Finite Group Databasebeta

← C2×#13072971 #14923222

Subgroups — C2×#13072971 #14923222

Order 2816.

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×#66948C211yesmaximal, characteristic
2#13072971C2116yesmaximal, direct product factor
3C2×#10622606C212yesmaximal, complemented
4C2×#10622604C212yesmaximal, complemented
5C2×#10622552C211yesmaximal, characteristic, complemented
6C2×#10622601C211yesmaximal, characteristic, complemented
7C2×#10622601C212yesmaximal, complemented
8C2×#10622603C212yesmaximal, complemented
9C2×#10622600C211yesmaximal, characteristic, complemented
10C2×C4⋊C4×Dic11C211yesmaximal, characteristic, complemented
11C2×C4×(C2×D11)⋊C4C211yesmaximal, characteristic, complemented
12C22×C11×#125C211yesmaximal, characteristic, complemented
13C23×D44C2211yescharacteristic
14C23×D44C2212yes—
15C23×C4×D11C2211yescharacteristic
16C2×C11×#125—28no—
17C2×C11×#125C2218yescomplemented
18C22×C11×C4⋊C4C2211yescharacteristic
19C22×C11×C4⋊C4C2212yes—
20C22×C11×C4⋊C4C2212yes—
21C22×C11×C4⋊C4C2211yescharacteristic
22C22×C42×C11C2211yescharacteristic
23C22×(C2×D11)⋊C4C2211yescharacteristic
24C22×(C2×D11)⋊C4C2211yescharacteristic
25C22×(C2×D11)⋊C4C2212yes—
26C22×(C2×D11)⋊C4C2212yes—
27C22×(C2×D11)⋊C4C2212yes—
28C22×(C2×D11)⋊C4C2212yes—
29C22×(C2×D11)⋊C4C2212yes—
30C22×(C2×D11)⋊C4C2212yes—
31C22×C4⋊Dic11C2211yescharacteristic
32C22×C4×Dic11C2211yescharacteristic
33C22×C4×Dic11C2212yes—
34C2×#66948—28no—
35C2×#66948C2218yes—
36C2×C11⋊#95C2211yescharacteristic
37C2×C11⋊#95C2211yescharacteristic
38C2×C11⋊#95C2212yes—
39C2×C11⋊#95C2212yes—
40#10622606C22116yescomplemented
41#10622604C22116yescomplemented
42#10622603C22116yescomplemented
43#10622601C2218yescomplemented
44#10622601C22116yescomplemented
45#10622600C2218yescomplemented
46C4⋊C4×Dic11C2218yescomplemented
47#10622552C2218yescomplemented
48C4×(C2×D11)⋊C4C2218yescomplemented
49C24×D11C2312yes—
50C24×D11C2311yescharacteristic
51C23×C4×C11C2311yescharacteristic
52C23×C4×C11C2311yescharacteristic
53C23×C4×C11C2312yes—
54C23×C4×C11C2312yes—
55C23×C4×C11C2311yescharacteristic
56C23×Dic11C2312yes—
57C23×Dic11C2311yescharacteristic
58C23×Dic11C2311yescharacteristic
59C23×Dic11C2311yescharacteristic
60C22×D44—28no—
61C22×D44—28no—
62C22×D44—24no—
63C22×D44—24no—
64C22×D44—24no—
65C22×D44—44no—
66C22×D44C2314yes—
67C22×D44C2318yes—
68C22×C4×D11—24no—
69C22×C4×D11—24no—
70C22×C4×D11—24no—
71C22×C4×D11C2314yes—
72C11×#125—232no—
73C2×C11×C4⋊C4—28no—
74C2×C11×C4⋊C4—28no—
75C2×C11×C4⋊C4—24no—
76C2×C11×C4⋊C4D418yes—
77C2×C11×C4⋊C4C2314yes—
78C2×C11×C4⋊C4C2318yes—
79C2×C11×C4⋊C4C2318yes—
80C2×C11×C4⋊C4C2314yes—
81C2×C42×C11—24no—
82C2×C42×C11C2314yes—
83C2×(C2×D11)⋊C4—24no—
84C2×(C2×D11)⋊C4—28no—
85C2×(C2×D11)⋊C4—24no—
86C2×(C2×D11)⋊C4—28no—
87C2×(C2×D11)⋊C4—44no—
88C2×(C2×D11)⋊C4—44no—
89C2×(C2×D11)⋊C4—44no—
90C2×(C2×D11)⋊C4—44no—
91C2×(C2×D11)⋊C4C2318yes—
92C2×(C2×D11)⋊C4C2314yes—
93C2×(C2×D11)⋊C4C2314yes—
94C2×(C2×D11)⋊C4C2318yes—
95C2×(C2×D11)⋊C4C2318yes—
96C2×(C2×D11)⋊C4C2318yes—
97C2×(C2×D11)⋊C4C2318yes—
98C2×(C2×D11)⋊C4C2318yes—
99C2×C4⋊Dic11—24no—
100C2×C4⋊Dic11C2314yes—
101C2×C4×Dic11—24no—
102C2×C4×Dic11—44no—
103C2×C4×Dic11C2314yes—
104C2×C4×Dic11C2318yes—
105#66948—232no—
106C11⋊#95C2314yes—
107C11⋊#95C2314yes—
108C11⋊#95C2318yes—
109C11⋊#95C2318yes—
110C2×#2490—111nomaximal, Sylow
111C24×C11C2411yescharacteristic
112C23×D11—22no—
113C23×D11—22no—
114C23×D11—22no—
115C23×D11—22no—
116C23×D11—22no—
117C23×D11—22no—
118C23×D11—24no—
119C23×D11—24no—
120C23×D11—24no—
121C23×D11—24no—
122C23×D11—24no—
123C23×D11—24no—
124C23×D11—24no—
125C23×D11#4114yes—
126C23×D11C2414yes—
127C23×D11C2412yes—
128C22×C4×C11—24no—
129C22×C4×C11—24no—
130C22×C4×C11—22no—
131C22×C4×C11—22no—
132C22×C4×C11—24no—
133C22×C4×C11—22no—
134C22×C4×C11—22no—
135C22×C4×C11—24no—
136C22×C4×C11—22no—
137C22×C4×C11—22no—
138C22×C4×C11—24no—
139C22×C4×C11C2×D418yes—
140C22×C4×C11C2×D414yes—
141C22×C4×C11#4114yes—
142C22×C4×C11#4114yes—
143C22×C4×C11C2412yes—
144C22×C4×C11C2412yes—
145C22×C4×C11C2414yes—
146C22×C4×C11C2414yes—
147C22×C4×C11C2412yes—
148C22×Dic11—22no—
149C22×Dic11—22no—
150C22×Dic11—22no—
151C22×Dic11—24no—
152C22×Dic11—22no—
153C22×Dic11—22no—
154C22×Dic11—22no—
155C22×Dic11—22no—
156C22×Dic11—24no—
157C22×Dic11—24no—
158C22×Dic11—22no—
159C22×Dic11#4114yes—
160C22×Dic11C2412yes—
161C22×Dic11C2412yes—
162C22×Dic11C2414yes—
163C22×Dic11C2412yes—
164C2×D44—216no—
165C2×D44—216no—
166C2×D44—28no—
167C2×D44—28no—
168C2×D44—28no—
169C2×D44—48no—
170C2×D44—432no—
171C2×D44—416no—
172C2×C4×D11—28no—
173C2×C4×D11—28no—
174C2×C4×D11—28no—
175C2×C4×D11—416no—
176C11×C4⋊C4—216no—
177C11×C4⋊C4—216no—
178C11×C4⋊C4—28no—
179C11×C4⋊C4C2×D4116yes—
180C42×C11—28no—
181(C2×D11)⋊C4—28no—
182(C2×D11)⋊C4—216no—
183(C2×D11)⋊C4—28no—
184(C2×D11)⋊C4—216no—
185(C2×D11)⋊C4—48no—
186(C2×D11)⋊C4—48no—
187(C2×D11)⋊C4—48no—
188(C2×D11)⋊C4—48no—
189C4⋊Dic11—28no—
190C4×Dic11—28no—
191C4×Dic11—48no—
192C22×#125D1111yescharacteristic, complemented
193C22×#121—111no—
194#2490—1116no—
195C2×#397—112no—
196C2×#394—112no—
197C2×#390—111no—
198C2×#390—112no—
199C2×#390—111no—
200C2×#388—112no—
201C2×#380—111no—
202C2×C4×C4⋊C4—111no—
203C2×C4×C22⋊C4—111no—
204C23×C11C22×D412yes—
205C23×C11C2×#4112yes—
206C23×C11C2×#4112yes—
207C23×C11C2×#4112yes—
208C23×C11C2×#4112yes—
209C23×C11H(5,𝔽2)12yes—
210C23×C11H(5,𝔽2)12yes—
211C23×C11C2511yescharacteristic
212C22×D11—28no—
213C22×D11—28no—
214C22×D11—28no—
215C22×D11—22no—
216C22×D11—22no—
217C22×D11—22no—
218C22×D11—22no—
219C22×D11—22no—
220C22×D11—22no—
221C22×D11—24no—
222C22×D11—24no—
223C22×D11—24no—
224C22×D11—24no—
225C22×D11—24no—
226C22×D11—24no—
227C22×D11—24no—
228C22×D11—44no—
229C22×D11—44no—
230C22×D11—44no—
231C22×D11—44no—
232C22×D11—48no—
233C22×D11—48no—
234C22×D11—48no—
235C22×D11—48no—
236C22×D11—48no—
237C22×D11—48no—
238C22×D11—48no—
239C22×D11C2×#4114yes—
240C2×C4×C11—216no—
241C2×C4×C11—24no—
242C2×C4×C11—24no—
243C2×C4×C11—22no—
244C2×C4×C11—22no—
245C2×C4×C11—24no—
246C2×C4×C11—22no—
247C2×C4×C11—22no—
248C2×C4×C11—24no—
249C2×C4×C11—22no—
250C2×C4×C11—22no—
251C2×C4×C11—24no—
252C2×C4×C11—44no—
253C2×C4×C11—44no—
254C2×C4×C11—48no—
255C2×C4×C11#124116yes—
256C2×C4×C11C22×D418yes—
257C2×C4×C11C22×D414yes—
258C2×C4×C11C2×#4114yes—
259C2×C4×C11C2×#4114yes—
260C2×Dic11—28no—
261C2×Dic11—22no—
262C2×Dic11—22no—
263C2×Dic11—22no—
264C2×Dic11—22no—
265C2×Dic11—24no—
266C2×Dic11—22no—
267C2×Dic11—22no—
268C2×Dic11—22no—
269C2×Dic11—24no—
270C2×Dic11—24no—
271C2×Dic11—22no—
272C2×Dic11—44no—
273C2×Dic11—48no—
274C2×Dic11—44no—
275C2×Dic11C2×#4114yes—
276D44—432no—
277D44—416no—
278C4×D11—416no—
279C23×D4—111no—
280C23×D4—112no—
281C24×C4—111no—
282C2×#125—28no—
283C2×#125C2×D1118yescomplemented
284C2×#121—118no—
285C2×#121—228no—
286C22×C4⋊C4—111no—
287C22×C4⋊C4C2×D1112yes—
288C22×C4⋊C4C2×D1112yes—
289C22×C4⋊C4C2×D1111yescharacteristic
290C22×C4⋊C4C2×D1111yescharacteristic
291C22×C22⋊C4—111no—
292C22×C22⋊C4—112no—
293C22×C22⋊C4—111no—
294C22×C22⋊C4—112no—
295C22×C22⋊C4—112no—
296C22×C22⋊C4—112no—
297C22×C22⋊C4—112no—
298C22×C22⋊C4—112no—
299C22×C42—111no—
300C22×C42—112no—
301C22×C42C2×D1111yescharacteristic
302#397—1116no—
303#394—1116no—
304#390—118no—
305#390—1116no—
306#390—118no—
307#388—1116no—
308#380—118no—
309C4×C4⋊C4—118no—
310C4×C22⋊C4—118no—
311C2×#95—112no—
312C2×#95—111no—
313C2×#95—112no—
314C2×#95—111no—
315C22×C11C2×#12414yes—
316C22×C11#52914yes—
317C22×C11#54014yes—
318C22×C11#54614yes—
319C22×C11#55014yes—
320C22×C11#55314yes—
321C22×C11#55314yes—
322C22×C11C23×D411yescharacteristic
323C22×C11C22×#4111yescharacteristic
324C22×C11C22×#4111yescharacteristic
325C22×C11C22×#4111yescharacteristic
326C22×C11C22×#4111yescharacteristic
327C22×C11C2×H(5,𝔽2)11yescharacteristic
328C22×C11C2×H(5,𝔽2)11yescharacteristic
329C2×D11—24no—
330C2×D11—24no—
331C2×D11—24no—
332C2×D11—42no—
333C2×D11—42no—
334C2×D11—42no—
335C2×D11—42no—
336C2×D11—416no—
337C2×D11—44no—
338C2×D11—44no—
339C2×D11—44no—
340C2×D11—44no—
341C2×D11—44no—
342C2×D11—44no—
343C2×D11—44no—
344C2×D11—816no—
345C4×C11—28no—
346C4×C11—42no—
347C4×C11—42no—
348C4×C11—44no—
349C4×C11C2×#12418yes—
350Dic11—24no—
351Dic11—42no—
352Dic11—44no—
353Dic11—42no—
354C25—111no—
355C25—112no—
356C22×D4—114no—
357C22×D4—118no—
358C22×D4—228no—
359C22×D4—224no—
360C22×D4—224no—
361C22×D4—228no—
362C22×D4—224no—
363C22×D4—444no—
364C23×C4—111no—
365C23×C4—112no—
366C23×C4—111no—
367C23×C4—111no—
368C23×C4—114no—
369C23×C4—224no—
370C23×C4—224no—
371C23×C4—224no—
372C23×C4C22×D1112yes—
373C23×C4C22×D1111yescharacteristic
374C23×C4C22×D1111yescharacteristic
375C23×C4C22×D1112yes—
376C23×C4C22×D1111yescharacteristic
377#125—232no—
378#121—2232no—
379C2×C4⋊C4—28no—
380C2×C4⋊C4—28no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—114no—
383C2×C4⋊C4—224no—
384C2×C4⋊C4C22⋊D1118yes—
385C2×C4⋊C4C22×D1114yes—
386C2×C4⋊C4C22×D1118yes—
387C2×C4⋊C4C22×D1118yes—
388C2×C4⋊C4C22×D1114yes—
389C2×C22⋊C4—118no—
390C2×C22⋊C4—114no—
391C2×C22⋊C4—114no—
392C2×C22⋊C4—118no—
393C2×C22⋊C4—118no—
394C2×C22⋊C4—118no—
395C2×C22⋊C4—118no—
396C2×C22⋊C4—118no—
397C2×C22⋊C4—224no—
398C2×C22⋊C4—228no—
399C2×C22⋊C4—224no—
400C2×C22⋊C4—228no—
401C2×C22⋊C4—444no—
402C2×C22⋊C4—444no—
403C2×C22⋊C4—444no—
404C2×C22⋊C4—444no—
405C2×C42—24no—
406C2×C42—114no—
407C2×C42—118no—
408C2×C42—224no—
409C2×C42—444no—
410C2×C42C22×D1114yes—
411#95—114no—
412#95—114no—
413#95—118no—
414#95—118no—
415C2×C11#249018yescomplemented
416C2×C11C22×#12411yescharacteristic
417C2×C11C2×#52911yescharacteristic
418C2×C11C2×#54011yescharacteristic
419C2×C11C2×#54611yescharacteristic
420C2×C11C2×#55011yescharacteristic
421C2×C11C2×#55311yescharacteristic
422C2×C11C2×#55311yescharacteristic
423D11—44no—
424D11—84no—
425C24—114no—
426C24—112no—
427C24—114no—
428C24—222no—
429C24—222no—
430C24—222no—
431C24—222no—
432C24—222no—
433C24—222no—
434C24—224no—
435C24—224no—
436C24—224no—
437C24—224no—
438C24—224no—
439C24—224no—
440C24—224no—
441C24C23×D1111yescharacteristic
442C2×D4—2216no—
443C2×D4—228no—
444C2×D4—228no—
445C2×D4—2216no—
446C2×D4—228no—
447C2×D4—448no—
448C2×D4—4432no—
449C2×D4—4416no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—114no—
462C22×C4—112no—
463C22×C4—112no—
464C22×C4—112no—
465C22×C4—114no—
466C22×C4—222no—
467C22×C4—222no—
468C22×C4—222no—
469C22×C4—224no—
470C22×C4—222no—
471C22×C4—222no—
472C22×C4—222no—
473C22×C4—224no—
474C22×C4—222no—
475C22×C4—224no—
476C22×C4—222no—
477C22×C4—228no—
478C22×C4—228no—
479C22×C4—228no—
480C22×C4—4416no—
481C22×C4#434314yes—
482C22×C4#434314yes—
483C22×C4C2×C22⋊D1118yes—
484C22×C4C2×C22⋊D1114yes—
485C22×C4C23×D1112yes—
486C22×C4C23×D1112yes—
487C22×C4C23×D1114yes—
488C22×C4C23×D1114yes—
489C22×C4C23×D1112yes—
490C4⋊C4—216no—
491C4⋊C4—216no—
492C4⋊C4—28no—
493C4⋊C4—228no—
494C4⋊C4C2×C22⋊D11116yes—
495C22⋊C4—228no—
496C22⋊C4—2216no—
497C22⋊C4—228no—
498C22⋊C4—2216no—
499C22⋊C4—448no—
500C22⋊C4—448no—
501C22⋊C4—448no—
502C22⋊C4—448no—
503C42—28no—
504C42—228no—
505C42—448no—
506C11C2×#249011yescharacteristic, complemented, Sylow
507C23—114no—
508C23—228no—
509C23—222no—
510C23—222no—
511C23—222no—
512C23—222no—
513C23—222no—
514C23—222no—
515C23—228no—
516C23—228no—
517C23—224no—
518C23—224no—
519C23—224no—
520C23—224no—
521C23—224no—
522C23—224no—
523C23—224no—
524C23—444no—
525C23—444no—
526C23—444no—
527C23—444no—
528C23—448no—
529C23—448no—
530C23—448no—
531C23—448no—
532C23—448no—
533C23—448no—
534C23—448no—
535C23C2×#434312yes—
536C23C2×#434312yes—
537C23#41×D1112yes—
538C23#41×D1112yes—
539C23#6704212yes—
540C23#6704212yes—
541C23C22×C22⋊D1112yes—
542C23C24×D1111yescharacteristic
543D4—4432no—
544D4—4416no—
545C2×C4—216no—
546C2×C4—24no—
547C2×C4—24no—
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—114no—
561C2×C4—222no—
562C2×C4—222no—
563C2×C4—228no—
564C2×C4—222no—
565C2×C4—224no—
566C2×C4—222no—
567C2×C4—222no—
568C2×C4—222no—
569C2×C4—224no—
570C2×C4—222no—
571C2×C4—224no—
572C2×C4—222no—
573C2×C4—444no—
574C2×C4—444no—
575C2×C4—448no—
576C2×C4—4416no—
577C2×C4#67000116yes—
578C2×C4C2×#434314yes—
579C2×C4C2×#434314yes—
580C2×C4C22×C22⋊D1118yes—
581C2×C4C22×C22⋊D1114yes—
582C22—224no—
583C22—224no—
584C22—224no—
585C22—442no—
586C22—442no—
587C22—4416no—
588C22—442no—
589C22—442no—
590C22—444no—
591C22—444no—
592C22—444no—
593C22—444no—
594C22—444no—
595C22—444no—
596C22—444no—
597C22—8816no—
598C22#1062322214yes—
599C22#1062322614yes—
600C22#1062322614yes—
601C22#1062322914yes—
602C22C2×#6700014yes—
603C22#1062336014yes—
604C22#1062336514yes—
605C22C22×#434311yescharacteristic
606C22C22×#434311yescharacteristic
607C22C2×#41×D1111yescharacteristic
608C22C2×#41×D1111yescharacteristic
609C22C2×#6704211yescharacteristic
610C22C2×#6704211yescharacteristic
611C22C23×C22⋊D1111yescharacteristic
612C4—28no—
613C4—42no—
614C4—42no—
615C4—44no—
616C4—224no—
617C4—442no—
618C4—442no—
619C4—444no—
620C4C2×#6700018yes—
621C2—444no—
622C2—884no—
623C2#1307297118yesdirect product factor
624C2C22×#6700011yescharacteristic
625C2C2×#1062322211yescharacteristic
626C2C2×#1062322911yescharacteristic
627C2C2×#1062336011yescharacteristic
628C2C2×#1062336511yescharacteristic
629C2C2×#1062322611yescharacteristic
630C2C2×#1062322611yescharacteristic
subgroupG/H
centerC24C23×D11
commutator, cosocleC23×C11C25
FrattiniC23C24×D11
FittingC22×C11×#125C2
socleC24×C11C24
prime radicalC22×#66948C2
hypercenterC22×#125D11
hypocenterC11C2×#2490
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
2. #13072971C2direct (N × H)
3. C2×#10622606C2semidirect
4. C2×#10622604C2semidirect
5. C2×#10622552C2semidirect
6. C2×#10622601C2semidirect
7. C2×#10622601C2semidirect
8. C2×#10622603C2semidirect
9. C2×#10622600C2semidirect
10. C2×C4⋊C4×Dic11C2semidirect
11. C2×C4×(C2×D11)⋊C4C2semidirect
12. C22×C11×#125C2semidirect
17. C2×C11×#125C22semidirect
40. #10622606C22semidirect
41. #10622604C22semidirect
42. #10622603C22semidirect
43. #10622601C22semidirect
44. #10622601C22semidirect
45. #10622600C22semidirect
46. C4⋊C4×Dic11C22semidirect
47. #10622552C22semidirect
48. C4×(C2×D11)⋊C4C22semidirect
192. C22×#125D11semidirect
283. C2×#125C2×D11semidirect
415. C2×C11#2490semidirect
506. C11C2×#2490semidirect
623. C2#13072971direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#669482
#130729712
C2×#106226062
C2×#106226042
C2×#106225522
C2×#106226012
C2×#106226032
C2×#106226002
C2×C4⋊C4×Dic112
C2×C4×(C2×D11)⋊C42
C22×C11×#1252
C2×#249011