Finite Group Databasebeta

← #30603105

Subgroups — #30603105

Order 15616.

1194 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519837C212yesmaximal
2C2×C4×#11758217C211yesmaximal, characteristic, complemented
3C2×#16840642C212yesmaximal, complemented
4#21461482C211yesmaximal, characteristic, complemented
5C2×#16839662C211yesmaximal, characteristic
6C22×#11758217C2212yes—
7C22×#11758217C2212yes—
8C2×#14519837—24no—
9C2×#14519837—24no—
10C2×#14519837C418yescomplemented
11C4×#11758217—24no—
12C2×#14519832—21no—
13C2×C4×#92420—21no—
14#16840642—24no—
15#16840628—21no—
16#16840628—21no—
17C23×C22⋊D61C2211yescharacteristic
18C22×#11758196C2211yescharacteristic
19C22×C22⋊Dic61C411yescharacteristic
20C22×#11758198C411yescharacteristic
21C22×#11758198C411yescharacteristic
22C22×C4×Dic61C2211yescharacteristic
23#16839662—21no—
24#16839662—21no—
25#16839662—21no—
26#16839662—21no—
27C22×C22⋊C4×C61C411yescharacteristic
28C2×#11758217—22no—
29C2×#11758217—22no—
30C2×#11758217—22no—
31C2×#11758217—22no—
32C2×#11758217—22no—
33C2×#11758217—22no—
34C2×#11758217—22no—
35C2×#11758217—22no—
36C2×#11758217C2×C414yes—
37C2×#11758217C2×C414yes—
38C2×#11758217D414yes—
39C2×#11758217D414yes—
40C22×#92420—22no—
41C22×#92420—22no—
42#14519837—216no—
43#14519832—24no—
44C4×#92420—24no—
45C24×D61C2×C411yescharacteristic
46C22×C22⋊D61—22no—
47C22×C22⋊D61—22no—
48C22×C22⋊D61—21no—
49C22×C22⋊D61—21no—
50C22×C22⋊D61—22no—
51C22×C22⋊D61—22no—
52C22×C22⋊D61—22no—
53C22×C22⋊D61C2×C414yes—
54C23×Dic61C2×C411yescharacteristic
55C23×Dic61C2×C411yescharacteristic
56C23×Dic61C2311yescharacteristic
57C2×#11758196—22no—
58C2×#11758196—21no—
59C2×#11758196—21no—
60C2×#11758196—22no—
61C2×C22⋊Dic61—21no—
62C2×C22⋊Dic61—21no—
63C2×C22⋊Dic61—42no—
64C2×#11758198—21no—
65C2×#11758198—21no—
66C2×#11758198—21no—
67C2×#11758198—21no—
68C2×#11758198—42no—
69C2×#11758198—42no—
70C2×C4×Dic61—22no—
71C2×C4×Dic61—22no—
72C2×C4×Dic61D411yescharacteristic
73C2×C4×Dic61D411yescharacteristic
74C2×C4×Dic61D411yescharacteristic
75C2×C4×Dic61D411yescharacteristic
76C25×C61C2×C411yescharacteristic
77C23×C4×C61C2×C411yescharacteristic
78C23×C4×C61C2×C411yescharacteristic
79C2×C22⋊C4×C61—21no—
80C2×C22⋊C4×C61—21no—
81C2×C22⋊C4×C61—42no—
82C24×C61—21no—
83C24×C61—21no—
84C24×C61—22no—
85C24×C61—22no—
86C24×C61—41no—
87C24×C61—41no—
88C24×C61—41no—
89C24×C61—41no—
90C24×C61C4212yes—
91C24×C61C22×C411yescharacteristic
92C23×D61—21no—
93C23×D61—21no—
94C23×D61—22no—
95C23×D61—22no—
96C23×D61—41no—
97C23×D61—41no—
98C23×D61—41no—
99C23×D61—41no—
100C23×D61C4212yes—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—21no—
104C22×C4×C61—22no—
105C22×C4×C61—41no—
106C22×C4×C61—41no—
107C22×C4×C61—41no—
108C22×C4×C61C22⋊C411yescharacteristic
109C22×C4×C61C22⋊C411yescharacteristic
110C2×C22⋊D61—24no—
111C2×C22⋊D61—24no—
112C2×C22⋊D61—24no—
113C2×C22⋊D61—24no—
114C2×C22⋊D61—24no—
115C2×C22⋊D61—24no—
116C2×C22⋊D61—41no—
117C2×C22⋊D61—42no—
118C2×C22⋊D61—42no—
119C2×C22⋊D61—42no—
120C2×C22⋊D61—42no—
121C2×C22⋊D61—41no—
122C2×C22⋊D61—41no—
123C2×C22⋊D61—42no—
124C2×C22⋊D61—42no—
125C2×C22⋊D61—41no—
126C22×Dic61—22no—
127C22×Dic61—22no—
128C22×Dic61—21no—
129C22×Dic61—22no—
130C22×Dic61—41no—
131C22×Dic61—41no—
132C22×Dic61—41no—
133C22×Dic61C22⋊C411yescharacteristic
134C22×Dic61C22⋊C411yescharacteristic
135C22×Dic61C22×C412yes—
136C22×Dic61C2×D412yes—
137C22×Dic61C2×D412yes—
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61#4112yes—
141C22×Dic61#4112yes—
142C22×Dic61#4112yes—
143#11758217—24no—
144#11758217—24no—
145#11758217—24no—
146#11758217—24no—
147#11758217—24no—
148#11758217—42no—
149#11758217—42no—
150#11758217C22⋊C418yescomplemented
151C2×#92420—24no—
152C2×#92420—24no—
153C2×#92420—24no—
154C2×#92420—24no—
155C2×#92420—42no—
156C2×#92420—42no—
157C22⋊C4×C61—42no—
158C22⋊C4×C61—42no—
159C22⋊Dic61—42no—
160C22⋊Dic61—42no—
161#11758198—42no—
162#11758198—42no—
163#11758198—42no—
164#11758198—42no—
165#11758196—42no—
166#11758196—41no—
167#11758196—41no—
168C4×Dic61—21no—
169C4×Dic61—21no—
170C4×Dic61—22no—
171C4×Dic61—22no—
172C4×Dic61—21no—
173C4×Dic61—21no—
174C23×C61—21no—
175C23×C61—21no—
176C23×C61—21no—
177C23×C61—21no—
178C23×C61—22no—
179C23×C61—24no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61—42no—
183C23×C61—42no—
184C23×C61—41no—
185C23×C61—42no—
186C23×C61—41no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61—82no—
192C23×C61—82no—
193C23×C61—82no—
194C23×C61—82no—
195C23×C61C2×C4211yescharacteristic
196C23×C61C2×C22⋊C411yescharacteristic
197C23×C61#11711yescharacteristic
198C23×C61C4×D412yes—
199C23×C61C4×D412yes—
200C22×D61—22no—
201C22×D61—24no—
202C22×D61—42no—
203C22×D61—42no—
204C22×D61—41no—
205C22×D61—42no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—41no—
211C22×D61—42no—
212C22×D61—42no—
213C22×D61—82no—
214C22×D61—82no—
215C22×D61—82no—
216C22×D61—82no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—44no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—82no—
227C2×C4×C61#10012yescomplemented
228C22⋊D61—44no—
229C22⋊D61—44no—
230C22⋊D61—44no—
231C22⋊D61—44no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—44no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—82no—
242C2×Dic61#10012yescomplemented
243C2×Dic61C2×C22⋊C412yes—
244C2×Dic61#11712yes—
245C2×Dic61C4×D412yes—
246C2×Dic61C4×D412yes—
247C2×Dic61C4×D412yes—
248C2×Dic61C4×D412yes—
249C2×Dic61#12011yescharacteristic
250C2×Dic61#12112yes—
251C2×Dic61#12112yes—
252C2×Dic61#12112yes—
253C2×Dic61#12312yes—
254C2×Dic61#12311yescharacteristic
255C2×Dic61#12411yescharacteristic
256C2×Dic61#12412yes—
257C2×Dic61#12612yes—
258C2×Dic61#12711yescharacteristic
259#92420—28no—
260#92420—44no—
261#10962—611nomaximal, Sylow
262C22×C61—22no—
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—22no—
266C22×C61—21no—
267C22×C61—21no—
268C22×C61—21no—
269C22×C61—21no—
270C22×C61—42no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—84no—
274C22×C61—84no—
275C22×C61—81no—
276C22×C61—81no—
277C22×C61—81no—
278C22×C61—81no—
279C22×C61C4×C22⋊C411yescharacteristic
280C22×C61C4×C22⋊C412yes—
281C22×C61#38611yescharacteristic
282C22×C61#38812yes—
283C22×C61#39011yescharacteristic
284C22×C61C2×#10011yescharacteristic
285C22×C61C2×#10011yescharacteristic
286C22×C61#41311yescharacteristic
287C22×C61#41311yescharacteristic
288C2×D61—42no—
289C2×D61—42no—
290C2×D61—42no—
291C2×D61—84no—
292C2×D61—81no—
293C2×D61—81no—
294C2×D61—81no—
295C2×D61—81no—
296C2×D61—84no—
297C4×C61—22no—
298C4×C61—41no—
299C4×C61—81no—
300Dic61—22no—
301Dic61—41no—
302Dic61—81no—
303Dic61#38612yes—
304Dic61#38812yes—
305Dic61#38812yes—
306Dic61#39012yes—
307C22×#100—612no—
308C2×C4×M16—611no—
309C2×#390—611no—
310C2×#343—612no—
311#1728—611no—
312C2×C61—22no—
313C2×C61—22no—
314C2×C61—82no—
315C2×C61#143711yescharacteristic
316C2×C61C4×#10011yescharacteristic
317C2×C61C4×#10011yescharacteristic
318C2×C61#190011yescharacteristic
319C2×C61#190011yescharacteristic
320C2×C61#196511yescharacteristic
321C2×C61#196711yescharacteristic
322D61—82no—
323C23×D4—611no—
324C22×M16—612no—
325C22×M16—612no—
326C22×C4⋊C4—611no—
327C22×C22⋊C4—611no—
328C22×C22⋊C4—611no—
329C22×C22⋊C4—611no—
330C22×C22⋊C4#692311yescharacteristic
331C22×C42—611no—
332C2×#100—618no—
333C2×#100—1224no—
334C2×#100—1224no—
335C4×M16—1224no—
336C2×#97—1221no—
337C2×C4×C8—1221no—
338#390—1221no—
339#390—1221no—
340#390—1221no—
341#390—1221no—
342#343—1224no—
343#336—1221no—
344#336—1221no—
345C61#1096211yescharacteristic, complemented, Sylow
346C25—611no—
347C25C2×#692311yescharacteristic
348C22×D4—614no—
349C22×D4—1222no—
350C22×D4—1222no—
351C22×D4—1222no—
352C22×D4—1221no—
353C22×D4—1221no—
354C22×D4—1222no—
355C22×D4—1222no—
356C23×C4—611no—
357C23×C4—611no—
358C23×C4—611no—
359C23×C4C2×#692311yescharacteristic
360C23×C4C2×#692311yescharacteristic
361C2×M16—614no—
362C2×M16—614no—
363C2×M16—614no—
364C2×M16—614no—
365C2×M16—1222no—
366C2×M16—1222no—
367C2×M16—1222no—
368C2×M16—1222no—
369C2×M16—1222no—
370C2×M16—1222no—
371C2×M16—1222no—
372C2×M16—1222no—
373C22×C8—1222no—
374C22×C8—1222no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1222no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—42no—
382C2×C22⋊C4—1221no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1221no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1221no—
388C2×C22⋊C4—2442no—
389C2×C22⋊C4—2442no—
390C2×C22⋊C4—2442no—
391C2×C42—611no—
392C2×C42—611no—
393C2×C42—611no—
394C2×C42—611no—
395C2×C42—1222no—
396C2×C42—1222no—
397#100—12216no—
398#97—1224no—
399C4×C8—1224no—
400C24—21no—
401C24—21no—
402C24—22no—
403C24—22no—
404C24—41no—
405C24—41no—
406C24—41no—
407C24—41no—
408C24—612no—
409C24—1222no—
410C24—1222no—
411C24—1221no—
412C24—1221no—
413C24—2441no—
414C24—2441no—
415C24—2441no—
416C24—2441no—
417C24C4×#692312yes—
418C24C22×#692311yescharacteristic
419C2×D4—1224no—
420C2×D4—1224no—
421C2×D4—1224no—
422C2×D4—1224no—
423C2×D4—1224no—
424C2×D4—1224no—
425C2×D4—2441no—
426C2×D4—2442no—
427C2×D4—2441no—
428C2×D4—2442no—
429C2×D4—2442no—
430C2×D4—2442no—
431C2×D4—2442no—
432C2×D4—2442no—
433C2×D4—2441no—
434C2×D4—2441no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—612no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—612no—
446C22×C4—612no—
447C22×C4—612no—
448C22×C4—612no—
449C22×C4—612no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1221no—
456C22×C4—2441no—
457C22×C4—2441no—
458C22×C4—2441no—
459C22×C4C22⋊#692311yescharacteristic
460C22×C4C22⋊#692311yescharacteristic
461M16—618no—
462M16—1224no—
463M16—1224no—
464M16—1224no—
465M16—1224no—
466M16—1224no—
467M16—2442no—
468M16—2442no—
469C2×C8—1224no—
470C2×C8—1224no—
471C2×C8—1224no—
472C2×C8—1224no—
473C2×C8—2442no—
474C2×C8—2442no—
475C4⋊C4—2442no—
476C4⋊C4—2441no—
477C4⋊C4—2441no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—2442no—
481C22⋊C4—2442no—
482C22⋊C4—2442no—
483C22⋊C4—2442no—
484C22⋊C4—2442no—
485C22⋊C4—2442no—
486C42—1221no—
487C42—1221no—
488C42—1222no—
489C42—1221no—
490C42—1222no—
491C42—1221no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—22no—
497C23—24no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—41no—
503C23—42no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—82no—
510C23—82no—
511C23—82no—
512C23—82no—
513C23—1224no—
514C23—1222no—
515C23—4882no—
516C23—4882no—
517C23—4882no—
518C23—4882no—
519C23—2442no—
520C23—2442no—
521C23—2442no—
522C23—2442no—
523C23—2441no—
524C23—2442no—
525C23—2441no—
526C23—2441no—
527C23—2441no—
528C23—2441no—
529C23—2441no—
530C23C2×C4×#692311yescharacteristic
531C23C2×C22⋊#692311yescharacteristic
532C23#1451985611yescharacteristic
533C23D4×#692312yes—
534C23D4×#692312yes—
535D4—2444no—
536D4—2444no—
537D4—2444no—
538D4—2444no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—44no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—82no—
549C2×C4—611no—
550C2×C4—612no—
551C2×C4—611no—
552C2×C4—612no—
553C2×C4—612no—
554C2×C4—612no—
555C2×C4—612no—
556C2×C4—612no—
557C2×C4—611no—
558C2×C4—611no—
559C2×C4—612no—
560C2×C4—612no—
561C2×C4—612no—
562C2×C4—612no—
563C2×C4—612no—
564C2×C4—612no—
565C2×C4—612no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—4882no—
572C2×C4—2444no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4#1451983712yescomplemented
577C8—1228no—
578C8—2444no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—84no—
591C22—84no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22—4884no—
597C22—4884no—
598C22—4881no—
599C22—4881no—
600C22—4881no—
601C22—4881no—
602C22—2442no—
603C22—2442no—
604C22—2442no—
605C22C4×C22⋊#692311yescharacteristic
606C22C22⋊C4×#692312yes—
607C22#1684071011yescharacteristic
608C22#1684071212yes—
609C22#1684071411yescharacteristic
610C22C2×#1451983711yescharacteristic
611C22D61⋊#10011yescharacteristic
612C22#1684074811yescharacteristic
613C22#1684074911yescharacteristic
614C4—22no—
615C4—41no—
616C4—81no—
617C4—612no—
618C4—612no—
619C4—612no—
620C4—612no—
621C4—1222no—
622C4—4881no—
623C4—2441no—
624C2—22no—
625C2—22no—
626C2—82no—
627C2—4882no—
628C2#2146100011yescharacteristic
629C2C4×#1451983711yescharacteristic
630C2#2146156911yescharacteristic
631C2#2146184411yescharacteristic
632C2#2146184511yescharacteristic
633C2#2146196111yescharacteristic
634C2#2146196311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC23×C61C2×C42
FrattiniC24C22×#6923
FittingC22×C22⋊C4×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC23×C22⋊D61C22
cosocleC23×Dic61C23
hypercenterC22×C22⋊C4#6923
hypocenterC61#10962
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11758217C2semidirect
3. C2×#16840642C2semidirect
4. #21461482C2semidirect
10. C2×#14519837C4semidirect
150. #11758217C22⋊C4semidirect
227. C2×C4×C61#100semidirect
242. C2×Dic61#100semidirect
345. C61#10962semidirect
576. C2×C4#14519837semidirect

Maximal subgroups

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

subgroupindex
C22×#145198372
C2×C4×#117582172
C2×#168406422
#214614822
C2×#168396622
#1096261