Finite Group Databasebeta

← C2×#13072321 #14921607

Subgroups — C2×#13072321 #14921607

Order 2816.

3550 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66941C211yesmaximal, characteristic, complemented
2C22×#66949C212yesmaximal
3C22×C4⋊C4×D11C211yesmaximal, characteristic
4#13072321C2116yesmaximal, direct product factor
5C2×#10622593C212yesmaximal, complemented
6C2×#10622592C212yesmaximal, complemented
7C2×#10622541C211yesmaximal, characteristic, complemented
8C2×#10622586C211yesmaximal, characteristic, complemented
9C2×#10622602C212yesmaximal
10C2×#10622600C211yesmaximal, characteristic
11C2×C4×(C2×D11)⋊C4C211yesmaximal, characteristic
12C2×C11×#384C211yesmaximal, characteristic, complemented
13C23×C4×D11C2212yes—
14C23×C4×D11C2211yescharacteristic
15C23×Dic22C2212yes—
16C22×C11×C4⋊C4C2211yescharacteristic
17C22×C11×C4⋊C4C2211yescharacteristic
18C22×C11×C4⋊C4C2212yes—
19C22×C11×C4⋊C4C2212yes—
20C22×C42×C11C2211yescharacteristic
21C22×(C2×D11)⋊C4C2212yes—
22C22×(C2×D11)⋊C4C2211yescharacteristic
23C22×(C2×D11)⋊C4C2211yescharacteristic
24C22×C4⋊Dic11C2211yescharacteristic
25C22×C4⋊Dic11C2211yescharacteristic
26C22×Dic11⋊C4C2212yes—
27C22×Dic11⋊C4C2212yes—
28C22×Dic11⋊C4C2212yes—
29C22×Dic11⋊C4C2212yes—
30C22×C4×Dic11C2211yescharacteristic
31C2×#66949—216no—
32C2×#66949C22116yes—
33C2×C4⋊C4×D11—28no—
34C2×C4⋊C4×D11C2218yes—
35C2×#66941—28no—
36C2×#66941C2218yescomplemented
37C11×#384C2218yescomplemented
38C2×C11⋊#95C2211yescharacteristic
39C2×C11⋊#95C2211yescharacteristic
40C2×C11⋊#95C2212yes—
41C2×C11⋊#95C2212yes—
42C2×C11⋊#95C2212yes—
43#10622602C22116yes—
44#10622600C2218yes—
45#10622593C22116yescomplemented
46#10622592C22116yescomplemented
47#10622586C2218yescomplemented
48C4×(C2×D11)⋊C4C2218yes—
49#10622541C2218yescomplemented
50C24×D11C2311yescharacteristic
51C23×C4×C11C2312yes—
52C23×C4×C11C2312yes—
53C23×C4×C11C2311yescharacteristic
54C23×C4×C11C2311yescharacteristic
55C23×C4×C11C2311yescharacteristic
56C23×Dic11C2312yes—
57C23×Dic11C2311yescharacteristic
58C23×Dic11C2312yes—
59C23×Dic11C2311yescharacteristic
60C23×Dic11C2311yescharacteristic
61C22×C4×D11—24no—
62C22×C4×D11—24no—
63C22×C4×D11—28no—
64C22×C4×D11—28no—
65C22×C4×D11—44no—
66C22×C4×D11Q818yescomplemented
67C22×C4×D11C2318yes—
68C22×C4×D11C2314yes—
69C22×Dic22—28no—
70C22×Dic22—28no—
71C22×Dic22—44no—
72C22×Dic22C2318yes—
73C2×C11×C4⋊C4—24no—
74C2×C11×C4⋊C4—24no—
75C2×C11×C4⋊C4—28no—
76C2×C11×C4⋊C4—44no—
77C2×C11×C4⋊C4C2314yes—
78C2×C11×C4⋊C4C2314yes—
79C2×C11×C4⋊C4C2318yes—
80C2×C11×C4⋊C4C2318yes—
81C2×C42×C11—24no—
82C2×C42×C11C2314yes—
83C2×(C2×D11)⋊C4—28no—
84C2×(C2×D11)⋊C4D418yes—
85C2×(C2×D11)⋊C4Q818yescomplemented
86C2×(C2×D11)⋊C4C2318yes—
87C2×(C2×D11)⋊C4C2314yes—
88C2×(C2×D11)⋊C4C2314yes—
89C2×C4⋊Dic11—24no—
90C2×C4⋊Dic11D418yes—
91C2×C4⋊Dic11C2314yes—
92C2×C4⋊Dic11C2314yes—
93C2×Dic11⋊C4—28no—
94C2×Dic11⋊C4—28no—
95C2×Dic11⋊C4—28no—
96C2×Dic11⋊C4—28no—
97C2×Dic11⋊C4C2318yes—
98C2×Dic11⋊C4C2318yes—
99C2×Dic11⋊C4C2318yes—
100C2×Dic11⋊C4C2318yes—
101C2×C4×Dic11—24no—
102C2×C4×Dic11C2314yes—
103#66949—264no—
104C4⋊C4×D11—232no—
105#66941—232no—
106C11⋊#95C2318yes—
107C11⋊#95C2314yes—
108C11⋊#95C2314yes—
109C11⋊#95C2318yes—
110C11⋊#95C2318yes—
111C2×#2440—111nomaximal, Sylow
112C24×C11C2411yescharacteristic
113C23×D11—22no—
114C23×D11—22no—
115C23×D11—22no—
116C23×D11—22no—
117C23×D11C2×D414yes—
118C23×D11C2×Q814yes—
119C23×D11C2×Q814yes—
120C23×D11C2412yes—
121C22×C4×C11—22no—
122C22×C4×C11—22no—
123C22×C4×C11—22no—
124C22×C4×C11—22no—
125C22×C4×C11—24no—
126C22×C4×C11—24no—
127C22×C4×C11—24no—
128C22×C4×C11—24no—
129C22×C4×C11—24no—
130C22×C4×C11—24no—
131C22×C4×C11C2×D414yes—
132C22×C4×C11C2×D414yes—
133C22×C4×C11C2×Q814yes—
134C22×C4×C11C2×Q814yes—
135C22×C4×C11#4114yes—
136C22×C4×C11C2414yes—
137C22×C4×C11C2414yes—
138C22×C4×C11C2412yes—
139C22×C4×C11C2412yes—
140C22×C4×C11C2412yes—
141C22×Dic11—22no—
142C22×Dic11—22no—
143C22×Dic11—22no—
144C22×Dic11—24no—
145C22×Dic11—24no—
146C22×Dic11—22no—
147C22×Dic11—24no—
148C22×Dic11—22no—
149C22×Dic11—24no—
150C22×Dic11—22no—
151C22×Dic11—24no—
152C22×Dic11C2×D418yes—
153C22×Dic11C2×D414yes—
154C22×Dic11C2×Q814yes—
155C22×Dic11C2×Q814yes—
156C22×Dic11C2414yes—
157C22×Dic11C2412yes—
158C22×Dic11C2414yes—
159C22×Dic11C2412yes—
160C22×Dic11C2412yes—
161C2×C4×D11—216no—
162C2×C4×D11—216no—
163C2×C4×D11—232no—
164C2×C4×D11—28no—
165C2×C4×D11—28no—
166C2×C4×D11—48no—
167C2×C4×D11—432no—
168C2×C4×D11C2×Q8116yescomplemented
169C2×Dic22—216no—
170C2×Dic22—216no—
171C2×Dic22—48no—
172C2×Dic22—432no—
173C11×C4⋊C4—28no—
174C11×C4⋊C4—28no—
175C11×C4⋊C4—216no—
176C11×C4⋊C4—48no—
177C42×C11—28no—
178(C2×D11)⋊C4—216no—
179(C2×D11)⋊C4C2×D4116yes—
180(C2×D11)⋊C4C2×Q8116yescomplemented
181C4⋊Dic11—28no—
182C4⋊Dic11C2×D4116yes—
183Dic11⋊C4—216no—
184Dic11⋊C4—216no—
185Dic11⋊C4—216no—
186Dic11⋊C4—216no—
187C4×Dic11—28no—
188C22×C4⋊Q8—111no—
189C22×C22⋊Q8—112no—
190C23×C4⋊C4—111no—
191#2440—1116no—
192C2×#398—112no—
193C2×#395—112no—
194C2×#389—111no—
195C2×#389—111no—
196C2×#385—112no—
197C2×#384D1111yescharacteristic, complemented
198C2×#380—111no—
199C2×C4×C22⋊C4—111no—
200C23×C11C22×D412yes—
201C23×C11C22×D412yes—
202C23×C11C22×Q812yes—
203C23×C11C22×Q812yes—
204C23×C11C2×#4112yes—
205C23×C11H(5,𝔽2)12yes—
206C23×C112−1+412yes—
207C23×C11C2511yescharacteristic
208C22×D11—28no—
209C22×D11—28no—
210C22×D11—28no—
211C22×D11—28no—
212C22×D11—28no—
213C22×D11—28no—
214C22×D11—22no—
215C22×D11—22no—
216C22×D11—22no—
217C22×D11—22no—
218C22×D11C4⋊Q8116yescomplemented
219C22×D11C22×D414yes—
220C22×D11C22×Q814yes—
221C22×D11C22×Q814yes—
222C2×C4×C11—28no—
223C2×C4×C11—28no—
224C2×C4×C11—22no—
225C2×C4×C11—22no—
226C2×C4×C11—22no—
227C2×C4×C11—22no—
228C2×C4×C11—24no—
229C2×C4×C11—24no—
230C2×C4×C11—24no—
231C2×C4×C11—24no—
232C2×C4×C11—24no—
233C2×C4×C11—24no—
234C2×C4×C11—48no—
235C2×C4×C11—48no—
236C2×C4×C11C22⋊Q8116yescomplemented
237C2×C4×C11C22×D414yes—
238C2×C4×C11C22×D414yes—
239C2×C4×C11C22×Q814yes—
240C2×C4×C11C22×Q814yes—
241C2×C4×C11C2×#4114yes—
242C2×Dic11—28no—
243C2×Dic11—28no—
244C2×Dic11—216no—
245C2×Dic11—28no—
246C2×Dic11—22no—
247C2×Dic11—22no—
248C2×Dic11—22no—
249C2×Dic11—24no—
250C2×Dic11—22no—
251C2×Dic11—22no—
252C2×Dic11—24no—
253C2×Dic11—22no—
254C2×Dic11—24no—
255C2×Dic11—24no—
256C2×Dic11—24no—
257C2×Dic11—48no—
258C2×Dic11C22×D418yes—
259C2×Dic11C22×D414yes—
260C2×Dic11C22×Q814yes—
261C2×Dic11C22×Q814yes—
262C4×D11—232no—
263C4×D11—432no—
264Dic22—432no—
265C23×Q8—112no—
266C24×C4—111no—
267C24×C4—112no—
268C2×C4⋊Q8—118no—
269C2×C4⋊Q8—228no—
270C2×C22⋊Q8—1116no—
271C2×C22⋊Q8—2216no—
272C22×C4⋊C4—111no—
273C22×C4⋊C4—112no—
274C22×C4⋊C4—112no—
275C22×C4⋊C4—112no—
276C22×C4⋊C4—112no—
277C22×C4⋊C4—111no—
278C22×C4⋊C4—118no—
279C22×C4⋊C4—228no—
280C22×C4⋊C4C2×D1112yes—
281C22×C4⋊C4C2×D1111yescharacteristic
282C22×C4⋊C4C2×D1111yescharacteristic
283C22×C4⋊C4C2×D1112yes—
284C22×C22⋊C4—111no—
285C22×C22⋊C4—111no—
286C22×C22⋊C4—112no—
287C22×C42—111no—
288C22×C42C2×D1111yescharacteristic
289#398—1116no—
290#395—1116no—
291#389—118no—
292#389—118no—
293#385—1116no—
294#384C2×D1118yescomplemented
295#380—118no—
296C4×C22⋊C4—118no—
297C2×#95—111no—
298C2×#95—112no—
299C2×#95—112no—
300C2×#95—111no—
301C2×#95—112no—
302C22×C11C2×C22⋊Q814yes—
303C22×C11C2×C4⋊Q814yes—
304C22×C11#54414yes—
305C22×C11D4214yes—
306C22×C11#54714yes—
307C22×C11D4×Q814yes—
308C22×C11#55414yes—
309C22×C11C23×D411yescharacteristic
310C22×C11C23×D411yescharacteristic
311C22×C11C23×Q811yescharacteristic
312C22×C11C23×Q811yescharacteristic
313C22×C11C22×#4111yescharacteristic
314C22×C11C2×H(5,𝔽2)11yescharacteristic
315C22×C11C2×2−1+411yescharacteristic
316C2×D11—24no—
317C2×D11—24no—
318C2×D11—24no—
319C2×D11—232no—
320C2×D11—24no—
321C2×D11—24no—
322C2×D11—24no—
323C2×D11C2×C4⋊Q818yescomplemented
324C4×C11—24no—
325C4×C11—24no—
326C4×C11—44no—
327C4×C11—44no—
328C4×C11C2×C22⋊Q818yescomplemented
329Dic11—24no—
330Dic11—24no—
331Dic11—28no—
332Dic11—24no—
333Dic11—44no—
334C25—111no—
335C22×Q8—118no—
336C22×Q8—228no—
337C22×Q8—228no—
338C22×Q8—444no—
339C23×C4—111no—
340C23×C4—111no—
341C23×C4—112no—
342C23×C4—111no—
343C23×C4—118no—
344C23×C4—114no—
345C23×C4—112no—
346C23×C4—118no—
347C23×C4—228no—
348C23×C4—228no—
349C23×C4—224no—
350C23×C4—224no—
351C23×C4—444no—
352C23×C4C22×D1112yes—
353C23×C4C22×D1112yes—
354C23×C4C22×D1111yescharacteristic
355C23×C4C22×D1111yescharacteristic
356C23×C4C22×D1111yescharacteristic
357C4⋊Q8—2232no—
358C22⋊Q8—2264no—
359C2×C4⋊C4—24no—
360C2×C4⋊C4—28no—
361C2×C4⋊C4—24no—
362C2×C4⋊C4—44no—
363C2×C4⋊C4—118no—
364C2×C4⋊C4—114no—
365C2×C4⋊C4—118no—
366C2×C4⋊C4—118no—
367C2×C4⋊C4—118no—
368C2×C4⋊C4—118no—
369C2×C4⋊C4—114no—
370C2×C4⋊C4—2232no—
371C2×C4⋊C4—228no—
372C2×C4⋊C4—228no—
373C2×C4⋊C4—228no—
374C2×C4⋊C4—224no—
375C2×C4⋊C4—228no—
376C2×C4⋊C4C22×D1114yes—
377C2×C4⋊C4C22×D1118yes—
378C2×C4⋊C4C22×D1114yes—
379C2×C4⋊C4C22×D1118yes—
380C2×C22⋊C4—118no—
381C2×C22⋊C4—118no—
382C2×C22⋊C4—114no—
383C2×C22⋊C4—114no—
384C2×C22⋊C4—118no—
385C2×C22⋊C4—228no—
386C2×C42—24no—
387C2×C42—114no—
388C2×C42—224no—
389C2×C42C22×D1114yes—
390#95—114no—
391#95—114no—
392#95—118no—
393#95—118no—
394#95—118no—
395C2×C11#244018yescomplemented
396C2×C11C22×C22⋊Q811yescharacteristic
397C2×C11C22×C4⋊Q811yescharacteristic
398C2×C11C2×#54411yescharacteristic
399C2×C11C2×D4211yescharacteristic
400C2×C11C2×#54711yescharacteristic
401C2×C11C2×D4×Q811yescharacteristic
402C2×C11C2×#55411yescharacteristic
403D11—28no—
404C24—114no—
405C24—114no—
406C24—114no—
407C24—112no—
408C24—222no—
409C24—222no—
410C24—222no—
411C24—222no—
412C24C23×D1111yescharacteristic
413C2×Q8—2216no—
414C2×Q8—2216no—
415C2×Q8—448no—
416C2×Q8—4432no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—114no—
428C22×C4—118no—
429C22×C4—114no—
430C22×C4—114no—
431C22×C4—112no—
432C22×C4—112no—
433C22×C4—112no—
434C22×C4—114no—
435C22×C4—1116no—
436C22×C4—114no—
437C22×C4—2216no—
438C22×C4—2216no—
439C22×C4—222no—
440C22×C4—222no—
441C22×C4—222no—
442C22×C4—222no—
443C22×C4—224no—
444C22×C4—224no—
445C22×C4—222no—
446C22×C4—222no—
447C22×C4—2232no—
448C22×C4—228no—
449C22×C4—228no—
450C22×C4—224no—
451C22×C4—224no—
452C22×C4—224no—
453C22×C4—448no—
454C22×C4—4432no—
455C22×C4#433914yes—
456C22×C4D4×D1114yes—
457C22×C4D4×D1114yes—
458C22×C4Q8×D1114yes—
459C22×C4Q8×D1114yes—
460C22×C4C23×D1114yes—
461C22×C4C23×D1114yes—
462C22×C4C23×D1112yes—
463C22×C4C23×D1112yes—
464C22×C4C23×D1112yes—
465C4⋊C4—28no—
466C4⋊C4—216no—
467C4⋊C4—28no—
468C4⋊C4—48no—
469C4⋊C4—1116no—
470C4⋊C4—2216no—
471C4⋊C4—2216no—
472C4⋊C4—2216no—
473C4⋊C4—228no—
474C4⋊C4—2216no—
475C22⋊C4—1116no—
476C22⋊C4—1116no—
477C22⋊C4—2216no—
478C42—28no—
479C42—228no—
480C11C2×#244011yescharacteristic, complemented, Sylow
481C23—1116no—
482C23—114no—
483C23—114no—
484C23—114no—
485C23—228no—
486C23—228no—
487C23—228no—
488C23—228no—
489C23—228no—
490C23—228no—
491C23—222no—
492C23—222no—
493C23—222no—
494C23—222no—
495C23C2×#433912yes—
496C23C2×D4×D1112yes—
497C23C2×D4×D1112yes—
498C23#6703712yes—
499C23C2×Q8×D1112yes—
500C23C2×Q8×D1112yes—
501C23#6704012yes—
502C23C24×D1111yescharacteristic
503Q8—4432no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—48no—
517C2×C4—48no—
518C2×C4—114no—
519C2×C4—118no—
520C2×C4—114no—
521C2×C4—114no—
522C2×C4—2216no—
523C2×C4—228no—
524C2×C4—228no—
525C2×C4—228no—
526C2×C4—222no—
527C2×C4—222no—
528C2×C4—222no—
529C2×C4—222no—
530C2×C4—224no—
531C2×C4—224no—
532C2×C4—222no—
533C2×C4—222no—
534C2×C4—2232no—
535C2×C4—224no—
536C2×C4—224no—
537C2×C4—224no—
538C2×C4—4432no—
539C2×C4—448no—
540C2×C4#66949116yescomplemented
541C2×C4C2×#433914yes—
542C2×C4C2×D4×D1114yes—
543C2×C4C2×D4×D1114yes—
544C2×C4C2×Q8×D1114yes—
545C2×C4C2×Q8×D1114yes—
546C22—118no—
547C22—224no—
548C22—224no—
549C22—2232no—
550C22—224no—
551C22—224no—
552C22—224no—
553C22—224no—
554C22#1062303814yes—
555C22C2×#6694914yes—
556C22#1062304214yes—
557C22#1062324814yes—
558C22D11×C4⋊Q814yes—
559C22#1062325814yes—
560C22#1062326014yes—
561C22C22×#433911yescharacteristic
562C22C22×D4×D1111yescharacteristic
563C22C22×D4×D1111yescharacteristic
564C22C2×#6703711yescharacteristic
565C22C22×Q8×D1111yescharacteristic
566C22C22×Q8×D1111yescharacteristic
567C22C2×#6704011yescharacteristic
568C4—24no—
569C4—24no—
570C4—44no—
571C4—44no—
572C4—228no—
573C4—224no—
574C4—224no—
575C4—224no—
576C4—444no—
577C4C2×#6694918yescomplemented
578C2—228no—
579C2#1307232118yesdirect product factor
580C2C22×#6694911yescharacteristic
581C2C2×D11×C4⋊Q811yescharacteristic
582C2C2×#1062326011yescharacteristic
583C2C2×#1062324811yescharacteristic
584C2C2×#1062303811yescharacteristic
585C2C2×#1062325811yescharacteristic
586C2C2×#1062304211yescharacteristic
subgroupG/H
centerC24C23×D11
commutator, cosocleC23×C11C25
FrattiniC23C24×D11
FittingC2×C11×#384C2
socleC24×C11C24
prime radicalC24×D11C23
hypercenterC2×#384D11
hypocenterC11C2×#2440
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ C2×#384

Complements

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

NH ≅ G/Ntype
1. C22×#66941C2semidirect
4. #13072321C2direct (N × H)
5. C2×#10622593C2semidirect
6. C2×#10622592C2semidirect
7. C2×#10622541C2semidirect
8. C2×#10622586C2semidirect
12. C2×C11×#384C2semidirect
36. C2×#66941C22semidirect
37. C11×#384C22semidirect
45. #10622593C22semidirect
46. #10622592C22semidirect
47. #10622586C22semidirect
49. #10622541C22semidirect
66. C22×C4×D11Q8semidirect
85. C2×(C2×D11)⋊C4Q8semidirect
168. C2×C4×D11C2×Q8semidirect
180. (C2×D11)⋊C4C2×Q8semidirect
197. C2×#384D11semidirect
218. C22×D11C4⋊Q8semidirect
236. C2×C4×C11C22⋊Q8semidirect
294. #384C2×D11semidirect
323. C2×D11C2×C4⋊Q8semidirect
328. C4×C11C2×C22⋊Q8semidirect
395. C2×C11#2440semidirect
480. C11C2×#2440semidirect
540. C2×C4#66949semidirect
577. C4C2×#66949semidirect
579. C2#13072321direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#669412
C22×#669492
C22×C4⋊C4×D112
#130723212
C2×#106225932
C2×#106225922
C2×#106225412
C2×#106225862
C2×#106226022
C2×#106226002
C2×C4×(C2×D11)⋊C42
C2×C11×#3842
C2×#244011