Finite Group Databasebeta

← C2×#21444135 #29645810

Subgroups — C2×#21444135 #29645810

Order 15616.

1826 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C22⋊D61C211yesmaximal, characteristic
2C22×C22⋊#92418C211yesmaximal, characteristic, complemented
3#21444135C218yesmaximal, direct product factor
4C2×#16839487C212yesmaximal, complemented
5C2×#16839485C212yesmaximal, complemented
6C22×C22⋊C8×C61C211yesmaximal, characteristic, complemented
7C23×C22⋊D61C411yescharacteristic
8C23×C4×D61C411yescharacteristic
9C23×#92418—21no—
10C2×C4×C22⋊D61—21no—
11C2×C4×C22⋊D61—21no—
12C2×C4×C22⋊D61—22no—
13C2×C4×C22⋊D61—21no—
14C2×C4×C22⋊D61—21no—
15C2×C4×C22⋊D61—22no—
16C2×C4×C22⋊D61C414yes—
17C2×C4×C22⋊D61C2214yes—
18C22×#11758196C2211yescharacteristic
19C22×C22⋊Dic61C411yescharacteristic
20C22×#11758198C2211yescharacteristic
21C22×C4×Dic61C411yescharacteristic
22C2×C22⋊#92418—22no—
23C2×C22⋊#92418—21no—
24C2×C22⋊#92418C2212yes—
25C2×C22⋊#92418C2214yescomplemented
26#16839487C2218yescomplemented
27#16839485C2218yescomplemented
28C24×C4×C61C2211yescharacteristic
29C23×C8×C61—21no—
30C2×C22⋊C8×C61—22no—
31C2×C22⋊C8×C61—21no—
32C2×C22⋊C8×C61C2212yes—
33C2×C22⋊C8×C61C2214yescomplemented
34C24×D61C2×C411yescharacteristic
35C22×C22⋊D61—21no—
36C22×C22⋊D61—21no—
37C22×C22⋊D61—22no—
38C22×C22⋊D61—24no—
39C22×C22⋊D61C818yescomplemented
40C22×C22⋊D61C2×C414yes—
41C23×Dic61C2×C411yescharacteristic
42C23×Dic61C2×C411yescharacteristic
43C23×Dic61C2×C411yescharacteristic
44C22×C4×D61—21no—
45C22×C4×D61—22no—
46C22×C4×D61—21no—
47C22×C4×D61—44no—
48C22×C4×D61C2×C414yes—
49C22×#92418—22no—
50C22×#92418—24no—
51C22×#92418—44no—
52C4×C22⋊D61—24no—
53C4×C22⋊D61—24no—
54C4×C22⋊D61—24no—
55C4×C22⋊D61—24no—
56C4×C22⋊D61—24no—
57C4×C22⋊D61—24no—
58C4×C22⋊D61—24no—
59C4×C22⋊D61—24no—
60C2×#11758196—21no—
61C2×#11758196—21no—
62C2×#11758196—21no—
63C2×#11758196—21no—
64C2×#11758196C2×C412yes—
65C2×#11758196C2312yes—
66C2×C22⋊Dic61—21no—
67C2×C22⋊Dic61—21no—
68C2×C22⋊Dic61—22no—
69C2×C22⋊Dic61C2×C414yes—
70C2×#11758198—21no—
71C2×#11758198—21no—
72C2×#11758198—21no—
73C2×#11758198—21no—
74C2×#11758198C2×C412yes—
75C2×#11758198C2312yes—
76C2×C4×Dic61—21no—
77C2×C4×Dic61—22no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61C2×C414yes—
80C22⋊#92418—22no—
81C22⋊#92418—24no—
82C22⋊#92418C2314yes—
83C25×C61C2×C411yescharacteristic
84C23×C4×C61—44no—
85C23×C4×C61C2×C411yescharacteristic
86C23×C4×C61C2×C412yes—
87C23×C4×C61D411yescharacteristic
88C23×C4×C61D412yes—
89C23×C4×C61D411yescharacteristic
90C23×C4×C61D411yescharacteristic
91C23×C4×C61D412yes—
92C23×C4×C61D411yescharacteristic
93C23×C4×C61C2311yescharacteristic
94C23×C4×C61C2312yes—
95C22×C8×C61—22no—
96C22×C8×C61—24no—
97C22×C8×C61—44no—
98C22⋊C8×C61—22no—
99C22⋊C8×C61—24no—
100C22⋊C8×C61C2314yes—
101C24×C61—42no—
102C24×C61—42no—
103C24×C61C22⋊C411yescharacteristic
104C24×C61C22⋊C411yescharacteristic
105C24×C61C22⋊C412yes—
106C24×C61C2×C814yes—
107C24×C61M1614yes—
108C24×C61C22×C411yescharacteristic
109C24×C61C22×C412yes—
110C23×D61—21no—
111C23×D61—21no—
112C23×D61—22no—
113C23×D61—42no—
114C23×D61—42no—
115C23×D61C2×C814yescomplemented
116C23×D61C22×C412yes—
117C22×C4×C61—24no—
118C22×C4×C61—24no—
119C22×C4×C61—24no—
120C22×C4×C61—44no—
121C22×C4×C61—44no—
122C22×C4×C61—44no—
123C22×C4×C61—44no—
124C22×C4×C61—42no—
125C22×C4×C61—42no—
126C22×C4×C61—42no—
127C22×C4×C61C22⋊C411yescharacteristic
128C22×C4×C61C22⋊C412yes—
129C22×C4×C61C22⋊C412yes—
130C22×C4×C61C22⋊C412yes—
131C22×C4×C61C22⋊C412yes—
132C22×C4×C61C22⋊C412yes—
133C22×C4×C61C22⋊C411yescharacteristic
134C22×C4×C61C22×C412yes—
135C22×C4×C61C22×C411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D412yes—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D412yes—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D412yes—
144C22×C4×C61C2411yescharacteristic
145C2×C22⋊D61—28no—
146C2×C22⋊D61—28no—
147C2×C22⋊D61—216no—
148C2×C22⋊D61—24no—
149C2×C22⋊D61—24no—
150C2×C22⋊D61—28no—
151C2×C22⋊D61C2×C8116yescomplemented
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—22no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—42no—
160C22×Dic61—42no—
161C22×Dic61C2×C814yescomplemented
162C22×Dic61C22×C412yes—
163C22×Dic61C22×C412yes—
164C22×Dic61C22×C412yes—
165C2×C4×D61—24no—
166C2×C4×D61—24no—
167C2×C4×D61—44no—
168C2×C4×D61—48no—
169C2×C4×D61—44no—
170C2×C4×D61—48no—
171C2×C8×C61—24no—
172C2×C8×C61—44no—
173C2×C8×C61—48no—
174C22⋊Dic61—24no—
175C22⋊Dic61—24no—
176#11758198—22no—
177#11758198—22no—
178#11758198—22no—
179#11758198—22no—
180#11758196—22no—
181#11758196—22no—
182#11758196—22no—
183#11758196—22no—
184C4×Dic61—24no—
185C4×Dic61—24no—
186C2×#92418—24no—
187C2×#92418—44no—
188C2×#92418—48no—
189C23×C61—22no—
190C23×C61—22no—
191C23×C61—22no—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—42no—
196C23×C61—42no—
197C23×C61—42no—
198C23×C61—48no—
199C23×C61—48no—
200C23×C61—42no—
201C23×C61C22⋊C814yes—
202C23×C61C22⋊C814yes—
203C23×C61C22⋊C818yes—
204C23×C61C2×C22⋊C412yes—
205C23×C61C2×C22⋊C412yes—
206C23×C61C2×C22⋊C411yescharacteristic
207C23×C61C2×C22⋊C411yescharacteristic
208C23×C61C22×C812yes—
209C23×C61C22×C814yes—
210C23×C61C2×M1612yes—
211C23×C61C2×M1614yes—
212C23×C61C23×C411yescharacteristic
213C22×D61—22no—
214C22×D61—22no—
215C22×D61—24no—
216C22×D61—24no—
217C22×D61—42no—
218C22×D61—42no—
219C22×D61—48no—
220C22×D61—42no—
221C22×D61—42no—
222C22×D61—42no—
223C22×D61—48no—
224C22×D61—42no—
225C22×D61C22×C814yescomplemented
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—24no—
232C2×C4×C61—24no—
233C2×C4×C61—24no—
234C2×C4×C61—24no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—41no—
238C2×C4×C61—42no—
239C2×C4×C61—41no—
240C2×C4×C61—42no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61#9914yes—
244C2×C4×C61C2×C22⋊C412yes—
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C2×C22⋊C412yes—
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C22⋊D61—216no—
251C22⋊D61—216no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—24no—
255C2×Dic61—24no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61—42no—
264C2×Dic61C22×C814yescomplemented
265C4×D61—48no—
266C4×D61—48no—
267C8×C61—44no—
268#92418—44no—
269C2×#1459—611nomaximal, Sylow
270C22×C61—21no—
271C22×C61—21no—
272C22×C61—22no—
273C22×C61—22no—
274C22×C61—21no—
275C22×C61—21no—
276C22×C61—22no—
277C22×C61—22no—
278C22×C61—24no—
279C22×C61—24no—
280C22×C61—28no—
281C22×C61—28no—
282C22×C61—48no—
283C22×C61—44no—
284C22×C61—44no—
285C22×C61C2×C22⋊C812yes—
286C22×C61C2×C22⋊C814yes—
287C22×C61C2×C22⋊C814yes—
288C22×C61C2×C22⋊C812yes—
289C22×C61C2×#9912yes—
290C22×C61#41312yes—
291C22×C61C22×C22⋊C411yescharacteristic
292C22×C61C23×C811yescharacteristic
293C22×C61C22×M1611yescharacteristic
294C2×D61—24no—
295C2×D61—44no—
296C2×D61—48no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—48no—
300C2×D61—44no—
301C4×C61—21no—
302C4×C61—21no—
303C4×C61—22no—
304C4×C61—21no—
305C4×C61—21no—
306C4×C61C2×#9914yes—
307Dic61—24no—
308Dic61—42no—
309Dic61—42no—
310C22×C4×D4—611no—
311C22×C22⋊C8—611no—
312C22×C22⋊C8D6111yescharacteristic, complemented
313#1459—618no—
314C2×#324—612no—
315C2×#323—612no—
316C2×C61—21no—
317C2×C61—21no—
318C2×C61—22no—
319C2×C61—24no—
320C2×C61—24no—
321C2×C61#145914yescomplemented
322C2×C61C22×C22⋊C811yescharacteristic
323C2×C61C22×#9911yescharacteristic
324C2×C61C2×#41311yescharacteristic
325D61—44no—
326C23×D4—611no—
327C24×C4—611no—
328C24×C4C2×D6111yescharacteristic
329C23×C8—21no—
330C23×C8—1221no—
331C2×C4×D4—614no—
332C2×C4×D4—614no—
333C2×C4×D4—1221no—
334C2×C4×D4—1222no—
335C2×C4×D4—1221no—
336C2×C4×D4—1221no—
337C2×C4×D4—1222no—
338C2×C4×D4—1221no—
339C22×C4⋊C4—611no—
340C22×C22⋊C4—611no—
341C22×C22⋊C4—611no—
342C22×C42—611no—
343C2×C22⋊C8—22no—
344C2×C22⋊C8—21no—
345C2×C22⋊C8—612no—
346C2×C22⋊C8—614no—
347C2×C22⋊C8—1222no—
348C2×C22⋊C8—1221no—
349C2×C22⋊C8C2×D6112yes—
350C2×C22⋊C8C2×D6114yescomplemented
351#324—618no—
352#323—618no—
353C61C2×#145911yescharacteristic, complemented, Sylow
354C25—611no—
355C25C4×D6111yescharacteristic
356C22×D4—618no—
357C22×D4—614no—
358C22×D4—1224no—
359C22×D4—1221no—
360C22×D4—1221no—
361C22×D4—1222no—
362C23×C4—44no—
363C23×C4—611no—
364C23×C4—611no—
365C23×C4—611no—
366C23×C4—614no—
367C23×C4—1222no—
368C23×C4—1221no—
369C23×C4—1221no—
370C23×C4—2444no—
371C23×C4C4×D6111yescharacteristic
372C23×C4C4×D6112yes—
373C23×C4D24411yescharacteristic
374C23×C4D24412yes—
375C23×C4D24411yescharacteristic
376C23×C4C22⋊D6111yescharacteristic
377C23×C4C22⋊D6112yes—
378C23×C4C22⋊D6111yescharacteristic
379C23×C4C22×D6111yescharacteristic
380C23×C4C22×D6112yes—
381C22×C8—22no—
382C22×C8—24no—
383C22×C8—44no—
384C22×C8—1222no—
385C22×C8—1224no—
386C22×C8—2444no—
387C4×D4—1224no—
388C4×D4—1224no—
389C4×D4—1224no—
390C4×D4—1224no—
391C4×D4—1224no—
392C4×D4—1224no—
393C4×D4—1224no—
394C4×D4—1224no—
395C2×C4⋊C4—612no—
396C2×C4⋊C4—612no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C22⋊C4—614no—
402C2×C22⋊C4—612no—
403C2×C22⋊C4—612no—
404C2×C22⋊C4—1221no—
405C2×C22⋊C4—1221no—
406C2×C22⋊C4—1222no—
407C2×C22⋊C4—1221no—
408C2×C22⋊C4—1221no—
409C2×C22⋊C4—1221no—
410C2×C22⋊C4—1221no—
411C2×C42—614no—
412C2×C42—1221no—
413C2×C42—1222no—
414C2×C42—1221no—
415C22⋊C8—22no—
416C22⋊C8—24no—
417C22⋊C8—614no—
418C22⋊C8—1222no—
419C22⋊C8—1224no—
420C22⋊C8C22×D6114yes—
421C24—42no—
422C24—42no—
423C24—612no—
424C24—614no—
425C24—1221no—
426C24—1221no—
427C24—1222no—
428C24—2442no—
429C24—2442no—
430C24C8×D6114yes—
431C24#1175818914yes—
432C24#1175819811yescharacteristic
433C24#1175819811yescharacteristic
434C24#1175819812yes—
435C24C2×C4×D6111yescharacteristic
436C24C2×C4×D6112yes—
437C2×D4—6116no—
438C2×D4—1228no—
439C2×D4—1228no—
440C2×D4—12216no—
441C2×D4—1224no—
442C2×D4—1224no—
443C2×D4—1228no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—612no—
455C22×C4—614no—
456C22×C4—612no—
457C22×C4—612no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1222no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—1224no—
466C22×C4—1224no—
467C22×C4—2444no—
468C22×C4—2448no—
469C22×C4—2448no—
470C22×C4—2444no—
471C22×C4—2442no—
472C22×C4—2442no—
473C22×C4#1175819811yescharacteristic
474C22×C4#1175819812yes—
475C22×C4#1175819812yes—
476C22×C4#1175819812yes—
477C22×C4#1175819812yes—
478C22×C4#1175819812yes—
479C22×C4#1175819811yescharacteristic
480C22×C4C2×C4×D6112yes—
481C22×C4C2×C4×D6111yescharacteristic
482C22×C4C2×D24411yescharacteristic
483C22×C4C2×D24412yes—
484C22×C4C2×D24411yescharacteristic
485C22×C4C2×D24412yes—
486C22×C4C2×C22⋊D6111yescharacteristic
487C22×C4C2×C22⋊D6112yes—
488C22×C4C2×C22⋊D6111yescharacteristic
489C22×C4C2×C22⋊D6112yes—
490C22×C4C23×D6111yescharacteristic
491C2×C8—24no—
492C2×C8—44no—
493C2×C8—48no—
494C2×C8—1224no—
495C2×C8—2444no—
496C2×C8—2448no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C22⋊C4—1224no—
502C22⋊C4—1224no—
503C22⋊C4—1222no—
504C22⋊C4—1222no—
505C22⋊C4—1222no—
506C22⋊C4—1222no—
507C42—1224no—
508C42—1224no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—42no—
516C23—42no—
517C23—48no—
518C23—48no—
519C23—42no—
520C23—42no—
521C23—614no—
522C23—1222no—
523C23—1224no—
524C23—1224no—
525C23—1222no—
526C23—2442no—
527C23—2448no—
528C23—2442no—
529C23—2442no—
530C23—2448no—
531C23—2442no—
532C23—2442no—
533C23—2442no—
534C23#1451968914yes—
535C23#1451968914yes—
536C23#1451968918yes—
537C23C2×#1175819812yes—
538C23C2×#1175819812yes—
539C23C2×#1175819811yescharacteristic
540C23C2×#1175819811yescharacteristic
541C23C2×C8×D6112yes—
542C23C2×C8×D6114yes—
543C23C2×#1175818912yes—
544C23C2×#1175818914yes—
545C23C22×C4×D6111yescharacteristic
546D4—12216no—
547D4—12216no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—614no—
566C2×C4—1222no—
567C2×C4—1224no—
568C2×C4—1224no—
569C2×C4—1222no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—2448no—
573C2×C4—2442no—
574C2×C4—2442no—
575C2×C4—2442no—
576C2×C4—2442no—
577C2×C4—2448no—
578C2×C4—2442no—
579C2×C4—2442no—
580C2×C4#1451969114yes—
581C2×C4C2×#1175819812yes—
582C2×C4C2×#1175819811yescharacteristic
583C2×C4C2×#1175819811yescharacteristic
584C2×C4C2×#1175819812yes—
585C2×C4C22×C22⋊D6111yescharacteristic
586C2×C4C22×D24411yescharacteristic
587C8—44no—
588C8—2444no—
589C22—24no—
590C22—24no—
591C22—28no—
592C22—28no—
593C22—21no—
594C22—21no—
595C22—22no—
596C22—22no—
597C22—21no—
598C22—21no—
599C22—22no—
600C22—22no—
601C22—48no—
602C22—44no—
603C22—44no—
604C22—1224no—
605C22—2444no—
606C22—2448no—
607C22—2444no—
608C22—2448no—
609C22—2442no—
610C22—2442no—
611C22C2×#1451968912yes—
612C22C2×#1451968914yes—
613C22C2×#1451968914yes—
614C22C2×#1451968912yes—
615C22C2×#1451969112yes—
616C22#1683974412yes—
617C22C22×#1175819811yescharacteristic
618C22C22×C8×D6111yescharacteristic
619C22C22×#1175818911yescharacteristic
620C4—21no—
621C4—21no—
622C4—22no—
623C4—21no—
624C4—21no—
625C4—1224no—
626C4—2442no—
627C4—2442no—
628C4C2×#1451969114yes—
629C2—22no—
630C2—21no—
631C2—21no—
632C2—24no—
633C2—24no—
634C2—2444no—
635C2#2144413514yesdirect product factor
636C2C22×#1451968911yescharacteristic
637C2C22×#1451969111yescharacteristic
638C2C2×#1683974411yescharacteristic
subgroupG/H
centerC22×C4#11758198
commutatorC22×C61C23×C8
FrattiniC22×C4C23×D61
FittingC22×C22⋊C8×C61C2
socleC23×C61C2×C22⋊C4
prime radicalC23×C22⋊D61C4
cosocleC22×C4×C61C24
hypercenterC22×C22⋊C8D61
hypocenterC61C2×#1459
LCSG ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C22×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C22×C22⋊#92418C2semidirect
3. #21444135C2direct (N × H)
4. C2×#16839487C2semidirect
5. C2×#16839485C2semidirect
6. C22×C22⋊C8×C61C2semidirect
25. C2×C22⋊#92418C22semidirect
26. #16839487C22semidirect
27. #16839485C22semidirect
33. C2×C22⋊C8×C61C22semidirect
39. C22×C22⋊D61C8semidirect
115. C23×D61C2×C8semidirect
151. C2×C22⋊D61C2×C8semidirect
161. C22×Dic61C2×C8semidirect
225. C22×D61C22×C8semidirect
264. C2×Dic61C22×C8semidirect
312. C22×C22⋊C8D61semidirect
321. C2×C61#1459semidirect
350. C2×C22⋊C8C2×D61semidirect
353. C61C2×#1459semidirect
635. C2#21444135direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C22⋊D612
C22×C22⋊#924182
#214441352
C2×#168394872
C2×#168394852
C22×C22⋊C8×C612
C2×#145961