Finite Group Databasebeta

← C2×#21456651 #30550633

Subgroups — C2×#21456651 #30550633

Order 15616.

2342 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840273C211yesmaximal, characteristic
2C22×#14519756C211yesmaximal, characteristic, complemented
3#21456651C2116yesmaximal, direct product factor
4C2×#16840098C211yesmaximal, characteristic, complemented
5C2×#16840081C211yesmaximal, characteristic, complemented
6C2×#16840104C212yesmaximal, complemented
7C2×#16840020C212yesmaximal, complemented
8C2×#16840008C212yesmaximal, complemented
9C2×#16839926C212yesmaximal, complemented
10C2×#16839920C211yesmaximal, characteristic, complemented
11C2×#16839903C211yesmaximal, characteristic, complemented
12C2×C61×#407C211yesmaximal, characteristic, complemented
13C22×#11758222C2211yescharacteristic
14C23×Dic122C2211yescharacteristic
15C22×Dic244C2212yescomplemented
16C22×#11758190C2212yescomplemented
17#16840273—24no—
18#16840273C2218yes—
19C2×#14519760—21no—
20C2×#14519756—21no—
21C2×#14519756—21no—
22C2×#14519756—24no—
23C2×#14519756C2212yes—
24C2×#14519756C2211yescharacteristic
25C2×#14519756C2212yes—
26C2×#14519756C2211yescharacteristic
27C2×#14519756C2218yescomplemented
28C2×#14519751C2211yescharacteristic
29C2×#14519751C2211yescharacteristic
30C2×#14519746—21no—
31C2×C4×C22⋊D61—21no—
32C2×C4×C22⋊D61—21no—
33C2×#14519736C2211yescharacteristic
34C22×C4⋊Dic61C2211yescharacteristic
35C22×#11758196—21no—
36C22×C22⋊Dic61—21no—
37#16840104C22116yescomplemented
38#16840098C2218yescomplemented
39#16840081C2218yescomplemented
40#16840020C22116yescomplemented
41#16840008C22116yescomplemented
42#16839926C22116yescomplemented
43#16839920C2218yescomplemented
44#16839903C2218yescomplemented
45C2×#14519711C2212yescomplemented
46C2×#14519709C2212yescomplemented
47C2×#14519702C2212yescomplemented
48C2×#14519702C2212yescomplemented
49C2×#14519702C2212yescomplemented
50C2×#14519699C2212yescomplemented
51C24×C4×C61C2211yescharacteristic
52C22×M16×C61C2212yes—
53C61×#407C2218yescomplemented
54C2×C22⋊C8×C61C2212yes—
55C2×C22⋊C8×C61C2211yescharacteristic
56C2×C22⋊C8×C61C2211yescharacteristic
57C2×#11758222—22no—
58C2×#11758222—44no—
59C2×#11758222C2314yes—
60C22×Dic122—21no—
61C22×Dic122—22no—
62C22×Dic122—44no—
63C22×Dic122C2312yes—
64C22×Dic122C2311yescharacteristic
65C22×Dic122C2314yes—
66C22×D244C2311yescharacteristic
67C22×C22⋊D61—21no—
68C22×C22⋊D61—21no—
69C22×C22⋊D61—21no—
70C23×Dic61—21no—
71C23×Dic61—21no—
72C22×C4×D61—21no—
73C2×Dic244—44no—
74C2×Dic244C2318yescomplemented
75C2×#11758190—44no—
76C2×#11758190C2318yescomplemented
77#14519760—28no—
78#14519756—24no—
79#14519756—24no—
80#14519756—28no—
81#14519756—28no—
82#14519756—22no—
83#14519756—22no—
84#14519756C2318yes—
85#14519756C2318yes—
86#14519756C2314yes—
87#14519756C2314yes—
88#14519751—22no—
89#14519751—22no—
90#14519751C2314yes—
91#14519751C2314yes—
92#14519746—28no—
93C4×C22⋊D61—28no—
94C4×C22⋊D61—28no—
95#14519736—22no—
96#14519736C2314yes—
97C2×C4⋊Dic61—21no—
98C2×C4⋊Dic61—22no—
99C2×C4⋊Dic61C2312yes—
100C2×C4⋊Dic61C2314yes—
101C2×C4⋊Dic61C2311yescharacteristic
102C2×C4⋊Dic61C2311yescharacteristic
103C2×#11758196—22no—
104C2×#11758196—22no—
105C2×#11758196—28no—
106C2×#11758196—21no—
107C2×#11758196—21no—
108C2×C22⋊Dic61—22no—
109C2×C22⋊Dic61—22no—
110C2×C22⋊Dic61—28no—
111C2×C22⋊Dic61—21no—
112C2×C22⋊Dic61—21no—
113C2×C22⋊Dic61—21no—
114C2×#11758198—21no—
115C2×#11758198—21no—
116C2×C4×Dic61—21no—
117#14519711C2318yescomplemented
118#14519709C2318yescomplemented
119#14519702C2318yescomplemented
120#14519702C2318yescomplemented
121#14519702C2318yescomplemented
122#14519699C2318yescomplemented
123C25×C61D411yescharacteristic
124C23×C4×C61—44no—
125C23×C4×C61D411yescharacteristic
126C23×C4×C61D411yescharacteristic
127C23×C4×C61D411yescharacteristic
128C23×C4×C61D414yes—
129C23×C4×C61C2311yescharacteristic
130C23×C4×C61C2311yescharacteristic
131C23×C4×C61C2311yescharacteristic
132C23×C4×C61C2314yes—
133C2×M16×C61—44no—
134C2×M16×C61C2318yes—
135C22×C8×C61C2312yes—
136C22×C8×C61C2312yes—
137C22⋊C8×C61C2314yes—
138C22⋊C8×C61C2314yes—
139C22⋊C8×C61C2318yes—
140C24×C61—24no—
141C24×C61—42no—
142C24×C61—42no—
143C24×C61C2×D411yescharacteristic
144C24×C61C2×D411yescharacteristic
145C24×C61C2×D411yescharacteristic
146C24×C61C2×D414yes—
147C23×D61—21no—
148C22×C4×C61—24no—
149C22×C4×C61—24no—
150C22×C4×C61—24no—
151C22×C4×C61—48no—
152C22×C4×C61—48no—
153C22×C4×C61—42no—
154C22×C4×C61—42no—
155C22×C4×C61—42no—
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D412yes—
159C22×C4×C61C2×D412yes—
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2×D412yes—
162C22×C4×C61C2×D414yes—
163C22×C4×C61C2×D414yes—
164C22×C4×C61C2×D414yes—
165C22×C4×C61C2412yes—
166C22×C4×C61C2412yes—
167C22×C4×C61C2412yes—
168C22×C4×C61C2411yescharacteristic
169C2×C22⋊D61—24no—
170C2×C22⋊D61—24no—
171C2×C22⋊D61—24no—
172C2×C22⋊D61—44no—
173C2×C22⋊D61—44no—
174C2×C22⋊D61—44no—
175C22×Dic61—21no—
176C22×Dic61—24no—
177C22×Dic61—22no—
178C22×Dic61—24no—
179C22×Dic61—22no—
180C22×Dic61—21no—
181C22×Dic61—21no—
182C22×Dic61—44no—
183C22×Dic61—44no—
184#11758222—48no—
185#11758222—48no—
186C2×D244—24no—
187C2×D244—21no—
188C2×D244C2412yes—
189C2×C4×D61—24no—
190C2×C4×D61—44no—
191C2×Dic122—24no—
192C2×Dic122—24no—
193C2×Dic122—24no—
194C2×Dic122—24no—
195C2×Dic122—22no—
196C2×Dic122—21no—
197C2×Dic122—48no—
198C2×Dic122—44no—
199C2×Dic122—48no—
200C2×Dic122C2414yes—
201C2×Dic122C2412yes—
202M16×C61—48no—
203C2×C8×C61—24no—
204C2×C8×C61—24no—
205C2×C8×C61C2414yes—
206C2×C8×C61C2414yes—
207C22⋊Dic61—28no—
208C22⋊Dic61—28no—
209C22⋊Dic61—24no—
210C22⋊Dic61—24no—
211C22⋊Dic61—24no—
212#11758198—24no—
213#11758198—24no—
214C4⋊Dic61—24no—
215C4⋊Dic61—22no—
216C4⋊Dic61—21no—
217C4⋊Dic61—21no—
218C4⋊Dic61C2414yes—
219C4⋊Dic61C2412yes—
220C4⋊Dic61C2412yes—
221#11758196—28no—
222#11758196—28no—
223#11758196—24no—
224#11758196—24no—
225C4×Dic61—24no—
226Dic244—48no—
227#11758190—48no—
228C23×C61—22no—
229C23×C61—22no—
230C23×C61—22no—
231C23×C61—22no—
232C23×C61—28no—
233C23×C61—22no—
234C23×C61—22no—
235C23×C61—22no—
236C23×C61—22no—
237C23×C61—44no—
238C23×C61—44no—
239C23×C61—416no—
240C23×C61#13714yes—
241C23×C61C22×D412yes—
242C23×C61C22×D412yes—
243C23×C61C22×D412yes—
244C23×C61C22×D411yescharacteristic
245C22×D61—22no—
246C22×D61—42no—
247C22×D61—42no—
248C22×D61—42no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—24no—
253C2×C4×C61—24no—
254C2×C4×C61—24no—
255C2×C4×C61—24no—
256C2×C4×C61—24no—
257C2×C4×C61—24no—
258C2×C4×C61—44no—
259C2×C4×C61—44no—
260C2×C4×C61—44no—
261C2×C4×C61C22×D412yes—
262C2×C4×C61C22×D412yes—
263C2×C4×C61C22×D412yes—
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61H(5,𝔽2)14yes—
266C2×C4×C61C2511yescharacteristic
267C22⋊D61—48no—
268C22⋊D61—48no—
269C22⋊D61—48no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—24no—
273C2×Dic61—24no—
274C2×Dic61—22no—
275C2×Dic61—48no—
276C2×Dic61—48no—
277C2×Dic61—44no—
278C2×Dic61—42no—
279C2×Dic61—42no—
280C2×Dic61—44no—
281C2×Dic61—42no—
282D244—24no—
283D244—42no—
284C4×D61—48no—
285Dic122—24no—
286Dic122—24no—
287Dic122—24no—
288Dic122—48no—
289Dic122—44no—
290Dic122—42no—
291C8×C61—24no—
292C8×C61—24no—
293C2×#3196—611nomaximal, Sylow
294C22×C61—22no—
295C22×C61—28no—
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—22no—
299C22×C61—22no—
300C22×C61—22no—
301C22×C61—22no—
302C22×C61—22no—
303C22×C61—416no—
304C22×C61—44no—
305C22×C61—44no—
306C22×C61#53412yes—
307C22×C61C2×#13714yes—
308C22×C61C2×#13712yes—
309C22×C61#57512yes—
310C22×C61C23×D411yescharacteristic
311C2×D61—42no—
312C2×D61—42no—
313C2×D61—42no—
314C2×D61—84no—
315C4×C61—22no—
316C4×C61—22no—
317C4×C61—22no—
318C4×C61#53412yes—
319C4×C61C2×H(5,𝔽2)12yes—
320Dic61—44no—
321Dic61—42no—
322Dic61—42no—
323Dic61—44no—
324Dic61—42no—
325C2×#525—611no—
326C22×C22⋊Q8—611no—
327#3196—6116no—
328C2×#484—611no—
329C2×#481—611no—
330C2×#485—612no—
331C2×#470—612no—
332C2×#468—612no—
333C2×#452—612no—
334C2×#451—611no—
335C2×#448—611no—
336C2×#407D6111yescharacteristic, complemented
337C2×C61—24no—
338C2×C61—42no—
339C2×C61—42no—
340C2×C61#319614yescomplemented
341C2×C61C2×#53411yescharacteristic
342C2×C61C22×#13711yescharacteristic
343C2×C61C2×#57511yescharacteristic
344D61—82no—
345C22×#41—611no—
346C23×Q8—611no—
347C24×C4C2×D6111yescharacteristic
348C22×Q16—612no—
349C22×SD16—612no—
350C22×M16C2×D6112yes—
351#525—618no—
352#525—1224no—
353C2×#123—1221no—
354C2×C22⋊Q8—612no—
355C2×C22⋊Q8—611no—
356C2×C22⋊Q8—612no—
357C2×C22⋊Q8—611no—
358C2×C22⋊Q8—618no—
359C2×C22⋊Q8—1221no—
360C2×C22⋊Q8—1221no—
361C2×C22⋊Q8—1224no—
362C2×#121—611no—
363C2×#121—611no—
364C2×#120—1221no—
365C2×C4×D4—1221no—
366C2×C4×D4—1221no—
367C2×#117—611no—
368C22×C4⋊C4—611no—
369C22×C4⋊C4—1221no—
370C22×C22⋊C4—1221no—
371#485—6116no—
372#484—618no—
373#481—618no—
374#470—6116no—
375#468—6116no—
376#452—6116no—
377#451—618no—
378#448—618no—
379C2×#107—612no—
380C2×#106—612no—
381C2×#103—612no—
382C2×#103—612no—
383C2×#103—612no—
384C2×#102—612no—
385#407C2×D6118yescomplemented
386C2×C22⋊C8C2×D6112yes—
387C2×C22⋊C8C2×D6111yescharacteristic
388C2×C22⋊C8C2×D6111yescharacteristic
389C61C2×#319611yescharacteristic, complemented, Sylow
390C25D24411yescharacteristic
391C2×#41—614no—
392C2×#41—1222no—
393C2×#41—2444no—
394C22×Q8—612no—
395C22×Q8—611no—
396C22×Q8—614no—
397C22×Q8—1222no—
398C22×Q8—1221no—
399C22×Q8—2444no—
400C22×D4—611no—
401C22×D4—1221no—
402C22×D4—1221no—
403C22×D4—1221no—
404C23×C4—44no—
405C23×C4—1221no—
406C23×C4—1221no—
407C23×C4—1221no—
408C23×C4D24411yescharacteristic
409C23×C4D24411yescharacteristic
410C23×C4D24411yescharacteristic
411C23×C4D24414yes—
412C23×C4C22×D6111yescharacteristic
413C23×C4C22×D6111yescharacteristic
414C23×C4C22×D6111yescharacteristic
415C23×C4C22×D6114yes—
416C2×Q16—618no—
417C2×Q16—2444no—
418C2×SD16—618no—
419C2×SD16—2444no—
420C2×M16—44no—
421C2×M16C22×D6118yes—
422C22×C8C22×D6112yes—
423C22×C8C22×D6112yes—
424#123—1228no—
425C22⋊Q8—618no—
426C22⋊Q8—618no—
427C22⋊Q8—614no—
428C22⋊Q8—614no—
429C22⋊Q8—1224no—
430C22⋊Q8—1224no—
431C22⋊Q8—1228no—
432C22⋊Q8—1228no—
433C22⋊Q8—1222no—
434C22⋊Q8—1222no—
435#121—614no—
436#121—614no—
437#121—1222no—
438#121—1222no—
439#120—1228no—
440C4×D4—1228no—
441C4×D4—1228no—
442#117—614no—
443#117—1222no—
444C2×C4⋊C4—612no—
445C2×C4⋊C4—614no—
446C2×C4⋊C4—611no—
447C2×C4⋊C4—611no—
448C2×C4⋊C4—1222no—
449C2×C4⋊C4—1221no—
450C2×C4⋊C4—1228no—
451C2×C4⋊C4—1222no—
452C2×C4⋊C4—1222no—
453C2×C4⋊C4—1221no—
454C2×C4⋊C4—1221no—
455C2×C22⋊C4—1228no—
456C2×C22⋊C4—1222no—
457C2×C22⋊C4—1222no—
458C2×C22⋊C4—1221no—
459C2×C22⋊C4—1221no—
460C2×C22⋊C4—1221no—
461C2×C22⋊C4—1221no—
462C2×C22⋊C4—1221no—
463C2×C42—1221no—
464#107—618no—
465#106—618no—
466#103—618no—
467#103—618no—
468#103—618no—
469#102—618no—
470C22⋊C8C22×D6114yes—
471C22⋊C8C22×D6114yes—
472C22⋊C8C22×D6118yes—
473C24—24no—
474C24—42no—
475C24—42no—
476C24—1221no—
477C24C2×D24411yescharacteristic
478C24C2×D24411yescharacteristic
479C24C2×D24411yescharacteristic
480C24C2×D24414yes—
481#41—2448no—
482#41—2448no—
483C2×Q8—614no—
484C2×Q8—612no—
485C2×Q8—1224no—
486C2×Q8—1222no—
487C2×Q8—1221no—
488C2×Q8—1224no—
489C2×Q8—1224no—
490C2×Q8—1224no—
491C2×Q8—2448no—
492C2×Q8—2444no—
493C2×Q8—2448no—
494C2×D4—612no—
495C2×D4—1224no—
496C2×D4—1224no—
497C2×D4—1224no—
498C2×D4—1221no—
499C2×D4—1224no—
500C2×D4—2444no—
501C2×D4—2444no—
502C2×D4—2444no—
503C22×C4—24no—
504C22×C4—24no—
505C22×C4—24no—
506C22×C4—48no—
507C22×C4—48no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4—1224no—
512C22×C4—1224no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1222no—
516C22×C4—1221no—
517C22×C4—1222no—
518C22×C4—1224no—
519C22×C4—2444no—
520C22×C4—2444no—
521C22×C4—2444no—
522C22×C4C2×D24412yes—
523C22×C4C2×D24412yes—
524C22×C4C2×D24412yes—
525C22×C4C2×D24411yescharacteristic
526C22×C4C2×D24411yescharacteristic
527C22×C4C2×D24411yescharacteristic
528C22×C4C2×D24414yes—
529C22×C4C2×D24414yes—
530C22×C4C2×D24414yes—
531C22×C4C23×D6112yes—
532C22×C4C23×D6112yes—
533C22×C4C23×D6112yes—
534C22×C4C23×D6111yescharacteristic
535Q16—2448no—
536SD16—2448no—
537M16—48no—
538C2×C8—24no—
539C2×C8—24no—
540C2×C8C23×D6114yes—
541C2×C8C23×D6114yes—
542C4⋊C4—614no—
543C4⋊C4—612no—
544C4⋊C4—612no—
545C4⋊C4—1224no—
546C4⋊C4—1222no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1228no—
550C4⋊C4—1228no—
551C4⋊C4—1224no—
552C4⋊C4—1224no—
553C22⋊C4—1228no—
554C22⋊C4—1228no—
555C22⋊C4—1224no—
556C22⋊C4—1224no—
557C22⋊C4—1224no—
558C22⋊C4—1224no—
559C22⋊C4—1224no—
560C42—1224no—
561C23—22no—
562C23—22no—
563C23—28no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—44no—
571C23—44no—
572C23—416no—
573C23—1222no—
574C23—2442no—
575C23—2442no—
576C23—2442no—
577C23#1451981014yes—
578C23C22×D24412yes—
579C23C22×D24412yes—
580C23C22×D24412yes—
581C23C22×D24411yescharacteristic
582Q8—1224no—
583Q8—1224no—
584Q8—1224no—
585Q8—2448no—
586Q8—2444no—
587Q8—2442no—
588D4—1224no—
589D4—2448no—
590D4—2448no—
591D4—2448no—
592D4—2442no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—1222no—
606C2×C4—1222no—
607C2×C4—1224no—
608C2×C4—1222no—
609C2×C4—1224no—
610C2×C4—2448no—
611C2×C4—2448no—
612C2×C4—2448no—
613C2×C4—2442no—
614C2×C4—2444no—
615C2×C4—2442no—
616C2×C4—2442no—
617C2×C4—2444no—
618C2×C4C22×D24412yes—
619C2×C4C22×D24412yes—
620C2×C4C22×D24412yes—
621C2×C4C22×D24411yescharacteristic
622C2×C4#1451982214yes—
623C2×C4C24×D6111yescharacteristic
624C8—24no—
625C8—24no—
626C22—28no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—416no—
636C22—44no—
637C22—44no—
638C22—4884no—
639C22—2442no—
640C22—2442no—
641C22—2442no—
642C22#1684032412yes—
643C22C2×#1451981014yes—
644C22C2×#1451981012yes—
645C22#1684056212yes—
646C22C23×D24411yescharacteristic
647C4—22no—
648C4—22no—
649C4—22no—
650C4—2442no—
651C4—2444no—
652C4—2442no—
653C4—2442no—
654C4—2444no—
655C4#1684032412yes—
656C4C2×#1451982212yes—
657C2—24no—
658C2—42no—
659C2—42no—
660C2—4882no—
661C2#2145665114yesdirect product factor
662C2C2×#1684032411yescharacteristic
663C2C22×#1451981011yescharacteristic
664C2C2×#1684056211yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#407C2
socleC23×C61C22×D4
prime radicalC2×#16840273C2
hypercenterC2×#407D61
hypocenterC61C2×#3196
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
2. C22×#14519756C2semidirect
3. #21456651C2direct (N × H)
4. C2×#16840098C2semidirect
5. C2×#16840081C2semidirect
6. C2×#16840104C2semidirect
7. C2×#16840020C2semidirect
8. C2×#16840008C2semidirect
9. C2×#16839926C2semidirect
10. C2×#16839920C2semidirect
11. C2×#16839903C2semidirect
12. C2×C61×#407C2semidirect
15. C22×Dic244C22semidirect
16. C22×#11758190C22semidirect
27. C2×#14519756C22semidirect
37. #16840104C22semidirect
38. #16840098C22semidirect
39. #16840081C22semidirect
40. #16840020C22semidirect
41. #16840008C22semidirect
42. #16839926C22semidirect
43. #16839920C22semidirect
44. #16839903C22semidirect
45. C2×#14519711C22semidirect
46. C2×#14519709C22semidirect
47. C2×#14519702C22semidirect
48. C2×#14519702C22semidirect
49. C2×#14519702C22semidirect
50. C2×#14519699C22semidirect
53. C61×#407C22semidirect
74. C2×Dic244C23semidirect
76. C2×#11758190C23semidirect
117. #14519711C23semidirect
118. #14519709C23semidirect
119. #14519702C23semidirect
120. #14519702C23semidirect
121. #14519702C23semidirect
122. #14519699C23semidirect
336. C2×#407D61semidirect
340. C2×C61#3196semidirect
385. #407C2×D61semidirect
389. C61C2×#3196semidirect
661. C2#21456651direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168402732
C22×#145197562
#214566512
C2×#168400982
C2×#168400812
C2×#168401042
C2×#168400202
C2×#168400082
C2×#168399262
C2×#168399202
C2×#168399032
C2×C61×#4072
C2×#319661