Finite Group Databasebeta

← C2×#13072977 #14922875

Subgroups — C2×#13072977 #14922875

Order 2816.

3678 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66922C211yesmaximal, characteristic, complemented
2C22×#66938C212yesmaximal, complemented
3C22×#66991C211yesmaximal, characteristic
4#13072977C2116yesmaximal, direct product factor
5C2×#10622574C212yesmaximal, complemented
6C2×#10622560C212yesmaximal, complemented
7C2×#10622834C211yesmaximal, characteristic, complemented
8C2×#10622564C212yesmaximal, complemented
9C2×#10622600C211yesmaximal, characteristic, complemented
10C2×C4⋊C4×Dic11C211yesmaximal, characteristic, complemented
11C2×C4×(C2×D11)⋊C4C211yesmaximal, characteristic, complemented
12C2×C11×#390C211yesmaximal, characteristic, complemented
13C23×D4×C11C2211yescharacteristic
14C23×C22⋊D11C2212yes—
15C23×C4×D11C2211yescharacteristic
16C23×Dic22C2212yes—
17C22×C11×C4⋊C4C2211yescharacteristic
18C22×C11×C22⋊C4C2212yes—
19C22×C11×C22⋊C4C2212yes—
20C22×C42×C11C2211yescharacteristic
21C22×C22⋊Dic11C2212yes—
22C22×C22⋊Dic11C2212yes—
23C22×C22⋊Dic11C2212yes—
24C2×#66991—28no—
25C2×#66991C2218yes—
26C22×(C2×D11)⋊C4C2212yes—
27C22×(C2×D11)⋊C4C2211yescharacteristic
28C22×(C2×D11)⋊C4C2211yescharacteristic
29C22×C4⋊Dic11C2211yescharacteristic
30C22×C4⋊Dic11C2212yes—
31C22×C4⋊Dic11C2211yescharacteristic
32C22×C4×Dic11C2212yes—
33C22×C4×Dic11C2211yescharacteristic
34C2×#66938—216no—
35C2×#66938C22116yescomplemented
36C2×#66922—28no—
37C2×#66922C2218yescomplemented
38C11×#390C2218yescomplemented
39#10622834C2218yescomplemented
40C2×C11⋊#95C2211yescharacteristic
41C2×C11⋊#95C2211yescharacteristic
42C2×C11⋊#95C2212yes—
43C2×C11⋊#95C2212yes—
44#10622600C2218yescomplemented
45C4⋊C4×Dic11C2218yescomplemented
46#10622574C22116yescomplemented
47#10622564C22116yescomplemented
48#10622560C22116yescomplemented
49C4×(C2×D11)⋊C4C2218yescomplemented
50C25×C11C2312yes—
51C24×D11C2311yescharacteristic
52C22×D4×C11—24no—
53C22×D4×C11—24no—
54C22×D4×C11—24no—
55C22×D4×C11C2314yes—
56C23×C4×C11C2312yes—
57C23×C4×C11C2311yescharacteristic
58C23×C4×C11C2311yescharacteristic
59C23×C4×C11C2311yescharacteristic
60C22×C22⋊D11—28no—
61C22×C22⋊D11—28no—
62C22×C22⋊D11—44no—
63C22×C22⋊D11C2318yes—
64C23×Dic11C2311yescharacteristic
65C23×Dic11C2312yes—
66C23×Dic11C2311yescharacteristic
67C23×Dic11C2311yescharacteristic
68C23×Dic11C2312yes—
69C22×C4×D11—24no—
70C22×C4×D11—24no—
71C22×C4×D11—24no—
72C22×C4×D11C2314yes—
73C22×Dic22—28no—
74C22×Dic22—28no—
75C22×Dic22—44no—
76C22×Dic22C2318yes—
77C2×C11×C4⋊C4—24no—
78C2×C11×C4⋊C4C2314yes—
79C2×C11×C22⋊C4—28no—
80C2×C11×C22⋊C4—28no—
81C2×C11×C22⋊C4C2318yes—
82C2×C11×C22⋊C4C2318yes—
83C2×C42×C11—24no—
84C2×C42×C11C2314yes—
85C2×C22⋊Dic11—28no—
86C2×C22⋊Dic11—28no—
87C2×C22⋊Dic11—44no—
88C2×C22⋊Dic11C2318yes—
89C2×C22⋊Dic11C2318yes—
90C2×C22⋊Dic11C2318yes—
91#66991—232no—
92C2×(C2×D11)⋊C4—28no—
93C2×(C2×D11)⋊C4—24no—
94C2×(C2×D11)⋊C4D418yescomplemented
95C2×(C2×D11)⋊C4C2318yes—
96C2×(C2×D11)⋊C4C2314yes—
97C2×(C2×D11)⋊C4C2314yes—
98C2×C4⋊Dic11—24no—
99C2×C4⋊Dic11—28no—
100C2×C4⋊Dic11D418yes—
101C2×C4⋊Dic11C2318yes—
102C2×C4⋊Dic11C2314yes—
103C2×C4⋊Dic11C2314yes—
104C2×C4×Dic11—24no—
105C2×C4×Dic11—28no—
106C2×C4×Dic11C2318yes—
107C2×C4×Dic11C2314yes—
108#66938—264no—
109#66922—232no—
110C11⋊#95C2314yes—
111C11⋊#95C2314yes—
112C11⋊#95C2318yes—
113C11⋊#95C2318yes—
114C2×#2491—111nomaximal, Sylow
115C24×C11—24no—
116C24×C11—24no—
117C24×C11—24no—
118C24×C11—24no—
119C24×C11—24no—
120C24×C11—24no—
121C24×C11—24no—
122C24×C11C2414yes—
123C24×C11C2411yescharacteristic
124C23×D11—22no—
125C23×D11—22no—
126C23×D11—22no—
127C23×D11—22no—
128C23×D11—22no—
129C23×D11—22no—
130C23×D11C2×D414yes—
131C23×D11C2412yes—
132C2×D4×C11—28no—
133C2×D4×C11—28no—
134C2×D4×C11—28no—
135C2×D4×C11—416no—
136C22×C4×C11—22no—
137C22×C4×C11—22no—
138C22×C4×C11—22no—
139C22×C4×C11—22no—
140C22×C4×C11—24no—
141C22×C4×C11—24no—
142C22×C4×C11—24no—
143C22×C4×C11C2×D414yes—
144C22×C4×C11C2×D414yes—
145C22×C4×C11#4114yes—
146C22×C4×C11#4114yes—
147C22×C4×C11#4114yes—
148C22×C4×C11C2414yes—
149C22×C4×C11C2412yes—
150C22×C4×C11C2412yes—
151C22×C4×C11C2412yes—
152C2×C22⋊D11—216no—
153C2×C22⋊D11—216no—
154C2×C22⋊D11—48no—
155C2×C22⋊D11—432no—
156C22×Dic11—24no—
157C22×Dic11—24no—
158C22×Dic11—24no—
159C22×Dic11—22no—
160C22×Dic11—22no—
161C22×Dic11—22no—
162C22×Dic11—22no—
163C22×Dic11—22no—
164C22×Dic11—22no—
165C22×Dic11—24no—
166C22×Dic11—22no—
167C22×Dic11—24no—
168C22×Dic11—22no—
169C22×Dic11C2×D414yes—
170C22×Dic11C2×D418yes—
171C22×Dic11C2412yes—
172C22×Dic11C2412yes—
173C22×Dic11C2414yes—
174C22×Dic11C2412yes—
175C22×Dic11C2414yes—
176C2×C4×D11—28no—
177C2×C4×D11—28no—
178C2×C4×D11—28no—
179C2×C4×D11—416no—
180C2×Dic22—216no—
181C2×Dic22—216no—
182C2×Dic22—48no—
183C2×Dic22—432no—
184C11×C4⋊C4—28no—
185C11×C22⋊C4—216no—
186C11×C22⋊C4—216no—
187C42×C11—28no—
188C22⋊Dic11—216no—
189C22⋊Dic11—216no—
190C22⋊Dic11—48no—
191(C2×D11)⋊C4—216no—
192(C2×D11)⋊C4—28no—
193(C2×D11)⋊C4C2×D4116yescomplemented
194C4⋊Dic11—216no—
195C4⋊Dic11—28no—
196C4⋊Dic11C2×D4116yes—
197C4×Dic11—216no—
198C4×Dic11—28no—
199C22×C4⋊Q8—111no—
200C22×#124—112no—
201C22×#121—111no—
202#2491—1116no—
203C2×#394—112no—
204C2×#393—112no—
205C2×#390—111no—
206C2×#390D1111yescharacteristic, complemented
207C2×#388—112no—
208C2×#380—111no—
209C2×C4×C4⋊C4—111no—
210C2×C4×C22⋊C4—111no—
211C23×C11—24no—
212C23×C11—24no—
213C23×C11—24no—
214C23×C11—24no—
215C23×C11—24no—
216C23×C11—24no—
217C23×C11—24no—
218C23×C11—48no—
219C23×C11—48no—
220C23×C11—48no—
221C23×C11—48no—
222C23×C11—48no—
223C23×C11—48no—
224C23×C11—48no—
225C23×C11C22×D412yes—
226C23×C11C22×D412yes—
227C23×C11C2×#4112yes—
228C23×C11C2×#4112yes—
229C23×C11C2×#4112yes—
230C23×C11H(5,𝔽2)12yes—
231C23×C112−1+412yes—
232C23×C11C2511yescharacteristic
233C22×D11—28no—
234C22×D11—28no—
235C22×D11—28no—
236C22×D11—22no—
237C22×D11—22no—
238C22×D11—22no—
239C22×D11—22no—
240C22×D11—22no—
241C22×D11—22no—
242C22×D11—44no—
243C22×D11—44no—
244C22×D11—44no—
245C22×D11—44no—
246C22×D11C22×D414yes—
247D4×C11—416no—
248C2×C4×C11—28no—
249C2×C4×C11—28no—
250C2×C4×C11—22no—
251C2×C4×C11—22no—
252C2×C4×C11—22no—
253C2×C4×C11—22no—
254C2×C4×C11—24no—
255C2×C4×C11—24no—
256C2×C4×C11—24no—
257C2×C4×C11—48no—
258C2×C4×C11#124116yescomplemented
259C2×C4×C11C22×D414yes—
260C2×C4×C11C22×D414yes—
261C2×C4×C11C2×#4114yes—
262C2×C4×C11C2×#4114yes—
263C2×C4×C11C2×#4114yes—
264C22⋊D11—432no—
265C2×Dic11—216no—
266C2×Dic11—28no—
267C2×Dic11—24no—
268C2×Dic11—22no—
269C2×Dic11—24no—
270C2×Dic11—22no—
271C2×Dic11—24no—
272C2×Dic11—24no—
273C2×Dic11—22no—
274C2×Dic11—22no—
275C2×Dic11—22no—
276C2×Dic11—22no—
277C2×Dic11—22no—
278C2×Dic11—22no—
279C2×Dic11—24no—
280C2×Dic11—44no—
281C2×Dic11—48no—
282C2×Dic11—44no—
283C2×Dic11C22×D414yes—
284C2×Dic11C22×D418yes—
285C4×D11—416no—
286Dic22—432no—
287C23×Q8—112no—
288C23×D4—112no—
289C23×D4C2×D1111yescharacteristic
290C24×C4—111no—
291C2×C4⋊Q8—118no—
292C2×C4⋊Q8—228no—
293C2×#124—1116no—
294C2×#124—2216no—
295C2×#121—118no—
296C2×#121—228no—
297C22×C4⋊C4—111no—
298C22×C4⋊C4—112no—
299C22×C4⋊C4—111no—
300C22×C4⋊C4C2×D1111yescharacteristic
301C22×C22⋊C4—111no—
302C22×C22⋊C4—112no—
303C22×C22⋊C4—112no—
304C22×C22⋊C4—111no—
305C22×C22⋊C4—112no—
306C22×C22⋊C4—112no—
307C22×C22⋊C4C2×D1112yes—
308C22×C22⋊C4C2×D1112yes—
309C22×C42—112no—
310C22×C42—111no—
311C22×C42C2×D1111yescharacteristic
312#394—1116no—
313#393—1116no—
314#390—118no—
315#390C2×D1118yescomplemented
316#388—1116no—
317#380—118no—
318C4×C4⋊C4—118no—
319C4×C22⋊C4—118no—
320C2×#95—111no—
321C2×#95—112no—
322C2×#95—112no—
323C2×#95—111no—
324C22×C11—44no—
325C22×C11—44no—
326C22×C11—44no—
327C22×C11—44no—
328C22×C11—44no—
329C22×C11—44no—
330C22×C11—44no—
331C22×C11—816no—
332C22×C11C2×#12414yes—
333C22×C11#53214yes—
334C22×C11#54214yes—
335C22×C11D4214yes—
336C22×C11#54714yes—
337C22×C11#54814yes—
338C22×C11#55514yes—
339C22×C11C23×D411yescharacteristic
340C22×C11C23×D411yescharacteristic
341C22×C11C22×#4111yescharacteristic
342C22×C11C22×#4111yescharacteristic
343C22×C11C22×#4111yescharacteristic
344C22×C11C2×H(5,𝔽2)11yescharacteristic
345C22×C11C2×2−1+411yescharacteristic
346C2×D11—24no—
347C2×D11—24no—
348C2×D11—24no—
349C2×D11—42no—
350C2×D11—42no—
351C2×D11—42no—
352C2×D11—42no—
353C2×D11—416no—
354C4×C11—24no—
355C4×C11—24no—
356C4×C11—44no—
357C4×C11C2×#12418yescomplemented
358Dic11—28no—
359Dic11—24no—
360Dic11—42no—
361Dic11—44no—
362Dic11—42no—
363C25—111no—
364C25C22×D1112yes—
365C22×Q8—118no—
366C22×Q8—228no—
367C22×Q8—228no—
368C22×Q8—444no—
369C22×D4—24no—
370C22×D4—24no—
371C22×D4—24no—
372C22×D4—118no—
373C22×D4—228no—
374C22×D4—228no—
375C22×D4—444no—
376C22×D4C22×D1114yes—
377C23×C4—111no—
378C23×C4—112no—
379C23×C4—112no—
380C23×C4—111no—
381C23×C4—111no—
382C23×C4—114no—
383C23×C4—224no—
384C23×C4—224no—
385C23×C4—224no—
386C23×C4C22×D1112yes—
387C23×C4C22×D1111yescharacteristic
388C23×C4C22×D1111yescharacteristic
389C23×C4C22×D1111yescharacteristic
390C4⋊Q8—2232no—
391#124—2264no—
392#121—2232no—
393C2×C4⋊C4—24no—
394C2×C4⋊C4—118no—
395C2×C4⋊C4—114no—
396C2×C4⋊C4—114no—
397C2×C4⋊C4—118no—
398C2×C4⋊C4—228no—
399C2×C4⋊C4—224no—
400C2×C4⋊C4C22×D1114yes—
401C2×C22⋊C4—28no—
402C2×C22⋊C4—28no—
403C2×C22⋊C4—118no—
404C2×C22⋊C4—114no—
405C2×C22⋊C4—118no—
406C2×C22⋊C4—118no—
407C2×C22⋊C4—118no—
408C2×C22⋊C4—114no—
409C2×C22⋊C4—118no—
410C2×C22⋊C4—228no—
411C2×C22⋊C4—228no—
412C2×C22⋊C4—228no—
413C2×C22⋊C4—224no—
414C2×C22⋊C4—444no—
415C2×C22⋊C4C22×D1118yes—
416C2×C22⋊C4C22×D1118yes—
417C2×C42—24no—
418C2×C42—118no—
419C2×C42—114no—
420C2×C42—228no—
421C2×C42—224no—
422C2×C42C22×D1114yes—
423#95—114no—
424#95—114no—
425#95—118no—
426#95—118no—
427C2×C11—84no—
428C2×C11#249118yescomplemented
429C2×C11C22×#12411yescharacteristic
430C2×C11C2×#53211yescharacteristic
431C2×C11C2×#54211yescharacteristic
432C2×C11C2×D4211yescharacteristic
433C2×C11C2×#54711yescharacteristic
434C2×C11C2×#54811yescharacteristic
435C2×C11C2×#55511yescharacteristic
436D11—44no—
437C24—24no—
438C24—24no—
439C24—24no—
440C24—24no—
441C24—24no—
442C24—24no—
443C24—24no—
444C24—114no—
445C24—112no—
446C24—222no—
447C24—222no—
448C24—222no—
449C24—222no—
450C24—222no—
451C24—222no—
452C24C23×D1114yes—
453C24C23×D1111yescharacteristic
454C2×Q8—2216no—
455C2×Q8—2216no—
456C2×Q8—448no—
457C2×Q8—4432no—
458C2×D4—28no—
459C2×D4—28no—
460C2×D4—28no—
461C2×D4—416no—
462C2×D4—2216no—
463C2×D4—2216no—
464C2×D4—448no—
465C2×D4—4432no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—114no—
474C22×C4—118no—
475C22×C4—112no—
476C22×C4—114no—
477C22×C4—112no—
478C22×C4—112no—
479C22×C4—114no—
480C22×C4—222no—
481C22×C4—224no—
482C22×C4—224no—
483C22×C4—222no—
484C22×C4—222no—
485C22×C4—222no—
486C22×C4—222no—
487C22×C4—224no—
488C22×C4—224no—
489C22×C4—222no—
490C22×C4—222no—
491C22×C4—222no—
492C22×C4—224no—
493C22×C4—228no—
494C22×C4—228no—
495C22×C4—228no—
496C22×C4—4416no—
497C22×C4#433914yes—
498C22×C4D4×D1114yes—
499C22×C4D4×D1114yes—
500C22×C4#434114yes—
501C22×C4#434114yes—
502C22×C4C23×D1114yes—
503C22×C4C23×D1112yes—
504C22×C4C23×D1112yes—
505C22×C4C23×D1112yes—
506C4⋊C4—28no—
507C4⋊C4—1116no—
508C4⋊C4—2216no—
509C4⋊C4—228no—
510C22⋊C4—216no—
511C22⋊C4—216no—
512C22⋊C4—1116no—
513C22⋊C4—2216no—
514C22⋊C4—2216no—
515C22⋊C4—2216no—
516C22⋊C4—228no—
517C22⋊C4—448no—
518C42—28no—
519C42—2216no—
520C42—228no—
521C11C2×#249111yescharacteristic, complemented, Sylow
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—48no—
530C23—48no—
531C23—48no—
532C23—48no—
533C23—48no—
534C23—48no—
535C23—48no—
536C23—114no—
537C23—228no—
538C23—228no—
539C23—228no—
540C23—222no—
541C23—222no—
542C23—222no—
543C23—222no—
544C23—222no—
545C23—222no—
546C23—444no—
547C23—444no—
548C23—444no—
549C23—444no—
550C23C2×#433912yes—
551C23C2×D4×D1112yes—
552C23C2×D4×D1112yes—
553C23C2×#434112yes—
554C23C2×#434112yes—
555C23#6704212yes—
556C23#6704312yes—
557C23C24×D1111yescharacteristic
558Q8—4432no—
559D4—416no—
560D4—4432no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—48no—
571C2×C4—114no—
572C2×C4—118no—
573C2×C4—228no—
574C2×C4—2216no—
575C2×C4—222no—
576C2×C4—224no—
577C2×C4—224no—
578C2×C4—222no—
579C2×C4—222no—
580C2×C4—222no—
581C2×C4—222no—
582C2×C4—224no—
583C2×C4—224no—
584C2×C4—222no—
585C2×C4—222no—
586C2×C4—222no—
587C2×C4—224no—
588C2×C4—444no—
589C2×C4—444no—
590C2×C4—448no—
591C2×C4—4416no—
592C2×C4#66938116yescomplemented
593C2×C4C2×#433914yes—
594C2×C4C2×D4×D1114yes—
595C2×C4C2×D4×D1114yes—
596C2×C4C2×#434114yes—
597C2×C4C2×#434114yes—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—816no—
606C22—224no—
607C22—224no—
608C22—224no—
609C22—442no—
610C22—442no—
611C22—4416no—
612C22—442no—
613C22—442no—
614C22C2×#6693814yes—
615C22#1062308214yes—
616C22#1062309414yes—
617C22#1062312914yes—
618C22#1062314014yes—
619C22#1062321314yes—
620C22#1062324714yes—
621C22C22×#433911yescharacteristic
622C22C22×D4×D1111yescharacteristic
623C22C22×D4×D1111yescharacteristic
624C22C22×#434111yescharacteristic
625C22C22×#434111yescharacteristic
626C22C2×#6704211yescharacteristic
627C22C2×#6704311yescharacteristic
628C4—24no—
629C4—24no—
630C4—44no—
631C4—228no—
632C4—224no—
633C4—442no—
634C4—442no—
635C4—444no—
636C4C2×#6693818yescomplemented
637C2—84no—
638C2—444no—
639C2#1307297718yesdirect product factor
640C2C22×#6693811yescharacteristic
641C2C2×#1062324711yescharacteristic
642C2C2×#1062321311yescharacteristic
643C2C2×#1062314011yescharacteristic
644C2C2×#1062308211yescharacteristic
645C2C2×#1062312911yescharacteristic
646C2C2×#1062309411yescharacteristic
subgroupG/H
centerC24C23×D11
commutator, cosocleC23×C11C25
FrattiniC23C24×D11
FittingC2×C11×#390C2
socleC24×C11C24
prime radicalC22×#66991C2
hypercenterC2×#390D11
hypocenterC11C2×#2491
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
1. C22×#66922C2semidirect
2. C22×#66938C2semidirect
4. #13072977C2direct (N × H)
5. C2×#10622574C2semidirect
6. C2×#10622560C2semidirect
7. C2×#10622834C2semidirect
8. C2×#10622564C2semidirect
9. C2×#10622600C2semidirect
10. C2×C4⋊C4×Dic11C2semidirect
11. C2×C4×(C2×D11)⋊C4C2semidirect
12. C2×C11×#390C2semidirect
35. C2×#66938C22semidirect
37. C2×#66922C22semidirect
38. C11×#390C22semidirect
39. #10622834C22semidirect
44. #10622600C22semidirect
45. C4⋊C4×Dic11C22semidirect
46. #10622574C22semidirect
47. #10622564C22semidirect
48. #10622560C22semidirect
49. C4×(C2×D11)⋊C4C22semidirect
94. C2×(C2×D11)⋊C4D4semidirect
193. (C2×D11)⋊C4C2×D4semidirect
206. C2×#390D11semidirect
258. C2×C4×C11#124semidirect
315. #390C2×D11semidirect
357. C4×C11C2×#124semidirect
428. C2×C11#2491semidirect
521. C11C2×#2491semidirect
592. C2×C4#66938semidirect
636. C4C2×#66938semidirect
639. C2#13072977direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#669222
C22×#669382
C22×#669912
#130729772
C2×#106225742
C2×#106225602
C2×#106228342
C2×#106225642
C2×#106226002
C2×C4⋊C4×Dic112
C2×C4×(C2×D11)⋊C42
C2×C11×#3902
C2×#249111