Finite Group Databasebeta

← C2×#21456648 #30550630

Subgroups — C2×#21456648 #30550630

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×#16840272C211yesmaximal, characteristic
2C22×#14519757C211yesmaximal, characteristic, complemented
3#21456648C2116yesmaximal, direct product factor
4C2×#16840097C211yesmaximal, characteristic, complemented
5C2×#16840106C212yesmaximal, complemented
6C2×#16840021C212yesmaximal, complemented
7C2×#16840007C212yesmaximal, complemented
8C2×#16839925C212yesmaximal, complemented
9C2×#16839920C211yesmaximal, characteristic, complemented
10C2×#16839900C211yesmaximal, characteristic, complemented
11C2×#16839724C211yesmaximal, characteristic, complemented
12C2×C61×#481C211yesmaximal, characteristic, complemented
13C22×#11758224C2211yescharacteristic
14C23×Dic122C2211yescharacteristic
15C23×C4×D61C2211yescharacteristic
16C22×Dic244C2212yescomplemented
17C22×#11758200C2212yescomplemented
18C22×#11758189C2212yes—
19#16840272—24no—
20#16840272C2218yes—
21C2×#14519759—21no—
22C2×#14519757—21no—
23C2×#14519757—21no—
24C2×#14519757—24no—
25C2×#14519757C2212yes—
26C2×#14519757C2211yescharacteristic
27C2×#14519757C2212yes—
28C2×#14519757C2218yescomplemented
29C2×#14519756C2211yescharacteristic
30C2×#14519748C2211yescharacteristic
31C2×#14519745—21no—
32C2×D4×Dic61—21no—
33C2×#14519737—21no—
34C2×#14519735C2211yescharacteristic
35C22×C4⋊Dic61—21no—
36C22×#11758198—21no—
37#16840106C22116yescomplemented
38#16840097C2218yescomplemented
39#16840021C22116yescomplemented
40#16840007C22116yescomplemented
41#16839925C22116yescomplemented
42#16839920C2218yescomplemented
43#16839900C2218yescomplemented
44C2×#14519710C2212yescomplemented
45C2×#14519702C2212yescomplemented
46C2×#14519700C2212yescomplemented
47C2×#14519700C2212yescomplemented
48#16839724C2218yescomplemented
49C2×C22⋊#92418C2211yescharacteristic
50C2×#14519689C2212yes—
51C2×#121×C61C2211yescharacteristic
52C22×C4⋊C4×C61C2211yescharacteristic
53C61×#481C2218yescomplemented
54C2×#106×C61C2212yescomplemented
55C2×#102×C61C2212yescomplemented
56C2×C22⋊C8×C61C2211yescharacteristic
57C24×D61D411yescharacteristic
58C2×#11758224—22no—
59C2×#11758224—44no—
60C2×#11758224C2314yes—
61C22×Dic122—21no—
62C22×Dic122—22no—
63C22×Dic122—44no—
64C22×Dic122C2311yescharacteristic
65C22×Dic122C2312yes—
66C22×Dic122C2314yes—
67C22×C22⋊D61—21no—
68C22×C22⋊D61—21no—
69C23×Dic61—21no—
70C23×Dic61—21no—
71C23×Dic61D411yescharacteristic
72C22×C4×D61—44no—
73C22×C4×D61D411yescharacteristic
74C22×C4×D61D411yescharacteristic
75C22×C4×D61D414yes—
76C22×C4×D61C2311yescharacteristic
77C22×C4×D61C2311yescharacteristic
78C22×C4×D61C2314yes—
79C2×Dic244—44no—
80C2×Dic244C2318yescomplemented
81C2×#11758200—44no—
82C2×#11758200C2318yescomplemented
83C2×#11758189—44no—
84C2×#11758189C2318yes—
85C22×#92418C2312yes—
86#14519759—28no—
87#14519757—24no—
88#14519757—24no—
89#14519757—28no—
90#14519757—28no—
91#14519757—22no—
92#14519757C2318yes—
93#14519757C2318yes—
94#14519757C2314yes—
95#14519756—22no—
96#14519756C2314yes—
97#14519748—22no—
98#14519748C2314yes—
99#14519745—28no—
100D4×Dic61—28no—
101#14519737—28no—
102#14519735—22no—
103#14519735C2314yes—
104C2×C4⋊Dic61—22no—
105C2×C4⋊Dic61—22no—
106C2×C4⋊Dic61—28no—
107C2×C4⋊Dic61—21no—
108C2×C4⋊Dic61C2311yescharacteristic
109C2×#11758196—21no—
110C2×C22⋊Dic61—21no—
111C2×C22⋊Dic61—21no—
112C2×#11758198—22no—
113C2×#11758198—22no—
114C2×#11758198—28no—
115C2×#11758198—21no—
116C2×#11758198—21no—
117C2×C4×Dic61—21no—
118#14519710C2318yescomplemented
119#14519702C2318yescomplemented
120#14519700C2318yescomplemented
121#14519700C2318yescomplemented
122C22⋊#92418C2314yes—
123#14519689C2318yes—
124C22×D4×C61—21no—
125C22×D4×C61C2311yescharacteristic
126C23×C4×C61—21no—
127C23×C4×C61C2311yescharacteristic
128C22×C8×C61C2312yes—
129#121×C61—22no—
130#121×C61C2314yes—
131C2×C4⋊C4×C61—21no—
132C2×C4⋊C4×C61—22no—
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C4⋊C4×C61C2312yes—
135C2×C4⋊C4×C61C2314yes—
136C2×C22⋊C4×C61—21no—
137#106×C61C2318yescomplemented
138#102×C61C2318yescomplemented
139C22⋊C8×C61C2314yes—
140C24×C61—21no—
141C24×C61C2×D411yescharacteristic
142C23×D61—24no—
143C23×D61—42no—
144C23×D61—42no—
145C23×D61C2×D411yescharacteristic
146C23×D61C2×D411yescharacteristic
147C23×D61C2×D414yes—
148C2×D4×C61—24no—
149C2×D4×C61—21no—
150C2×D4×C61—24no—
151C2×D4×C61—44no—
152C2×D4×C61C2412yes—
153C22×C4×C61—24no—
154C22×C4×C61—21no—
155C22×C4×C61—24no—
156C22×C4×C61—22no—
157C22×C4×C61—44no—
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D412yes—
160C22×C4×C61C2412yes—
161C22×C4×C61C2411yescharacteristic
162C2×C22⋊D61—24no—
163C2×C22⋊D61—24no—
164C2×C22⋊D61—44no—
165C2×C22⋊D61—44no—
166C22×Dic61—21no—
167C22×Dic61—24no—
168C22×Dic61—22no—
169C22×Dic61—21no—
170C22×Dic61—24no—
171C22×Dic61—42no—
172C22×Dic61—44no—
173C22×Dic61—44no—
174C22×Dic61C2×D411yescharacteristic
175C22×Dic61C2×D411yescharacteristic
176C22×Dic61C2×D414yes—
177#11758224—48no—
178#11758224—48no—
179C2×C4×D61—24no—
180C2×C4×D61—24no—
181C2×C4×D61—48no—
182C2×C4×D61—48no—
183C2×C4×D61—42no—
184C2×C4×D61—42no—
185C2×C4×D61C2×D412yes—
186C2×C4×D61C2×D412yes—
187C2×C4×D61C2×D414yes—
188C2×C4×D61C2×D414yes—
189C2×C4×D61C2412yes—
190C2×C4×D61C2412yes—
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—
202C2×C8×C61—24no—
203C2×C8×C61C2414yes—
204C4⋊C4×C61—24no—
205C4⋊C4×C61—22no—
206C4⋊C4×C61—21no—
207C4⋊C4×C61C2414yes—
208C4⋊C4×C61C2412yes—
209C22⋊C4×C61—24no—
210C22⋊Dic61—24no—
211C22⋊Dic61—24no—
212#11758200—48no—
213#11758198—28no—
214#11758198—28no—
215#11758198—24no—
216#11758198—24no—
217C4⋊Dic61—21no—
218C4⋊Dic61—28no—
219C4⋊Dic61—28no—
220C4⋊Dic61—24no—
221C4⋊Dic61C2412yes—
222#11758196—24no—
223C4×Dic61—24no—
224C2×#92418—24no—
225C2×#92418C2414yes—
226Dic244—48no—
227#11758189—48no—
228C23×C61—22no—
229C23×C61—22no—
230C23×C61—22no—
231C23×C61—42no—
232C23×C61—42no—
233C23×C61—42no—
234C23×C61#13714yes—
235C23×C61C22×D412yes—
236C23×C61C22×D411yescharacteristic
237C22×D61—22no—
238C22×D61—22no—
239C22×D61—28no—
240C22×D61—22no—
241C22×D61—22no—
242C22×D61—22no—
243C22×D61—22no—
244C22×D61—44no—
245C22×D61—44no—
246C22×D61—416no—
247C22×D61C22×D412yes—
248C22×D61C22×D412yes—
249D4×C61—24no—
250D4×C61—48no—
251D4×C61—42no—
252C2×C4×C61—22no—
253C2×C4×C61—24no—
254C2×C4×C61—24no—
255C2×C4×C61—22no—
256C2×C4×C61—24no—
257C2×C4×C61—48no—
258C2×C4×C61—42no—
259C2×C4×C61—44no—
260C2×C4×C61C22×D412yes—
261C2×C4×C61C22×D411yescharacteristic
262C2×C4×C61H(5,𝔽2)14yes—
263C2×C4×C61C2511yescharacteristic
264C22⋊D61—48no—
265C22⋊D61—48no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—24no—
271C2×Dic61—44no—
272C2×Dic61—48no—
273C2×Dic61—48no—
274C2×Dic61—44no—
275C2×Dic61—42no—
276C2×Dic61—42no—
277C2×Dic61C22×D412yes—
278C2×Dic61C22×D412yes—
279C4×D61—24no—
280C4×D61—24no—
281C4×D61—24no—
282C4×D61—24no—
283C4×D61—44no—
284C4×D61—44no—
285Dic122—24no—
286Dic122—24no—
287Dic122—24no—
288Dic122—48no—
289Dic122—44no—
290Dic122—42no—
291C8×C61—24no—
292#92418—24no—
293C2×#3196—611nomaximal, Sylow
294C22×C61—22no—
295C22×C61—28no—
296C22×C61—22no—
297C22×C61—42no—
298C22×C61—42no—
299C22×C61—42no—
300C22×C61—84no—
301C22×C61#53412yes—
302C22×C61C2×#13714yes—
303C22×C61C2×#13712yes—
304C22×C61#57512yes—
305C22×C61C23×D411yescharacteristic
306C2×D61—22no—
307C2×D61—22no—
308C2×D61—22no—
309C2×D61—22no—
310C2×D61—22no—
311C2×D61—22no—
312C2×D61—416no—
313C2×D61—44no—
314C2×D61—44no—
315C4×C61—22no—
316C4×C61—42no—
317C4×C61—44no—
318C4×C61#53412yes—
319C4×C61C2×H(5,𝔽2)12yes—
320Dic61—22no—
321Dic61—22no—
322Dic61—44no—
323Dic61—42no—
324Dic61—42no—
325C2×#525—611no—
326C22×C22⋊Q8—611no—
327#3196—6116no—
328C2×#484—611no—
329C2×#481D6111yescharacteristic, complemented
330C2×#485—612no—
331C2×#470—612no—
332C2×#468—612no—
333C2×#452—612no—
334C2×#451—611no—
335C2×#448—611no—
336C2×#407—611no—
337C2×C61—24no—
338C2×C61—82no—
339C2×C61#319614yescomplemented
340C2×C61C2×#53411yescharacteristic
341C2×C61C22×#13711yescharacteristic
342C2×C61C2×#57511yescharacteristic
343D61—42no—
344D61—42no—
345C22×#41—611no—
346C23×Q8—611no—
347C24×C4—611no—
348C22×Q16—612no—
349C22×SD16—612no—
350C22×M16—612no—
351#525—618no—
352#525—1224no—
353C2×#123—1221no—
354C2×C22⋊Q8—611no—
355C2×C22⋊Q8—612no—
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×#121C2×D6111yescharacteristic
364C2×#120—1221no—
365C2×C4×D4—1221no—
366C2×C4×D4—1221no—
367C2×#117—611no—
368C22×C4⋊C4—1221no—
369C22×C4⋊C4C2×D6111yescharacteristic
370C22×C22⋊C4—1221no—
371#485—6116no—
372#484—618no—
373#481C2×D6118yescomplemented
374#470—6116no—
375#468—6116no—
376#452—6116no—
377#451—618no—
378#448—618no—
379C2×#107—612no—
380C2×#106C2×D6112yescomplemented
381C2×#103—612no—
382C2×#103—612no—
383C2×#103—612no—
384C2×#102C2×D6112yescomplemented
385#407—618no—
386C2×C22⋊C8—611no—
387C2×C22⋊C8—612no—
388C2×C22⋊C8C2×D6111yescharacteristic
389C61C2×#319611yescharacteristic, complemented, Sylow
390C25—611no—
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—21no—
401C22×D4—1221no—
402C22×D4—1221no—
403C22×D4C22×D6111yescharacteristic
404C23×C4—21no—
405C23×C4—611no—
406C23×C4—611no—
407C23×C4—611no—
408C23×C4—611no—
409C23×C4—611no—
410C23×C4—614no—
411C23×C4—614no—
412C23×C4—1221no—
413C23×C4—1221no—
414C23×C4—2444no—
415C23×C4C22×D6111yescharacteristic
416C2×Q16—618no—
417C2×Q16—2444no—
418C2×SD16—618no—
419C2×SD16—2444no—
420C2×M16—618no—
421C2×M16—2444no—
422C22×C8—612no—
423C22×C8C22×D6112yes—
424#123—1228no—
425C22⋊Q8—614no—
426C22⋊Q8—618no—
427C22⋊Q8—618no—
428C22⋊Q8—614no—
429C22⋊Q8—1224no—
430C22⋊Q8—1224no—
431C22⋊Q8—1228no—
432C22⋊Q8—1228no—
433C22⋊Q8—1222no—
434C22⋊Q8—1222no—
435#121—22no—
436#121—614no—
437#121—1222no—
438#121C22×D6114yes—
439#120—1228no—
440C4×D4—1228no—
441C4×D4—1228no—
442#117—614no—
443#117—1222no—
444C2×C4⋊C4—22no—
445C2×C4⋊C4—21no—
446C2×C4⋊C4—611no—
447C2×C4⋊C4—1228no—
448C2×C4⋊C4—1222no—
449C2×C4⋊C4—1222no—
450C2×C4⋊C4—1221no—
451C2×C4⋊C4—1221no—
452C2×C4⋊C4C22×D6112yes—
453C2×C4⋊C4C22×D6114yes—
454C2×C4⋊C4C22×D6111yescharacteristic
455C2×C22⋊C4—21no—
456C2×C22⋊C4—1228no—
457C2×C22⋊C4—1222no—
458C2×C22⋊C4—1222no—
459C2×C22⋊C4—1221no—
460C2×C22⋊C4—1221no—
461C2×C22⋊C4—1221no—
462C2×C22⋊C4—1221no—
463C2×C42—1221no—
464#107—618no—
465#106C22×D6118yescomplemented
466#103—618no—
467#103—618no—
468#103—618no—
469#102C22×D6118yescomplemented
470C22⋊C8—614no—
471C22⋊C8—618no—
472C22⋊C8C22×D6114yes—
473C24—21no—
474C24—611no—
475C24—611no—
476C24—614no—
477C24—1224no—
478C24—2442no—
479C24—2442no—
480C24D4×D6111yescharacteristic
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—24no—
495C2×D4—21no—
496C2×D4—24no—
497C2×D4—44no—
498C2×D4—1224no—
499C2×D4—1224no—
500C2×D4—2444no—
501C2×D4—2444no—
502C2×D4C23×D6112yes—
503C22×C4—24no—
504C22×C4—24no—
505C22×C4—21no—
506C22×C4—22no—
507C22×C4—44no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—614no—
511C22×C4—612no—
512C22×C4—612no—
513C22×C4—612no—
514C22×C4—612no—
515C22×C4—614no—
516C22×C4—614no—
517C22×C4—1224no—
518C22×C4—1224no—
519C22×C4—1224no—
520C22×C4—1224no—
521C22×C4—1221no—
522C22×C4—1221no—
523C22×C4—1222no—
524C22×C4—2448no—
525C22×C4—2448no—
526C22×C4—2442no—
527C22×C4—2442no—
528C22×C4—2442no—
529C22×C4—2444no—
530C22×C4—2444no—
531C22×C4D4×D6112yes—
532C22×C4D4×D6111yescharacteristic
533C22×C4C23×D6112yes—
534C22×C4C23×D6111yescharacteristic
535Q16—2448no—
536SD16—2448no—
537M16—2448no—
538C2×C8—24no—
539C2×C8—614no—
540C2×C8—1224no—
541C2×C8C23×D6114yes—
542C4⋊C4—24no—
543C4⋊C4—22no—
544C4⋊C4—21no—
545C4⋊C4—612no—
546C4⋊C4—1228no—
547C4⋊C4—1228no—
548C4⋊C4—1224no—
549C4⋊C4—1221no—
550C4⋊C4—1224no—
551C4⋊C4C23×D6114yes—
552C4⋊C4C23×D6112yes—
553C22⋊C4—24no—
554C22⋊C4—1228no—
555C22⋊C4—1228no—
556C22⋊C4—1224no—
557C22⋊C4—1224no—
558C22⋊C4—1224no—
559C22⋊C4—1224no—
560C42—1224no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—612no—
568C23—612no—
569C23—1228no—
570C23—1222no—
571C23—1222no—
572C23—1222no—
573C23—1222no—
574C23—1222no—
575C23—1222no—
576C23—2444no—
577C23—2444no—
578C23—24416no—
579C23#1451980614yes—
580C23C2×D4×D6112yes—
581C23C2×D4×D6111yescharacteristic
582Q8—1224no—
583Q8—1224no—
584Q8—1224no—
585Q8—2448no—
586Q8—2444no—
587Q8—2442no—
588D4—24no—
589D4—48no—
590D4—42no—
591D4—2448no—
592D4—2448no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—48no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—612no—
602C2×C4—612no—
603C2×C4—1222no—
604C2×C4—1222no—
605C2×C4—1224no—
606C2×C4—1224no—
607C2×C4—1224no—
608C2×C4—1224no—
609C2×C4—1222no—
610C2×C4—1222no—
611C2×C4—1224no—
612C2×C4—2444no—
613C2×C4—2444no—
614C2×C4—2444no—
615C2×C4—2448no—
616C2×C4—2448no—
617C2×C4—2442no—
618C2×C4—2442no—
619C2×C4—2444no—
620C2×C4C2×D4×D6112yes—
621C2×C4C2×D4×D6111yescharacteristic
622C2×C4#1451982314yes—
623C2×C4C24×D6111yescharacteristic
624C8—24no—
625C8—1224no—
626C22—28no—
627C22—22no—
628C22—22no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—84no—
633C22—1222no—
634C22—1222no—
635C22—1222no—
636C22—1222no—
637C22—1222no—
638C22—1222no—
639C22—24416no—
640C22—2444no—
641C22—2444no—
642C22#1684032312yes—
643C22C2×#1451980614yes—
644C22C2×#1451980612yes—
645C22#1684055812yes—
646C22C22×D4×D6111yescharacteristic
647C4—22no—
648C4—42no—
649C4—44no—
650C4—1222no—
651C4—1222no—
652C4—2442no—
653C4—2442no—
654C4—2444no—
655C4#1684032312yes—
656C4C2×#1451982312yes—
657C2—24no—
658C2—82no—
659C2—2442no—
660C2—2442no—
661C2#2145664814yesdirect product factor
662C2C2×#1684032311yescharacteristic
663C2C22×#1451980611yescharacteristic
664C2C2×#1684055811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#481C2
socleC23×C61C22×D4
prime radicalC2×#16840272C2
hypercenterC2×#481D61
hypocenterC61C2×#3196
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#481

Complements

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

NH ≅ G/Ntype
2. C22×#14519757C2semidirect
3. #21456648C2direct (N × H)
4. C2×#16840097C2semidirect
5. C2×#16840106C2semidirect
6. C2×#16840021C2semidirect
7. C2×#16840007C2semidirect
8. C2×#16839925C2semidirect
9. C2×#16839920C2semidirect
10. C2×#16839900C2semidirect
11. C2×#16839724C2semidirect
12. C2×C61×#481C2semidirect
16. C22×Dic244C22semidirect
17. C22×#11758200C22semidirect
28. C2×#14519757C22semidirect
37. #16840106C22semidirect
38. #16840097C22semidirect
39. #16840021C22semidirect
40. #16840007C22semidirect
41. #16839925C22semidirect
42. #16839920C22semidirect
43. #16839900C22semidirect
44. C2×#14519710C22semidirect
45. C2×#14519702C22semidirect
46. C2×#14519700C22semidirect
47. C2×#14519700C22semidirect
48. #16839724C22semidirect
53. C61×#481C22semidirect
54. C2×#106×C61C22semidirect
55. C2×#102×C61C22semidirect
80. C2×Dic244C23semidirect
82. C2×#11758200C23semidirect
118. #14519710C23semidirect
119. #14519702C23semidirect
120. #14519700C23semidirect
121. #14519700C23semidirect
137. #106×C61C23semidirect
138. #102×C61C23semidirect
329. C2×#481D61semidirect
339. C2×C61#3196semidirect
373. #481C2×D61semidirect
380. C2×#106C2×D61semidirect
384. C2×#102C2×D61semidirect
389. C61C2×#3196semidirect
465. #106C22×D61semidirect
469. #102C22×D61semidirect
661. C2#21456648direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168402722
C22×#145197572
#214566482
C2×#168400972
C2×#168401062
C2×#168400212
C2×#168400072
C2×#168399252
C2×#168399202
C2×#168399002
C2×#168397242
C2×C61×#4812
C2×#319661