Finite Group Databasebeta

← #30608773

Subgroups — #30608773

Order 15616.

1326 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840711C212yesmaximal
2#21460999C212yesmaximal
3#21460998C212yesmaximal
4#21460998C212yesmaximal
5#21460996C212yesmaximal
6#21460995C211yesmaximal, characteristic
7#21460995C211yesmaximal, characteristic
8#21460994C212yesmaximal
9#389×D61C211yesmaximal, characteristic
10C22×C4⋊#6923C2212yes—
11C22×C4×#6923C2212yes—
12C22×C4×#6923C2212yes—
13#16840711C4116yescomplemented
14C2×(C2×#6923)⋊C4C2211yescharacteristic
15C2×(C2×#6923)⋊C4C2212yes—
16C2×(C2×#6923)⋊C4C2212yes—
17C2×(C2×#6923)⋊C4C2212yes—
18C2×(C2×#6923)⋊C4C2211yescharacteristic
19C2×(C2×#6923)⋊C4C2212yes—
20C2×(C2×#6923)⋊C4C2211yescharacteristic
21C2×(C2×#6923)⋊C4C2211yescharacteristic
22C2×(C2×#6923)⋊C4C2212yes—
23C2×(C2×#6923)⋊C4C2212yes—
24C2×(C2×#6923)⋊C4C2212yes—
25C2×(C2×#6923)⋊C4C2212yes—
26C2×(C2×#6923)⋊C4C2212yes—
27C2×C4⋊C4×D61C2212yes—
28C2×C4⋊C4×D61C2211yescharacteristic
29C2×C4⋊C4×D61C2211yescharacteristic
30C2×C42×D61C2211yescharacteristic
31#16839661C411yescharacteristic, complemented
32#16839660C411yescharacteristic, complemented
33#16839659C412yescomplemented
34#16839658C411yescharacteristic, complemented
35#16839657C412yescomplemented
36#95×D61C2212yes—
37C61×#389C411yescharacteristic, complemented
38C23×#6923C2312yes—
39C23×#6923C2312yes—
40C23×#6923C2312yes—
41C23×#6923C2312yes—
42C2×C4⋊#6923—42no—
43C2×C4⋊#6923—42no—
44C2×C4⋊#6923C2×C418yes—
45C2×C4×#6923—42no—
46C2×C4×#6923—42no—
47C2×C4×#6923—42no—
48C2×C4×#6923—42no—
49C2×C4×#6923—42no—
50C2×C4×#6923C2×C418yes—
51(C2×#6923)⋊C4—41no—
52(C2×#6923)⋊C4—41no—
53(C2×#6923)⋊C4—41no—
54(C2×#6923)⋊C4—41no—
55(C2×#6923)⋊C4—42no—
56(C2×#6923)⋊C4—42no—
57(C2×#6923)⋊C4—42no—
58(C2×#6923)⋊C4—42no—
59(C2×#6923)⋊C4—42no—
60(C2×#6923)⋊C4C2×C418yes—
61(C2×#6923)⋊C4C2×C418yes—
62(C2×#6923)⋊C4C2×C418yes—
63(C2×#6923)⋊C4C2×C418yes—
64C22×C4×D61C2311yescharacteristic
65C22×C4×D61C2311yescharacteristic
66C22×C4×D61C2312yes—
67C22×C4×D61C2311yescharacteristic
68C22×C4×D61C2312yes—
69C2×C4⋊Dic61C2×C412yes—
70C2×C4⋊Dic61C2×C411yescharacteristic
71C2×C4⋊Dic61C2×C411yescharacteristic
72C2×#11758196C2×C412yes—
73C2×#11758196C2×C412yes—
74C2×#11758196C2×C412yes—
75C2×#11758196C2×C412yes—
76C4⋊C4×D61—42no—
77C4⋊C4×D61—42no—
78C4⋊C4×D61—42no—
79C4⋊C4×D61C2×C418yes—
80C2×C4×Dic61C2×C411yescharacteristic
81C2×C4×Dic61C2×C411yescharacteristic
82C2×C4×Dic61C2×C411yescharacteristic
83C42×D61—42no—
84#14519684C2×C412yes—
85#14519684C2×C412yes—
86#14519684C2×C412yes—
87C2×C4⋊C4×C61C2×C412yes—
88C2×C4⋊C4×C61C2×C411yescharacteristic
89C2×C4⋊C4×C61C2×C411yescharacteristic
90C2×C42×C61C2×C411yescharacteristic
91#95×C61C2×C412yes—
92C23×D61C2411yescharacteristic
93C22×#6923—22no—
94C22×#6923—22no—
95C22×#6923—22no—
96C22×#6923—22no—
97C22×#6923—22no—
98C22×#6923—22no—
99C22×#6923—22no—
100C22×#6923—22no—
101C22×#6923—22no—
102C22×#6923—22no—
103C22×#6923—22no—
104C22×#6923—22no—
105C22×#6923—22no—
106C22×#6923—22no—
107C22×#6923—22no—
108C22×#6923—22no—
109C22×#6923—22no—
110C22×#6923—22no—
111C22×#6923—22no—
112C22×#6923—22no—
113C22×#6923—22no—
114C22×#6923—22no—
115C22×#6923—22no—
116C22×#6923—22no—
117C22×#6923C22×C414yes—
118C22×#6923C22×C414yes—
119C22×#6923C22×C414yes—
120C22×#6923C22×C414yes—
121C22×C4×C61C22×C411yescharacteristic
122C22×C4×C61C22×C412yes—
123C22×C4×C61C22×C411yescharacteristic
124C22×C4×C61C22×C412yes—
125C22×C4×C61C22×C411yescharacteristic
126C22×Dic61C22×C411yescharacteristic
127C22×Dic61C22×C412yes—
128C22×Dic61C22×C411yescharacteristic
129C22×Dic61C22×C412yes—
130C22×Dic61C22×C411yescharacteristic
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—21no—
136C2×C4×D61—21no—
137C2×C4×D61—22no—
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—21no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—21no—
152C2×C4×D61C22×C414yes—
153C2×C4×D61C22×C414yes—
154C2×C4×D61C22×C414yes—
155C4⋊#6923—48no—
156C4×#6923—48no—
157C4×#6923—48no—
158C4⋊C4×C61—41no—
159C4⋊C4×C61—42no—
160C4⋊C4×C61C4214yescomplemented
161C42×C61—41no—
162C4⋊Dic61—42no—
163C4⋊Dic61—41no—
164C4⋊Dic61C4214yescomplemented
165#11758196—42no—
166#11758196—42no—
167#11758196—42no—
168#11758196C4218yescomplemented
169C4×Dic61—41no—
170C4×Dic61—41no—
171C4×Dic61—41no—
172C23×C61C23×C411yescharacteristic
173C22×D61C23×C412yes—
174C22×D61H(5,𝔽2)11yescharacteristic
175C22×D61H(5,𝔽2)12yes—
176C22×D61H(5,𝔽2)11yescharacteristic
177C22×D61H(5,𝔽2)12yes—
178C22×D612−1+411yescharacteristic
179C22×D612−1+412yes—
180C22×D612−1+411yescharacteristic
181C22×D612−1+411yescharacteristic
182C22×D612−1+411yescharacteristic
183C2×#6923—24no—
184C2×#6923—24no—
185C2×#6923—24no—
186C2×#6923—24no—
187C2×#6923—24no—
188C2×#6923—24no—
189C2×#6923—24no—
190C2×#6923—24no—
191C2×#6923—24no—
192C2×#6923—24no—
193C2×#6923—24no—
194C2×#6923—24no—
195C2×#6923—42no—
196C2×#6923—42no—
197C2×#6923—42no—
198C2×#6923—42no—
199C2×#6923—42no—
200C2×#6923—42no—
201C2×#6923—42no—
202C2×#6923—42no—
203C2×#6923—42no—
204C2×#6923—42no—
205C2×#6923—42no—
206C2×#6923—42no—
207C2×#6923—42no—
208C2×#6923—42no—
209C2×#6923—42no—
210C2×#6923—42no—
211C2×C4×C61—21no—
212C2×C4×C61—22no—
213C2×C4×C61—21no—
214C2×C4×C61—22no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—22no—
223C2×C4×C61—21no—
224C2×C4×C61C2×C4214yes—
225C2×C4×C61C2×C4212yes—
226C2×Dic61—21no—
227C2×Dic61—22no—
228C2×Dic61—21no—
229C2×Dic61—22no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—22no—
238C2×Dic61—21no—
239C2×Dic61C2×C4212yes—
240C2×Dic61C2×C4214yes—
241C4×D61—24no—
242C4×D61—24no—
243C4×D61—24no—
244C4×D61—24no—
245C4×D61—24no—
246C4×D61—24no—
247C4×D61—42no—
248C4×D61—42no—
249C4×D61—42no—
250C4×D61—42no—
251C4×D61—42no—
252C4×D61—42no—
253C4×D61—42no—
254C4×D61—42no—
255#13833—611nomaximal, Sylow
256C22×C61C22×C4211yescharacteristic
257C22×C61#51811yescharacteristic
258C22×C61#51811yescharacteristic
259C22×C61#51911yescharacteristic
260C22×C61#51911yescharacteristic
261C22×C61#52011yescharacteristic
262C22×C61#52011yescharacteristic
263C2×D61#51812yes—
264C2×D61#51912yes—
265C2×D61#52012yes—
266C2×D61#52012yes—
267C2×D61#52012yes—
268C2×D61#52012yes—
269C2×D61#56012yes—
270C2×D61#56211yescharacteristic
271C2×D61#56212yes—
272C2×D61#56212yes—
273C2×D61#56312yes—
274C2×D61#56311yescharacteristic
275C2×D61#56312yes—
276C2×D61#56312yes—
277C2×D61#56311yescharacteristic
278C2×D61#56411yescharacteristic
279#6923—44no—
280#6923—44no—
281#6923—44no—
282#6923—44no—
283C4×C61—22no—
284C4×C61—24no—
285C4×C61—42no—
286C4×C61—41no—
287C4×C61—41no—
288Dic61—22no—
289Dic61—24no—
290Dic61—42no—
291Dic61—41no—
292Dic61—41no—
293C2×#389—611no—
294C2×#387—612no—
295#1436—612no—
296#1436—612no—
297#1436—612no—
298C4⋊#95—612no—
299#1434—612no—
300#1434—611no—
301#1434—611no—
302C2×C61#227411yescharacteristic
303C2×C61#227511yescharacteristic
304C2×C61#227611yescharacteristic
305C2×C61#237711yescharacteristic
306C2×C61#237811yescharacteristic
307C2×C61#238111yescharacteristic
308C2×C61#238211yescharacteristic
309D61#237612yes—
310D61#237812yes—
311D61#238212yes—
312D61#238312yes—
313C22×C4⋊C4—611no—
314C22×C4⋊C4—612no—
315C22×C4⋊C4—612no—
316C22×C4⋊C4—611no—
317C22×C42—612no—
318C22×C42—611no—
319C22×C42—612no—
320#389—612no—
321#389—611no—
322#389—611no—
323#389—611no—
324#389—612no—
325#389#692311yescharacteristic, complemented
326#387—6116no—
327C2×#95—612no—
328C2×#95—612no—
329C2×#95—612no—
330C2×#95—612no—
331C2×#95—612no—
332C2×#95—611no—
333C2×#95—612no—
334C2×#95—611no—
335C2×#95—612no—
336C2×#95—612no—
337C2×#95—611no—
338C2×#95—612no—
339C2×#95—611no—
340C2×#95—612no—
341C61#1383311yescharacteristic, complemented, Sylow
342C23×C4—612no—
343C23×C4—611no—
344C23×C4—612no—
345C23×C4—612no—
346C23×C4—612no—
347C23×C4—612no—
348C23×C4—612no—
349C23×C4—611no—
350C23×C4—611no—
351C2×C4⋊C4—612no—
352C2×C4⋊C4—611no—
353C2×C4⋊C4—618no—
354C2×C4⋊C4—618no—
355C2×C4⋊C4—612no—
356C2×C4⋊C4—612no—
357C2×C4⋊C4—612no—
358C2×C4⋊C4—611no—
359C2×C4⋊C4—612no—
360C2×C4⋊C4—2442no—
361C2×C4⋊C4—2442no—
362C2×C4⋊C4—2442no—
363C2×C4⋊C4—2442no—
364C2×C4⋊C4—2442no—
365C2×C4⋊C4C2×#692311yescharacteristic
366C2×C4⋊C4C2×#692312yes—
367C2×C4⋊C4C2×#692311yescharacteristic
368C2×C42—618no—
369C2×C42—611no—
370C2×C42—611no—
371C2×C42—611no—
372C2×C42—2442no—
373C2×C42—2442no—
374C2×C42—2442no—
375C2×C42—2442no—
376C2×C42—2442no—
377C2×C42—2442no—
378C2×C42C2×#692311yescharacteristic
379#95—618no—
380#95—618no—
381#95—618no—
382#95—618no—
383#95—612no—
384#95—612no—
385#95—612no—
386#95—2441no—
387#95—2441no—
388#95—2441no—
389#95—2441no—
390#95—2442no—
391#95—2442no—
392#95—2442no—
393#95—2442no—
394#95—2442no—
395#95C2×#692312yes—
396C24—611no—
397C22×C4—612no—
398C22×C4—611no—
399C22×C4—614no—
400C22×C4—614no—
401C22×C4—614no—
402C22×C4—614no—
403C22×C4—614no—
404C22×C4—614no—
405C22×C4—614no—
406C22×C4—611no—
407C22×C4—612no—
408C22×C4—611no—
409C22×C4—1222no—
410C22×C4—1222no—
411C22×C4—1222no—
412C22×C4—1222no—
413C22×C4—1221no—
414C22×C4—1221no—
415C22×C4—1222no—
416C22×C4—1221no—
417C22×C4—1221no—
418C22×C4—1222no—
419C22×C4—1222no—
420C22×C4—1222no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1221no—
428C22×C4—1222no—
429C22×C4—1221no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4C22×#692311yescharacteristic
455C22×C4C22×#692312yes—
456C22×C4C22×#692311yescharacteristic
457C22×C4C22×#692312yes—
458C22×C4C22×#692311yescharacteristic
459C4⋊C4—41no—
460C4⋊C4—42no—
461C4⋊C4—618no—
462C4⋊C4—614no—
463C4⋊C4—2442no—
464C4⋊C4—2442no—
465C4⋊C4—2442no—
466C4⋊C4—2441no—
467C4⋊C4—2442no—
468C4⋊C4—2448no—
469C4⋊C4C4×#692314yescomplemented
470C42—41no—
471C42—2441no—
472C42—2441no—
473C42—2441no—
474C42—2448no—
475C42—2448no—
476C23—612no—
477C23—612no—
478C23—612no—
479C23—612no—
480C23—611no—
481C23—611no—
482C23—611no—
483C23—611no—
484C23—611no—
485C23—611no—
486C23C23×#692311yescharacteristic
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—612no—
501C2×C4—614no—
502C2×C4—1224no—
503C2×C4—1224no—
504C2×C4—1224no—
505C2×C4—1224no—
506C2×C4—1224no—
507C2×C4—1224no—
508C2×C4—1221no—
509C2×C4—1222no—
510C2×C4—1221no—
511C2×C4—1222no—
512C2×C4—1224no—
513C2×C4—1224no—
514C2×C4—1224no—
515C2×C4—1224no—
516C2×C4—1224no—
517C2×C4—1224no—
518C2×C4—1224no—
519C2×C4—1224no—
520C2×C4—1224no—
521C2×C4—1224no—
522C2×C4—1224no—
523C2×C4—1224no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1221no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—1221no—
530C2×C4—1222no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4—2442no—
536C2×C4—2442no—
537C2×C4—2442no—
538C2×C4—2442no—
539C2×C4—2442no—
540C2×C4—2442no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4C2×C4×#692312yes—
558C2×C4C2×C4×#692314yes—
559C22—612no—
560C22—612no—
561C22—612no—
562C22—612no—
563C22—612no—
564C22—612no—
565C22—611no—
566C22—611no—
567C22—611no—
568C22—611no—
569C22—612no—
570C22—612no—
571C22—612no—
572C22—612no—
573C22—612no—
574C22—612no—
575C22C22×C4×#692311yescharacteristic
576C22#1684082911yescharacteristic
577C22#1684082911yescharacteristic
578C22#1684083011yescharacteristic
579C22#1684083011yescharacteristic
580C22#1684083111yescharacteristic
581C22#1684083111yescharacteristic
582C4—22no—
583C4—24no—
584C4—41no—
585C4—42no—
586C4—41no—
587C4—1222no—
588C4—1224no—
589C4—2444no—
590C4—2444no—
591C4—2444no—
592C4—2444no—
593C4—2441no—
594C4—2442no—
595C4—2441no—
596C2—612no—
597C2—612no—
598C2—612no—
599C2—612no—
600C2#2146225711yescharacteristic
601C2#2146225811yescharacteristic
602C2#2146225911yescharacteristic
603C2#2146236011yescharacteristic
604C2#2146236111yescharacteristic
605C2#2146236411yescharacteristic
606C2#2146236511yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutatorC22×C61C22×C42
FittingC61×#389C4
socleC23×C61C23×C4
prime radical, cosocleC23×D61C24
hypercenter#389#6923
hypocenterC61#13833
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
13. #16840711C4semidirect
31. #16839661C4semidirect
32. #16839660C4semidirect
33. #16839659C4semidirect
34. #16839658C4semidirect
35. #16839657C4semidirect
37. C61×#389C4semidirect
160. C4⋊C4×C61C42semidirect
164. C4⋊Dic61C42semidirect
168. #11758196C42semidirect
325. #389#6923semidirect
341. C61#13833semidirect
469. C4⋊C4C4×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168407112
#214609992
#214609982
#214609962
#214609952
#214609942
#389×D612
#1383361