Finite Group Databasebeta

← #30603436

Subgroups — #30603436

Order 15616.

958 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840708C211yesmaximal, characteristic
2C2×#16840635C212yesmaximal
3C2×#16840635C212yesmaximal
4#21461480C211yesmaximal, characteristic
5#389×D61C211yesmaximal, characteristic
6C22×D61⋊C8C2212yes—
7C22×C4⋊#6923—21no—
8C22×C4⋊#6923—21no—
9C22×C4⋊#6923C2212yes—
10#16840708—24no—
11#16840635—24no—
12#16840635—24no—
13#16840628—21no—
14#16840628—21no—
15C2×C4⋊C4×D61—21no—
16C2×C4⋊C4×D61C2211yescharacteristic
17C2×C4⋊C4×D61C2211yescharacteristic
18C2×C42×D61C2211yescharacteristic
19#16839661C411yescharacteristic, complemented
20#16839660C411yescharacteristic, complemented
21#16839659—21no—
22#16839658C411yescharacteristic, complemented
23#16839657—21no—
24#95×D61—21no—
25C61×#389C411yescharacteristic, complemented
26C23×#6923—22no—
27C2×D61⋊C8—22no—
28C2×D61⋊C8—22no—
29C2×D61⋊C8—22no—
30C2×D61⋊C8—22no—
31C2×D61⋊C8—22no—
32C2×D61⋊C8Q814yes—
33C22×#92420—22no—
34C2×C4⋊#6923—22no—
35C2×C4⋊#6923—22no—
36C2×C4⋊#6923—22no—
37C2×C4⋊#6923—22no—
38C2×C4⋊#6923—24no—
39C2×C4⋊#6923—24no—
40C2×C4⋊#6923—22no—
41C2×C4⋊#6923—42no—
42C2×C4⋊#6923—42no—
43C2×C4⋊#6923—42no—
44C2×C4⋊#6923—42no—
45C2×C4⋊#6923D414yes—
46C22×C4×D61—21no—
47C22×C4×D61—21no—
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61C2×C411yescharacteristic
53C2×C4⋊Dic61C2×C411yescharacteristic
54C2×#11758196—21no—
55C2×#11758196—21no—
56C2×#11758196—21no—
57C2×#11758196—21no—
58C4⋊C4×D61—21no—
59C4⋊C4×D61—21no—
60C4⋊C4×D61—22no—
61C4⋊C4×D61—22no—
62C4⋊C4×D61—42no—
63C4⋊C4×D61—42no—
64C4⋊C4×D61D412yes—
65C4⋊C4×D61Q812yes—
66C2×C4×Dic61C2×C411yescharacteristic
67C2×C4×Dic61D411yescharacteristic
68C2×C4×Dic61D411yescharacteristic
69C42×D61—22no—
70C42×D61—22no—
71#14519684—21no—
72#14519684—21no—
73#14519684—21no—
74C2×C4⋊C4×C61—21no—
75C2×C4⋊C4×C61C2×C411yescharacteristic
76C2×C4⋊C4×C61C2×C411yescharacteristic
77C2×C42×C61C2×C411yescharacteristic
78#95×C61—21no—
79C23×D61C2×D411yescharacteristic
80C22×#6923—24no—
81C22×#6923—42no—
82C22×#6923—42no—
83C22×#6923—42no—
84C22×#6923—42no—
85C22×#6923—42no—
86C22×#6923—42no—
87C22×C4×C61—21no—
88C22×C4×C61—21no—
89C22×C4×C61C22⋊C411yescharacteristic
90C22×C4×C61C22⋊C411yescharacteristic
91C22×C4×C61C22×C411yescharacteristic
92C22×Dic61—21no—
93C22×Dic61—21no—
94C22×Dic61C22⋊C411yescharacteristic
95C22×Dic61C22⋊C411yescharacteristic
96C22×Dic61C2×D411yescharacteristic
97C2×C4×D61—21no—
98C2×C4×D61—21no—
99C2×C4×D61—21no—
100C2×C4×D61—21no—
101C2×C4×D61—21no—
102C2×C4×D61—21no—
103C2×C4×D61—21no—
104C2×C4×D61—21no—
105C2×C4×D61—22no—
106C2×C4×D61—22no—
107C2×C4×D61—41no—
108C2×C4×D61—41no—
109C2×C4×D61—41no—
110C2×C4×D61—41no—
111C2×C4×D61—41no—
112C2×C4×D61—41no—
113C2×C4×D61—41no—
114C2×C4×D61—41no—
115C2×C4×D61—41no—
116C2×C4×D61—41no—
117C2×C4×D61D812yes—
118C2×C4×D61SD1612yes—
119C2×C4×D61SD1612yes—
120C2×C4×D61Q1612yes—
121C2×C4×D61C2×D411yescharacteristic
122C2×C4×D61C2×D412yes—
123C2×C4×D61C2×Q812yes—
124C2×C4×D61C2×Q811yescharacteristic
125C2×C4×D61#4112yes—
126C2×C4×D61#4111yescharacteristic
127C2×C4×D61#4112yes—
128C2×C4×D61#4111yescharacteristic
129C2×#92420—24no—
130C2×#92420—42no—
131C2×#92420—42no—
132C4⋊#6923—24no—
133C4⋊#6923—24no—
134C4⋊#6923—48no—
135C4⋊#6923—48no—
136C4⋊#6923—42no—
137C4⋊#6923—42no—
138D61⋊C8—24no—
139D61⋊C8—24no—
140D61⋊C8—42no—
141D61⋊C8—42no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—42no—
145C4⋊C4×C61C4⋊C412yescomplemented
146C42×C61—22no—
147C4⋊Dic61—21no—
148C4⋊Dic61—22no—
149C4⋊Dic61—42no—
150C4⋊Dic61C4⋊C412yescomplemented
151#11758196—22no—
152#11758196—22no—
153#11758196—42no—
154#11758196—42no—
155#11758196—42no—
156C4×Dic61—22no—
157C4×Dic61—22no—
158C4×Dic61—22no—
159C23×C61C2×C22⋊C411yescharacteristic
160C22×D61#12011yescharacteristic
161C22×D61#12111yescharacteristic
162C22×D61C22⋊Q811yescharacteristic
163C22×D61#12311yescharacteristic
164C22×D61#12411yescharacteristic
165C22×D61C4⋊Q811yescharacteristic
166C22×D61C2×D811yescharacteristic
167C22×D61C2×SD1611yescharacteristic
168C22×D61C2×SD1611yescharacteristic
169C22×D61C2×Q1611yescharacteristic
170C22×D61AGL(1,ℤ/8)11yescharacteristic
171C22×D61AGL(1,ℤ/8)11yescharacteristic
172C22×D61#13711yescharacteristic
173C22×D61#13711yescharacteristic
174C2×#6923—44no—
175C2×#6923—44no—
176C2×#6923—44no—
177C2×#6923—82no—
178C2×#6923—82no—
179C2×#6923—82no—
180C2×#6923—82no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—22no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61#10212yes—
192C2×C4×C61#10312yes—
193C2×C4×C61C2×C4⋊C411yescharacteristic
194C2×C4×C61#11711yescharacteristic
195C2×C4×C61C4×D412yes—
196C2×C4×C61C4×Q812yes—
197C2×Dic61—21no—
198C2×Dic61—21no—
199C2×Dic61—21no—
200C2×Dic61—21no—
201C2×Dic61—22no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61#10212yes—
208C2×Dic61#10312yes—
209C2×Dic61#12111yescharacteristic
210C2×Dic61#12112yes—
211C2×Dic61C22⋊Q812yes—
212C2×Dic61C22⋊Q811yescharacteristic
213C4×D61—22no—
214C4×D61—22no—
215C4×D61—22no—
216C4×D61—22no—
217C4×D61—22no—
218C4×D61—22no—
219C4×D61—22no—
220C4×D61—22no—
221C4×D61—42no—
222C4×D61—42no—
223C4×D61—42no—
224C4×D61—42no—
225C4×D61—81no—
226C4×D61—81no—
227C4×D61—81no—
228C4×D61—81no—
229C4×D61#12112yes—
230C4×D61C22⋊Q812yes—
231C4×D61C22⋊Q812yes—
232C4×D61#12312yes—
233C4×D61#12512yes—
234C4×D61#12512yes—
235C4×D61#12612yes—
236C4×D61C4⋊Q812yes—
237#92420—44no—
238#11207—611nomaximal, Sylow
239C22×C61#38011yescharacteristic
240C22×C61#38611yescharacteristic
241C22×C61#39111yescharacteristic
242C22×C61C2×#10211yescharacteristic
243C22×C61C2×#10311yescharacteristic
244C22×C61#41711yescharacteristic
245C22×C61#41711yescharacteristic
246C2×D61#39411yescharacteristic
247C2×D61#39511yescharacteristic
248C2×D61#39811yescharacteristic
249C2×D61#40011yescharacteristic
250C2×D61#44711yescharacteristic
251C2×D61#44811yescharacteristic
252C2×D61#45011yescharacteristic
253C2×D61#45111yescharacteristic
254C2×D61#45911yescharacteristic
255C2×D61#46011yescharacteristic
256C2×D61#46111yescharacteristic
257C2×D61#46211yescharacteristic
258C2×D61#47411yescharacteristic
259C2×D61#47511yescharacteristic
260C2×D61#47611yescharacteristic
261C2×D61#47711yescharacteristic
262C2×D61#48011yescharacteristic
263C2×D61#48111yescharacteristic
264C2×D61#48311yescharacteristic
265C2×D61#48411yescharacteristic
266C2×D61#48611yescharacteristic
267C2×D61#48711yescharacteristic
268C2×D61#49011yescharacteristic
269C2×D61#49111yescharacteristic
270C2×D61#49811yescharacteristic
271C2×D61#50011yescharacteristic
272C2×D61#50111yescharacteristic
273C2×D61#50111yescharacteristic
274#6923—84no—
275C4×C61—22no—
276C4×C61—22no—
277C4×C61—42no—
278C4×C61—81no—
279C4×C61#38712yes—
280C4×C61#38912yes—
281Dic61—22no—
282Dic61—22no—
283Dic61—42no—
284Dic61—81no—
285Dic61#39612yes—
286Dic61#39812yes—
287C2×#389—611no—
288C2×#384—611no—
289C2×#340—612no—
290C2×#340—612no—
291C2×#336—611no—
292C2×C61#144111yescharacteristic
293C2×C61#180211yescharacteristic
294C2×C61#180311yescharacteristic
295C2×C61#187211yescharacteristic
296C2×C61#187411yescharacteristic
297C2×C61#199011yescharacteristic
298C2×C61#199211yescharacteristic
299D61#204111yescharacteristic
300D61#204311yescharacteristic
301D61#205711yescharacteristic
302D61#205811yescharacteristic
303D61#208311yescharacteristic
304D61#208411yescharacteristic
305D61#209911yescharacteristic
306D61#210011yescharacteristic
307C23×C8—612no—
308C22×C4⋊C4—611no—
309C22×C4⋊C4—612no—
310C22×C4⋊C4—611no—
311C22×C4⋊C4—1221no—
312C22×C4⋊C4—1221no—
313C22×C4⋊C4—1221no—
314C22×C42—611no—
315#389—611no—
316#389—611no—
317#389—611no—
318#389—1221no—
319#389—1221no—
320#389#692311yescharacteristic, complemented
321#384—1224no—
322C2×#95—1221no—
323#340—1224no—
324#340—1224no—
325#336—1221no—
326#336—1221no—
327C61#1120711yescharacteristic, complemented, Sylow
328C23×C4—611no—
329C23×C4—611no—
330C23×C4—611no—
331C23×C4—1221no—
332C23×C4—1221no—
333C23×C4—1222no—
334C22×C8—614no—
335C22×C8—1222no—
336C22×C8—1222no—
337C22×C8—1222no—
338C22×C8—1222no—
339C22×C8—1222no—
340C22×C8—1222no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—611no—
343C2×C4⋊C4—612no—
344C2×C4⋊C4—612no—
345C2×C4⋊C4—614no—
346C2×C4⋊C4—611no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4—1222no—
350C2×C4⋊C4—1222no—
351C2×C4⋊C4—1221no—
352C2×C4⋊C4—1222no—
353C2×C4⋊C4—1222no—
354C2×C4⋊C4—1222no—
355C2×C4⋊C4—1222no—
356C2×C4⋊C4—1221no—
357C2×C4⋊C4—1221no—
358C2×C4⋊C4—1221no—
359C2×C4⋊C4—1222no—
360C2×C4⋊C4—1224no—
361C2×C4⋊C4—1224no—
362C2×C4⋊C4—1221no—
363C2×C4⋊C4—2442no—
364C2×C4⋊C4—2442no—
365C2×C4⋊C4—2442no—
366C2×C4⋊C4—2442no—
367C2×C4⋊C4—2442no—
368C2×C4⋊C4—2442no—
369C2×C4⋊C4C2×#692311yescharacteristic
370C2×C4⋊C4C2×#692311yescharacteristic
371C2×C42—611no—
372C2×C42—611no—
373C2×C42—611no—
374C2×C42—1222no—
375C2×C42—1222no—
376C2×C42C2×#692311yescharacteristic
377#95—21no—
378#95—1221no—
379#95—1221no—
380#95—1221no—
381C24—611no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—611no—
385C22×C4—611no—
386C22×C4—611no—
387C22×C4—611no—
388C22×C4—611no—
389C22×C4—612no—
390C22×C4—612no—
391C22×C4—612no—
392C22×C4—612no—
393C22×C4—611no—
394C22×C4—611no—
395C22×C4—612no—
396C22×C4—612no—
397C22×C4—612no—
398C22×C4—612no—
399C22×C4—1221no—
400C22×C4—1221no—
401C22×C4—1221no—
402C22×C4—1221no—
403C22×C4—1221no—
404C22×C4—1221no—
405C22×C4—1221no—
406C22×C4—1221no—
407C22×C4—1222no—
408C22×C4—1222no—
409C22×C4—1221no—
410C22×C4—1224no—
411C22×C4—1221no—
412C22×C4—2441no—
413C22×C4—2441no—
414C22×C4—2441no—
415C22×C4—2441no—
416C22×C4—2441no—
417C22×C4—2441no—
418C22×C4—2441no—
419C22×C4—2441no—
420C22×C4—2441no—
421C22×C4—2441no—
422C22×C4—2442no—
423C22×C4—2442no—
424C22×C4—2442no—
425C22×C4—2442no—
426C22×C4—2442no—
427C22×C4—2442no—
428C22×C4C22⋊#692311yescharacteristic
429C22×C4C22⋊#692311yescharacteristic
430C22×C4C22×#692311yescharacteristic
431C2×C8—1224no—
432C2×C8—1224no—
433C2×C8—1224no—
434C2×C8—2442no—
435C2×C8—2442no—
436C2×C8—2442no—
437C2×C8—2442no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—42no—
441C4⋊C4—612no—
442C4⋊C4—1222no—
443C4⋊C4—1222no—
444C4⋊C4—1224no—
445C4⋊C4—1224no—
446C4⋊C4—1221no—
447C4⋊C4—1222no—
448C4⋊C4—2442no—
449C4⋊C4—2442no—
450C4⋊C4—2442no—
451C4⋊C4—2442no—
452C4⋊C4—2442no—
453C4⋊C4—2448no—
454C4⋊C4—2448no—
455C4⋊C4—2442no—
456C4⋊C4C4⋊#692312yescomplemented
457C42—22no—
458C42—1222no—
459C42—1222no—
460C42—1222no—
461C23—611no—
462C23—611no—
463C23—611no—
464C23—611no—
465C23—611no—
466C23—611no—
467C23—611no—
468C23—611no—
469C23—611no—
470C23—611no—
471C23—611no—
472C23—611no—
473C23—611no—
474C23—611no—
475C23C2×C22⋊#692311yescharacteristic
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—611no—
487C2×C4—611no—
488C2×C4—612no—
489C2×C4—612no—
490C2×C4—612no—
491C2×C4—612no—
492C2×C4—612no—
493C2×C4—612no—
494C2×C4—612no—
495C2×C4—612no—
496C2×C4—612no—
497C2×C4—612no—
498C2×C4—612no—
499C2×C4—612no—
500C2×C4—1222no—
501C2×C4—1222no—
502C2×C4—1222no—
503C2×C4—1222no—
504C2×C4—1222no—
505C2×C4—1222no—
506C2×C4—1222no—
507C2×C4—1222no—
508C2×C4—1221no—
509C2×C4—1221no—
510C2×C4—1221no—
511C2×C4—1221no—
512C2×C4—1222no—
513C2×C4—4881no—
514C2×C4—4881no—
515C2×C4—4881no—
516C2×C4—4881no—
517C2×C4—4882no—
518C2×C4—4882no—
519C2×C4—4882no—
520C2×C4—4882no—
521C2×C4—2442no—
522C2×C4—2442no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—2444no—
526C2×C4—2444no—
527C2×C4—2444no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4#1451983912yes—
534C2×C4#1451984012yes—
535C2×C4C2×C4⋊#692311yescharacteristic
536C2×C4#1451985611yescharacteristic
537C2×C4D4×#692312yes—
538C2×C4Q8×#692312yes—
539C8—2444no—
540C22—611no—
541C22—611no—
542C22—611no—
543C22—611no—
544C22—611no—
545C22—611no—
546C22—611no—
547C22—611no—
548C22—611no—
549C22—611no—
550C22—611no—
551C22—611no—
552C22—611no—
553C22—611no—
554C22—611no—
555C22—611no—
556C22—611no—
557C22—611no—
558C22—611no—
559C22—611no—
560C22—611no—
561C22—611no—
562C22—611no—
563C22—611no—
564C22—611no—
565C22—611no—
566C22—611no—
567C22—611no—
568C22#1684070411yescharacteristic
569C22#1684071011yescharacteristic
570C22(C2×#6923)⋊Q811yescharacteristic
571C22C2×#1451983911yescharacteristic
572C22C2×#1451984011yescharacteristic
573C22#1684075311yescharacteristic
574C22#1684075311yescharacteristic
575C4—22no—
576C4—22no—
577C4—42no—
578C4—81no—
579C4—612no—
580C4—612no—
581C4—1222no—
582C4—1222no—
583C4—4884no—
584C4—4881no—
585C4—2442no—
586C4#1684071112yes—
587C4#1684071312yes—
588C2—611no—
589C2—611no—
590C2—611no—
591C2—611no—
592C2—611no—
593C2—611no—
594C2—611no—
595C2—611no—
596C2#2146100411yescharacteristic
597C2#2146168711yescharacteristic
598C2#2146168811yescharacteristic
599C2#2146180411yescharacteristic
600C2#2146180611yescharacteristic
601C2#2146200011yescharacteristic
602C2#2146200211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#389C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#389#6923
hypocenterC61#11207
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
19. #16839661C4semidirect
20. #16839660C4semidirect
22. #16839658C4semidirect
25. C61×#389C4semidirect
145. C4⋊C4×C61C4⋊C4semidirect
150. C4⋊Dic61C4⋊C4semidirect
320. #389#6923semidirect
327. C61#11207semidirect
456. C4⋊C4C4⋊#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168407082
C2×#168406352
#214614802
#389×D612
#1120761