Finite Group Databasebeta

← C2×#13071294 #14912511

Subgroups — C2×#13071294 #14912511

Order 2816.

3198 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66977C211yesmaximal, characteristic
2C22×#66940C211yesmaximal, characteristic, complemented
3C22×#66979C211yesmaximal, characteristic
4#13071294C2116yesmaximal, direct product factor
5C2×#10622587C211yesmaximal, characteristic, complemented
6C2×#10622260C212yesmaximal, complemented
7C2×#10622586C211yesmaximal, characteristic, complemented
8C2×#10622558C212yesmaximal
9C2×#10622557C212yesmaximal
10C2×#10622268C212yesmaximal, complemented
11C2×C4⋊C4×Dic11C211yesmaximal, characteristic, complemented
12C2×C11×#380C211yesmaximal, characteristic
13C24×C4×C11C2211yescharacteristic
14C23×Dic22C2211yescharacteristic
15C22×C11×C4⋊C4C2211yescharacteristic
16C22×C11×C4⋊C4C2211yescharacteristic
17C22×C11×C22⋊C4C2212yes—
18C22×C22⋊Dic11C2211yescharacteristic
19C22×C22⋊Dic11C2211yescharacteristic
20C22×C22⋊Dic11C2212yes—
21C2×#66979—28no—
22C2×#66979C2218yes—
23C22×C4⋊Dic11C2211yescharacteristic
24C22×C4⋊Dic11C2211yescharacteristic
25C22×C4⋊Dic11C2211yescharacteristic
26C2×#66977C4116yescomplemented
27C2×#66977C2218yes—
28C22×Dic11⋊C4C2212yes—
29C22×Dic11⋊C4C2212yes—
30C22×Dic11⋊C4C2212yes—
31C22×C4×Dic11C2211yescharacteristic
32C22×C4×Dic11C2212yes—
33C22×C4×Dic11C2212yes—
34C22×C4×Dic11C2211yescharacteristic
35C2×#66940—28no—
36C2×#66940C2218yescomplemented
37C11×#380C2218yes—
38C2×C11×#95C2212yes—
39C2×C11⋊#95C2212yes—
40C2×C11⋊#95C2212yes—
41C2×C11⋊#95C2212yes—
42C2×C11⋊#95C2212yes—
43C4⋊C4×Dic11C2218yescomplemented
44#10622587C2218yescomplemented
45#10622586C2218yescomplemented
46#10622558C22116yes—
47#10622557C22116yes—
48#10622268C22116yescomplemented
49#10622260C22116yescomplemented
50C25×C11C2311yescharacteristic
51C23×C4×C11—24no—
52C23×C4×C11—24no—
53C23×C4×C11C2×C418yes—
54C23×C4×C11C2312yes—
55C23×C4×C11C2312yes—
56C23×C4×C11C2311yescharacteristic
57C23×C4×C11C2314yes—
58C23×C4×C11C2311yescharacteristic
59C23×Dic11C2311yescharacteristic
60C23×Dic11C2311yescharacteristic
61C23×Dic11C2312yes—
62C23×Dic11C2312yes—
63C23×Dic11C2311yescharacteristic
64C23×Dic11C2311yescharacteristic
65C22×Dic22—24no—
66C22×Dic22—24no—
67C22×Dic22C2×C418yes—
68C22×Dic22C2314yes—
69C2×C11×C4⋊C4—24no—
70C2×C11×C4⋊C4—24no—
71C2×C11×C4⋊C4C2314yes—
72C2×C11×C4⋊C4C2314yes—
73C2×C11×C22⋊C4—44no—
74C2×C11×C22⋊C4C2318yes—
75C2×C22⋊Dic11—24no—
76C2×C22⋊Dic11—24no—
77C2×C22⋊Dic11C2×C4116yes—
78C2×C22⋊Dic11C2314yes—
79C2×C22⋊Dic11C2314yes—
80C2×C22⋊Dic11C2318yes—
81#66979—232no—
82C2×C4⋊Dic11—24no—
83C2×C4⋊Dic11—24no—
84C2×C4⋊Dic11C2×C418yes—
85C2×C4⋊Dic11C2314yes—
86C2×C4⋊Dic11C2314yes—
87C2×C4⋊Dic11C2314yes—
88#66977C2×C4164yescomplemented
89C2×Dic11⋊C4—28no—
90C2×Dic11⋊C4—44no—
91C2×Dic11⋊C4C2×C4116yes—
92C2×Dic11⋊C4C2318yes—
93C2×Dic11⋊C4C2318yes—
94C2×Dic11⋊C4C2318yes—
95C2×C4×Dic11—24no—
96C2×C4×Dic11—28no—
97C2×C4×Dic11—44no—
98C2×C4×Dic11D418yes—
99C2×C4×Dic11C2318yes—
100C2×C4×Dic11C2314yes—
101C2×C4×Dic11C2318yes—
102C2×C4×Dic11C2314yes—
103#66940—232no—
104C11×#95C2318yes—
105C11⋊#95C2318yes—
106C11⋊#95C2318yes—
107C11⋊#95C2318yes—
108C11⋊#95C2318yes—
109C2×#2323—111nomaximal, Sylow
110C24×C11—22no—
111C24×C11—22no—
112C24×C11—22no—
113C24×C11—22no—
114C24×C11—22no—
115C24×C11—22no—
116C24×C11C22×C414yes—
117C24×C11C2412yes—
118C24×C11C2411yescharacteristic
119C22×C4×C11—22no—
120C22×C4×C11—22no—
121C22×C4×C11—232no—
122C22×C4×C11—28no—
123C22×C4×C11—28no—
124C22×C4×C11—24no—
125C22×C4×C11—24no—
126C22×C4×C11—24no—
127C22×C4×C11—24no—
128C22×C4×C11—24no—
129C22×C4×C11—24no—
130C22×C4×C11C22×C414yes—
131C22×C4×C11C22×C4116yes—
132C22×C4×C11C22×C414yes—
133C22×C4×C11C2×D414yes—
134C22×C4×C11#4114yes—
135C22×C4×C11C2414yes—
136C22×C4×C11C2414yes—
137C22×C4×C11C2412yes—
138C22×C4×C11C2412yes—
139C22×Dic11—24no—
140C22×Dic11—22no—
141C22×Dic11—22no—
142C22×Dic11—24no—
143C22×Dic11—22no—
144C22×Dic11—24no—
145C22×Dic11—22no—
146C22×Dic11—24no—
147C22×Dic11—22no—
148C22×Dic11—22no—
149C22×Dic11—22no—
150C22×Dic11—22no—
151C22×Dic11—22no—
152C22×Dic11—22no—
153C22×Dic11C22×C418yes—
154C22×Dic11C22×C418yes—
155C22×Dic11C2×D414yes—
156C22×Dic11C2×D414yes—
157C22×Dic11C2412yes—
158C22×Dic11C2412yes—
159C22×Dic11C2414yes—
160C22×Dic11C2414yes—
161C22×Dic11C2412yes—
162C22×Dic11C2412yes—
163C2×Dic22—232no—
164C2×Dic22—28no—
165C2×Dic22—28no—
166C2×Dic22C22×C4116yes—
167C11×C4⋊C4—28no—
168C11×C4⋊C4—28no—
169C11×C22⋊C4—48no—
170C22⋊Dic11—28no—
171C22⋊Dic11—28no—
172C22⋊Dic11C22×C4132yes—
173C4⋊Dic11—28no—
174C4⋊Dic11—28no—
175C4⋊Dic11C22×C4116yes—
176Dic11⋊C4—216no—
177Dic11⋊C4—48no—
178Dic11⋊C4C22×C4132yes—
179C4×Dic11—216no—
180C4×Dic11—28no—
181C4×Dic11—48no—
182C4×Dic11C2×D4116yes—
183C22×C22⋊Q8—111no—
184C22×C4×Q8—111no—
185C22×#117—111no—
186#2323—1116no—
187C2×#391—111no—
188C2×#389—112no—
189C2×#389—111no—
190C2×#388—112no—
191C2×#388—112no—
192C2×#387—112no—
193C2×#380D1111yescharacteristic
194C2×C4×C4⋊C4—111no—
195C23×C11—28no—
196C23×C11—28no—
197C23×C11—28no—
198C23×C11—22no—
199C23×C11—22no—
200C23×C11—22no—
201C23×C11—22no—
202C23×C11—22no—
203C23×C11—22no—
204C23×C11—44no—
205C23×C11—44no—
206C23×C11—44no—
207C23×C11—44no—
208C23×C11C23×C414yes—
209C23×C11C23×C412yes—
210C23×C11C22×D412yes—
211C23×C11C2×#4112yes—
212C23×C11H(5,𝔽2)12yes—
213C23×C112−1+412yes—
214C23×C112−1+412yes—
215C23×C112−1+412yes—
216C23×C11C2511yescharacteristic
217C2×C4×C11—28no—
218C2×C4×C11—22no—
219C2×C4×C11—22no—
220C2×C4×C11—232no—
221C2×C4×C11—24no—
222C2×C4×C11—24no—
223C2×C4×C11—24no—
224C2×C4×C11—24no—
225C2×C4×C11—24no—
226C2×C4×C11—24no—
227C2×C4×C11—48no—
228C2×C4×C11—48no—
229C2×C4×C11C4×D4116yes—
230C2×C4×C11C23×C414yes—
231C2×C4×C11C23×C414yes—
232C2×C4×C11C22×D414yes—
233C2×C4×C11C2×#4114yes—
234C2×Dic11—216no—
235C2×Dic11—216no—
236C2×Dic11—28no—
237C2×Dic11—28no—
238C2×Dic11—24no—
239C2×Dic11—24no—
240C2×Dic11—22no—
241C2×Dic11—24no—
242C2×Dic11—24no—
243C2×Dic11—22no—
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—44no—
253C2×Dic11—44no—
254C2×Dic11C23×C418yes—
255C2×Dic11C23×C418yes—
256C2×Dic11C22×D414yes—
257C2×Dic11C22×D414yes—
258Dic22—232no—
259C23×Q8—111no—
260C24×C4C2×D1111yescharacteristic
261C2×C22⋊Q8—118no—
262C2×C22⋊Q8—1116no—
263C2×C4×Q8—118no—
264C2×C4×Q8—228no—
265C2×#117—118no—
266C2×#117—228no—
267C22×C4⋊C4—112no—
268C22×C4⋊C4—111no—
269C22×C4⋊C4—112no—
270C22×C4⋊C4—111no—
271C22×C4⋊C4—111no—
272C22×C4⋊C4—112no—
273C22×C4⋊C4C2×D1111yescharacteristic
274C22×C4⋊C4C2×D1111yescharacteristic
275C22×C22⋊C4—112no—
276C22×C22⋊C4—111no—
277C22×C22⋊C4—111no—
278C22×C22⋊C4C2×D1112yes—
279C22×C42—111no—
280C22×C42—111no—
281C22×C42—112no—
282C22×C42—112no—
283#391—118no—
284#389—118no—
285#389—1116no—
286#388—1116no—
287#388—1116no—
288#387—1116no—
289#380C2×D1118yes—
290C4×C4⋊C4—118no—
291C2×#95—112no—
292C2×#95—112no—
293C2×#95—112no—
294C2×#95—112no—
295C2×#95C2×D1112yes—
296C22×C11—24no—
297C22×C11—24no—
298C22×C11—24no—
299C22×C11—416no—
300C22×C11—42no—
301C22×C11—42no—
302C22×C11—42no—
303C22×C11—42no—
304C22×C11C2×C4×D414yes—
305C22×C11#51914yes—
306C22×C11#52014yes—
307C22×C11#53614yes—
308C22×C11#53714yes—
309C22×C11#54114yes—
310C22×C11#54214yes—
311C22×C11C24×C411yescharacteristic
312C22×C11C23×D411yescharacteristic
313C22×C11C22×#4111yescharacteristic
314C22×C11C2×H(5,𝔽2)11yescharacteristic
315C22×C11C2×2−1+411yescharacteristic
316C22×C11C2×2−1+411yescharacteristic
317C22×C11C2×2−1+411yescharacteristic
318C4×C11—24no—
319C4×C11—44no—
320C4×C11—44no—
321C4×C11C2×C4×D418yes—
322Dic11—28no—
323Dic11—28no—
324Dic11—24no—
325Dic11—24no—
326Dic11—42no—
327Dic11—42no—
328C25C22×D1111yescharacteristic
329C22×Q8—118no—
330C22×Q8—114no—
331C22×Q8—224no—
332C22×Q8—224no—
333C23×C4—24no—
334C23×C4—24no—
335C23×C4—112no—
336C23×C4—111no—
337C23×C4—112no—
338C23×C4—111no—
339C23×C4—111no—
340C23×C4—111no—
341C23×C4C4×D1118yes—
342C23×C4C22×D1112yes—
343C23×C4C22×D1112yes—
344C23×C4C22×D1114yes—
345C23×C4C22×D1111yescharacteristic
346C23×C4C22×D1111yescharacteristic
347C22⋊Q8—1164no—
348C4×Q8—2232no—
349#117—2232no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—1116no—
353C2×C4⋊C4—118no—
354C2×C4⋊C4—114no—
355C2×C4⋊C4—118no—
356C2×C4⋊C4—114no—
357C2×C4⋊C4—114no—
358C2×C4⋊C4—118no—
359C2×C4⋊C4—118no—
360C2×C4⋊C4—224no—
361C2×C4⋊C4—224no—
362C2×C4⋊C4—228no—
363C2×C4⋊C4—444no—
364C2×C4⋊C4C22×D1114yes—
365C2×C4⋊C4C22×D1114yes—
366C2×C22⋊C4—44no—
367C2×C22⋊C4—1116no—
368C2×C22⋊C4—118no—
369C2×C22⋊C4—114no—
370C2×C22⋊C4—114no—
371C2×C22⋊C4—224no—
372C2×C22⋊C4—224no—
373C2×C22⋊C4C22×D1118yes—
374C2×C42—118no—
375C2×C42—114no—
376C2×C42—114no—
377C2×C42—118no—
378C2×C42—118no—
379C2×C42—228no—
380C2×C42—224no—
381C2×C42—444no—
382#95—118no—
383#95—118no—
384#95—118no—
385#95—118no—
386#95C22×D1118yes—
387C2×C11—44no—
388C2×C11#232318yescomplemented
389C2×C11C22×C4×D411yescharacteristic
390C2×C11C2×#51911yescharacteristic
391C2×C11C2×#52011yescharacteristic
392C2×C11C2×#53611yescharacteristic
393C2×C11C2×#53711yescharacteristic
394C2×C11C2×#54111yescharacteristic
395C2×C11C2×#54211yescharacteristic
396C24—22no—
397C24—22no—
398C24—22no—
399C24—22no—
400C24—22no—
401C24—22no—
402C24C2×C4×D1114yes—
403C24C23×D1112yes—
404C24C23×D1111yescharacteristic
405C2×Q8—1116no—
406C2×Q8—2232no—
407C2×Q8—228no—
408C2×Q8—228no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—232no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—118no—
421C22×C4—118no—
422C22×C4—114no—
423C22×C4—114no—
424C22×C4—112no—
425C22×C4—114no—
426C22×C4—114no—
427C22×C4—112no—
428C22×C4—112no—
429C22×C4—112no—
430C22×C4—224no—
431C22×C4—224no—
432C22×C4—222no—
433C22×C4—224no—
434C22×C4—224no—
435C22×C4—222no—
436C22×C4—222no—
437C22×C4—222no—
438C22×C4—222no—
439C22×C4—222no—
440C22×C4—222no—
441C22×C4—222no—
442C22×C4—222no—
443C22×C4—222no—
444C22×C4C2×C4×D1114yes—
445C22×C4C2×C4×D11116yes—
446C22×C4C2×C4×D1114yes—
447C22×C4D4×D1114yes—
448C22×C4#434114yes—
449C22×C4C23×D1114yes—
450C22×C4C23×D1114yes—
451C22×C4C23×D1112yes—
452C22×C4C23×D1112yes—
453C4⋊C4—28no—
454C4⋊C4—28no—
455C4⋊C4—1132no—
456C4⋊C4—1116no—
457C4⋊C4—228no—
458C4⋊C4—228no—
459C4⋊C4—2216no—
460C4⋊C4—448no—
461C22⋊C4—48no—
462C22⋊C4—1132no—
463C22⋊C4—228no—
464C22⋊C4—228no—
465C42—1116no—
466C42—2216no—
467C42—228no—
468C42—448no—
469C11C2×#232311yescharacteristic, complemented, Sylow
470C23—28no—
471C23—28no—
472C23—28no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23C22×C4×D1114yes—
484C23C22×C4×D1112yes—
485C23C2×D4×D1112yes—
486C23C2×#434112yes—
487C23#6703712yes—
488C23#6704012yes—
489C23#6704312yes—
490C23#6704312yes—
491C23C24×D1111yescharacteristic
492Q8—2232no—
493C2×C4—28no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—232no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—48no—
504C2×C4—48no—
505C2×C4—118no—
506C2×C4—118no—
507C2×C4—114no—
508C2×C4—114no—
509C2×C4—2216no—
510C2×C4—2216no—
511C2×C4—228no—
512C2×C4—228no—
513C2×C4—224no—
514C2×C4—224no—
515C2×C4—222no—
516C2×C4—224no—
517C2×C4—222no—
518C2×C4—224no—
519C2×C4—222no—
520C2×C4—222no—
521C2×C4—222no—
522C2×C4—222no—
523C2×C4—222no—
524C2×C4—222no—
525C2×C4—222no—
526C2×C4—222no—
527C2×C4—444no—
528C2×C4—444no—
529C2×C4(C22⋊D11)⋊C4116yes—
530C2×C4C22×C4×D1114yes—
531C2×C4C22×C4×D1114yes—
532C2×C4C2×D4×D1114yes—
533C2×C4C2×#434114yes—
534C22—24no—
535C22—24no—
536C22—24no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—416no—
542C22C2×(C22⋊D11)⋊C414yes—
543C22#1062303514yes—
544C22#1062304714yes—
545C22#1062313414yes—
546C22#1062313914yes—
547C22#1062315614yes—
548C22#1062315914yes—
549C22C23×C4×D1111yescharacteristic
550C22C22×D4×D1111yescharacteristic
551C22C22×#434111yescharacteristic
552C22C2×#6703711yescharacteristic
553C22C2×#6704011yescharacteristic
554C22C2×#6704311yescharacteristic
555C22C2×#6704311yescharacteristic
556C4—24no—
557C4—44no—
558C4—44no—
559C4—224no—
560C4—224no—
561C4—228no—
562C4—228no—
563C4—442no—
564C4—442no—
565C4C2×(C22⋊D11)⋊C418yes—
566C2—44no—
567C2#1307129418yesdirect product factor
568C2C22×(C22⋊D11)⋊C411yescharacteristic
569C2C2×#1062304711yescharacteristic
570C2C2×#1062303511yescharacteristic
571C2C2×#1062315911yescharacteristic
572C2C2×#1062313911yescharacteristic
573C2C2×#1062315611yescharacteristic
574C2C2×#1062313411yescharacteristic
subgroupG/H
centerC24C23×D11
commutatorC22×C11C24×C4
FrattiniC23C24×D11
FittingC2×C11×#380C2
socleC24×C11C24
prime radicalC25×C11C23
cosocleC23×C11C25
hypercenterC2×#380D11
hypocenterC11C2×#2323
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C22×#66940C2semidirect
4. #13071294C2direct (N × H)
5. C2×#10622587C2semidirect
6. C2×#10622260C2semidirect
7. C2×#10622586C2semidirect
10. C2×#10622268C2semidirect
11. C2×C4⋊C4×Dic11C2semidirect
26. C2×#66977C4semidirect
36. C2×#66940C22semidirect
43. C4⋊C4×Dic11C22semidirect
44. #10622587C22semidirect
45. #10622586C22semidirect
48. #10622268C22semidirect
49. #10622260C22semidirect
88. #66977C2×C4semidirect
388. C2×C11#2323semidirect
469. C11C2×#2323semidirect
567. C2#13071294direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#669772
C22×#669402
C22×#669792
#130712942
C2×#106225872
C2×#106222602
C2×#106225862
C2×#106225582
C2×#106225572
C2×#106222682
C2×C4⋊C4×Dic112
C2×C11×#3802
C2×#232311