Finite Group Databasebeta

← C2×#21450174 #29849462

Subgroups — C2×#21450174 #29849462

Order 15616.

3158 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519752C211yesmaximal, characteristic
2#21450174C2116yesmaximal, direct product factor
3C2×#16839688C212yesmaximal, complemented
4C2×#16839679C212yesmaximal, complemented
5C2×#16839664C211yesmaximal, characteristic, complemented
6C2×#16839663C211yesmaximal, characteristic, complemented
7C2×#16839663C212yesmaximal, complemented
8C2×#16839652C212yesmaximal, complemented
9C2×#16839619C211yesmaximal, characteristic, complemented
10C2×C4⋊C4×Dic61C211yesmaximal, characteristic, complemented
11C2×C4×#11758198C211yesmaximal, characteristic, complemented
12C22×#125×C61C211yesmaximal, characteristic, complemented
13C23×D244C2211yescharacteristic
14C23×D244C2212yes—
15C23×C4×D61C2211yescharacteristic
16C2×#14519752—28no—
17C2×#14519752C2218yes—
18C22×C4⋊Dic61C2211yescharacteristic
19C22×#11758198C2212yes—
20C22×#11758198C2212yes—
21C22×#11758198C2211yescharacteristic
22C22×#11758198C2211yescharacteristic
23C22×#11758198C2212yes—
24C22×#11758198C2212yes—
25C22×#11758198C2212yes—
26C22×#11758198C2212yes—
27C22×C4×Dic61C2211yescharacteristic
28C22×C4×Dic61C2212yes—
29#16839688C22116yescomplemented
30#16839679C22116yescomplemented
31#16839664C2218yescomplemented
32#16839663C2218yescomplemented
33#16839663C22116yescomplemented
34#16839652C22116yescomplemented
35#16839619C2218yescomplemented
36C4⋊C4×Dic61C2218yescomplemented
37C4×#11758198C2218yescomplemented
38C2×#14519684C2211yescharacteristic
39C2×#14519684C2212yes—
40C2×#14519684C2212yes—
41C2×#14519684C2211yescharacteristic
42C2×#125×C61—28no—
43C2×#125×C61C2218yescomplemented
44C22×C4⋊C4×C61C2212yes—
45C22×C4⋊C4×C61C2212yes—
46C22×C4⋊C4×C61C2211yescharacteristic
47C22×C4⋊C4×C61C2211yescharacteristic
48C22×C42×C61C2211yescharacteristic
49C24×D61C2312yes—
50C24×D61C2311yescharacteristic
51C22×D244—28no—
52C22×D244—28no—
53C22×D244—24no—
54C22×D244—24no—
55C22×D244—24no—
56C22×D244—44no—
57C22×D244C2314yes—
58C22×D244C2318yes—
59C23×Dic61C2312yes—
60C23×Dic61C2311yescharacteristic
61C23×Dic61C2311yescharacteristic
62C23×Dic61C2311yescharacteristic
63C22×C4×D61—24no—
64C22×C4×D61—24no—
65C22×C4×D61—24no—
66C22×C4×D61C2314yes—
67#14519752—232no—
68C2×C4⋊Dic61—24no—
69C2×C4⋊Dic61C2314yes—
70C2×#11758198—24no—
71C2×#11758198—28no—
72C2×#11758198—24no—
73C2×#11758198—28no—
74C2×#11758198—44no—
75C2×#11758198—44no—
76C2×#11758198—44no—
77C2×#11758198—44no—
78C2×#11758198C2318yes—
79C2×#11758198C2314yes—
80C2×#11758198C2314yes—
81C2×#11758198C2318yes—
82C2×#11758198C2318yes—
83C2×#11758198C2318yes—
84C2×#11758198C2318yes—
85C2×#11758198C2318yes—
86C2×C4×Dic61—24no—
87C2×C4×Dic61—44no—
88C2×C4×Dic61C2314yes—
89C2×C4×Dic61C2318yes—
90#14519684C2314yes—
91#14519684C2318yes—
92#14519684C2318yes—
93#14519684C2314yes—
94C23×C4×C61C2311yescharacteristic
95C23×C4×C61C2311yescharacteristic
96C23×C4×C61C2312yes—
97C23×C4×C61C2312yes—
98C23×C4×C61C2311yescharacteristic
99#125×C61—232no—
100C2×C4⋊C4×C61—28no—
101C2×C4⋊C4×C61—28no—
102C2×C4⋊C4×C61—24no—
103C2×C4⋊C4×C61D418yes—
104C2×C4⋊C4×C61C2318yes—
105C2×C4⋊C4×C61C2314yes—
106C2×C4⋊C4×C61C2318yes—
107C2×C4⋊C4×C61C2314yes—
108C2×C42×C61—24no—
109C2×C42×C61C2314yes—
110C24×C61C2411yescharacteristic
111C23×D61—22no—
112C23×D61—22no—
113C23×D61—22no—
114C23×D61—22no—
115C23×D61—22no—
116C23×D61—22no—
117C23×D61—24no—
118C23×D61—24no—
119C23×D61—24no—
120C23×D61—24no—
121C23×D61—24no—
122C23×D61—24no—
123C23×D61—24no—
124C23×D61#4114yes—
125C23×D61C2414yes—
126C23×D61C2412yes—
127C22×C4×C61—24no—
128C22×C4×C61—24no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—24no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—24no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—24no—
138C22×C4×C61C2×D418yes—
139C22×C4×C61C2×D414yes—
140C22×C4×C61#4114yes—
141C22×C4×C61#4114yes—
142C22×C4×C61C2412yes—
143C22×C4×C61C2412yes—
144C22×C4×C61C2414yes—
145C22×C4×C61C2414yes—
146C22×C4×C61C2412yes—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—24no—
155C22×Dic61—24no—
156C22×Dic61—24no—
157C22×Dic61—22no—
158C22×Dic61#4114yes—
159C22×Dic61C2412yes—
160C22×Dic61C2412yes—
161C22×Dic61C2414yes—
162C22×Dic61C2412yes—
163C2×D244—216no—
164C2×D244—216no—
165C2×D244—28no—
166C2×D244—28no—
167C2×D244—28no—
168C2×D244—48no—
169C2×D244—432no—
170C2×D244—416no—
171C2×C4×D61—28no—
172C2×C4×D61—28no—
173C2×C4×D61—28no—
174C2×C4×D61—416no—
175C4⋊C4×C61—216no—
176C4⋊C4×C61—216no—
177C4⋊C4×C61—28no—
178C4⋊C4×C61C2×D4116yes—
179C42×C61—28no—
180#11758198—28no—
181#11758198—216no—
182#11758198—28no—
183#11758198—216no—
184#11758198—48no—
185#11758198—48no—
186#11758198—48no—
187#11758198—48no—
188C4⋊Dic61—28no—
189C4×Dic61—28no—
190C4×Dic61—48no—
191C23×C61C22×D412yes—
192C23×C61C2×#4112yes—
193C23×C61C2×#4112yes—
194C23×C61C2×#4112yes—
195C23×C61C2×#4112yes—
196C23×C61H(5,𝔽2)12yes—
197C23×C61H(5,𝔽2)12yes—
198C23×C61C2511yescharacteristic
199C22×D61—28no—
200C22×D61—28no—
201C22×D61—28no—
202C22×D61—22no—
203C22×D61—22no—
204C22×D61—22no—
205C22×D61—22no—
206C22×D61—22no—
207C22×D61—22no—
208C22×D61—24no—
209C22×D61—24no—
210C22×D61—24no—
211C22×D61—24no—
212C22×D61—24no—
213C22×D61—24no—
214C22×D61—24no—
215C22×D61—44no—
216C22×D61—44no—
217C22×D61—44no—
218C22×D61—44no—
219C22×D61—48no—
220C22×D61—48no—
221C22×D61—48no—
222C22×D61—48no—
223C22×D61—48no—
224C22×D61—48no—
225C22×D61—48no—
226C22×D61C2×#4114yes—
227C2×C4×C61—216no—
228C2×C4×C61—24no—
229C2×C4×C61—24no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—24no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—24no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—24no—
239C2×C4×C61—44no—
240C2×C4×C61—44no—
241C2×C4×C61—48no—
242C2×C4×C61#124116yes—
243C2×C4×C61C22×D418yes—
244C2×C4×C61C22×D414yes—
245C2×C4×C61C2×#4114yes—
246C2×C4×C61C2×#4114yes—
247C2×Dic61—28no—
248C2×Dic61—22no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—24no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—24no—
257C2×Dic61—24no—
258C2×Dic61—22no—
259C2×Dic61—44no—
260C2×Dic61—48no—
261C2×Dic61—44no—
262C2×Dic61C2×#4114yes—
263D244—432no—
264D244—416no—
265C4×D61—416no—
266C2×#2490—611nomaximal, Sylow
267C22×C61C2×#12414yes—
268C22×C61#52914yes—
269C22×C61#54014yes—
270C22×C61#54614yes—
271C22×C61#55014yes—
272C22×C61#55314yes—
273C22×C61#55314yes—
274C22×C61C23×D411yescharacteristic
275C22×C61C22×#4111yescharacteristic
276C22×C61C22×#4111yescharacteristic
277C22×C61C22×#4111yescharacteristic
278C22×C61C22×#4111yescharacteristic
279C22×C61C2×H(5,𝔽2)11yescharacteristic
280C22×C61C2×H(5,𝔽2)11yescharacteristic
281C2×D61—24no—
282C2×D61—24no—
283C2×D61—24no—
284C2×D61—42no—
285C2×D61—42no—
286C2×D61—42no—
287C2×D61—42no—
288C2×D61—416no—
289C2×D61—44no—
290C2×D61—44no—
291C2×D61—44no—
292C2×D61—44no—
293C2×D61—44no—
294C2×D61—44no—
295C2×D61—44no—
296C2×D61—816no—
297C4×C61—28no—
298C4×C61—42no—
299C4×C61—42no—
300C4×C61—44no—
301C4×C61C2×#12418yes—
302Dic61—24no—
303Dic61—42no—
304Dic61—44no—
305Dic61—42no—
306C22×#125D6111yescharacteristic, complemented
307C22×#121—611no—
308#2490—6116no—
309C2×#397—612no—
310C2×#394—612no—
311C2×#390—611no—
312C2×#390—612no—
313C2×#390—611no—
314C2×#388—612no—
315C2×#380—611no—
316C2×C4×C4⋊C4—611no—
317C2×C4×C22⋊C4—611no—
318C2×C61#249018yescomplemented
319C2×C61C22×#12411yescharacteristic
320C2×C61C2×#52911yescharacteristic
321C2×C61C2×#54011yescharacteristic
322C2×C61C2×#54611yescharacteristic
323C2×C61C2×#55011yescharacteristic
324C2×C61C2×#55311yescharacteristic
325C2×C61C2×#55311yescharacteristic
326D61—44no—
327D61—84no—
328C23×D4—611no—
329C23×D4—612no—
330C24×C4—611no—
331C2×#125—28no—
332C2×#125C2×D6118yescomplemented
333C2×#121—618no—
334C2×#121—1228no—
335C22×C4⋊C4—611no—
336C22×C4⋊C4C2×D6112yes—
337C22×C4⋊C4C2×D6112yes—
338C22×C4⋊C4C2×D6111yescharacteristic
339C22×C4⋊C4C2×D6111yescharacteristic
340C22×C22⋊C4—611no—
341C22×C22⋊C4—612no—
342C22×C22⋊C4—611no—
343C22×C22⋊C4—612no—
344C22×C22⋊C4—612no—
345C22×C22⋊C4—612no—
346C22×C22⋊C4—612no—
347C22×C22⋊C4—612no—
348C22×C42—611no—
349C22×C42—612no—
350C22×C42C2×D6111yescharacteristic
351#397—6116no—
352#394—6116no—
353#390—618no—
354#390—6116no—
355#390—618no—
356#388—6116no—
357#380—618no—
358C4×C4⋊C4—618no—
359C4×C22⋊C4—618no—
360C2×#95—612no—
361C2×#95—611no—
362C2×#95—612no—
363C2×#95—611no—
364C61C2×#249011yescharacteristic, complemented, Sylow
365C25—611no—
366C25—612no—
367C22×D4—614no—
368C22×D4—618no—
369C22×D4—1228no—
370C22×D4—1224no—
371C22×D4—1224no—
372C22×D4—1228no—
373C22×D4—1224no—
374C22×D4—2444no—
375C23×C4—611no—
376C23×C4—612no—
377C23×C4—611no—
378C23×C4—611no—
379C23×C4—614no—
380C23×C4—1224no—
381C23×C4—1224no—
382C23×C4—1224no—
383C23×C4C22×D6112yes—
384C23×C4C22×D6111yescharacteristic
385C23×C4C22×D6111yescharacteristic
386C23×C4C22×D6112yes—
387C23×C4C22×D6111yescharacteristic
388#125—232no—
389#121—12232no—
390C2×C4⋊C4—28no—
391C2×C4⋊C4—28no—
392C2×C4⋊C4—24no—
393C2×C4⋊C4—614no—
394C2×C4⋊C4—1224no—
395C2×C4⋊C4C22⋊D6118yes—
396C2×C4⋊C4C22×D6114yes—
397C2×C4⋊C4C22×D6118yes—
398C2×C4⋊C4C22×D6118yes—
399C2×C4⋊C4C22×D6114yes—
400C2×C22⋊C4—618no—
401C2×C22⋊C4—614no—
402C2×C22⋊C4—614no—
403C2×C22⋊C4—618no—
404C2×C22⋊C4—618no—
405C2×C22⋊C4—618no—
406C2×C22⋊C4—618no—
407C2×C22⋊C4—618no—
408C2×C22⋊C4—1224no—
409C2×C22⋊C4—1228no—
410C2×C22⋊C4—1224no—
411C2×C22⋊C4—1228no—
412C2×C22⋊C4—2444no—
413C2×C22⋊C4—2444no—
414C2×C22⋊C4—2444no—
415C2×C22⋊C4—2444no—
416C2×C42—24no—
417C2×C42—614no—
418C2×C42—618no—
419C2×C42—1224no—
420C2×C42—2444no—
421C2×C42C22×D6114yes—
422#95—614no—
423#95—614no—
424#95—618no—
425#95—618no—
426C24—614no—
427C24—612no—
428C24—614no—
429C24—1222no—
430C24—1222no—
431C24—1222no—
432C24—1222no—
433C24—1222no—
434C24—1222no—
435C24—1224no—
436C24—1224no—
437C24—1224no—
438C24—1224no—
439C24—1224no—
440C24—1224no—
441C24—1224no—
442C24C23×D6111yescharacteristic
443C2×D4—12216no—
444C2×D4—12216no—
445C2×D4—1228no—
446C2×D4—1228no—
447C2×D4—1228no—
448C2×D4—2448no—
449C2×D4—24432no—
450C2×D4—24416no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—614no—
463C22×C4—612no—
464C22×C4—612no—
465C22×C4—612no—
466C22×C4—614no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1224no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1224no—
476C22×C4—1222no—
477C22×C4—1224no—
478C22×C4—1228no—
479C22×C4—1228no—
480C22×C4—1228no—
481C22×C4—24416no—
482C22×C4#1175822614yes—
483C22×C4#1175822614yes—
484C22×C4C2×C22⋊D6118yes—
485C22×C4C2×C22⋊D6114yes—
486C22×C4C23×D6112yes—
487C22×C4C23×D6112yes—
488C22×C4C23×D6114yes—
489C22×C4C23×D6114yes—
490C22×C4C23×D6112yes—
491C4⋊C4—216no—
492C4⋊C4—216no—
493C4⋊C4—28no—
494C4⋊C4—1228no—
495C4⋊C4C2×C22⋊D61116yes—
496C22⋊C4—1228no—
497C22⋊C4—12216no—
498C22⋊C4—1228no—
499C22⋊C4—12216no—
500C22⋊C4—2448no—
501C22⋊C4—2448no—
502C22⋊C4—2448no—
503C22⋊C4—2448no—
504C42—28no—
505C42—1228no—
506C42—2448no—
507C23—614no—
508C23—1228no—
509C23—1228no—
510C23—1222no—
511C23—1222no—
512C23—1222no—
513C23—1222no—
514C23—1222no—
515C23—1222no—
516C23—1228no—
517C23—1224no—
518C23—1224no—
519C23—1224no—
520C23—1224no—
521C23—1224no—
522C23—1224no—
523C23—1224no—
524C23—2448no—
525C23—2448no—
526C23—2448no—
527C23—2448no—
528C23—2448no—
529C23—2448no—
530C23—2448no—
531C23—2444no—
532C23—2444no—
533C23—2444no—
534C23—2444no—
535C23C22×C22⋊D6112yes—
536C23#41×D6112yes—
537C23#41×D6112yes—
538C23C2×#1175822612yes—
539C23C2×#1175822612yes—
540C23#1451982312yes—
541C23#1451982312yes—
542C23C24×D6111yescharacteristic
543D4—24432no—
544D4—24416no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—216no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—48no—
560C2×C4—614no—
561C2×C4—1228no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1222no—
567C2×C4—1224no—
568C2×C4—1222no—
569C2×C4—1222no—
570C2×C4—1224no—
571C2×C4—1222no—
572C2×C4—1224no—
573C2×C4—2444no—
574C2×C4—2444no—
575C2×C4—2448no—
576C2×C4—24416no—
577C2×C4#14519766116yes—
578C2×C4C22×C22⋊D6118yes—
579C2×C4C22×C22⋊D6114yes—
580C2×C4C2×#1175822614yes—
581C2×C4C2×#1175822614yes—
582C22—1224no—
583C22—1224no—
584C22—1224no—
585C22—48816no—
586C22—2444no—
587C22—2444no—
588C22—2444no—
589C22—2444no—
590C22—2444no—
591C22—2444no—
592C22—2444no—
593C22—2442no—
594C22—2442no—
595C22—2442no—
596C22—24416no—
597C22—2442no—
598C22C2×#1451976614yes—
599C22#1684029714yes—
600C22#1684036814yes—
601C22#1684040014yes—
602C22#1684043114yes—
603C22#1684046514yes—
604C22#1684046514yes—
605C22C23×C22⋊D6111yescharacteristic
606C22C2×#41×D6111yescharacteristic
607C22C2×#41×D6111yescharacteristic
608C22C22×#1175822611yescharacteristic
609C22C22×#1175822611yescharacteristic
610C22C2×#1451982311yescharacteristic
611C22C2×#1451982311yescharacteristic
612C4—28no—
613C4—42no—
614C4—42no—
615C4—44no—
616C4—1224no—
617C4—2442no—
618C4—2442no—
619C4—2444no—
620C4C2×#1451976618yes—
621C2—4884no—
622C2—2444no—
623C2#2145017418yesdirect product factor
624C2C22×#1451976611yescharacteristic
625C2C2×#1684029711yescharacteristic
626C2C2×#1684036811yescharacteristic
627C2C2×#1684040011yescharacteristic
628C2C2×#1684043111yescharacteristic
629C2C2×#1684046511yescharacteristic
630C2C2×#1684046511yescharacteristic
subgroupG/H
centerC24C23×D61
commutator, cosocleC23×C61C25
FrattiniC23C24×D61
FittingC22×#125×C61C2
socleC24×C61C24
prime radicalC22×#14519752C2
hypercenterC22×#125D61
hypocenterC61C2×#2490
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
2. #21450174C2direct (N × H)
3. C2×#16839688C2semidirect
4. C2×#16839679C2semidirect
5. C2×#16839664C2semidirect
6. C2×#16839663C2semidirect
7. C2×#16839663C2semidirect
8. C2×#16839652C2semidirect
9. C2×#16839619C2semidirect
10. C2×C4⋊C4×Dic61C2semidirect
11. C2×C4×#11758198C2semidirect
12. C22×#125×C61C2semidirect
29. #16839688C22semidirect
30. #16839679C22semidirect
31. #16839664C22semidirect
32. #16839663C22semidirect
33. #16839663C22semidirect
34. #16839652C22semidirect
35. #16839619C22semidirect
36. C4⋊C4×Dic61C22semidirect
37. C4×#11758198C22semidirect
43. C2×#125×C61C22semidirect
306. C22×#125D61semidirect
318. C2×C61#2490semidirect
332. C2×#125C2×D61semidirect
364. C61C2×#2490semidirect
623. C2#21450174direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145197522
#214501742
C2×#168396882
C2×#168396792
C2×#168396642
C2×#168396632
C2×#168396522
C2×#168396192
C2×C4⋊C4×Dic612
C2×C4×#117581982
C22×#125×C612
C2×#249061