Finite Group Databasebeta

← #30600182

Subgroups — #30600182

Order 15616.

1136 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D61⋊(C22⋊C8)C211yesmaximal, characteristic, complemented
2#21461642C211yesmaximal, characteristic, complemented
3C2×#16840605C212yesmaximal, complemented
4#21460657C212yesmaximal, complemented
5C2×#16839638C211yesmaximal, characteristic
6C22×D61⋊C8—21no—
7C2×#14519834—21no—
8C2×#14519834—21no—
9C2×#14519834—21no—
10D61⋊(C22⋊C8)—24no—
11C2×C22⋊#92420C2212yes—
12C2×C22⋊#92420C2212yes—
13#16840605—24no—
14C23×C22⋊D61C411yescharacteristic
15C23×C4×D61C2211yescharacteristic
16C22×C22⋊Dic61C2211yescharacteristic
17C22×#11758198C411yescharacteristic
18#16839638—21no—
19#16839638—21no—
20#16839638—21no—
21#16839638—21no—
22C2×#14519684C411yescharacteristic
23C2×#14519684C2211yescharacteristic
24C22×C22⋊C4×C61C411yescharacteristic
25C2×D61⋊C8—24no—
26C2×D61⋊C8—42no—
27C2×D61⋊C8—42no—
28C22×#92420—22no—
29C22×#92420—22no—
30#14519834—24no—
31#14519834—42no—
32#14519834—42no—
33C22⋊#92420—22no—
34C22⋊#92420—22no—
35C22⋊#92420—22no—
36C22⋊#92420D414yescomplemented
37C24×D61C2×C411yescharacteristic
38C22×C22⋊D61—21no—
39C22×C22⋊D61—21no—
40C22×C22⋊D61—22no—
41C22×C22⋊D61—42no—
42C22×C22⋊D61—42no—
43C22×C22⋊D61C814yescomplemented
44C23×Dic61C2×C411yescharacteristic
45C23×Dic61C2×C411yescharacteristic
46C23×Dic61C2311yescharacteristic
47C22×C4×D61—22no—
48C22×C4×D61—22no—
49C22×C4×D61—42no—
50C22×C4×D61—42no—
51C22×C4×D61D411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61D411yescharacteristic
54C22×C4×D61D411yescharacteristic
55C2×C22⋊Dic61—22no—
56C2×C22⋊Dic61—21no—
57C2×C22⋊Dic61—21no—
58C2×C22⋊Dic61—22no—
59C2×#11758198—21no—
60C2×#11758198—21no—
61C2×#11758198—42no—
62#14519684—21no—
63#14519684—21no—
64#14519684—21no—
65#14519684—21no—
66C25×C61C2×C411yescharacteristic
67C23×C4×C61C2×C411yescharacteristic
68C23×C4×C61C2×C411yescharacteristic
69C2×C22⋊C4×C61—21no—
70C2×C22⋊C4×C61—21no—
71C2×C22⋊C4×C61—42no—
72C24×C61—21no—
73C24×C61—21no—
74C24×C61—22no—
75C24×C61—22no—
76C24×C61—41no—
77C24×C61—41no—
78C24×C61—41no—
79C24×C61—41no—
80C24×C61C2×C812yescomplemented
81C24×C61C22×C411yescharacteristic
82C23×D61—22no—
83C23×D61—22no—
84C23×D61—22no—
85C23×D61—22no—
86C23×D61—41no—
87C23×D61—41no—
88C23×D61C22⋊C411yescharacteristic
89C23×D61C22⋊C411yescharacteristic
90C23×D61C2×C812yes—
91C23×D61M1612yes—
92C22×C4×C61—22no—
93C22×C4×C61—22no—
94C22×C4×C61—21no—
95C22×C4×C61—22no—
96C22×C4×C61—41no—
97C22×C4×C61—41no—
98C22×C4×C61—41no—
99C22×C4×C61C22⋊C411yescharacteristic
100C22×C4×C61C22⋊C411yescharacteristic
101C2×C22⋊D61—24no—
102C2×C22⋊D61—24no—
103C2×C22⋊D61—48no—
104C2×C22⋊D61—42no—
105C2×C22⋊D61—42no—
106C2×C22⋊D61—42no—
107C2×C22⋊D61—42no—
108C2×C22⋊D61—48no—
109C22×Dic61—21no—
110C22×Dic61—21no—
111C22×Dic61—21no—
112C22×Dic61—22no—
113C22×Dic61—41no—
114C22×Dic61—41no—
115C22×Dic61—41no—
116C22×Dic61—41no—
117C22×Dic61—41no—
118C22×Dic61—41no—
119C22×Dic61C2×C812yescomplemented
120C22×Dic61C2×D411yescharacteristic
121C22×Dic61C2×D411yescharacteristic
122C22×Dic61C2×D412yes—
123C22×Dic61#4112yes—
124C2×C4×D61—22no—
125C2×C4×D61—22no—
126C2×C4×D61—21no—
127C2×C4×D61—21no—
128C2×C4×D61—22no—
129C2×C4×D61—22no—
130C2×C4×D61—21no—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—21no—
134C2×C4×D61—48no—
135C2×C4×D61—48no—
136C2×C4×D61—41no—
137C2×C4×D61—41no—
138C2×C4×D61—41no—
139C2×C4×D61—41no—
140C2×#92420—24no—
141C2×#92420—42no—
142C2×#92420—42no—
143C2×#92420—42no—
144C2×#92420—42no—
145C2×#92420—42no—
146D61⋊C8—48no—
147C22⋊C4×C61—42no—
148C22⋊C4×C61—42no—
149C22⋊Dic61—41no—
150C22⋊Dic61—42no—
151C22⋊Dic61—41no—
152#11758198—42no—
153#11758198—42no—
154C23×C61—21no—
155C23×C61—21no—
156C23×C61—21no—
157C23×C61—21no—
158C23×C61—22no—
159C23×C61—24no—
160C23×C61—42no—
161C23×C61—42no—
162C23×C61—42no—
163C23×C61—42no—
164C23×C61—41no—
165C23×C61—41no—
166C23×C61—41no—
167C23×C61—41no—
168C23×C61—41no—
169C23×C61—41no—
170C23×C61—42no—
171C23×C61—82no—
172C23×C61—82no—
173C23×C61—82no—
174C23×C61—82no—
175C23×C61C2×C22⋊C411yescharacteristic
176C23×C61C4×D412yes—
177C23×C61C22×C811yescharacteristic
178C23×C61C2×M1611yescharacteristic
179C23×C61#13112yes—
180C22×D61—21no—
181C22×D61—21no—
182C22×D61—21no—
183C22×D61—21no—
184C22×D61—21no—
185C22×D61—21no—
186C22×D61—21no—
187C22×D61—21no—
188C22×D61—21no—
189C22×D61—21no—
190C22×D61—24no—
191C22×D61—24no—
192C22×D61—24no—
193C22×D61—24no—
194C22×D61—42no—
195C22×D61—42no—
196C22×D61—44no—
197C22×D61—44no—
198C22×D61—44no—
199C22×D61—44no—
200C22×D61C22⋊C812yes—
201C22×D61C22⋊C812yes—
202C22×D61#9912yes—
203C22×D61#10012yes—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—44no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—82no—
214C2×C4×C61#9912yes—
215C22⋊D61—48no—
216C22⋊D61—48no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—21no—
220C2×Dic61—21no—
221C2×Dic61—22no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—42no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—82no—
232C2×Dic61#12011yescharacteristic
233C2×Dic61#12112yes—
234C2×Dic61#12311yescharacteristic
235C4×D61—24no—
236C4×D61—24no—
237C4×D61—44no—
238C4×D61—44no—
239C4×D61—42no—
240C4×D61—42no—
241#92420—44no—
242#92420—82no—
243#9079—611nomaximal, Sylow
244C22×C61—22no—
245C22×C61—22no—
246C22×C61—21no—
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—21no—
250C22×C61—22no—
251C22×C61—22no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—84no—
256C22×C61—84no—
257C22×C61—81no—
258C22×C61—81no—
259C22×C61—81no—
260C22×C61—81no—
261C22×C61#38611yescharacteristic
262C22×C61C2×C22⋊C811yescharacteristic
263C22×C61#40811yescharacteristic
264C22×C61C2×#9911yescharacteristic
265C22×C61C2×#9911yescharacteristic
266C22×C61C2×#10011yescharacteristic
267C22×C61#41311yescharacteristic
268C22×C61C8×D412yes—
269C22×C61#43512yes—
270C2×D61—22no—
271C2×D61—22no—
272C2×D61—22no—
273C2×D61—22no—
274C2×D61—22no—
275C2×D61—22no—
276C2×D61—22no—
277C2×D61—22no—
278C2×D61—22no—
279C2×D61—48no—
280C2×D61—48no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61—41no—
284C2×D61—41no—
285C2×D61#32314yescomplemented
286C4×C61—22no—
287C4×C61—41no—
288C4×C61—81no—
289Dic61—21no—
290Dic61—22no—
291Dic61—21no—
292Dic61—42no—
293Dic61—81no—
294C22×C22⋊C8—611no—
295C2×#386—611no—
296#1779—611no—
297C2×#323—612no—
298#1280—612no—
299C2×C61—22no—
300C2×C61—22no—
301C2×C61—82no—
302C2×C61C2×#32311yescharacteristic
303C2×C61#145911yescharacteristic
304C2×C61#146811yescharacteristic
305C2×C61#147111yescharacteristic
306C2×C61#186911yescharacteristic
307C2×C61#189811yescharacteristic
308C2×C61#190011yescharacteristic
309D61—24no—
310D61—42no—
311C23×D4—611no—
312C24×C4—611no—
313C23×C8—1221no—
314C22×C22⋊C4—611no—
315C22×C22⋊C4—611no—
316C22×C22⋊C4#692311yescharacteristic
317C2×C22⋊C8—612no—
318C2×C22⋊C8—612no—
319C2×C22⋊C8—1224no—
320C2×C22⋊C8—1221no—
321C2×C22⋊C8—1221no—
322C2×C22⋊C8—1221no—
323#386—1221no—
324#386—1221no—
325#386—1221no—
326#386—1221no—
327C2×#95—611no—
328C2×#95—611no—
329#323—1224no—
330C61#907911yescharacteristic, complemented, Sylow
331C25—611no—
332C25C2×#692311yescharacteristic
333C22×D4—614no—
334C22×D4—1222no—
335C22×D4—1221no—
336C22×D4—1221no—
337C22×D4—2442no—
338C22×D4—2442no—
339C23×C4—611no—
340C23×C4—611no—
341C23×C4—611no—
342C23×C4—611no—
343C23×C4—611no—
344C23×C4—611no—
345C23×C4—611no—
346C23×C4—1222no—
347C23×C4—1222no—
348C23×C4—2442no—
349C23×C4—2442no—
350C23×C4C2×#692311yescharacteristic
351C23×C4C2×#692311yescharacteristic
352C22×C8—1222no—
353C22×C8—1224no—
354C22×C8—1222no—
355C22×C8—2442no—
356C22×C8—2442no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—42no—
360C2×C22⋊C4—1221no—
361C2×C22⋊C4—1221no—
362C2×C22⋊C4—1222no—
363C2×C22⋊C4—1221no—
364C2×C22⋊C4—1221no—
365C2×C22⋊C4—1222no—
366C2×C22⋊C4—2442no—
367C22⋊C8—614no—
368C22⋊C8—1222no—
369C22⋊C8—1224no—
370C22⋊C8—1222no—
371C22⋊C8—1222no—
372C22⋊C8—2442no—
373C22⋊C8—2442no—
374#95—1221no—
375#95—1221no—
376#95—1221no—
377#95—1221no—
378C24—21no—
379C24—21no—
380C24—22no—
381C24—22no—
382C24—41no—
383C24—41no—
384C24—41no—
385C24—41no—
386C24—611no—
387C24—611no—
388C24—612no—
389C24—612no—
390C24—1222no—
391C24—1222no—
392C24—1222no—
393C24—1222no—
394C24—2441no—
395C24—2441no—
396C24D61⋊C812yescomplemented
397C24C22×#692311yescharacteristic
398C2×D4—1224no—
399C2×D4—1224no—
400C2×D4—2442no—
401C2×D4—2442no—
402C2×D4—2448no—
403C2×D4—2442no—
404C2×D4—2442no—
405C2×D4—2448no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—611no—
414C22×C4—611no—
415C22×C4—612no—
416C22×C4—612no—
417C22×C4—612no—
418C22×C4—1221no—
419C22×C4—1221no—
420C22×C4—1221no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1221no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—1221no—
432C22×C4—2441no—
433C22×C4—2441no—
434C22×C4—2441no—
435C22×C4—2441no—
436C22×C4—2448no—
437C22×C4—2448no—
438C22×C4—2441no—
439C22×C4—2441no—
440C22×C4—2441no—
441C22×C4—2441no—
442C22×C4—2441no—
443C22×C4—2441no—
444C22×C4C22⋊#692311yescharacteristic
445C22×C4C22⋊#692311yescharacteristic
446C2×C8—1224no—
447C2×C8—2442no—
448C2×C8—2442no—
449C2×C8—2442no—
450C2×C8—2448no—
451C2×C8—2442no—
452C2×C8—2442no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—2442no—
456C22⋊C4—2442no—
457C22⋊C4—2441no—
458C22⋊C4—2442no—
459C22⋊C4—2441no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—22no—
465C23—24no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—42no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—612no—
482C23—612no—
483C23—612no—
484C23—612no—
485C23—1224no—
486C23—1224no—
487C23—1221no—
488C23—1221no—
489C23—1221no—
490C23—1221no—
491C23—1221no—
492C23—1221no—
493C23—1221no—
494C23—1221no—
495C23—1221no—
496C23—1221no—
497C23—1224no—
498C23—1224no—
499C23—2442no—
500C23—2442no—
501C23—2444no—
502C23—2444no—
503C23—2444no—
504C23—2444no—
505C23C2×C22⋊#692311yescharacteristic
506C23D4×#692312yes—
507C23C2×D61⋊C811yescharacteristic
508C23D61⋊M1611yescharacteristic
509C23#1451986412yes—
510D4—2448no—
511D4—2448no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—44no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—82no—
522C2×C4—611no—
523C2×C4—612no—
524C2×C4—611no—
525C2×C4—1224no—
526C2×C4—1224no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1222no—
532C2×C4—4882no—
533C2×C4—2444no—
534C2×C4—2444no—
535C2×C4—2442no—
536C2×C4—2442no—
537C2×C4—2442no—
538C2×C4—2442no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2442no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2442no—
546C2×C4#1451983512yes—
547C8—4882no—
548C8—2444no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—84no—
561C22—84no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—614no—
567C22—1222no—
568C22—1222no—
569C22—1222no—
570C22—1222no—
571C22—1222no—
572C22—1222no—
573C22—1222no—
574C22—1222no—
575C22—1222no—
576C22—2448no—
577C22—2448no—
578C22—2441no—
579C22—2441no—
580C22—2441no—
581C22—2441no—
582C22#1684071011yescharacteristic
583C22D61⋊(C22⋊C8)11yescharacteristic
584C22#1684073611yescharacteristic
585C22C2×#1451983511yescharacteristic
586C22D61⋊#9911yescharacteristic
587C22D61⋊#10011yescharacteristic
588C22#1684074811yescharacteristic
589C22#1684078912yes—
590C22#1684079412yes—
591C4—22no—
592C4—41no—
593C4—81no—
594C4—1222no—
595C4—1221no—
596C4—1221no—
597C4—4881no—
598C4—2442no—
599C2—22no—
600C2—22no—
601C2—82no—
602C2—1224no—
603C2—2442no—
604C2D61⋊#32311yescharacteristic
605C2#2146105411yescharacteristic
606C2#2146108511yescharacteristic
607C2#2146109411yescharacteristic
608C2#2146179311yescharacteristic
609C2#2146184011yescharacteristic
610C2#2146184511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC23×C61C22×C8
FrattiniC24C22×#6923
FittingC22×C22⋊C4×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC23×C22⋊D61C4
cosocleC23×Dic61C23
hypercenterC22×C22⋊C4#6923
hypocenterC61#9079
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
1. C2×D61⋊(C22⋊C8)C2semidirect
2. #21461642C2semidirect
3. C2×#16840605C2semidirect
4. #21460657C2semidirect
36. C22⋊#92420D4semidirect
43. C22×C22⋊D61C8semidirect
80. C24×C61C2×C8semidirect
119. C22×Dic61C2×C8semidirect
285. C2×D61#323semidirect
330. C61#9079semidirect
396. C24D61⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×D61⋊(C22⋊C8)2
#214616422
C2×#168406052
#214606572
C2×#168396382
#907961