2×#11758189—22no—
69C22×#92418—21no—
70C22×#92418C2×C414yes—
71C22×#92418D411yescharacteristic
72C22×#92418D411yescharacteristic
73C22×#92418D411yescharacteristic
74C22×#92418D411yescharacteristic
75C22×#92418D414yes—
76C22×#92418C2311yescharacteristic
77C22×#92418C2311yescharacteristic
78C2×C8×D61—28no—
79C2×C8×D61—22no—
80C2×C8×D61—22no—
81#14519736—21no—
82#14519736—21no—
83#14519736—21no—
84#14519736—21no—
85C2×C4⋊Dic61C2312yes—
86C2×C22⋊Dic61—21no—
87C2×C4×Dic61—21no—
88#14519712—24no—
89#14519712—24no—
90#14519710—22no—
91#14519710—22no—
92#14519708—22no—
93#14519708—22no—
94#14519702—24no—
95#14519700—24no—
96#14519699—24no—
97#14519697—24no—
98C23×C4×C61C2311yescharacteristic
99C2×M16×C61—22no—
100C2×M16×C61—21no—
101C2×M16×C61—21no—
102C2×M16×C61—22no—
103C2×M16×C61—22no—
104C2×M16×C61C2×C414yes—
105C2×M16×C61C2312yes—
106C2×M16×C61C2312yes—
107C22×C8×C61—21no—
108C22×C8×C61—21no—
109#117×C61—21no—
110#117×C61—21no—
111#117×C61—21no—
112#117×C61—21no—
113C2×C4⋊C4×C61C2312yes—
114C2×C22⋊C4×C61—21no—
115C2×C42×C61—21no—
116C24×C61#4111yescharacteristic
117C23×D61—21no—
118C22×C4×C61—22no—
119C22×C4×C61C22×C412yes—
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×Q812yes—
126C22×C4×C61#4111yescharacteristic
127C22×C4×C61#4112yes—
128C22×C4×C61C2411yescharacteristic
129C2×C22⋊D61—24no—
130C2×C22⋊D61—22no—
131C2×C22⋊D61—24no—
132C2×C22⋊D61—22no—
133C22×Dic61—21no—
134C22×Dic61—22no—
135#11758222—24no—
136#11758222—24no—
137#11758222—24no—
138#11758222—24no—
139#11758222—24no—
140#11758222—22no—
141#11758222—22no—
142#11758222C4⋊C418yes—
143#11758222C22×C414yes—
144#11758222C2×Q814yes—
145C2×D244—22no—
146C2×D244—21no—
147C2×D244C4⋊C414yes—
148C2×D244C2×D412yes—
149C2×C4×D61—24no—
150C2×C4×D61—22no—
151C2×C4×D61—24no—
152C2×C4×D61—22no—
153C2×Dic122—22no—
154C2×Dic122—21no—
155C2×Dic122C4⋊C414yes—
156C2×Dic122C2×D412yes—
157M16×C61—24no—
158M16×C61—22no—
159M16×C61—22no—
160M16×C61—22no—
161M16×C61—42no—
162M16×C61C22×C414yes—
163C2×C8×C61—24no—
164C2×C8×C61—22no—
165C2×C8×C61—22no—
166C2×C8×C61—42no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C22⋊C4×C61—41no—
170C22⋊C4×C61—41no—
171C42×C61—41no—
172C42×C61—41no—
173C22⋊Dic61—41no—
174C22⋊Dic61—41no—
175C4⋊Dic61—22no—
176C4⋊Dic61—22no—
177C4×Dic61—41no—
178C4×Dic61—41no—
179#11758194—22no—
180#11758194—22no—
181#11758194—42no—
182C2×#92418—22no—
183C2×#92418—42no—
184C2×#92418C22⋊C414yes—
185C2×#92418C22⋊C414yes—
186C2×#92418C22⋊C414yes—
187C2×#92418C22⋊C414yes—
188C2×#92418C22×C412yes—
189C2×#92418C22×C412yes—
190C2×#92418C2×D411yescharacteristic
191C2×#92418C2×D411yescharacteristic
192C2×#92418C2×D411yescharacteristic
193C2×#92418C2×D411yescharacteristic
194C2×#92418C2×D412yes—
195C2×#92418C2×D412yes—
196#11758189—28no—
197#11758189—28no—
198C8×D61—28no—
199C8×D61—28no—
200C23×C61—42no—
201C23×C61#11712yes—
202C23×C61#12111yescharacteristic
203C23×C61#12111yescharacteristic
204C23×C61C22⋊Q812yes—
205C23×C61C2×#4111yescharacteristic
206C22×D61—22no—
207C22×D61—24no—
208C22×D61—44no—
209C2×C4×C61—42no—
210C2×C4×C61—82no—
211C2×C4×C61C2×C22⋊C412yes—
212C2×C4×C61C2×C22⋊C412yes—
213C2×C4×C61C2×C4⋊C412yes—
214C2×C4×C61C2×C4⋊C412yes—
215C2×C4×C61#11712yes—
216C2×C4×C61#12112yes—
217C2×C4×C61#12112yes—
218C2×C4×C61#12111yescharacteristic
219C2×C4×C61#12111yescharacteristic
220C2×C4×C61C22⋊Q812yes—
221C2×C4×C61C22⋊Q812yes—
222C2×C4×C61C22⋊Q812yes—
223C2×C4×C61C23×C411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×Q811yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C22⋊D61—24no—
230C22⋊D61—24no—
231C22⋊D61—28no—
232C2×Dic61—22no—
233C2×Dic61—24no—
234C2×Dic61—42no—
235C2×Dic61—82no—
236D244—22no—
237D244—24no—
238D244C2×C4⋊C414yes—
239C4×D61—24no—
240C4×D61—24no—
241C4×D61—28no—
242Dic122—22no—
243Dic122—24no—
244Dic122C2×C4⋊C414yes—
245C8×C61—24no—
246C8×C61—42no—
247#92418—42no—
248#92418C2×C22⋊C412yes—
249#92418C2×C22⋊C412yes—
250#92418C2×C22⋊C412yes—
251#92418C2×C22⋊C412yes—
252#20678—611nomaximal, Sylow
253C22×C61—22no—
254C22×C61—22no—
255C22×C61—22no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61#38012yes—
259C22×C61#38012yes—
260C22×C61C2×#11711yescharacteristic
261C22×C61C2×#12111yescharacteristic
262C22×C61C2×C22⋊Q811yescharacteristic
263C2×D61—24no—
264C2×D61—48no—
265C2×D61—44no—
266C4×C61—82no—
267C4×C61#38012yes—
268C4×C61#38012yes—
269C4×C61C22×C22⋊C411yescharacteristic
270C4×C61C22×C4⋊C411yescharacteristic
271C4×C61C2×#12111yescharacteristic
272C4×C61C2×C22⋊Q811yescharacteristic
273Dic61—24no—
274Dic61—82no—
275C22×#131—611no—
276C22×#108—611no—
277C2×#427—611no—
278#2896—611no—
279#2896—611no—
280#1816—614no—
281#1815—614no—
282#1742—611no—
283#1742D6111yescharacteristic, complemented
284C2×C61—22no—
285C2×C61—21no—
286C2×C61—21no—
287C2×C61—22no—
288C2×C61C2×#38011yescharacteristic
289C2×C61#306712yes—
290D61—44no—
291C22×#41—611no—
292C2×#131—618no—
293C2×#131—612no—
294C2×#131—612no—
295C2×#131—1221no—
296C2×#131—1221no—
297C2×#131—1224no—
298C22×M16—611no—
299C22×M16—1221no—
300C22×M16C2×D6111yescharacteristic
301C22×M16C2×D6111yescharacteristic
302C23×C8—611no—
303C23×C8—1221no—
304C2×#117—611no—
305C2×#117C2×D6111yescharacteristic
306C2×#108—612no—
307C2×#108—612no—
308C2×#108—1224no—
309#427—1221no—
310#427—1221no—
311#427—1221no—
312#427—1221no—
313C2×#107—612no—
314C2×#106—612no—
315#419—1221no—
316#419—1221no—
317#418—1221no—
318#418—1221no—
319#417—614no—
320#417—614no—
321#416—1222no—
322#416—1222no—
323#343—614no—
324#343C2×D6114yescomplemented
325C61#2067811yescharacteristic, complemented, Sylow
326C2×#41—614no—
327C2×#41—612no—
328C2×#41—614no—
329C2×#41—612no—
330C2×#41—1221no—
331C2×#41—1222no—
332C2×#41—1221no—
333C2×#41—1222no—
334C22×Q8—611no—
335C22×D4—611no—
336C22×D4—1221no—
337C23×C4—1221no—
338C23×C4C22×D6111yescharacteristic
339#131—618no—
340#131—618no—
341#131—1224no—
342#131—1224no—
343#131—1228no—
344#131—1228no—
345C2×M16—21no—
346C2×M16—22no—
347C2×M16—21no—
348C2×M16—22no—
349C2×M16—22no—
350C2×M16—612no—
351C2×M16—1222no—
352C2×M16—1222no—
353C2×M16—1228no—
354C2×M16—1221no—
355C2×M16—1222no—
356C2×M16—1222no—
357C2×M16C4×D6114yes—
358C2×M16C22×D6112yes—
359C2×M16C22×D6112yes—
360C22×C8—21no—
361C22×C8—21no—
362C22×C8—614no—
363C22×C8—611no—
364C22×C8—611no—
365C22×C8—611no—
366C22×C8—611no—
367C22×C8—614no—
368C22×C8—611no—
369C22×C8—611no—
370C22×C8—1222no—
371C22×C8—1222no—
372C22×C8—1228no—
373C22×C8—1221no—
374#117—21no—
375#117—21no—
376#117—21no—
377#117—21no—
378#117—1221no—
379#117—1221no—
380#117—1221no—
381#117—1221no—
382C2×C4⋊C4—612no—
383C2×C4⋊C4C22×D6112yes—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—1221no—
386C2×C42—21no—
387C2×C42—1221no—
388#108—1224no—
389#108—1224no—
390#107—1222no—
391#107—1222no—
392#106—1222no—
393#106—1222no—
394#103—1224no—
395#103—1224no—
396#102—1224no—
397#102—1224no—
398C24—1221no—
399C24#1175822211yescharacteristic
400#41—614no—
401#41—614no—
402#41—618no—
403#41—1222no—
404#41—1222no—
405#41—1224no—
406#41—1224no—
407#41—1224no—
408#41—1224no—
409#41—1224no—
410C2×Q8—612no—
411C2×Q8—614no—
412C2×Q8—1221no—
413C2×Q8—1222no—
414C2×D4—612no—
415C2×D4—614no—
416C2×D4—1222no—
417C2×D4—1224no—
418C2×D4—1222no—
419C2×D4—1224no—
420C2×D4—1222no—
421C2×D4—1221no—
422C22×C4—22no—
423C22×C4—1221no—
424C22×C4—1222no—
425C22×C4—1224no—
426C22×C4—1222no—
427C22×C4—1224no—
428C22×C4—1222no—
429C22×C4C2×C4×D6112yes—
430C22×C4#1175822212yes—
431C22×C4#1175822211yescharacteristic
432C22×C4D4×D6112yes—
433C22×C4D4×D6111yescharacteristic
434C22×C4D4×D6111yescharacteristic
435C22×C4D4×D6111yescharacteristic
436C22×C4D4×D6111yescharacteristic
437C22×C4Q8×D6112yes—
438C22×C4C23×D6111yescharacteristic
439M16—22no—
440M16—24no—
441M16—22no—
442M16—22no—
443M16—42no—
444M16—1228no—
445M16—1228no—
446M16—1222no—
447M16—1222no—
448M16—2442no—
449M16C2×C4×D6114yes—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—42no—
454C2×C8—612no—
455C2×C8—612no—
456C2×C8—611no—
457C2×C8—611no—
458C2×C8—611no—
459C2×C8—611no—
460C2×C8—612no—
461C2×C8—612no—
462C2×C8—614no—
463C2×C8—614no—
464C2×C8—614no—
465C2×C8—614no—
466C2×C8—1228no—
467C2×C8—1228no—
468C2×C8—1222no—
469C2×C8—2442no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—1222no—
473C4⋊C4—1222no—
474C22⋊C4—41no—
475C22⋊C4—41no—
476C22⋊C4—2441no—
477C22⋊C4—2441no—
478C42—41no—
479C42—41no—
480C42—2441no—
481C42—2441no—
482C23—42no—
483C23—1222no—
484C23—1224no—
485C23—2444no—
486C23#1451973412yes—
487C23#1451975011yescharacteristic
488C23#1451975011yescharacteristic
489C23#1451975512yes—
490C23C2×#1175822211yescharacteristic
491Q8—614no—
492Q8—1222no—
493Q8—1224no—
494D4—614no—
495D4—1224no—
496D4—1224no—
497D4—1228no—
498D4—1222no—
499D4—1224no—
500C2×C4—42no—
501C2×C4—82no—
502C2×C4—1222no—
503C2×C4—1224no—
504C2×C4—1224no—
505C2×C4—1224no—
506C2×C4—1228no—
507C2×C4—4882no—
508C2×C4—2442no—
509C2×C4C22⋊C4×D6112yes—
510C2×C4C22⋊C4×D6112yes—
511C2×C4C4⋊C4×D6112yes—
512C2×C4C4⋊C4×D6112yes—
513C2×C4#1451973412yes—
514C2×C4#1451975012yes—
515C2×C4#1451975012yes—
516C2×C4#1451975011yescharacteristic
517C2×C4#1451975011yescharacteristic
518C2×C4#1451975512yes—
519C2×C4#1451975512yes—
520C2×C4#1451975512yes—
521C2×C4C22×C4×D6111yescharacteristic
522C2×C4C2×D4×D6111yescharacteristic
523C2×C4C2×D4×D6111yescharacteristic
524C2×C4C2×D4×D6111yescharacteristic
525C2×C4C2×Q8×D6111yescharacteristic
526C2×C4C2×#1175822211yescharacteristic
527C8—24no—
528C8—42no—
529C8—612no—
530C8—612no—
531C8—612no—
532C8—612no—
533C8—2442no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—42no—
538C22—42no—
539C22—1224no—
540C22—2448no—
541C22—2444no—
542C22#1683962112yes—
543C22#1683962112yes—
544C22C2×#1451973411yescharacteristic
545C22C2×#1451975011yescharacteristic
546C22C2×#1451975511yescharacteristic
547C4—82no—
548C4—1224no—
549C4—4882no—
550C4#1683962112yes—
551C4#1683962112yes—
552C4C2×C22⋊C4×D6111yescharacteristic
553C4C2×C4⋊C4×D6111yescharacteristic
554C4C2×#1451975011yescharacteristic
555C4C2×#1451975511yescharacteristic
556C2—22no—
557C2—21no—
558C2—21no—
559C2—22no—
560C2—2444no—
561C2C2×#1683962111yescharacteristic
562C2#2145553812yes—
subgroupG/H
centerC2×C4#14519750
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1742C2
socleC22×C61C2×#121
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenter#1742D61
hypocenterC61#20678
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1742

Complements

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

NH ≅ G/Ntype
2. C22×#14519712C2semidirect
3. C2×#16839798C2semidirect
6. #21445882C2semidirect
7. #21445877C2semidirect
8. #21445557C2semidirect
9. C61×#1742C2semidirect
14. C2×#14519785C4semidirect
23. C2×#14519712C22semidirect
24. C2×#14519712C22semidirect
29. C2×#14519710C22semidirect
30. C2×#14519708C22semidirect
39. #16839530C22semidirect
43. C61×#343C22semidirect
283. #1742D61semidirect
324. #343C2×D61semidirect
325. C61#20678semidirect

Maximal subgroups

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

subgroupindex
C22×#145197852
C22×#145197122
C2×#168397982
#214543392
#214543362
#214458822
#214458772
#214455572
C61×#17422
#2067861