Finite Group Databasebeta

← C2×#13072312 #14921600

Subgroups — C2×#13072312 #14921600

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×#66977C212yesmaximal
3#13072312C2116yesmaximal, direct product factor
4C2×#10622597C212yesmaximal, complemented
5C2×#10622592C212yesmaximal, complemented
6C2×#10622598C211yesmaximal, characteristic, complemented
7C2×#10622586C211yesmaximal, characteristic, complemented
8C2×#10622823C212yesmaximal
9C2×#10622589C211yesmaximal, characteristic, complemented
10C2×#10622825C211yesmaximal, characteristic
11C2×C4×C22⋊Dic11C211yesmaximal, characteristic
12C23×C11×C4⋊C4C211yesmaximal, characteristic
13C24×C4×C11C2212yes—
14C24×C4×C11C2211yescharacteristic
15C23×Dic22C2212yes—
16C22×C11×C4⋊C4—28no—
17C22×C11×C4⋊C4C2211yescharacteristic
18C22×C11×C4⋊C4C2218yes—
19C22×C11×C4⋊C4C2212yes—
20C22×C11×C4⋊C4C2211yescharacteristic
21C22×C22⋊Dic11C2212yes—
22C22×C22⋊Dic11C2211yescharacteristic
23C22×C22⋊Dic11C2211yescharacteristic
24C22×C4⋊Dic11C2211yescharacteristic
25C22×C4⋊Dic11C2211yescharacteristic
26C22×C4⋊Dic11C2212yes—
27C2×#66977—216no—
28C2×#66977C22116yes—
29C22×Dic11⋊C4C2212yes—
30C22×Dic11⋊C4C2212yes—
31C22×Dic11⋊C4C2212yes—
32C22×Dic11⋊C4C2212yes—
33C22×C4×Dic11C2211yescharacteristic
34C22×C4×Dic11C2211yescharacteristic
35C2×#66941—28no—
36C2×#66941C2218yescomplemented
37#10622825C2218yes—
38#10622823C22116yes—
39C4×C22⋊Dic11C2218yes—
40C2×C11⋊#95C2211yescharacteristic
41C2×C11⋊#95C2211yescharacteristic
42C2×C11⋊#95C2212yes—
43C2×C11⋊#95C2212yes—
44C2×C11⋊#95C2212yes—
45#10622598C2218yescomplemented
46#10622597C22116yescomplemented
47#10622592C22116yescomplemented
48#10622589C2218yescomplemented
49#10622586C2218yescomplemented
50C25×C11C2311yescharacteristic
51C23×C4×C11—24no—
52C23×C4×C11—24no—
53C23×C4×C11—28no—
54C23×C4×C11—28no—
55C23×C4×C11—44no—
56C23×C4×C11Q818yescomplemented
57C23×C4×C11C2312yes—
58C23×C4×C11C2318yes—
59C23×C4×C11C2312yes—
60C23×C4×C11C2311yescharacteristic
61C23×C4×C11C2314yes—
62C23×C4×C11C2311yescharacteristic
63C23×Dic11C2312yes—
64C23×Dic11C2312yes—
65C23×Dic11C2311yescharacteristic
66C23×Dic11C2311yescharacteristic
67C23×Dic11C2311yescharacteristic
68C23×Dic11C2311yescharacteristic
69C22×Dic22—28no—
70C22×Dic22—28no—
71C22×Dic22—44no—
72C22×Dic22C2318yes—
73C2×C11×C4⋊C4—28no—
74C2×C11×C4⋊C4—24no—
75C2×C11×C4⋊C4—232no—
76C2×C11×C4⋊C4D418yes—
77C2×C11×C4⋊C4C2314yes—
78C2×C11×C4⋊C4C2318yes—
79C2×C11×C4⋊C4C2314yes—
80C2×C22⋊Dic11—28no—
81C2×C22⋊Dic11D418yes—
82C2×C22⋊Dic11Q818yescomplemented
83C2×C22⋊Dic11C2318yes—
84C2×C22⋊Dic11C2314yes—
85C2×C22⋊Dic11C2314yes—
86C2×C4⋊Dic11—24no—
87C2×C4⋊Dic11—24no—
88C2×C4⋊Dic11—28no—
89C2×C4⋊Dic11C2314yes—
90C2×C4⋊Dic11C2318yes—
91C2×C4⋊Dic11C2314yes—
92#66977—264no—
93C2×Dic11⋊C4—28no—
94C2×Dic11⋊C4—28no—
95C2×Dic11⋊C4—28no—
96C2×Dic11⋊C4—44no—
97C2×Dic11⋊C4C2318yes—
98C2×Dic11⋊C4C2318yes—
99C2×Dic11⋊C4C2318yes—
100C2×Dic11⋊C4C2318yes—
101C2×C4×Dic11—24no—
102C2×C4×Dic11—24no—
103C2×C4×Dic11C2314yes—
104C2×C4×Dic11C2314yes—
105#66941—232no—
106C11⋊#95C2314yes—
107C11⋊#95C2314yes—
108C11⋊#95C2318yes—
109C11⋊#95C2318yes—
110C11⋊#95C2318yes—
111C2×#2440—111nomaximal, Sylow
112C24×C11—22no—
113C24×C11—22no—
114C24×C11—22no—
115C24×C11—22no—
116C24×C11C2×D414yes—
117C24×C11C2×Q814yes—
118C24×C11C2×Q814yes—
119C24×C11C2412yes—
120C24×C11C2411yescharacteristic
121C22×C4×C11—24no—
122C22×C4×C11—22no—
123C22×C4×C11—22no—
124C22×C4×C11—24no—
125C22×C4×C11—232no—
126C22×C4×C11—28no—
127C22×C4×C11—28no—
128C22×C4×C11—24no—
129C22×C4×C11—216no—
130C22×C4×C11—24no—
131C22×C4×C11—24no—
132C22×C4×C11—216no—
133C22×C4×C11—48no—
134C22×C4×C11—432no—
135C22×C4×C11C2×D418yes—
136C22×C4×C11C2×D414yes—
137C22×C4×C11C2×Q814yes—
138C22×C4×C11C2×Q814yes—
139C22×C4×C11C2×Q8116yescomplemented
140C22×C4×C11#4114yes—
141C22×C4×C11C2414yes—
142C22×C4×C11C2414yes—
143C22×C4×C11C2412yes—
144C22×C4×C11C2412yes—
145C22×Dic11—22no—
146C22×Dic11—22no—
147C22×Dic11—24no—
148C22×Dic11—24no—
149C22×Dic11—24no—
150C22×Dic11—22no—
151C22×Dic11—22no—
152C22×Dic11—24no—
153C22×Dic11—22no—
154C22×Dic11—22no—
155C22×Dic11—24no—
156C22×Dic11—22no—
157C22×Dic11—22no—
158C22×Dic11—24no—
159C22×Dic11C2×D414yes—
160C22×Dic11C2×D414yes—
161C22×Dic11C2×Q814yes—
162C22×Dic11C2×Q814yes—
163C22×Dic11C2414yes—
164C22×Dic11C2414yes—
165C22×Dic11C2412yes—
166C22×Dic11C2412yes—
167C22×Dic11C2412yes—
168C22×Dic11C2412yes—
169C2×Dic22—216no—
170C2×Dic22—216no—
171C2×Dic22—48no—
172C2×Dic22—432no—
173C11×C4⋊C4—216no—
174C11×C4⋊C4—28no—
175C11×C4⋊C4C2×D4116yes—
176C22⋊Dic11—216no—
177C22⋊Dic11C2×D4116yes—
178C22⋊Dic11C2×Q8116yescomplemented
179C4⋊Dic11—216no—
180C4⋊Dic11—28no—
181C4⋊Dic11—28no—
182Dic11⋊C4—216no—
183Dic11⋊C4—216no—
184Dic11⋊C4—216no—
185Dic11⋊C4—48no—
186C4×Dic11—28no—
187C4×Dic11—28no—
188C22×C4⋊Q8—111no—
189C22×C22⋊Q8—112no—
190C23×C4⋊C4D1111yescharacteristic
191#2440—1116no—
192C2×#398—112no—
193C2×#395—112no—
194C2×#389—111no—
195C2×#389—111no—
196C2×#385—112no—
197C2×#384—111no—
198C2×#380—111no—
199C2×C4×C22⋊C4—111no—
200C23×C11—28no—
201C23×C11—28no—
202C23×C11—28no—
203C23×C11—28no—
204C23×C11—28no—
205C23×C11—28no—
206C23×C11—22no—
207C23×C11—22no—
208C23×C11—22no—
209C23×C11—22no—
210C23×C11C4⋊Q8116yescomplemented
211C23×C11C22×D414yes—
212C23×C11C22×D412yes—
213C23×C11C22×D412yes—
214C23×C11C22×Q814yes—
215C23×C11C22×Q812yes—
216C23×C11C22×Q814yes—
217C23×C11C22×Q812yes—
218C23×C11C2×#4112yes—
219C23×C11H(5,𝔽2)12yes—
220C23×C112−1+412yes—
221C23×C11C2511yescharacteristic
222C2×C4×C11—28no—
223C2×C4×C11—216no—
224C2×C4×C11—24no—
225C2×C4×C11—22no—
226C2×C4×C11—22no—
227C2×C4×C11—24no—
228C2×C4×C11—232no—
229C2×C4×C11—24no—
230C2×C4×C11—24no—
231C2×C4×C11—24no—
232C2×C4×C11—432no—
233C2×C4×C11—48no—
234C2×C4×C11C22⋊Q8116yescomplemented
235C2×C4×C11C22×D418yes—
236C2×C4×C11C22×D414yes—
237C2×C4×C11C22×Q814yes—
238C2×C4×C11C22×Q814yes—
239C2×C4×C11C2×#4114yes—
240C2×Dic11—28no—
241C2×Dic11—28no—
242C2×Dic11—28no—
243C2×Dic11—28no—
244C2×Dic11—22no—
245C2×Dic11—22no—
246C2×Dic11—22no—
247C2×Dic11—22no—
248C2×Dic11—22no—
249C2×Dic11—22no—
250C2×Dic11—22no—
251C2×Dic11—22no—
252C2×Dic11—24no—
253C2×Dic11—24no—
254C2×Dic11—24no—
255C2×Dic11—24no—
256C2×Dic11—24no—
257C2×Dic11—24no—
258C2×Dic11—48no—
259C2×Dic11—48no—
260C2×Dic11C22×D414yes—
261C2×Dic11C22×D414yes—
262C2×Dic11C22×Q814yes—
263C2×Dic11C22×Q814yes—
264Dic22—432no—
265C23×Q8—112no—
266C24×C4C2×D1112yes—
267C24×C4C2×D1111yescharacteristic
268C2×C4⋊Q8—118no—
269C2×C4⋊Q8—228no—
270C2×C22⋊Q8—1116no—
271C2×C22⋊Q8—2216no—
272C22×C4⋊C4—28no—
273C22×C4⋊C4—112no—
274C22×C4⋊C4—112no—
275C22×C4⋊C4—112no—
276C22×C4⋊C4—112no—
277C22×C4⋊C4—111no—
278C22×C4⋊C4—111no—
279C22×C4⋊C4—112no—
280C22×C4⋊C4C2×D1112yes—
281C22×C4⋊C4C2×D1118yes—
282C22×C4⋊C4C2×D1111yescharacteristic
283C22×C4⋊C4C2×D1111yescharacteristic
284C22×C22⋊C4—111no—
285C22×C22⋊C4—111no—
286C22×C22⋊C4—112no—
287C22×C42—111no—
288C22×C42—111no—
289#398—1116no—
290#395—1116no—
291#389—118no—
292#389—118no—
293#385—1116no—
294#384—118no—
295#380—118no—
296C4×C22⋊C4—118no—
297C2×#95—111no—
298C2×#95—112no—
299C2×#95—112no—
300C2×#95—111no—
301C2×#95—112no—
302C22×C11—24no—
303C22×C11—24no—
304C22×C11—24no—
305C22×C11—232no—
306C22×C11—24no—
307C22×C11—24no—
308C22×C11—24no—
309C22×C11C2×C22⋊Q814yes—
310C22×C11C2×C4⋊Q814yes—
311C22×C11C2×C4⋊Q818yescomplemented
312C22×C11#54414yes—
313C22×C11D4214yes—
314C22×C11#54714yes—
315C22×C11D4×Q814yes—
316C22×C11#55414yes—
317C22×C11C23×D411yescharacteristic
318C22×C11C23×D411yescharacteristic
319C22×C11C23×Q811yescharacteristic
320C22×C11C23×Q811yescharacteristic
321C22×C11C22×#4111yescharacteristic
322C22×C11C2×H(5,𝔽2)11yescharacteristic
323C22×C11C2×2−1+411yescharacteristic
324C4×C11—24no—
325C4×C11—28no—
326C4×C11—44no—
327C4×C11C2×C22⋊Q818yescomplemented
328Dic11—24no—
329Dic11—24no—
330Dic11—24no—
331Dic11—24no—
332Dic11—44no—
333Dic11—44no—
334C25C22×D1111yescharacteristic
335C22×Q8—118no—
336C22×Q8—228no—
337C22×Q8—228no—
338C22×Q8—444no—
339C23×C4—24no—
340C23×C4—24no—
341C23×C4—28no—
342C23×C4—28no—
343C23×C4—44no—
344C23×C4—111no—
345C23×C4—111no—
346C23×C4—111no—
347C23×C4—111no—
348C23×C4—112no—
349C23×C4—112no—
350C23×C4Dic2218yescomplemented
351C23×C4C22×D1112yes—
352C23×C4C22×D1118yes—
353C23×C4C22×D1111yescharacteristic
354C23×C4C22×D1114yes—
355C23×C4C22×D1112yes—
356C23×C4C22×D1111yescharacteristic
357C4⋊Q8—2232no—
358C22⋊Q8—2264no—
359C2×C4⋊C4—28no—
360C2×C4⋊C4—24no—
361C2×C4⋊C4—232no—
362C2×C4⋊C4—118no—
363C2×C4⋊C4—118no—
364C2×C4⋊C4—118no—
365C2×C4⋊C4—114no—
366C2×C4⋊C4—118no—
367C2×C4⋊C4—114no—
368C2×C4⋊C4—118no—
369C2×C4⋊C4—228no—
370C2×C4⋊C4—228no—
371C2×C4⋊C4—228no—
372C2×C4⋊C4—224no—
373C2×C4⋊C4—228no—
374C2×C4⋊C4—224no—
375C2×C4⋊C4—444no—
376C2×C4⋊C4C22⋊D1118yes—
377C2×C4⋊C4C22×D1114yes—
378C2×C4⋊C4C22×D1118yes—
379C2×C4⋊C4C22×D1114yes—
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—114no—
387C2×C42—114no—
388C2×C42—224no—
389C2×C42—224no—
390#95—114no—
391#95—114no—
392#95—118no—
393#95—118no—
394#95—118no—
395C2×C11—28no—
396C2×C11#244018yescomplemented
397C2×C11C22×C22⋊Q811yescharacteristic
398C2×C11C22×C4⋊Q811yescharacteristic
399C2×C11C2×#54411yescharacteristic
400C2×C11C2×D4211yescharacteristic
401C2×C11C2×#54711yescharacteristic
402C2×C11C2×D4×Q811yescharacteristic
403C2×C11C2×#55411yescharacteristic
404C24—22no—
405C24—22no—
406C24—22no—
407C24—22no—
408C24C2×Dic2214yes—
409C24D4×D1114yes—
410C24Q8×D1114yes—
411C24C23×D1112yes—
412C24C23×D1111yescharacteristic
413C2×Q8—2216no—
414C2×Q8—2216no—
415C2×Q8—448no—
416C2×Q8—4432no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—232no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—24no—
425C22×C4—216no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—216no—
429C22×C4—48no—
430C22×C4—432no—
431C22×C4—114no—
432C22×C4—114no—
433C22×C4—114no—
434C22×C4—114no—
435C22×C4—112no—
436C22×C4—112no—
437C22×C4—112no—
438C22×C4—112no—
439C22×C4—114no—
440C22×C4—114no—
441C22×C4—222no—
442C22×C4—222no—
443C22×C4—222no—
444C22×C4—222no—
445C22×C4—222no—
446C22×C4—222no—
447C22×C4—222no—
448C22×C4—222no—
449C22×C4—224no—
450C22×C4—224no—
451C22×C4—224no—
452C22×C4—224no—
453C22×C4—224no—
454C22×C4—224no—
455C22×C4C2×Dic2214yes—
456C22×C4C2×Dic2214yes—
457C22×C4C2×Dic22116yescomplemented
458C22×C4#433914yes—
459C22×C4C2×C22⋊D1118yes—
460C22×C4C2×C22⋊D1114yes—
461C22×C4C23×D1114yes—
462C22×C4C23×D1112yes—
463C22×C4C23×D1114yes—
464C22×C4C23×D1112yes—
465C4⋊C4—216no—
466C4⋊C4—28no—
467C4⋊C4—2216no—
468C4⋊C4—2216no—
469C4⋊C4—2216no—
470C4⋊C4—228no—
471C4⋊C4—2216no—
472C4⋊C4—228no—
473C4⋊C4—448no—
474C4⋊C4C2×C22⋊D11116yes—
475C22⋊C4—1116no—
476C22⋊C4—1116no—
477C22⋊C4—2216no—
478C42—228no—
479C42—228no—
480C11C2×#244011yescharacteristic, complemented, Sylow
481C23—28no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—28no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23#66941116yescomplemented
492C23C22×Dic2214yes—
493C23C22×Dic2212yes—
494C23C2×#433912yes—
495C23C2×D4×D1114yes—
496C23C2×D4×D1112yes—
497C23#6703712yes—
498C23C2×Q8×D1114yes—
499C23C2×Q8×D1112yes—
500C23#6704012yes—
501C23C22×C22⋊D1112yes—
502C23C24×D1111yescharacteristic
503Q8—4432no—
504C2×C4—28no—
505C2×C4—216no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—232no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—432no—
515C2×C4—48no—
516C2×C4—114no—
517C2×C4—114no—
518C2×C4—114no—
519C2×C4—114no—
520C2×C4—228no—
521C2×C4—228no—
522C2×C4—228no—
523C2×C4—228no—
524C2×C4—222no—
525C2×C4—222no—
526C2×C4—222no—
527C2×C4—222no—
528C2×C4—222no—
529C2×C4—222no—
530C2×C4—222no—
531C2×C4—222no—
532C2×C4—224no—
533C2×C4—224no—
534C2×C4—224no—
535C2×C4—224no—
536C2×C4—224no—
537C2×C4—224no—
538C2×C4—448no—
539C2×C4—448no—
540C2×C4#66977116yescomplemented
541C2×C4C22×Dic2214yes—
542C2×C4C22×Dic2214yes—
543C2×C4C2×#433914yes—
544C2×C4C22×C22⋊D1118yes—
545C2×C4C22×C22⋊D1114yes—
546C22—24no—
547C22—24no—
548C22—232no—
549C22—24no—
550C22—24no—
551C22—24no—
552C22—24no—
553C22C2×#6694114yes—
554C22C2×#6694118yescomplemented
555C22#1062303114yes—
556C22#1062303814yes—
557C22#1062304214yes—
558C22C2×#6697714yes—
559C22D4×C22⋊D1114yes—
560C22Q8×C22⋊D1114yes—
561C22C23×Dic2211yescharacteristic
562C22C22×#433911yescharacteristic
563C22C22×D4×D1111yescharacteristic
564C22C2×#6703711yescharacteristic
565C22C22×Q8×D1111yescharacteristic
566C22C2×#6704011yescharacteristic
567C22C23×C22⋊D1111yescharacteristic
568C4—28no—
569C4—24no—
570C4—44no—
571C4—224no—
572C4—224no—
573C4—224no—
574C4—224no—
575C4—444no—
576C4—444no—
577C4C2×#6697718yescomplemented
578C2—28no—
579C2#1307231218yesdirect product factor
580C2C22×#6697711yescharacteristic
581C2C22×#6694111yescharacteristic
582C2C2×#1062303111yescharacteristic
583C2C2×D4×C22⋊D1111yescharacteristic
584C2C2×#1062303811yescharacteristic
585C2C2×Q8×C22⋊D1111yescharacteristic
586C2C2×#1062304211yescharacteristic
subgroupG/H
centerC24C23×D11
commutator, cosocleC23×C11C25
FrattiniC23C24×D11
FittingC23×C11×C4⋊C4C2
socleC24×C11C24
prime radicalC25×C11C23
hypercenterC23×C4⋊C4D11
hypocenterC11C2×#2440
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#66941C2semidirect
3. #13072312C2direct (N × H)
4. C2×#10622597C2semidirect
5. C2×#10622592C2semidirect
6. C2×#10622598C2semidirect
7. C2×#10622586C2semidirect
9. C2×#10622589C2semidirect
36. C2×#66941C22semidirect
45. #10622598C22semidirect
46. #10622597C22semidirect
47. #10622592C22semidirect
48. #10622589C22semidirect
49. #10622586C22semidirect
56. C23×C4×C11Q8semidirect
82. C2×C22⋊Dic11Q8semidirect
139. C22×C4×C11C2×Q8semidirect
178. C22⋊Dic11C2×Q8semidirect
210. C23×C11C4⋊Q8semidirect
234. C2×C4×C11C22⋊Q8semidirect
311. C22×C11C2×C4⋊Q8semidirect
327. C4×C11C2×C22⋊Q8semidirect
350. C23×C4Dic22semidirect
396. C2×C11#2440semidirect
457. C22×C4C2×Dic22semidirect
480. C11C2×#2440semidirect
491. C23#66941semidirect
540. C2×C4#66977semidirect
554. C22C2×#66941semidirect
577. C4C2×#66977semidirect
579. C2#13072312direct (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×#669772
#130723122
C2×#106225972
C2×#106225922
C2×#106225982
C2×#106225862
C2×#106228232
C2×#106225892
C2×#106228252
C2×C4×C22⋊Dic112
C23×C11×C4⋊C42
C2×#244011