Finite Group Databasebeta

← #398×Dic61 #29601852

Subgroups — #398×Dic61 #29601852

Order 15616.

1574 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×Dic61C211yesmaximal, characteristic
2C2×C4⋊C4×Dic61C211yesmaximal, characteristic
3C2×C4⋊C4×Dic61C212yesmaximal
4#21444087C211yesmaximal, characteristic
5#21444084C211yesmaximal, characteristic
6#21444076C212yesmaximal
7#21444072C212yesmaximal
8#21444072C211yesmaximal, characteristic
9#95×Dic61C212yesmaximal
10#95×Dic61C211yesmaximal, characteristic
11C2×C61×#398C211yesmaximal, characteristic
12C22×C4⋊Dic61C2211yescharacteristic
13C22×C4⋊Dic61C2211yescharacteristic
14C22×C4⋊Dic61C2212yes—
15C22×C4×Dic61C2211yescharacteristic
16C22×C4×Dic61C2212yes—
17C22×C4×Dic61C2211yescharacteristic
18C22×C4×Dic61C2211yescharacteristic
19C22×C4×Dic61C2212yes—
20C4⋊C4×Dic61—28no—
21C4⋊C4×Dic61—24no—
22C4⋊C4×Dic61—24no—
23C2×#14519684C2212yes—
24C2×#14519684C2212yes—
25C2×#14519684C2212yes—
26C2×#14519684C2211yescharacteristic
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2211yescharacteristic
31C2×#14519684C2212yes—
32C2×#14519684C2211yescharacteristic
33C22×C4⋊C4×C61C2211yescharacteristic
34C22×C4⋊C4×C61C2212yes—
35C22×C4⋊C4×C61C2211yescharacteristic
36C61×#398C418yescomplemented
37C2×#95×C61C2212yes—
38C2×#95×C61C2211yescharacteristic
39C23×Dic61C2311yescharacteristic
40C23×Dic61C2311yescharacteristic
41C23×Dic61C2312yes—
42C23×Dic61C2312yes—
43C23×Dic61C2311yescharacteristic
44C23×Dic61C2311yescharacteristic
45C2×C4⋊Dic61—22no—
46C2×C4⋊Dic61—22no—
47C2×C4⋊Dic61—24no—
48C2×C4⋊Dic61—22no—
49C2×C4⋊Dic61—24no—
50C2×C4⋊Dic61—22no—
51C2×C4⋊Dic61—24no—
52C2×C4⋊Dic61—24no—
53C2×C4×Dic61—22no—
54C2×C4×Dic61—24no—
55C2×C4×Dic61—22no—
56C2×C4×Dic61—24no—
57C2×C4×Dic61—24no—
58C2×C4×Dic61—24no—
59C2×C4×Dic61—24no—
60C2×C4×Dic61—22no—
61C2×C4×Dic61—24no—
62C2×C4×Dic61—24no—
63C2×C4×Dic61D414yes—
64C2×C4×Dic61D414yes—
65C2×C4×Dic61Q818yes—
66#14519684—24no—
67#14519684—22no—
68#14519684—22no—
69#14519684—24no—
70#14519684—24no—
71#14519684—22no—
72#14519684—24no—
73#14519684—24no—
74#14519684—24no—
75#14519684—24no—
76C23×C4×C61C2311yescharacteristic
77C23×C4×C61C2312yes—
78C23×C4×C61C2311yescharacteristic
79C23×C4×C61C2311yescharacteristic
80C23×C4×C61C2312yes—
81C2×C4⋊C4×C61—24no—
82C2×C4⋊C4×C61—22no—
83C2×C4⋊C4×C61—22no—
84C2×C4⋊C4×C61—22no—
85C2×C4⋊C4×C61—24no—
86C2×C4⋊C4×C61—22no—
87C2×C4⋊C4×C61C2×C414yes—
88C2×C4⋊C4×C61C2×C414yes—
89C2×C4⋊C4×C61C2×C418yes—
90#95×C61C2×C418yes—
91#95×C61C2×C414yes—
92C24×C61C2411yescharacteristic
93C22×C4×C61—21no—
94C22×C4×C61—22no—
95C22×C4×C61—21no—
96C22×C4×C61—22no—
97C22×C4×C61—21no—
98C22×C4×C61—22no—
99C22×C4×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61—21no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—22no—
105C22×C4×C61—21no—
106C22×C4×C61—22no—
107C22×C4×C61—21no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61C22×C412yes—
114C22×C4×C61C22×C414yes—
115C22×C4×C61C22×C412yes—
116C22×C4×C61C22×C412yes—
117C22×C4×C61C22×C414yes—
118C22×C4×C61C2×D412yes—
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2×Q814yes—
121C22×C4×C61#4112yes—
122C22×C4×C61#4112yes—
123C22×C4×C61#4114yes—
124C22×Dic61—21no—
125C22×Dic61—22no—
126C22×Dic61—22no—
127C22×Dic61—22no—
128C22×Dic61—21no—
129C22×Dic61—22no—
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—21no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—21no—
147C22×Dic61—22no—
148C22×Dic61C2×D412yes—
149C22×Dic61C2×D412yes—
150C22×Dic61C2×D412yes—
151C22×Dic61C2×D412yes—
152C22×Dic61C2×Q814yes—
153C22×Dic61C2×Q814yes—
154C22×Dic61#4112yes—
155C22×Dic61#4114yes—
156C22×Dic61#4112yes—
157C22×Dic61#4114yes—
158C4⋊C4×C61—28no—
159C4⋊C4×C61—216no—
160C4⋊C4×C61—28no—
161C4⋊Dic61—216no—
162C4⋊Dic61—28no—
163C4⋊Dic61—28no—
164C4×Dic61—28no—
165C4×Dic61—216no—
166C4×Dic61—28no—
167C4×Dic61—48no—
168C4×Dic61—44no—
169C23×C61C23×C411yescharacteristic
170C23×C61C22×D411yescharacteristic
171C23×C61C22×D411yescharacteristic
172C23×C61C22×Q812yes—
173C23×C61C2×#4111yescharacteristic
174C23×C61C2×#4111yescharacteristic
175C23×C61C2×#4112yes—
176C23×C61C2×#4111yescharacteristic
177C23×C61C2×#4112yes—
178C23×C61C2×#4112yes—
179C23×C61C2×#4111yescharacteristic
180C2×C4×C61—22no—
181C2×C4×C61—24no—
182C2×C4×C61—22no—
183C2×C4×C61—24no—
184C2×C4×C61—24no—
185C2×C4×C61—22no—
186C2×C4×C61—24no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—24no—
191C2×C4×C61—22no—
192C2×C4×C61—24no—
193C2×C4×C61—24no—
194C2×C4×C61—22no—
195C2×C4×C61—24no—
196C2×C4×C61—24no—
197C2×C4×C61—24no—
198C2×C4×C61—22no—
199C2×C4×C61—24no—
200C2×C4×C61—24no—
201C2×C4×C61—42no—
202C2×C4×C61—41no—
203C2×C4×C61—42no—
204C2×C4×C61—41no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61C4×D414yes—
209C2×C4×C61C4×D414yes—
210C2×C4×C61C4×Q818yes—
211C2×Dic61—22no—
212C2×Dic61—24no—
213C2×Dic61—22no—
214C2×Dic61—24no—
215C2×Dic61—22no—
216C2×Dic61—24no—
217C2×Dic61—24no—
218C2×Dic61—22no—
219C2×Dic61—24no—
220C2×Dic61—24no—
221C2×Dic61—22no—
222C2×Dic61—24no—
223C2×Dic61—24no—
224C2×Dic61—42no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—41no—
237C2×Dic61—42no—
238C2×Dic61—41no—
239C2×Dic61—42no—
240C2×Dic61#12114yes—
241C2×Dic61C22⋊Q818yes—
242C2×Dic61#12314yes—
243C2×Dic61#12518yes—
244C2×Dic61C4⋊Q814yes—
245C4×#398—611nomaximal, Sylow
246C22×C61C2×C4×D411yescharacteristic
247C22×C61C2×C4×D411yescharacteristic
248C22×C61C2×C4×Q812yes—
249C22×C61C4×#4111yescharacteristic
250C22×C61C4×#4112yes—
251C22×C61C2×#12111yescharacteristic
252C22×C61C2×C22⋊Q812yes—
253C22×C61C2×#12311yescharacteristic
254C22×C61#52511yescharacteristic
255C22×C61#52511yescharacteristic
256C22×C61#52512yes—
257C22×C61C2×#12512yes—
258C22×C61#52912yes—
259C22×C61#52911yescharacteristic
260C22×C61#52912yes—
261C22×C61#52912yes—
262C22×C61#52912yes—
263C22×C61C2×C4⋊Q811yescharacteristic
264C22×C61#53211yescharacteristic
265C22×C61#53211yescharacteristic
266C22×C61#53312yes—
267C22×C61#53312yes—
268C22×C61#53312yes—
269C4×C61—24no—
270C4×C61—24no—
271C4×C61—28no—
272C4×C61—44no—
273C4×C61—42no—
274Dic61—42no—
275Dic61—44no—
276Dic61—42no—
277Dic61—42no—
278Dic61—44no—
279Dic61#39818yesdirect product factor
280C2×#398D6111yescharacteristic
281C2×C4×C4⋊C4—611no—
282C2×C4×C4⋊C4—611no—
283C2×C4×C4⋊C4—612no—
284#1445—611no—
285#1444—611no—
286#1442—612no—
287#1441—612no—
288#1441—611no—
289C4×#95—611no—
290C4×#95—612no—
291C2×C61C4×#12111yescharacteristic
292C2×C61C4×#12311yescharacteristic
293C2×C61C4×C22⋊Q812yes—
294C2×C61C4×#12512yes—
295C2×C61C4×C4⋊Q811yescharacteristic
296C2×C61C2×#39811yescharacteristic
297C2×C61#239211yescharacteristic
298C2×C61#239511yescharacteristic
299C2×C61#239612yes—
300C2×C61#239911yescharacteristic
301C2×C61#240212yes—
302C22×C4⋊C4—611no—
303C22×C4⋊C4—611no—
304C22×C4⋊C4—612no—
305C22×C4⋊C4C2×D6112yes—
306C22×C4⋊C4C2×D6111yescharacteristic
307C22×C4⋊C4C2×D6111yescharacteristic
308C22×C42—612no—
309C22×C42—611no—
310C22×C42—611no—
311C22×C42—611no—
312C22×C42—612no—
313#398Dic6118yesdirect product factor
314C4×C4⋊C4—1224no—
315C4×C4⋊C4—1224no—
316C4×C4⋊C4—1228no—
317C2×#95—612no—
318C2×#95—612no—
319C2×#95—612no—
320C2×#95—612no—
321C2×#95—611no—
322C2×#95—612no—
323C2×#95—612no—
324C2×#95—611no—
325C2×#95—612no—
326C2×#95—611no—
327C2×#95C2×D6112yes—
328C2×#95C2×D6111yescharacteristic
329C61C4×#39811yescharacteristic, complemented, Sylow
330C23×C4—611no—
331C23×C4—611no—
332C23×C4—612no—
333C23×C4—611no—
334C23×C4—611no—
335C23×C4—612no—
336C23×C4C22×D6112yes—
337C23×C4C22×D6111yescharacteristic
338C23×C4C22×D6111yescharacteristic
339C23×C4C22×D6111yescharacteristic
340C23×C4C22×D6112yes—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—1222no—
348C2×C4⋊C4—1224no—
349C2×C4⋊C4—1222no—
350C2×C4⋊C4—1224no—
351C2×C4⋊C4—1224no—
352C2×C4⋊C4—1222no—
353C2×C4⋊C4—1222no—
354C2×C4⋊C4—1224no—
355C2×C4⋊C4C2×Dic6114yes—
356C2×C4⋊C4C2×Dic6114yes—
357C2×C4⋊C4C2×Dic6118yes—
358C2×C42—614no—
359C2×C42—614no—
360C2×C42—618no—
361C2×C42—1224no—
362C2×C42—1222no—
363C2×C42—1224no—
364C2×C42—1224no—
365C2×C42—1222no—
366C2×C42—1222no—
367C2×C42—1224no—
368C2×C42—1224no—
369C2×C42—1224no—
370C2×C42—1224no—
371#95—1222no—
372#95—1224no—
373#95—1224no—
374#95—1224no—
375#95—1224no—
376#95—1224no—
377#95—1222no—
378#95—1222no—
379#95—1224no—
380#95—1224no—
381#95C2×Dic6118yes—
382#95C2×Dic6114yes—
383C24C23×D6111yescharacteristic
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—612no—
405C22×C4—612no—
406C22×C4—614no—
407C22×C4—612no—
408C22×C4—614no—
409C22×C4—612no—
410C22×C4—612no—
411C22×C4—614no—
412C22×C4—612no—
413C22×C4—614no—
414C22×C4—1221no—
415C22×C4—1222no—
416C22×C4—1222no—
417C22×C4—1221no—
418C22×C4—1222no—
419C22×C4—1222no—
420C22×C4—1222no—
421C22×C4—1221no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1221no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1221no—
431C22×C4—1221no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4D4×D6112yes—
439C22×C4D4×D6112yes—
440C22×C4#1175822412yes—
441C22×C4#1175822412yes—
442C22×C4Q8×D6114yes—
443C22×C4#1175822614yes—
444C22×C4C22×Dic6112yes—
445C22×C4C22×Dic6114yes—
446C22×C4C22×Dic6112yes—
447C22×C4C22×Dic6112yes—
448C22×C4C22×Dic6114yes—
449C4⋊C4—28no—
450C4⋊C4—216no—
451C4⋊C4—28no—
452C4⋊C4—1228no—
453C4⋊C4—1228no—
454C4⋊C4—12216no—
455C42—1228no—
456C42—12216no—
457C42—1228no—
458C42—2448no—
459C42—2444no—
460C23C23×Dic6111yescharacteristic
461C23C2×D4×D6111yescharacteristic
462C23C2×D4×D6111yescharacteristic
463C23C2×Q8×D6112yes—
464C23#41×D6111yescharacteristic
465C23#41×D6112yes—
466C23#41×D6112yes—
467C23#41×D6111yescharacteristic
468C23C2×#1175822411yescharacteristic
469C23C2×#1175822411yescharacteristic
470C23C2×#1175822612yes—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—614no—
500C2×C4—618no—
501C2×C4—614no—
502C2×C4—618no—
503C2×C4—614no—
504C2×C4—1222no—
505C2×C4—1224no—
506C2×C4—1222no—
507C2×C4—1224no—
508C2×C4—1222no—
509C2×C4—1222no—
510C2×C4—1224no—
511C2×C4—1224no—
512C2×C4—1224no—
513C2×C4—1224no—
514C2×C4—1222no—
515C2×C4—1224no—
516C2×C4—1224no—
517C2×C4—2442no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4—2442no—
521C2×C4—2442no—
522C2×C4—2442no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4—2442no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4—2441no—
531C2×C4—2442no—
532C2×C4—2441no—
533C2×C4D4×Dic6114yes—
534C2×C4D4×Dic6114yes—
535C2×C4Q8×Dic6118yes—
536C22C2×D4×Dic6111yescharacteristic
537C22C2×D4×Dic6111yescharacteristic
538C22C2×Q8×Dic6112yes—
539C22#41×Dic6111yescharacteristic
540C22#41×Dic6112yes—
541C22#121×D6111yescharacteristic
542C22C22⋊Q8×D6112yes—
543C22#123×D6111yescharacteristic
544C22#1684027011yescharacteristic
545C22#1684027211yescharacteristic
546C22#1684027512yes—
547C22#125×D6112yes—
548C22#1684028912yes—
549C22#1684029311yescharacteristic
550C22#1684029712yes—
551C22#1684029712yes—
552C22#1684029812yes—
553C22C4⋊Q8×D6111yescharacteristic
554C22#1684030711yescharacteristic
555C22#1684031311yescharacteristic
556C22#1684031412yes—
557C22#1684031712yes—
558C22#1684031812yes—
559C4—24no—
560C4—24no—
561C4—28no—
562C4—44no—
563C4—42no—
564C4—618no—
565C4—2442no—
566C4—2444no—
567C4—2442no—
568C4—2442no—
569C4—2444no—
570C2#121×Dic6111yescharacteristic
571C2#123×Dic6111yescharacteristic
572C2C22⋊Q8×Dic6112yes—
573C2#125×Dic6112yes—
574C2C4⋊Q8×Dic6111yescharacteristic
575C2#398×D6111yescharacteristic
576C2#2144906011yescharacteristic
577C2#2144908311yescharacteristic
578C2#2144909612yes—
579C2#2144911111yescharacteristic
580C2#2144912312yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#398C2
socle, prime radical, cosocleC24×C61C24
hypercenterC2×#398D61
hypocenterC61C4×#398
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
36. C61×#398C4semidirect
279. Dic61#398direct (N × H)
313. #398Dic61direct (N × H)
329. C61C4×#398semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×Dic612
#214440872
#214440842
#214440762
#214440722
#95×Dic612
C2×C61×#3982
C4×#39861