Finite Group Databasebeta

← #29708697

Subgroups — #29708697

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×#16839789C211yesmaximal, characteristic, complemented
3C2×#16839763C211yesmaximal, characteristic
4C2×#16839758C211yesmaximal, characteristic
5#21448439C211yesmaximal, characteristic, complemented
6#21447214C212yesmaximal, complemented
7#21447207C212yesmaximal, complemented
8#21447200C212yesmaximal, complemented
9#21447193C212yesmaximal, complemented
10C2×#16839524C211yesmaximal, characteristic, complemented
11C61×#2311C211yesmaximal, characteristic
12C22×#11758194C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14C2×#14519736C2211yescharacteristic
15C22×C4⋊Dic61C2211yescharacteristic
16#16839802—24no—
17C2×#14519710C2212yes—
18C2×#14519708C2212yes—
19#16839789—24no—
20C2×C4⋊#92418C2212yes—
21C2×C4⋊#92418C2212yes—
22#16839763—24no—
23#16839758—24no—
24C2×#14519701C2211yescharacteristic
25C2×#14519701C2212yes—
26C2×#14519701C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519698C2212yes—
29C2×#14519698C2211yescharacteristic
30#16839660C2212yes—
31#16839660C2212yes—
32#16839634—22no—
33#16839631—21no—
34#16839627—21no—
35#16839524C2212yes—
36#16839524C2212yes—
37C22×#41×C61C2211yescharacteristic
38C2×#117×C61C2211yescharacteristic
39C22×C4⋊C4×C61C2211yescharacteristic
40C61×#391C2212yes—
41C61×#390C2212yes—
42C61×#386—22no—
43C61×#380—21no—
44C61×#380—21no—
45C23×Dic61—21no—
46C2×#11758194—22no—
47C2×#11758194—22no—
48C2×#11758194D414yes—
49C22×#92418—22no—
50C22×#92418—22no—
51C22×#92418—22no—
52C22×#92418C2311yescharacteristic
53C22×#92418C2311yescharacteristic
54C22×#92418C2312yes—
55#14519736—22no—
56#14519736D414yes—
57C2×C4⋊Dic61—22no—
58C2×C4⋊Dic61—22no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61C2312yes—
61C2×C4⋊Dic61C2311yescharacteristic
62C2×C4⋊Dic61C2311yescharacteristic
63C2×#11758196—22no—
64C2×#11758196—22no—
65C2×C22⋊Dic61—21no—
66C2×C4×Dic61—21no—
67#14519710—24no—
68#14519708—24no—
69C4⋊#92418—24no—
70C4⋊#92418—24no—
71#14519701—22no—
72#14519701—22no—
73#14519701—24no—
74#14519698—22no—
75#14519698—22no—
76#14519698—24no—
77#14519684—22no—
78#14519684—22no—
79C2×#41×C61—22no—
80C2×#41×C61—22no—
81C2×#41×C61—22no—
82C2×#41×C61—22no—
83C2×#41×C61—22no—
84C2×#41×C61Q814yes—
85C22×Q8×C61C2311yescharacteristic
86C22×D4×C61—21no—
87C22×D4×C61C2311yescharacteristic
88C23×C4×C61—21no—
89C23×C4×C61—21no—
90C23×C4×C61C2311yescharacteristic
91#117×C61—22no—
92#117×C61—22no—
93C2×C4⋊C4×C61—22no—
94C2×C4⋊C4×C61—22no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61D411yescharacteristic
97C2×C4⋊C4×C61D411yescharacteristic
98C2×C4⋊C4×C61C2312yes—
99C2×C22⋊C4×C61—22no—
100C2×C22⋊C4×C61—22no—
101C2×C22⋊C4×C61D412yes—
102C2×C42×C61C2312yes—
103#95×C61—22no—
104#95×C61—22no—
105C24×C61—21no—
106C24×C61C2×D411yescharacteristic
107#41×C61—28no—
108#41×C61—28no—
109#41×C61—44no—
110#41×C61—44no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61C2×Q812yes—
117C2×D4×C61—24no—
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—42no—
124C2×D4×C61—42no—
125C2×D4×C61C2×Q812yes—
126C22×C4×C61—21no—
127C22×C4×C61—24no—
128C22×C4×C61—22no—
129C22×C4×C61—42no—
130C22×C4×C61—42no—
131C22×C4×C61—42no—
132C22×C4×C61—81no—
133C22×C4×C61—81no—
134C22×C4×C61C2×D412yes—
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2×D412yes—
139C22×C4×C61C2×Q812yes—
140C22×C4×C61#4112yes—
141C22×C4×C61#4112yes—
142C22×C4×C61#4112yes—
143C22×C4×C61C2411yescharacteristic
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—22no—
147C22×Dic61—42no—
148C22×Dic61—81no—
149C22×Dic61—81no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—42no—
155C22⋊C4×C61—24no—
156C22⋊C4×C61—82no—
157C22⋊C4×C61—82no—
158C42×C61—22no—
159C42×C61—22no—
160C22⋊Dic61—24no—
161C4⋊Dic61—21no—
162C4⋊Dic61—21no—
163C4⋊Dic61—22no—
164C4⋊Dic61—22no—
165C4⋊Dic61—42no—
166C4⋊Dic61C2×D412yes—
167C4⋊Dic61C2×D412yes—
168#11758196—82no—
169#11758196—82no—
170C4×Dic61—24no—
171#11758194—28no—
172C2×#92418—22no—
173C2×#92418—22no—
174C2×#92418—21no—
175C2×#92418—21no—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418—21no—
179C2×#92418—21no—
180C2×#92418—44no—
181C2×#92418C2×D414yes—
182C23×C61—21no—
183C23×C61—21no—
184C23×C61—21no—
185C23×C61—21no—
186C23×C61—22no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61—41no—
192C23×C61—41no—
193C23×C61#12112yes—
194C23×C61#12112yes—
195C23×C61C22⋊Q812yes—
196C23×C61C22×D411yescharacteristic
197Q8×C61—22no—
198Q8×C61—22no—
199Q8×C61—41no—
200Q8×C61—41no—
201D4×C61—22no—
202D4×C61—22no—
203D4×C61—48no—
204D4×C61—41no—
205D4×C61—41no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—48no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—84no—
220C2×C4×C61#12114yes—
221C2×C4×C61#12112yes—
222C2×C4×C61#12112yes—
223C2×C4×C61C22⋊Q814yes—
224C2×C4×C61C22⋊Q814yes—
225C2×C4×C61C22⋊Q812yes—
226C2×C4×C61#12614yes—
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×Q811yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61—84no—
243#92418—24no—
244#92418—41no—
245#92418—41no—
246#23616—611nomaximal, Sylow
247C22×C61—22no—
248C22×C61—22no—
249C22×C61—22no—
250C22×C61—22no—
251C22×C61—22no—
252C22×C61—22no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—81no—
257C22×C61—81no—
258C22×C61—81no—
259C22×C61—81no—
260C22×C61#39614yes—
261C22×C61C2×#12111yescharacteristic
262C22×C61C2×#12111yescharacteristic
263C22×C61C2×C22⋊Q811yescharacteristic
264C22×C61#57611yescharacteristic
265C22×C61#57711yescharacteristic
266C22×C61#57711yescharacteristic
267C22×C61#57811yescharacteristic
268C4×C61—42no—
269C4×C61—42no—
270C4×C61—42no—
271C4×C61—82no—
272C4×C61#39614yes—
273C4×C61C2×#12111yescharacteristic
274C4×C61C2×C22⋊Q811yescharacteristic
275C4×C61C2×C22⋊Q811yescharacteristic
276C4×C61C2×#12611yescharacteristic
277Dic61—42no—
278Dic61—42no—
279Dic61—82no—
280C2×#428—611no—
281C2×#424—611no—
282C2×#417—611no—
283C2×#416—611no—
284#2317—611no—
285#2311D6111yescharacteristic
286#2068—612no—
287#2067—612no—
288#2066—612no—
289#2065—612no—
290C2×#340—611no—
291C2×C61—24no—
292C2×C61—82no—
293C2×C61C2×#39611yescharacteristic
294C2×C61#304511yescharacteristic
295C2×C61#304611yescharacteristic
296C2×C61#306211yescharacteristic
297C2×C61#306311yescharacteristic
298C2×C61#308311yescharacteristic
299C2×C61#308411yescharacteristic
300C22×#41C2×D6111yescharacteristic
301C22×M16—611no—
302C23×C8—611no—
303C2×#117—611no—
304C2×#117C2×D6111yescharacteristic
305C22×C4⋊C4—611no—
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—612no—
317C2×#103—611no—
318C2×#102—611no—
319C2×#102—612no—
320C2×#102—611no—
321#391C2×D6112yes—
322#390C2×D6112yes—
323#389—612no—
324#389—612no—
325#386—22no—
326#385—1222no—
327#384—1221no—
328#382—1221no—
329#380—21no—
330#380—21no—
331#340—612no—
332#340—612no—
333C61#2361611yescharacteristic, complemented, Sylow
334C2×#41—22no—
335C2×#41—22no—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41—22no—
339C2×#41Dic12214yes—
340C22×Q8C22×D6111yescharacteristic
341C22×D4—21no—
342C22×D4C22×D6111yescharacteristic
343C23×C4—21no—
344C23×C4—21no—
345C23×C4—1221no—
346C23×C4C22×D6111yescharacteristic
347C2×M16—614no—
348C2×M16—1222no—
349C2×M16—1222no—
350C22×C8—612no—
351C22×C8—611no—
352C22×C8—611no—
353C22×C8—1222no—
354C22×C8—1222no—
355C22×C8—1222no—
356#117—22no—
357#117—22no—
358#117—614no—
359#117—1222no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—611no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4—612no—
366C2×C4⋊C4—1222no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1222no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4C22⋊D6111yescharacteristic
372C2×C4⋊C4C22⋊D6111yescharacteristic
373C2×C4⋊C4C22×D6112yes—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—1221no—
377C2×C22⋊C4C22⋊D6112yes—
378C2×C42—1221no—
379C2×C42C22×D6112yes—
380#107—1224no—
381#106—1224no—
382C4⋊C8—1224no—
383C4⋊C8—1224no—
384#103—1222no—
385#103—1222no—
386#103—1224no—
387#102—1222no—
388#102—1222no—
389#102—1224no—
390#95—22no—
391#95—22no—
392#95—1222no—
393#95—1222no—
394C24—21no—
395C24C2×C22⋊D6111yescharacteristic
396#41—28no—
397#41—28no—
398#41—44no—
399#41—44no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8C2×Dic12212yes—
406C2×D4—24no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4C2×Dic12212yes—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—1221no—
424C22×C4—1221no—
425C22×C4—1222no—
426C22×C4—4881no—
427C22×C4—4881no—
428C22×C4—2442no—
429C22×C4C2×Dic12212yes—
430C22×C4#1175822212yes—
431C22×C4D4×D6112yes—
432C22×C4D4×D6112yes—
433C22×C4#1175822412yes—
434C22×C4#1175822412yes—
435C22×C4C2×C22⋊D6112yes—
436C22×C4C2×C22⋊D6112yes—
437C22×C4C2×C22⋊D6111yescharacteristic
438C22×C4C23×D6111yescharacteristic
439M16—1228no—
440C2×C8—614no—
441C2×C8—1222no—
442C2×C8—1222no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—1221no—
446C2×C8—1221no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8—2444no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—42no—
455C4⋊C4—612no—
456C4⋊C4—612no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C4⋊C4—1222no—
460C4⋊C4—1222no—
461C4⋊C4—4882no—
462C4⋊C4—4882no—
463C4⋊C4—2442no—
464C22⋊C4—24no—
465C22⋊C4—82no—
466C22⋊C4—82no—
467C22⋊C4—1224no—
468C42—22no—
469C42—22no—
470C42—1224no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—22no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23#1451974812yes—
483C23#1451974912yes—
484C23#1451975612yes—
485C23C22×C22⋊D6111yescharacteristic
486Q8—22no—
487Q8—22no—
488Q8—41no—
489Q8—41no—
490D4—22no—
491D4—22no—
492D4—48no—
493D4—41no—
494D4—41no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—48no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—84no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—4884no—
512C2×C4—2442no—
513C2×C4—2442no—
514C2×C4—2442no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4#1451974812yes—
520C2×C4#1451974912yes—
521C2×C4#1451975014yes—
522C2×C4#1451975414yes—
523C2×C4#1451975414yes—
524C2×C4#1451975612yes—
525C2×C4#1451977014yes—
526C2×C4C2×D4×D6111yescharacteristic
527C2×C4C2×D4×D6111yescharacteristic
528C2×C4C22×Dic12211yescharacteristic
529C2×C4C2×#1175822411yescharacteristic
530C2×C4C2×#1175822411yescharacteristic
531C2×C4C2×#1175822211yescharacteristic
532C8—1224no—
533C8—2441no—
534C8—2441no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22#1683968514yes—
549C22C2×#1451974811yescharacteristic
550C22C2×#1451974911yescharacteristic
551C22C2×#1451975611yescharacteristic
552C22#1684056511yescharacteristic
553C22#1684057011yescharacteristic
554C22#1684057311yescharacteristic
555C22#1684057711yescharacteristic
556C4—42no—
557C4—42no—
558C4—42no—
559C4—82no—
560C4—4882no—
561C4—2442no—
562C4—2442no—
563C4#1683968514yes—
564C4C2×#1451975011yescharacteristic
565C4C2×#1451975411yescharacteristic
566C4C2×#1451975411yescharacteristic
567C4C2×#1451977011yescharacteristic
568C2—24no—
569C2—82no—
570C2C2×#1683968511yescharacteristic
571C2#2145535411yescharacteristic
572C2#2145536511yescharacteristic
573C2#2145550111yescharacteristic
574C2#2145551211yescharacteristic
575C2#2145567011yescharacteristic
576C2#2145568111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2311C2
socleC23×C61C22×D4
prime radicalC22×#41×C61C22
hypercenter#2311D61
hypocenterC61#23616
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2311

Complements

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

NH ≅ G/Ntype
1. C2×#16839802C2semidirect
2. C2×#16839789C2semidirect
5. #21448439C2semidirect
6. #21447214C2semidirect
7. #21447207C2semidirect
8. #21447200C2semidirect
9. #21447193C2semidirect
10. C2×#16839524C2semidirect
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×#168397892
C2×#168397632
C2×#168397582
#214484392
#214472142
#214472072
#214472002
#214471932
C2×#168395242
C61×#23112
#2361661