Finite Group Databasebeta

← #29708698

Subgroups — #29708698

Order 15616.

1102 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839802C211yesmaximal, characteristic, complemented
2C2×#16839788C211yesmaximal, characteristic, complemented
3C2×#16839761C211yesmaximal, characteristic
4C2×#16839759C211yesmaximal, characteristic
5#21448443C211yesmaximal, characteristic, complemented
6#21448412C211yesmaximal, characteristic
7#21447215C212yesmaximal, complemented
8#21447208C212yesmaximal, complemented
9#21447201C212yesmaximal, complemented
10#21447194C212yesmaximal, complemented
11C2×C61×#340C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C22×#11758194C2211yescharacteristic
14C2×#14519736C2211yescharacteristic
15C2×#14519736C2211yescharacteristic
16#16839802—24no—
17C2×#14519710C2212yes—
18C2×#14519708C2212yes—
19#16839788—24no—
20C2×#14519706C2212yes—
21C2×#14519706C2212yes—
22#16839761—24no—
23#16839759—24no—
24C2×#14519702C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519700C2212yes—
27C2×#14519699C2211yescharacteristic
28C2×#14519699C2211yescharacteristic
29C2×#14519697C2212yes—
30#16839667C2212yes—
31#16839663C2212yes—
32#16839658C2212yes—
33#16839658C2212yes—
34#16839641—22no—
35#16839635—22no—
36#16839631—21no—
37#16839627—21no—
38#16839619—21no—
39#16839619—21no—
40C23×C8×C61C2211yescharacteristic
41C22×C4⋊C4×C61C2211yescharacteristic
42C22×C4⋊C4×C61C2211yescharacteristic
43C61×#340C2212yes—
44C61×#340C2212yes—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222Q814yes—
51C22×Dic122C2311yescharacteristic
52C22×D244C2311yescharacteristic
53C22×C22⋊D61—21no—
54C22×C4×D61—21no—
55C2×#11758194—22no—
56C2×#11758194—22no—
57C2×#11758194D414yes—
58C22×#92418C2312yes—
59#14519736—22no—
60#14519736—22no—
61#14519736—22no—
62#14519736D414yes—
63C2×C4⋊Dic61D411yescharacteristic
64C2×C4⋊Dic61D411yescharacteristic
65C2×C4⋊Dic61C2311yescharacteristic
66C2×C4⋊Dic61C2311yescharacteristic
67C2×#11758196—22no—
68C2×#11758196—22no—
69C2×C22⋊Dic61—21no—
70C2×C22⋊Dic61D412yes—
71C2×#11758198—22no—
72C2×#11758198—22no—
73C2×C4×Dic61—21no—
74C2×C4×Dic61C2312yes—
75#14519710—24no—
76#14519708—24no—
77#14519706—24no—
78#14519706—24no—
79#14519702—22no—
80#14519702—22no—
81#14519700—24no—
82#14519699—22no—
83#14519699—22no—
84#14519697—24no—
85#14519684—22no—
86#14519684—22no—
87#14519684—22no—
88#14519684—22no—
89C23×C4×C61—21no—
90C23×C4×C61—21no—
91C23×C4×C61C2311yescharacteristic
92C22×C8×C61—22no—
93C22×C8×C61—22no—
94C22×C8×C61—22no—
95C22×C8×C61C2311yescharacteristic
96C22×C8×C61C2311yescharacteristic
97C2×C4⋊C4×C61—22no—
98C2×C4⋊C4×C61—22no—
99C2×C4⋊C4×C61—22no—
100C2×C4⋊C4×C61—22no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61C2312yes—
104C2×C4⋊C4×C61C2312yes—
105C24×C61C2×D411yescharacteristic
106C23×D61—21no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—42no—
110C22×C4×C61—42no—
111C22×C4×C61—81no—
112C22×C4×C61—81no—
113C22×C4×C61—81no—
114C22×C4×C61—81no—
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D412yes—
117C22×C4×C61C2×D412yes—
118C22×C4×C61C2×Q812yes—
119C22×C4×C61#4112yes—
120C22×C4×C61#4112yes—
121C22×C4×C61#4112yes—
122C22×C4×C61C2411yescharacteristic
123C2×C22⋊D61—24no—
124C2×C22⋊D61—42no—
125C2×C22⋊D61—42no—
126C22×Dic61—21no—
127C22×Dic61—21no—
128C22×Dic61—21no—
129C22×Dic61C2×D412yes—
130C22×Dic61C2×D412yes—
131#11758222—28no—
132#11758222—28no—
133#11758222—44no—
134#11758222—44no—
135C2×D244—21no—
136C2×D244—21no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244C2×Q812yes—
141C2×C4×D61—24no—
142C2×C4×D61—42no—
143C2×C4×D61—42no—
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122—21no—
148C2×Dic122—21no—
149C2×Dic122C2×Q812yes—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—44no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—42no—
162C4⋊C4×C61—42no—
163C22⋊Dic61—24no—
164C22⋊Dic61—24no—
165#11758198—82no—
166#11758198—82no—
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61C2×D412yes—
172C4⋊Dic61C2×D412yes—
173#11758196—82no—
174#11758196—82no—
175C4×Dic61—24no—
176C4×Dic61—22no—
177C4×Dic61—22no—
178#11758194—28no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418C2×D414yes—
182C23×C61—21no—
183C23×C61—21no—
184C23×C61—21no—
185C23×C61—21no—
186C23×C61#12112yes—
187C23×C61#12112yes—
188C23×C61C22⋊Q812yes—
189C23×C61C22×D411yescharacteristic
190C22×D61—22no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—41no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—84no—
204C2×C4×C61—84no—
205C2×C4×C61#12114yes—
206C2×C4×C61#12112yes—
207C2×C4×C61#12112yes—
208C2×C4×C61C22⋊Q814yes—
209C2×C4×C61C22⋊Q814yes—
210C2×C4×C61C22⋊Q812yes—
211C2×C4×C61#12614yes—
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×Q811yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216C2×C4×C61C2×#4111yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C22⋊D61—48no—
219C2×Dic61—22no—
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234D244—22no—
235D244—22no—
236D244—41no—
237D244—41no—
238C4×D61—48no—
239Dic122—22no—
240Dic122—22no—
241Dic122—41no—
242Dic122—41no—
243C8×C61—41no—
244C8×C61—41no—
245#92418—24no—
246#23616—611nomaximal, Sylow
247C22×C61—22no—
248C22×C61—22no—
249C22×C61—22no—
250C22×C61—22no—
251C22×C61—22no—
252C22×C61—22no—
253C22×C61#39614yes—
254C22×C61C2×#12111yescharacteristic
255C22×C61C2×#12111yescharacteristic
256C22×C61C2×C22⋊Q811yescharacteristic
257C22×C61#57611yescharacteristic
258C22×C61#57711yescharacteristic
259C22×C61#57711yescharacteristic
260C22×C61#57811yescharacteristic
261C2×D61—42no—
262C2×D61—42no—
263C2×D61—42no—
264C2×D61—81no—
265C2×D61—81no—
266C2×D61—81no—
267C2×D61—81no—
268C4×C61—82no—
269C4×C61—82no—
270C4×C61#39614yes—
271C4×C61C2×#12111yescharacteristic
272C4×C61C2×C22⋊Q811yescharacteristic
273C4×C61C2×C22⋊Q811yescharacteristic
274C4×C61C2×#12611yescharacteristic
275Dic61—42no—
276Dic61—42no—
277Dic61—42no—
278Dic61—42no—
279Dic61—42no—
280C2×#428—611no—
281C2×#424—611no—
282C2×#417—611no—
283C2×#416—611no—
284#2317—611no—
285#2311—611no—
286#2068—612no—
287#2067—612no—
288#2066—612no—
289#2065—612no—
290C2×#340D6111yescharacteristic, complemented
291C2×C61—24no—
292C2×C61C2×#39611yescharacteristic
293C2×C61#304511yescharacteristic
294C2×C61#304611yescharacteristic
295C2×C61#306211yescharacteristic
296C2×C61#306311yescharacteristic
297C2×C61#308311yescharacteristic
298C2×C61#308411yescharacteristic
299D61—82no—
300C22×#41—611no—
301C22×M16—611no—
302C23×C8C2×D6111yescharacteristic
303C2×#117—611no—
304C2×#117—611no—
305C22×C4⋊C4C2×D6111yescharacteristic
306C22×C4⋊C4C2×D6111yescharacteristic
307#428—1224no—
308C2×#107—612no—
309C2×#106—612no—
310#424—1224no—
311C2×C4⋊C8—612no—
312C2×C4⋊C8—612no—
313#417—1224no—
314#416—1224no—
315C2×#103—611no—
316C2×#103—611no—
317C2×#103—612no—
318C2×#102—611no—
319C2×#102—611no—
320C2×#102—612no—
321#391—612no—
322#390—612no—
323#389—612no—
324#389—612no—
325#386—1222no—
326#385—1222no—
327#384—1221no—
328#382—1221no—
329#380—1221no—
330#380—1221no—
331#340C2×D6112yes—
332#340C2×D6112yes—
333C61#2361611yescharacteristic, complemented, Sylow
334C2×#41—614no—
335C2×#41—1222no—
336C2×#41—1222no—
337C2×#41—1222no—
338C2×#41—1222no—
339C2×#41—1222no—
340C22×Q8—611no—
341C22×D4—611no—
342C22×D4—1221no—
343C23×C4—21no—
344C23×C4—21no—
345C23×C4—1221no—
346C23×C4C22×D6111yescharacteristic
347C2×M16—614no—
348C2×M16—1222no—
349C2×M16—1222no—
350C22×C8—22no—
351C22×C8—22no—
352C22×C8—22no—
353C22×C8—612no—
354C22×C8C22×D6111yescharacteristic
355C22×C8C22×D6111yescharacteristic
356#117—614no—
357#117—1222no—
358#117—1222no—
359#117—1222no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1222no—
372C2×C4⋊C4C22×D6112yes—
373C2×C4⋊C4C22×D6112yes—
374C2×C22⋊C4—612no—
375C2×C22⋊C4—1221no—
376C2×C22⋊C4—1222no—
377C2×C22⋊C4—1222no—
378C2×C42—612no—
379C2×C42—1221no—
380#107—1224no—
381#106—1224no—
382C4⋊C8—1224no—
383C4⋊C8—1224no—
384#103—1224no—
385#103—1222no—
386#103—1222no—
387#102—1224no—
388#102—1222no—
389#102—1222no—
390#95—1222no—
391#95—1222no—
392#95—1222no—
393#95—1222no—
394C24—1221no—
395C24D4×D6111yescharacteristic
396#41—1228no—
397#41—1228no—
398#41—2444no—
399#41—2444no—
400C2×Q8—612no—
401C2×Q8—1221no—
402C2×Q8—1221no—
403C2×Q8—1221no—
404C2×Q8—1221no—
405C2×Q8—1221no—
406C2×D4—612no—
407C2×D4—1224no—
408C2×D4—1221no—
409C2×D4—1221no—
410C2×D4—1221no—
411C2×D4—1221no—
412C2×D4—1221no—
413C2×D4—2442no—
414C2×D4—2442no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—612no—
424C22×C4—612no—
425C22×C4—1221no—
426C22×C4—1221no—
427C22×C4—1221no—
428C22×C4—1224no—
429C22×C4—2442no—
430C22×C4—2442no—
431C22×C4#1175822212yes—
432C22×C4#1175822212yes—
433C22×C4#1175822212yes—
434C22×C4D4×D6112yes—
435C22×C4D4×D6112yes—
436C22×C4D4×D6111yescharacteristic
437C22×C4Q8×D6112yes—
438C22×C4C23×D6111yescharacteristic
439M16—1228no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—44no—
447C2×C8—614no—
448C2×C8—1222no—
449C2×C8—1222no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—612no—
457C4⋊C4—612no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—1222no—
461C4⋊C4—1222no—
462C4⋊C4—4882no—
463C4⋊C4—4882no—
464C22⋊C4—1224no—
465C22⋊C4—1224no—
466C22⋊C4—4882no—
467C22⋊C4—4882no—
468C42—1222no—
469C42—1222no—
470C42—1224no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—1222no—
476C23—2441no—
477C23—2441no—
478C23—2441no—
479C23—2441no—
480C23—2441no—
481C23—2441no—
482C23#1451975012yes—
483C23#1451975012yes—
484C23#1451975512yes—
485C23C2×D4×D6111yescharacteristic
486Q8—1222no—
487Q8—1222no—
488Q8—2441no—
489Q8—2441no—
490D4—1222no—
491D4—1222no—
492D4—2448no—
493D4—2441no—
494D4—2441no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—84no—
502C2×C4—84no—
503C2×C4—1222no—
504C2×C4—1222no—
505C2×C4—1222no—
506C2×C4—1222no—
507C2×C4—1222no—
508C2×C4—1222no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—1222no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4—2448no—
519C2×C4#1451975012yes—
520C2×C4#1451975012yes—
521C2×C4#1451975014yes—
522C2×C4#1451975512yes—
523C2×C4#1451975514yes—
524C2×C4#1451975514yes—
525C2×C4#1451977114yes—
526C2×C4C2×D4×D6111yescharacteristic
527C2×C4C2×D4×D6111yescharacteristic
528C2×C4C2×Q8×D6111yescharacteristic
529C2×C4C2×#1175822211yescharacteristic
530C2×C4C2×#1175822211yescharacteristic
531C2×C4C2×#1175822211yescharacteristic
532C8—41no—
533C8—41no—
534C8—1224no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—4881no—
542C22—4881no—
543C22—4881no—
544C22—4881no—
545C22—2442no—
546C22—2442no—
547C22—2442no—
548C22#1683968614yes—
549C22C2×#1451975011yescharacteristic
550C22C2×#1451975011yescharacteristic
551C22C2×#1451975511yescharacteristic
552C22#1684056311yescharacteristic
553C22#1684056811yescharacteristic
554C22#1684056811yescharacteristic
555C22#1684057511yescharacteristic
556C4—82no—
557C4—82no—
558C4—2442no—
559C4—2442no—
560C4—2442no—
561C4—2442no—
562C4—2442no—
563C4#1683968614yes—
564C4C2×#1451975011yescharacteristic
565C4C2×#1451975511yescharacteristic
566C4C2×#1451975511yescharacteristic
567C4C2×#1451977111yescharacteristic
568C2—24no—
569C2—4882no—
570C2C2×#1683968611yescharacteristic
571C2#2145536311yescharacteristic
572C2#2145537411yescharacteristic
573C2#2145549711yescharacteristic
574C2#2145550811yescharacteristic
575C2#2145567511yescharacteristic
576C2#2145568611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#340C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenterC2×#340D61
hypocenterC61#23616
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
1. C2×#16839802C2semidirect
2. C2×#16839788C2semidirect
5. #21448443C2semidirect
7. #21447215C2semidirect
8. #21447208C2semidirect
9. #21447201C2semidirect
10. #21447194C2semidirect
11. C2×C61×#340C2semidirect
290. C2×#340D61semidirect
333. C61#23616semidirect

Maximal subgroups

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

subgroupindex
C2×#168398022
C2×#168397882
C2×#168397612
C2×#168397592
#214484432
#214484122
#214472152
#214472082
#214472012
#214471942
C2×C61×#3402
#2361661