Finite Group Databasebeta

← C2×#21456603 #30550568

Subgroups — C2×#21456603 #30550568

Order 15616.

2374 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×#16840275C211yesmaximal, characteristic
2C22×#14519753C211yesmaximal, characteristic, complemented
3#21456603C2116yesmaximal, direct product factor
4C2×#16840102C212yesmaximal, complemented
5C2×#16840090C211yesmaximal, characteristic, complemented
6C2×#16840076C211yesmaximal, characteristic, complemented
7C2×#16840000C212yesmaximal, complemented
8C2×#16839994C212yesmaximal, complemented
9C2×#16839923C212yesmaximal, complemented
10C2×#16839901C211yesmaximal, characteristic, complemented
11C2×C22⋊C8×D61C211yesmaximal, characteristic, complemented
12C2×C61×#451C211yesmaximal, characteristic, complemented
13C22×#11758226C2211yescharacteristic
14C23×C4×D61C2211yescharacteristic
15C22×#11758201C2212yescomplemented
16C22×C8×D61C2212yes—
17#16840275—24no—
18#16840275C2218yes—
19C2×#14519762—21no—
20C2×#14519753—21no—
21C2×#14519753—21no—
22C2×#14519753—24no—
23C2×#14519753C2212yes—
24C2×#14519753C2211yescharacteristic
25C2×#14519753C2212yes—
26C2×#14519753C2218yescomplemented
27C2×#14519752C2211yescharacteristic
28C2×#14519751C2211yescharacteristic
29C2×#14519747—21no—
30C2×#14519739—21no—
31C2×#14519737—21no—
32C2×#14519735C2211yescharacteristic
33C22×C4⋊Dic61C2211yescharacteristic
34C22×#11758196—21no—
35C22×#11758198—21no—
36#16840102C22116yescomplemented
37#16840090C2218yescomplemented
38#16840076C2218yescomplemented
39#16840000C22116yescomplemented
40#16839994C22116yescomplemented
41#16839923C22116yescomplemented
42#16839901C2218yescomplemented
43C2×#14519711C2212yescomplemented
44C2×#14519708C2212yescomplemented
45C2×#14519701C2212yescomplemented
46C2×#14519701C2212yescomplemented
47C2×#14519699C2212yescomplemented
48C2×C22⋊#92418C2211yescharacteristic
49C2×#14519689C2212yes—
50C22⋊C8×D61C2218yescomplemented
51C23×Q8×C61C2211yescharacteristic
52C22×Q16×C61C2212yescomplemented
53C2×C22⋊Q8×C61C2211yescharacteristic
54C61×#451C2218yescomplemented
55C2×#103×C61C2212yescomplemented
56C2×C22⋊C8×C61C2211yescharacteristic
57C24×D61D411yescharacteristic
58C2×#11758226—22no—
59C2×#11758226—44no—
60C2×#11758226C2314yes—
61C22×D244—21no—
62C22×D244—21no—
63C22×D244C2311yescharacteristic
64C22×C22⋊D61—21no—
65C23×Dic61—21no—
66C23×Dic61D411yescharacteristic
67C22×C4×D61—21no—
68C22×C4×D61—44no—
69C22×C4×D61D411yescharacteristic
70C22×C4×D61D411yescharacteristic
71C22×C4×D61D414yes—
72C22×C4×D61C2311yescharacteristic
73C22×C4×D61C2311yescharacteristic
74C22×C4×D61C2314yes—
75C2×#11758201—44no—
76C2×#11758201C2318yescomplemented
77C22×#92418C2312yes—
78C2×C8×D61—44no—
79C2×C8×D61C2318yes—
80#14519762—28no—
81#14519753—24no—
82#14519753—24no—
83#14519753—28no—
84#14519753—28no—
85#14519753—22no—
86#14519753C2318yes—
87#14519753C2314yes—
88#14519753C2318yes—
89#14519752—22no—
90#14519752C2314yes—
91#14519751—22no—
92#14519751C2314yes—
93#14519747—28no—
94#14519739—28no—
95#14519737—28no—
96#14519735—22no—
97#14519735C2314yes—
98C2×C4⋊Dic61—21no—
99C2×C4⋊Dic61—22no—
100C2×C4⋊Dic61C2312yes—
101C2×C4⋊Dic61C2314yes—
102C2×C4⋊Dic61C2311yescharacteristic
103C2×#11758196—22no—
104C2×#11758196—22no—
105C2×#11758196—28no—
106C2×#11758196—21no—
107C2×#11758198—22no—
108C2×#11758198—22no—
109C2×#11758198—28no—
110C2×#11758198—21no—
111C2×#11758198—21no—
112C2×#11758198—21no—
113C2×#11758198—21no—
114C2×C4×Dic61—21no—
115#14519711C2318yescomplemented
116#14519708C2318yescomplemented
117#14519701C2318yescomplemented
118#14519701C2318yescomplemented
119#14519699C2318yescomplemented
120C22⋊#92418C2314yes—
121#14519689C2318yes—
122C22×Q8×C61—21no—
123C22×Q8×C61—22no—
124C22×Q8×C61—44no—
125C22×Q8×C61C2311yescharacteristic
126C22×Q8×C61C2312yes—
127C22×Q8×C61C2314yes—
128C23×C4×C61—21no—
129C23×C4×C61C2311yescharacteristic
130C2×Q16×C61—44no—
131C2×Q16×C61C2318yescomplemented
132C22×C8×C61C2312yes—
133C22⋊Q8×C61—22no—
134C22⋊Q8×C61C2314yes—
135C2×C4⋊C4×C61—21no—
136C2×C4⋊C4×C61C2311yescharacteristic
137C2×C22⋊C4×C61—21no—
138#103×C61C2318yescomplemented
139C22⋊C8×C61C2314yes—
140C24×C61C2×D411yescharacteristic
141C23×D61—24no—
142C23×D61—24no—
143C23×D61—21no—
144C23×D61—42no—
145C23×D61C2×D411yescharacteristic
146C23×D61C2×D414yes—
147C23×D61C2×D411yescharacteristic
148C2×Q8×C61—21no—
149C2×Q8×C61—24no—
150C2×Q8×C61—22no—
151C2×Q8×C61—24no—
152C2×Q8×C61—24no—
153C2×Q8×C61—24no—
154C2×Q8×C61—48no—
155C2×Q8×C61—44no—
156C2×Q8×C61—48no—
157C2×Q8×C61C2412yes—
158C2×Q8×C61C2414yes—
159C22×C4×C61—24no—
160C22×C4×C61—21no—
161C22×C4×C61—21no—
162C22×C4×C61—24no—
163C22×C4×C61—22no—
164C22×C4×C61—44no—
165C22×C4×C61C2×D411yescharacteristic
166C22×C4×C61C2×D412yes—
167C22×C4×C61C2412yes—
168C22×C4×C61C2411yescharacteristic
169C2×C22⋊D61—24no—
170C2×C22⋊D61—44no—
171C22×Dic61—24no—
172C22×Dic61—22no—
173C22×Dic61—21no—
174C22×Dic61—42no—
175C22×Dic61—44no—
176C22×Dic61C2×D411yescharacteristic
177C22×Dic61C2×D411yescharacteristic
178C22×Dic61C2×D414yes—
179#11758226—48no—
180#11758226—48no—
181C2×D244—24no—
182C2×D244—24no—
183C2×D244—24no—
184C2×D244—21no—
185C2×D244—44no—
186C2×D244—44no—
187C2×D244C2412yes—
188C2×C4×D61—24no—
189C2×C4×D61—24no—
190C2×C4×D61—24no—
191C2×C4×D61—48no—
192C2×C4×D61—48no—
193C2×C4×D61—42no—
194C2×C4×D61—42no—
195C2×C4×D61—44no—
196C2×C4×D61C2×D412yes—
197C2×C4×D61C2×D412yes—
198C2×C4×D61C2×D414yes—
199C2×C4×D61C2×D414yes—
200C2×C4×D61C2412yes—
201C2×C4×D61C2412yes—
202Q16×C61—48no—
203C2×C8×C61—24no—
204C2×C8×C61C2414yes—
205C4⋊C4×C61—24no—
206C4⋊C4×C61—21no—
207C4⋊C4×C61C2412yes—
208C22⋊C4×C61—24no—
209#11758201—48no—
210#11758198—28no—
211#11758198—28no—
212#11758198—24no—
213#11758198—24no—
214#11758198—24no—
215#11758198—24no—
216C4⋊Dic61—24no—
217C4⋊Dic61—22no—
218C4⋊Dic61—21no—
219C4⋊Dic61C2414yes—
220C4⋊Dic61C2412yes—
221#11758196—28no—
222#11758196—28no—
223#11758196—24no—
224C4×Dic61—24no—
225C2×#92418—24no—
226C2×#92418C2414yes—
227C8×D61—48no—
228C23×C61—22no—
229C23×C61—22no—
230C23×C61#13514yes—
231C23×C61C22×D412yes—
232C23×C61C22×D411yescharacteristic
233C22×D61—22no—
234C22×D61—22no—
235C22×D61—22no—
236C22×D61—28no—
237C22×D61—22no—
238C22×D61—22no—
239C22×D61—28no—
240C22×D61—22no—
241C22×D61—44no—
242C22×D61—416no—
243C22×D61—42no—
244C22×D61—42no—
245C22×D61—42no—
246C22×D61#13714yes—
247C22×D61C22×D412yes—
248C22×D61C22×D412yes—
249Q8×C61—24no—
250Q8×C61—24no—
251Q8×C61—24no—
252Q8×C61—48no—
253Q8×C61—42no—
254Q8×C61—44no—
255C2×C4×C61—22no—
256C2×C4×C61—24no—
257C2×C4×C61—24no—
258C2×C4×C61—22no—
259C2×C4×C61—24no—
260C2×C4×C61—22no—
261C2×C4×C61—48no—
262C2×C4×C61—42no—
263C2×C4×C61—44no—
264C2×C4×C61—42no—
265C2×C4×C61C22×D412yes—
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61H(5,𝔽2)14yes—
268C2×C4×C61C2511yescharacteristic
269C22⋊D61—48no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—24no—
274C2×Dic61—44no—
275C2×Dic61—48no—
276C2×Dic61—44no—
277C2×Dic61—42no—
278C2×Dic61C22×D412yes—
279C2×Dic61C22×D412yes—
280D244—24no—
281D244—48no—
282D244—48no—
283D244—42no—
284C4×D61—24no—
285C4×D61—24no—
286C4×D61—24no—
287C4×D61—24no—
288C4×D61—44no—
289C4×D61—44no—
290C4×D61—48no—
291C8×C61—24no—
292#92418—24no—
293C2×#3190—611nomaximal, Sylow
294C22×C61—22no—
295C22×C61—28no—
296C22×C61—22no—
297C22×C61#53412yes—
298C22×C61C2×#13514yes—
299C22×C61C2×#13512yes—
300C22×C61C2×#13712yes—
301C22×C61C23×D411yescharacteristic
302C2×D61—22no—
303C2×D61—28no—
304C2×D61—22no—
305C2×D61—22no—
306C2×D61—22no—
307C2×D61—22no—
308C2×D61—416no—
309C2×D61—44no—
310C2×D61—42no—
311C2×D61—42no—
312C2×D61—42no—
313C2×D61—84no—
314C2×D61C2×#13714yes—
315C4×C61—22no—
316C4×C61—42no—
317C4×C61—44no—
318C4×C61—42no—
319C4×C61#53412yes—
320C4×C61C2×H(5,𝔽2)12yes—
321Dic61—22no—
322Dic61—22no—
323Dic61—44no—
324Dic61—42no—
325C2×#525—611no—
326C22×C22⋊Q8—611no—
327#3190—6116no—
328C2×#485—612no—
329C2×#483—611no—
330C2×#480—611no—
331C2×#467—612no—
332C2×#465—612no—
333C2×#452—612no—
334C2×#451D6111yescharacteristic, complemented
335C2×#448—611no—
336C22×C22⋊C8—611no—
337C2×C61—24no—
338C2×C61#319014yescomplemented
339C2×C61C2×#53411yescharacteristic
340C2×C61C22×#13511yescharacteristic
341C2×C61C22×#13711yescharacteristic
342D61—24no—
343D61—42no—
344D61—82no—
345C22×#41—611no—
346C23×Q8C2×D6111yescharacteristic
347C24×C4—611no—
348C22×Q16C2×D6112yescomplemented
349C22×SD16—612no—
350C23×C8—612no—
351#525—618no—
352#525—1224no—
353C2×#123—1221no—
354C2×C22⋊Q8—612no—
355C2×C22⋊Q8—612no—
356C2×C22⋊Q8—611no—
357C2×C22⋊Q8—618no—
358C2×C22⋊Q8—1221no—
359C2×C22⋊Q8—1221no—
360C2×C22⋊Q8—1224no—
361C2×C22⋊Q8C2×D6111yescharacteristic
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#483—618no—
373#480—618no—
374#467—6116no—
375#465—6116no—
376#452—6116no—
377#451C2×D6118yescomplemented
378#448—618no—
379C2×#107—612no—
380C2×#106—612no—
381C2×#103—612no—
382C2×#103—612no—
383C2×#103C2×D6112yescomplemented
384C2×#102—612no—
385C2×C22⋊C8—611no—
386C2×C22⋊C8—612no—
387C2×C22⋊C8—618no—
388C2×C22⋊C8C2×D6111yescharacteristic
389C61C2×#319011yescharacteristic, complemented, Sylow
390C25—611no—
391C2×#41—614no—
392C2×#41—1222no—
393C2×#41—2444no—
394C22×Q8—22no—
395C22×Q8—21no—
396C22×Q8—44no—
397C22×Q8C22×D6112yes—
398C22×Q8C22×D6111yescharacteristic
399C22×Q8C22×D6114yes—
400C22×D4—611no—
401C22×D4—1221no—
402C22×D4—1221no—
403C22×D4—1221no—
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—44no—
417C2×Q16C22×D6118yescomplemented
418C2×SD16—618no—
419C2×SD16—2444no—
420C22×C8—612no—
421C22×C8—618no—
422C22×C8—2444no—
423C22×C8C22×D6112yes—
424#123—1228no—
425C22⋊Q8—22no—
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⋊Q8C22×D6114yes—
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—21no—
445C2×C4⋊C4—612no—
446C2×C4⋊C4—614no—
447C2×C4⋊C4—611no—
448C2×C4⋊C4—1228no—
449C2×C4⋊C4—1222no—
450C2×C4⋊C4—1222no—
451C2×C4⋊C4—1221no—
452C2×C4⋊C4—1222no—
453C2×C4⋊C4—1221no—
454C2×C4⋊C4C22×D6111yescharacteristic
455C2×C22⋊C4—21no—
456C2×C22⋊C4—1221no—
457C2×C22⋊C4—1228no—
458C2×C22⋊C4—1222no—
459C2×C22⋊C4—1222no—
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#103C22×D6118yescomplemented
469#102—618no—
470C22⋊C8—614no—
471C22⋊C8—618no—
472C22⋊C8C22×D6114yes—
473C24—611no—
474C24—611no—
475C24—614no—
476C24—1224no—
477C24—1224no—
478C24—1221no—
479C24—2442no—
480C24D4×D6111yescharacteristic
481#41—2448no—
482#41—2448no—
483C2×Q8—21no—
484C2×Q8—24no—
485C2×Q8—22no—
486C2×Q8—24no—
487C2×Q8—24no—
488C2×Q8—24no—
489C2×Q8—48no—
490C2×Q8—44no—
491C2×Q8—48no—
492C2×Q8C23×D6112yes—
493C2×Q8C23×D6114yes—
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—21no—
506C22×C4—21no—
507C22×C4—22no—
508C22×C4—44no—
509C22×C4—611no—
510C22×C4—611no—
511C22×C4—614no—
512C22×C4—612no—
513C22×C4—612no—
514C22×C4—612no—
515C22×C4—612no—
516C22×C4—614no—
517C22×C4—614no—
518C22×C4—1224no—
519C22×C4—1224no—
520C22×C4—1224no—
521C22×C4—1224no—
522C22×C4—1221no—
523C22×C4—1222no—
524C22×C4—2448no—
525C22×C4—2448no—
526C22×C4—2442no—
527C22×C4—2442no—
528C22×C4—2444no—
529C22×C4—2442no—
530C22×C4—2444no—
531C22×C4D4×D6112yes—
532C22×C4D4×D6111yescharacteristic
533C22×C4C23×D6112yes—
534C22×C4C23×D6111yescharacteristic
535Q16—48no—
536SD16—2448no—
537C2×C8—24no—
538C2×C8—614no—
539C2×C8—1224no—
540C2×C8—2448no—
541C2×C8C23×D6114yes—
542C4⋊C4—24no—
543C4⋊C4—21no—
544C4⋊C4—614no—
545C4⋊C4—612no—
546C4⋊C4—1228no—
547C4⋊C4—1228no—
548C4⋊C4—1224no—
549C4⋊C4—1224no—
550C4⋊C4—1222no—
551C4⋊C4—1221no—
552C4⋊C4C23×D6112yes—
553C22⋊C4—24no—
554C22⋊C4—1224no—
555C22⋊C4—1228no—
556C22⋊C4—1228no—
557C22⋊C4—1224no—
558C22⋊C4—1224no—
559C22⋊C4—1224no—
560C42—1224no—
561C23—22no—
562C23—22no—
563C23—614no—
564C23—612no—
565C23—612no—
566C23—1228no—
567C23—1222no—
568C23—1222no—
569C23—1228no—
570C23—1222no—
571C23—1222no—
572C23—1222no—
573C23—1222no—
574C23—2444no—
575C23—24416no—
576C23—2442no—
577C23—2442no—
578C23—2442no—
579C23#1451980014yes—
580C23C2×D4×D6112yes—
581C23C2×D4×D6111yescharacteristic
582Q8—24no—
583Q8—24no—
584Q8—24no—
585Q8—48no—
586Q8—44no—
587Q8—42no—
588D4—1224no—
589D4—2448no—
590D4—2448no—
591D4—2448no—
592D4—2442no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—48no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—612no—
604C2×C4—612no—
605C2×C4—1222no—
606C2×C4—1222no—
607C2×C4—1224no—
608C2×C4—1224no—
609C2×C4—1224no—
610C2×C4—1224no—
611C2×C4—1222no—
612C2×C4—1224no—
613C2×C4—2444no—
614C2×C4—2444no—
615C2×C4—2448no—
616C2×C4—2444no—
617C2×C4—2448no—
618C2×C4—2444no—
619C2×C4—2442no—
620C2×C4C2×D4×D6112yes—
621C2×C4C2×D4×D6111yescharacteristic
622C2×C4#1451982214yes—
623C2×C4C24×D6111yescharacteristic
624C8—24no—
625C8—1224no—
626C22—22no—
627C22—22no—
628C22—28no—
629C22—614no—
630C22—1222no—
631C22—1222no—
632C22—1228no—
633C22—1222no—
634C22—1222no—
635C22—1222no—
636C22—4884no—
637C22—24416no—
638C22—2442no—
639C22—2442no—
640C22—2442no—
641C22—2444no—
642C22#1684032212yes—
643C22C2×#1451980014yes—
644C22C2×#1451980012yes—
645C22#137×D6112yes—
646C22C22×D4×D6111yescharacteristic
647C4—22no—
648C4—42no—
649C4—44no—
650C4—42no—
651C4—1222no—
652C4—1222no—
653C4—2444no—
654C4—2442no—
655C4#1684032212yes—
656C4C2×#1451982212yes—
657C2—24no—
658C2—1224no—
659C2—4882no—
660C2—2442no—
661C2#2145660314yesdirect product factor
662C2C2×#1684032211yescharacteristic
663C2C22×#1451980011yescharacteristic
664C2C2×#137×D6111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#451C2
socleC23×C61C22×D4
prime radicalC2×#16840275C2
hypercenterC2×#451D61
hypocenterC61C2×#3190
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
2. C22×#14519753C2semidirect
3. #21456603C2direct (N × H)
4. C2×#16840102C2semidirect
5. C2×#16840090C2semidirect
6. C2×#16840076C2semidirect
7. C2×#16840000C2semidirect
8. C2×#16839994C2semidirect
9. C2×#16839923C2semidirect
10. C2×#16839901C2semidirect
11. C2×C22⋊C8×D61C2semidirect
12. C2×C61×#451C2semidirect
15. C22×#11758201C22semidirect
26. C2×#14519753C22semidirect
36. #16840102C22semidirect
37. #16840090C22semidirect
38. #16840076C22semidirect
39. #16840000C22semidirect
40. #16839994C22semidirect
41. #16839923C22semidirect
42. #16839901C22semidirect
43. C2×#14519711C22semidirect
44. C2×#14519708C22semidirect
45. C2×#14519701C22semidirect
46. C2×#14519701C22semidirect
47. C2×#14519699C22semidirect
50. C22⋊C8×D61C22semidirect
52. C22×Q16×C61C22semidirect
54. C61×#451C22semidirect
55. C2×#103×C61C22semidirect
76. C2×#11758201C23semidirect
115. #14519711C23semidirect
116. #14519708C23semidirect
117. #14519701C23semidirect
118. #14519701C23semidirect
119. #14519699C23semidirect
131. C2×Q16×C61C23semidirect
138. #103×C61C23semidirect
334. C2×#451D61semidirect
338. C2×C61#3190semidirect
348. C22×Q16C2×D61semidirect
377. #451C2×D61semidirect
383. C2×#103C2×D61semidirect
389. C61C2×#3190semidirect
417. C2×Q16C22×D61semidirect
468. #103C22×D61semidirect
661. C2#21456603direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168402752
C22×#145197532
#214566032
C2×#168401022
C2×#168400902
C2×#168400762
C2×#168400002
C2×#168399942
C2×#168399232
C2×#168399012
C2×C22⋊C8×D612
C2×C61×#4512
C2×#319061