Finite Group Databasebeta

← C2×#323×D61 #29645797

Subgroups — C2×#323×D61 #29645797

Order 15616.

2878 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊C4×D61C211yesmaximal, characteristic
2C2×C22⋊C8×D61C212yesmaximal, complemented
3C2×#16839486C211yesmaximal, characteristic, complemented
4C2×#16839485C212yesmaximal, complemented
5#323×D61C218yesmaximal, direct product factor
6C2×C61×#323C211yesmaximal, characteristic, complemented
7C25×D61C411yescharacteristic
8C23×C4×D61C411yescharacteristic
9C23×C4×D61C2211yescharacteristic
10C22×C8×D61—22no—
11C22×C22⋊Dic61C411yescharacteristic
12C22×#11758198C411yescharacteristic
13C22×#11758198C2211yescharacteristic
14C2×C22⋊C4×D61—22no—
15C2×C22⋊C4×D61—24no—
16C2×C22⋊C4×D61—22no—
17C2×C22⋊C4×D61C414yes—
18C2×C22⋊C4×D61C2214yes—
19C2×C22⋊#92418C2212yescomplemented
20C2×#14519689—22no—
21C22⋊C8×D61—24no—
22C22⋊C8×D61C2218yescomplemented
23#16839486C2214yescomplemented
24#16839485C2218yescomplemented
25C22×C22⋊C4×C61C2211yescharacteristic
26C2×C22⋊C8×C61C2212yescomplemented
27C61×#323C2214yescomplemented
28C24×D61—22no—
29C24×D61—21no—
30C24×D61—21no—
31C24×D61—24no—
32C24×D61—44no—
33C24×D61—44no—
34C24×D61C818yescomplemented
35C24×D61C2×C411yescharacteristic
36C24×D61C2×C414yes—
37C24×D61C2×C411yescharacteristic
38C23×Dic61C2×C411yescharacteristic
39C23×Dic61C2×C411yescharacteristic
40C22×C4×D61—22no—
41C22×C4×D61—21no—
42C22×C4×D61—21no—
43C22×C4×D61—44no—
44C22×C4×D61—44no—
45C22×C4×D61C2×C414yes—
46C22×C4×D61C2×C412yes—
47C22×C4×D61D412yes—
48C22×C4×D61D414yes—
49C22×C4×D61D412yes—
50C22×C4×D61C2312yes—
51C22×#92418—22no—
52C2×C8×D61—28no—
53C2×C8×D61—48no—
54C2×C22⋊Dic61—22no—
55C2×C22⋊Dic61—22no—
56C2×C22⋊Dic61C2×C414yes—
57C2×#11758198—22no—
58C2×#11758198—22no—
59C2×#11758198—22no—
60C2×#11758198—22no—
61C2×#11758198C2×C412yes—
62C2×#11758198C2×C414yes—
63C2×#11758198C2312yes—
64C22⋊C4×D61—28no—
65C22⋊C4×D61—28no—
66C22⋊C4×D61—28no—
67C22⋊C4×D61—28no—
68C22⋊#92418—22no—
69C22⋊#92418C2314yescomplemented
70#14519689—28no—
71C25×C61C2×C411yescharacteristic
72C23×C4×C61C2×C411yescharacteristic
73C23×C4×C61C2311yescharacteristic
74C22×C8×C61—22no—
75C2×C22⋊C4×C61—22no—
76C2×C22⋊C4×C61—22no—
77C2×C22⋊C4×C61C2×C412yes—
78C2×C22⋊C4×C61C2312yes—
79C22⋊C8×C61—22no—
80C22⋊C8×C61C2314yescomplemented
81C24×C61—21no—
82C24×C61—21no—
83C24×C61—22no—
84C24×C61—42no—
85C24×C61—42no—
86C24×C61C2×C814yescomplemented
87C24×C61C22×C411yescharacteristic
88C24×C61C22×C412yes—
89C23×D61—24no—
90C23×D61—24no—
91C23×D61—21no—
92C23×D61—21no—
93C23×D61—22no—
94C23×D61—216no—
95C23×D61—28no—
96C23×D61—28no—
97C23×D61—24no—
98C23×D61—24no—
99C23×D61—28no—
100C23×D61—44no—
101C23×D61—44no—
102C23×D61—432no—
103C23×D61—48no—
104C23×D61—48no—
105C23×D61—432no—
106C23×D61—48no—
107C23×D61—48no—
108C23×D61—44no—
109C23×D61—44no—
110C23×D61—42no—
111C23×D61—42no—
112C23×D61C22⋊C411yescharacteristic
113C23×D61C22⋊C412yes—
114C23×D61C22⋊C411yescharacteristic
115C23×D61C2×C814yes—
116C23×D61C2×C8116yescomplemented
117C23×D61C2×C814yescomplemented
118C23×D61M1614yes—
119C23×D61C22×C412yes—
120C23×D61C22×C412yes—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—42no—
124C22×C4×C61—42no—
125C22×C4×C61C22×C412yes—
126C22×C4×C61C22×C411yescharacteristic
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2411yescharacteristic
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—42no—
133C22×Dic61—42no—
134C22×Dic61C22⋊C412yes—
135C22×Dic61C22⋊C412yes—
136C22×Dic61C22×C412yes—
137C22×Dic61C22×C412yes—
138C2×C4×D61—28no—
139C2×C4×D61—24no—
140C2×C4×D61—24no—
141C2×C4×D61—44no—
142C2×C4×D61—48no—
143C2×C4×D61—44no—
144C2×C4×D61—44no—
145C2×C4×D61—44no—
146C2×C4×D61—48no—
147C2×C4×D61—44no—
148C2×C4×D61—48no—
149C2×C4×D61C22⋊C414yes—
150C2×C4×D61C22⋊C414yes—
151C2×C4×D61C2×D414yes—
152C2×C4×D61C2×D414yes—
153C2×C8×C61—24no—
154C2×C8×C61—44no—
155C22⋊C4×C61—24no—
156C22⋊C4×C61—24no—
157C22⋊Dic61—28no—
158#11758198—24no—
159#11758198—24no—
160#11758198—28no—
161C2×#92418—24no—
162C2×#92418—44no—
163C8×D61—416no—
164C23×C61—22no—
165C23×C61—22no—
166C23×C61—22no—
167C23×C61—22no—
168C23×C61—24no—
169C23×C61—24no—
170C23×C61—42no—
171C23×C61—42no—
172C23×C61—48no—
173C23×C61—48no—
174C23×C61—42no—
175C23×C61—42no—
176C23×C61—42no—
177C23×C61—42no—
178C23×C61C2×C22⋊C411yescharacteristic
179C23×C61C2×C22⋊C411yescharacteristic
180C23×C61C22×C812yes—
181C23×C61C22×C814yescomplemented
182C23×C61C2×M1612yes—
183C23×C61C23×C411yescharacteristic
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—216no—
187C22×D61—216no—
188C22×D61—24no—
189C22×D61—24no—
190C22×D61—24no—
191C22×D61—24no—
192C22×D61—22no—
193C22×D61—22no—
194C22×D61—24no—
195C22×D61—24no—
196C22×D61—216no—
197C22×D61—216no—
198C22×D61—48no—
199C22×D61—48no—
200C22×D61—432no—
201C22×D61—464no—
202C22×D61—432no—
203C22×D61—432no—
204C22×D61—42no—
205C22×D61—42no—
206C22×D61—48no—
207C22×D61—42no—
208C22×D61—42no—
209C22×D61—432no—
210C22×D61—48no—
211C22×D61—48no—
212C22×D61—48no—
213C22×D61—42no—
214C22×D61—42no—
215C22×D61C22⋊C814yes—
216C22×D61C22⋊C818yes—
217C22×D61C22⋊C814yes—
218C22×D61#9914yes—
219C22×D61#10014yes—
220C22×D61C2×C22⋊C412yes—
221C22×D61C2×C22⋊C412yes—
222C22×D61C22×C814yes—
223C22×D61C22×C814yescomplemented
224C22×D61C2×M1614yes—
225C2×C4×C61—24no—
226C2×C4×C61—22no—
227C2×C4×C61—41no—
228C2×C4×C61—42no—
229C2×C4×C61—41no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61C2×C22⋊C412yes—
234C2×C4×C61C22×D412yes—
235C2×Dic61—24no—
236C2×Dic61—22no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61C2×C22⋊C414yes—
243C4×D61—28no—
244C4×D61—28no—
245C4×D61—44no—
246C4×D61—44no—
247C4×D61—48no—
248C4×D61—48no—
249C8×C61—44no—
250#92418—44no—
251C22×#323—611nomaximal, Sylow
252C22×C61—21no—
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—21no—
256C22×C61—24no—
257C22×C61—24no—
258C22×C61—24no—
259C22×C61—44no—
260C22×C61—44no—
261C22×C61—48no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—48no—
265C22×C61C2×C22⋊C812yes—
266C22×C61C2×C22⋊C812yes—
267C22×C61C2×#9912yes—
268C22×C61C2×#10012yes—
269C22×C61C22×C22⋊C411yescharacteristic
270C22×C61C23×C811yescharacteristic
271C22×C61C22×M1611yescharacteristic
272C2×D61—22no—
273C2×D61—22no—
274C2×D61—28no—
275C2×D61—216no—
276C2×D61—28no—
277C2×D61—28no—
278C2×D61—24no—
279C2×D61—432no—
280C2×D61—44no—
281C2×D61—48no—
282C2×D61—42no—
283C2×D61—42no—
284C2×D61—48no—
285C2×D61—432no—
286C2×D61—44no—
287C2×D61#323116yesdirect product factor
288C2×D61C2×C22⋊C814yes—
289C2×D61C2×C22⋊C814yes—
290C2×D61C2×#9914yes—
291C2×D61C2×#10014yes—
292C4×C61—22no—
293C4×C61—22no—
294C4×C61—42no—
295Dic61—24no—
296Dic61—42no—
297C23×C22⋊C4—611no—
298C22×C22⋊C8—612no—
299C2×#323—612no—
300C2×#323—618no—
301C2×#323—611no—
302C2×#323D6111yescharacteristic, direct product factor
303C2×C61—21no—
304C2×C61—21no—
305C2×C61—22no—
306C2×C61—44no—
307C2×C61C2×#32314yescomplemented
308C2×C61C22×C22⋊C811yescharacteristic
309C2×C61C22×#9911yescharacteristic
310C2×C61C22×#10011yescharacteristic
311D61—24no—
312D61—44no—
313D61C2×#32318yesdirect product factor
314C26—611no—
315C24×C4—611no—
316C24×C4—611no—
317C23×C8—1222no—
318C22×C22⋊C4—611no—
319C22×C22⋊C4—611no—
320C22×C22⋊C4—611no—
321C22×C22⋊C4—614no—
322C22×C22⋊C4—614no—
323C22×C22⋊C4—1222no—
324C22×C22⋊C4—1224no—
325C22×C22⋊C4—1222no—
326C22×C22⋊C4C2×D6111yescharacteristic
327C2×C22⋊C8—612no—
328C2×C22⋊C8—618no—
329C2×C22⋊C8—1224no—
330C2×C22⋊C8—1222no—
331C2×C22⋊C8C2×D6112yescomplemented
332#323—614no—
333#323—618no—
334#323C2×D6114yesdirect product factor
335C61C22×#32311yescharacteristic, complemented, Sylow
336C25—611no—
337C25—618no—
338C25—614no—
339C25—611no—
340C25—1222no—
341C25—1221no—
342C25—1221no—
343C25—1224no—
344C25—2444no—
345C25—2444no—
346C25C4×D6111yescharacteristic
347C23×C4—611no—
348C23×C4—612no—
349C23×C4—612no—
350C23×C4—612no—
351C23×C4—614no—
352C23×C4—612no—
353C23×C4—611no—
354C23×C4—614no—
355C23×C4—1222no—
356C23×C4—1221no—
357C23×C4—1221no—
358C23×C4—2444no—
359C23×C4—2444no—
360C23×C4C4×D6111yescharacteristic
361C23×C4C22×D6111yescharacteristic
362C22×C8—22no—
363C22×C8—1228no—
364C22×C8—1222no—
365C22×C8—2448no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—614no—
369C2×C22⋊C4—614no—
370C2×C22⋊C4—612no—
371C2×C22⋊C4—612no—
372C2×C22⋊C4—1228no—
373C2×C22⋊C4—1228no—
374C2×C22⋊C4—1228no—
375C2×C22⋊C4—1228no—
376C2×C22⋊C4—1222no—
377C2×C22⋊C4—1222no—
378C2×C22⋊C4—1222no—
379C2×C22⋊C4—1222no—
380C2×C22⋊C4—1222no—
381C2×C22⋊C4—1222no—
382C2×C22⋊C4C4×D6112yes—
383C2×C22⋊C4C22×D6112yes—
384C22⋊C8—22no—
385C22⋊C8—614no—
386C22⋊C8—1228no—
387C22⋊C8—1222no—
388C22⋊C8C22×D6114yescomplemented
389C24—21no—
390C24—21no—
391C24—22no—
392C24—42no—
393C24—42no—
394C24—611no—
395C24—612no—
396C24—612no—
397C24—611no—
398C24—614no—
399C24—614no—
400C24—612no—
401C24—614no—
402C24—6116no—
403C24—1224no—
404C24—1224no—
405C24—1224no—
406C24—1224no—
407C24—1228no—
408C24—12216no—
409C24—1228no—
410C24—1228no—
411C24—1221no—
412C24—1221no—
413C24—1222no—
414C24—2442no—
415C24—2442no—
416C24—2444no—
417C24—2444no—
418C24—24432no—
419C24—2448no—
420C24—2448no—
421C24—2448no—
422C24—2448no—
423C24—24432no—
424C24—2444no—
425C24—2444no—
426C24C8×D6114yescomplemented
427C24C2×C4×D6111yescharacteristic
428C24C2×C4×D6112yes—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—612no—
434C22×C4—612no—
435C22×C4—612no—
436C22×C4—614no—
437C22×C4—614no—
438C22×C4—614no—
439C22×C4—614no—
440C22×C4—612no—
441C22×C4—1228no—
442C22×C4—1224no—
443C22×C4—1224no—
444C22×C4—1221no—
445C22×C4—1221no—
446C22×C4—2444no—
447C22×C4—2444no—
448C22×C4—2448no—
449C22×C4—2444no—
450C22×C4—2444no—
451C22×C4—2448no—
452C22×C4—2444no—
453C22×C4—2448no—
454C22×C4—2442no—
455C22×C4—2442no—
456C22×C4C2×C4×D6112yes—
457C22×C4C2×C4×D6111yescharacteristic
458C22×C4D4×D6112yes—
459C22×C4D4×D6112yes—
460C22×C4C23×D6111yescharacteristic
461C2×C8—24no—
462C2×C8—44no—
463C2×C8—1224no—
464C2×C8—24416no—
465C2×C8—2444no—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—1228no—
469C22⋊C4—1228no—
470C22⋊C4—1224no—
471C22⋊C4—1224no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—24no—
476C23—24no—
477C23—22no—
478C23—48no—
479C23—42no—
480C23—42no—
481C23—48no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—612no—
487C23—614no—
488C23—614no—
489C23—614no—
490C23—612no—
491C23—618no—
492C23—614no—
493C23—614no—
494C23—614no—
495C23—614no—
496C23—1224no—
497C23—1224no—
498C23—1224no—
499C23—1224no—
500C23—12216no—
501C23—12216no—
502C23—12216no—
503C23—12216no—
504C23—1222no—
505C23—1222no—
506C23—1222no—
507C23—1224no—
508C23—1224no—
509C23—1222no—
510C23—2448no—
511C23—2442no—
512C23—2442no—
513C23—2448no—
514C23—2442no—
515C23—2442no—
516C23—24432no—
517C23—2448no—
518C23—2448no—
519C23—24432no—
520C23—24464no—
521C23—24432no—
522C23—24432no—
523C23—2448no—
524C23—2448no—
525C23—2442no—
526C23—2442no—
527C23C22⋊C4×D6111yescharacteristic
528C23C22⋊C4×D6111yescharacteristic
529C23C2×C8×D6112yes—
530C23C2×C8×D6114yescomplemented
531C23M16×D6112yes—
532C23C22×C4×D6111yescharacteristic
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—614no—
542C2×C4—1228no—
543C2×C4—1228no—
544C2×C4—1224no—
545C2×C4—1222no—
546C2×C4—2444no—
547C2×C4—2444no—
548C2×C4—2448no—
549C2×C4—2448no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4C22⋊C4×D6112yes—
556C2×C4C2×D4×D6112yes—
557C8—44no—
558C8—2444no—
559C22—24no—
560C22—24no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—24no—
566C22—44no—
567C22—48no—
568C22—44no—
569C22—48no—
570C22—42no—
571C22—42no—
572C22—6116no—
573C22—614no—
574C22—614no—
575C22—614no—
576C22—614no—
577C22—12216no—
578C22—1228no—
579C22—1228no—
580C22—1228no—
581C22—1222no—
582C22—1222no—
583C22—1224no—
584C22—2444no—
585C22—2448no—
586C22—2444no—
587C22—24432no—
588C22—24432no—
589C22—2448no—
590C22—2442no—
591C22—2442no—
592C22C22⋊C8×D6112yes—
593C22C22⋊C8×D6112yes—
594C22#99×D6112yes—
595C22#100×D6112yes—
596C22C2×C22⋊C4×D6111yescharacteristic
597C22C22×C8×D6111yescharacteristic
598C22C2×M16×D6111yescharacteristic
599C4—22no—
600C4—22no—
601C4—42no—
602C4—1224no—
603C4—2442no—
604C2—22no—
605C2—21no—
606C2—21no—
607C2—44no—
608C2—618no—
609C2—1224no—
610C2—2444no—
611C2#323×D6114yesdirect product factor
612C2C2×C22⋊C8×D6111yescharacteristic
613C2C2×#99×D6111yescharacteristic
614C2C2×#100×D6111yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC22×C61C23×C8
FrattiniC22×C4C23×D61
FittingC2×C61×#323C2
socleC23×C61C2×C22⋊C4
prime radicalC25×D61C4
cosocleC22×C4×C61C24
hypercenterC2×#323D61
hypocenterC61C22×#323
LCSG ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊C8×D61C2semidirect
3. C2×#16839486C2semidirect
4. C2×#16839485C2semidirect
5. #323×D61C2direct (N × H)
6. C2×C61×#323C2semidirect
19. C2×C22⋊#92418C22semidirect
22. C22⋊C8×D61C22semidirect
23. #16839486C22semidirect
24. #16839485C22semidirect
26. C2×C22⋊C8×C61C22semidirect
27. C61×#323C22semidirect
34. C24×D61C8semidirect
69. C22⋊#92418C23semidirect
80. C22⋊C8×C61C23semidirect
86. C24×C61C2×C8semidirect
116. C23×D61C2×C8semidirect
117. C23×D61C2×C8semidirect
181. C23×C61C22×C8semidirect
223. C22×D61C22×C8semidirect
287. C2×D61#323direct (N × H)
302. C2×#323D61direct (N × H)
307. C2×C61C2×#323semidirect
313. D61C2×#323direct (N × H)
331. C2×C22⋊C8C2×D61semidirect
334. #323C2×D61direct (N × H)
335. C61C22×#323semidirect
388. C22⋊C8C22×D61semidirect
426. C24C8×D61semidirect
530. C23C2×C8×D61semidirect
611. C2#323×D61direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C22⋊C4×D612
C2×C22⋊C8×D612
C2×#168394862
C2×#168394852
#323×D612
C2×C61×#3232
C22×#32361