Finite Group Databasebeta

← C2×#13072984 #14922993

Subgroups — C2×#13072984 #14922993

Order 2816.

2854 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66941C211yesmaximal, characteristic, complemented
2C22×#66950C211yesmaximal, characteristic
3#13072984C2116yesmaximal, direct product factor
4C2×#10622272C212yesmaximal, complemented
5C2×#10622286C212yesmaximal
6C2×#10622542C211yesmaximal, characteristic, complemented
7C2×#10622542C212yesmaximal, complemented
8C2×#10622587C211yesmaximal, characteristic, complemented
9C2×#10622553C212yesmaximal
10C2×#10622600C211yesmaximal, characteristic
11C2×C4×(C2×D11)⋊C4C211yesmaximal, characteristic
12C2×C4×C11×C4⋊C4C211yesmaximal, characteristic, complemented
13C23×C4×D11C2211yescharacteristic
14C23×Dic22C2211yescharacteristic
15C23×Dic22C2212yes—
16C22×C11×C4⋊C4C2211yescharacteristic
17C22×C11×C4⋊C4C2211yescharacteristic
18C22×C42×C11C2211yescharacteristic
19C22×C42×C11C2212yes—
20C22×(C2×D11)⋊C4C2211yescharacteristic
21C22×(C2×D11)⋊C4C2212yes—
22C22×(C2×D11)⋊C4C2212yes—
23C22×(C2×D11)⋊C4C2211yescharacteristic
24C22×C4⋊Dic11C2211yescharacteristic
25C22×C4⋊Dic11C2212yes—
26C22×Dic11⋊C4C2212yes—
27C22×C4×Dic11C2211yescharacteristic
28C2×#66950—28no—
29C2×#66950C2218yes—
30C2×#66941—28no—
31C2×#66941C2218yescomplemented
32C4×C11×C4⋊C4C2218yescomplemented
33C2×C11×#95C2212yes—
34C2×C11⋊#95C2211yescharacteristic
35C2×C11⋊#95C2211yescharacteristic
36C2×C11⋊#95C2212yes—
37C2×C11⋊#95C2212yes—
38C2×C11⋊#95C2212yes—
39C2×C11⋊#95C2212yes—
40C2×C11⋊#95C2212yes—
41#10622600C2218yes—
42#10622587C2218yescomplemented
43#10622553C22116yes—
44C4×(C2×D11)⋊C4C2218yes—
45#10622542C2218yescomplemented
46#10622542C22116yescomplemented
47#10622286C22116yes—
48#10622272C22116yescomplemented
49C24×D11C2311yescharacteristic
50C23×C4×C11C2311yescharacteristic
51C23×C4×C11C2312yes—
52C23×C4×C11C2312yes—
53C23×C4×C11C2311yescharacteristic
54C23×C4×C11C2311yescharacteristic
55C23×Dic11C2312yes—
56C23×Dic11C2311yescharacteristic
57C23×Dic11C2312yes—
58C23×Dic11C2311yescharacteristic
59C23×Dic11C2311yescharacteristic
60C22×C4×D11—24no—
61C22×C4×D11—24no—
62C22×C4×D11—24no—
63C22×C4×D11C2314yes—
64C22×Dic22—28no—
65C22×Dic22—24no—
66C22×Dic22—24no—
67C22×Dic22—24no—
68C22×Dic22—28no—
69C22×Dic22—44no—
70C22×Dic22C2314yes—
71C22×Dic22C2318yes—
72C2×C11×C4⋊C4—24no—
73C2×C11×C4⋊C4D418yes—
74C2×C11×C4⋊C4C2314yes—
75C2×C11×C4⋊C4C2314yes—
76C2×C42×C11—24no—
77C2×C42×C11—44no—
78C2×C42×C11C2314yes—
79C2×C42×C11C2318yes—
80C2×(C2×D11)⋊C4—28no—
81C2×(C2×D11)⋊C4—24no—
82C2×(C2×D11)⋊C4—44no—
83C2×(C2×D11)⋊C4Q818yescomplemented
84C2×(C2×D11)⋊C4C2318yes—
85C2×(C2×D11)⋊C4C2314yes—
86C2×(C2×D11)⋊C4C2318yes—
87C2×(C2×D11)⋊C4C2314yes—
88C2×C4⋊Dic11—24no—
89C2×C4⋊Dic11—28no—
90C2×C4⋊Dic11C2314yes—
91C2×C4⋊Dic11C2318yes—
92C2×Dic11⋊C4—28no—
93C2×Dic11⋊C4C2318yes—
94C2×C4×Dic11—24no—
95C2×C4×Dic11C2314yes—
96#66950—232no—
97#66941—232no—
98C11×#95C2318yes—
99C11⋊#95C2314yes—
100C11⋊#95C2318yes—
101C11⋊#95C2318yes—
102C11⋊#95C2318yes—
103C11⋊#95C2318yes—
104C11⋊#95C2314yes—
105C11⋊#95C2318yes—
106C2×#2492—111nomaximal, Sylow
107C24×C11C2411yescharacteristic
108C23×D11—22no—
109C23×D11—22no—
110C23×D11—22no—
111C23×D11—22no—
112C23×D11—22no—
113C23×D11—22no—
114C23×D11C2×Q814yes—
115C23×D11C2412yes—
116C22×C4×C11—22no—
117C22×C4×C11—24no—
118C22×C4×C11—22no—
119C22×C4×C11—24no—
120C22×C4×C11—22no—
121C22×C4×C11—22no—
122C22×C4×C11—24no—
123C22×C4×C11—24no—
124C22×C4×C11—24no—
125C22×C4×C11—22no—
126C22×C4×C11C2×D418yes—
127C22×C4×C11C2×D414yes—
128C22×C4×C11C2×Q814yes—
129C22×C4×C11#4114yes—
130C22×C4×C11#4114yes—
131C22×C4×C11C2412yes—
132C22×C4×C11C2414yes—
133C22×C4×C11C2414yes—
134C22×C4×C11C2412yes—
135C22×C4×C11C2412yes—
136C22×Dic11—22no—
137C22×Dic11—22no—
138C22×Dic11—24no—
139C22×Dic11—22no—
140C22×Dic11—24no—
141C22×Dic11—24no—
142C22×Dic11—24no—
143C22×Dic11—24no—
144C22×Dic11—22no—
145C22×Dic11—22no—
146C22×Dic11—22no—
147C22×Dic11—24no—
148C22×Dic11—22no—
149C22×Dic11C2×Q814yes—
150C22×Dic11#4114yes—
151C22×Dic11C2412yes—
152C22×Dic11C2414yes—
153C22×Dic11C2414yes—
154C22×Dic11C2412yes—
155C22×Dic11C2412yes—
156C2×C4×D11—28no—
157C2×C4×D11—28no—
158C2×C4×D11—28no—
159C2×C4×D11—416no—
160C2×Dic22—216no—
161C2×Dic22—216no—
162C2×Dic22—28no—
163C2×Dic22—28no—
164C2×Dic22—28no—
165C2×Dic22—48no—
166C2×Dic22—432no—
167C2×Dic22—416no—
168C11×C4⋊C4—28no—
169C11×C4⋊C4C2×D4116yes—
170C42×C11—28no—
171C42×C11—48no—
172(C2×D11)⋊C4—28no—
173(C2×D11)⋊C4—216no—
174(C2×D11)⋊C4—48no—
175(C2×D11)⋊C4C2×Q8116yescomplemented
176C4⋊Dic11—28no—
177C4⋊Dic11—216no—
178Dic11⋊C4—216no—
179C4×Dic11—28no—
180C22×C4⋊Q8—111no—
181C22×C22⋊Q8—111no—
182#2492—1116no—
183C2×#400—112no—
184C2×#393—112no—
185C2×#391—112no—
186C2×#391—111no—
187C2×#391—111no—
188C2×#388—112no—
189C2×#380—111no—
190C2×C4×C4⋊C4D1111yescharacteristic, complemented
191C2×C4×C22⋊C4—111no—
192C23×C11C22×D412yes—
193C23×C11C22×Q812yes—
194C23×C11C2×#4112yes—
195C23×C11C2×#4112yes—
196C23×C11C2×#4112yes—
197C23×C11H(5,𝔽2)12yes—
198C23×C112−1+412yes—
199C23×C11C2511yescharacteristic
200C22×D11—28no—
201C22×D11—28no—
202C22×D11—28no—
203C22×D11—22no—
204C22×D11—22no—
205C22×D11—22no—
206C22×D11—22no—
207C22×D11—22no—
208C22×D11—22no—
209C22×D11—44no—
210C22×D11—44no—
211C22×D11—44no—
212C22×D11—44no—
213C22×D11C22×Q814yes—
214C2×C4×C11—216no—
215C2×C4×C11—28no—
216C2×C4×C11—22no—
217C2×C4×C11—24no—
218C2×C4×C11—22no—
219C2×C4×C11—24no—
220C2×C4×C11—22no—
221C2×C4×C11—22no—
222C2×C4×C11—24no—
223C2×C4×C11—24no—
224C2×C4×C11—24no—
225C2×C4×C11—22no—
226C2×C4×C11—48no—
227C2×C4×C11—44no—
228C2×C4×C11#124116yes—
229C2×C4×C11C22×D418yes—
230C2×C4×C11C22×D414yes—
231C2×C4×C11C22×Q814yes—
232C2×C4×C11C2×#4114yes—
233C2×C4×C11C2×#4114yes—
234C2×Dic11—28no—
235C2×Dic11—28no—
236C2×Dic11—22no—
237C2×Dic11—22no—
238C2×Dic11—22no—
239C2×Dic11—22no—
240C2×Dic11—24no—
241C2×Dic11—22no—
242C2×Dic11—24no—
243C2×Dic11—24no—
244C2×Dic11—24no—
245C2×Dic11—24no—
246C2×Dic11—22no—
247C2×Dic11—22no—
248C2×Dic11—24no—
249C2×Dic11—48no—
250C2×Dic11—44no—
251C2×Dic11—48no—
252C2×Dic11C22×Q814yes—
253C2×Dic11C2×#4114yes—
254C4×D11—416no—
255Dic22—432no—
256Dic22—416no—
257C23×Q8—111no—
258C23×Q8—112no—
259C24×C4—111no—
260C2×C4⋊Q8—118no—
261C2×C4⋊Q8—228no—
262C2×C22⋊Q8—118no—
263C2×C22⋊Q8—228no—
264C22×C4⋊C4—112no—
265C22×C4⋊C4—112no—
266C22×C4⋊C4—111no—
267C22×C4⋊C4C2×D1111yescharacteristic
268C22×C4⋊C4C2×D1111yescharacteristic
269C22×C22⋊C4—111no—
270C22×C22⋊C4—111no—
271C22×C22⋊C4—112no—
272C22×C22⋊C4—112no—
273C22×C42—111no—
274C22×C42C2×D1111yescharacteristic
275C22×C42C2×D1112yes—
276#400—1116no—
277#393—1116no—
278#391—118no—
279#391—1116no—
280#391—118no—
281#388—1116no—
282#380—118no—
283C4×C4⋊C4C2×D1118yescomplemented
284C4×C22⋊C4—118no—
285C2×#95—112no—
286C2×#95—112no—
287C2×#95—111no—
288C2×#95—112no—
289C2×#95—112no—
290C2×#95—111no—
291C2×#95—112no—
292C2×#95C2×D1112yes—
293C22×C11C2×#12414yes—
294C22×C11#53314yes—
295C22×C11#54214yes—
296C22×C11#54614yes—
297C22×C11D4×Q814yes—
298C22×C11#55414yes—
299C22×C11#55614yes—
300C22×C11C23×D411yescharacteristic
301C22×C11C23×Q811yescharacteristic
302C22×C11C22×#4111yescharacteristic
303C22×C11C22×#4111yescharacteristic
304C22×C11C22×#4111yescharacteristic
305C22×C11C2×H(5,𝔽2)11yescharacteristic
306C22×C11C2×2−1+411yescharacteristic
307C2×D11—24no—
308C2×D11—24no—
309C2×D11—24no—
310C2×D11—42no—
311C2×D11—42no—
312C2×D11—42no—
313C2×D11—42no—
314C2×D11—416no—
315C4×C11—28no—
316C4×C11—24no—
317C4×C11—44no—
318C4×C11—42no—
319C4×C11C2×#12418yes—
320Dic11—24no—
321Dic11—24no—
322Dic11—44no—
323Dic11—42no—
324Dic11—44no—
325C25—111no—
326C22×Q8—114no—
327C22×Q8—118no—
328C22×Q8—228no—
329C22×Q8—224no—
330C22×Q8—228no—
331C22×Q8—224no—
332C22×Q8—224no—
333C22×Q8—444no—
334C23×C4—111no—
335C23×C4—111no—
336C23×C4—112no—
337C23×C4—112no—
338C23×C4—111no—
339C23×C4—114no—
340C23×C4—224no—
341C23×C4—224no—
342C23×C4—224no—
343C23×C4C22×D1112yes—
344C23×C4C22×D1111yescharacteristic
345C23×C4C22×D1112yes—
346C23×C4C22×D1111yescharacteristic
347C23×C4C22×D1111yescharacteristic
348C4⋊Q8—2232no—
349C22⋊Q8—2232no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—118no—
352C2×C4⋊C4—114no—
353C2×C4⋊C4—118no—
354C2×C4⋊C4—228no—
355C2×C4⋊C4—228no—
356C2×C4⋊C4—224no—
357C2×C4⋊C4D4418yes—
358C2×C4⋊C4C22×D1114yes—
359C2×C4⋊C4C22×D1114yes—
360C2×C22⋊C4—118no—
361C2×C22⋊C4—114no—
362C2×C22⋊C4—118no—
363C2×C22⋊C4—114no—
364C2×C22⋊C4—118no—
365C2×C22⋊C4—224no—
366C2×C22⋊C4—228no—
367C2×C22⋊C4—444no—
368C2×C42—24no—
369C2×C42—44no—
370C2×C42—114no—
371C2×C42—224no—
372C2×C42C22×D1114yes—
373C2×C42C22×D1118yes—
374#95—114no—
375#95—114no—
376#95—118no—
377#95—118no—
378#95—118no—
379#95—118no—
380#95—118no—
381#95C22×D1118yes—
382C2×C11#249218yescomplemented
383C2×C11C22×#12411yescharacteristic
384C2×C11C2×#53311yescharacteristic
385C2×C11C2×#54211yescharacteristic
386C2×C11C2×#54611yescharacteristic
387C2×C11C2×D4×Q811yescharacteristic
388C2×C11C2×#55411yescharacteristic
389C2×C11C2×#55611yescharacteristic
390D11—44no—
391C24—114no—
392C24—112no—
393C24—222no—
394C24—222no—
395C24—222no—
396C24—222no—
397C24—222no—
398C24—222no—
399C24C23×D1111yescharacteristic
400C2×Q8—2216no—
401C2×Q8—228no—
402C2×Q8—2216no—
403C2×Q8—228no—
404C2×Q8—228no—
405C2×Q8—448no—
406C2×Q8—4432no—
407C2×Q8—4416no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—114no—
419C22×C4—114no—
420C22×C4—112no—
421C22×C4—112no—
422C22×C4—114no—
423C22×C4—112no—
424C22×C4—114no—
425C22×C4—222no—
426C22×C4—222no—
427C22×C4—222no—
428C22×C4—222no—
429C22×C4—224no—
430C22×C4—222no—
431C22×C4—224no—
432C22×C4—222no—
433C22×C4—222no—
434C22×C4—224no—
435C22×C4—224no—
436C22×C4—224no—
437C22×C4—224no—
438C22×C4—228no—
439C22×C4—228no—
440C22×C4—228no—
441C22×C4—4416no—
442C22×C4C2×D4418yes—
443C22×C4C2×D4414yes—
444C22×C4#433914yes—
445C22×C4#433914yes—
446C22×C4Q8×D1114yes—
447C22×C4C23×D1112yes—
448C22×C4C23×D1114yes—
449C22×C4C23×D1114yes—
450C22×C4C23×D1112yes—
451C22×C4C23×D1112yes—
452C4⋊C4—28no—
453C4⋊C4—2216no—
454C4⋊C4—2216no—
455C4⋊C4—228no—
456C4⋊C4C2×D44116yes—
457C22⋊C4—1116no—
458C22⋊C4—228no—
459C22⋊C4—2216no—
460C22⋊C4—448no—
461C42—28no—
462C42—48no—
463C42—228no—
464C11C2×#249211yescharacteristic, complemented, Sylow
465C23—114no—
466C23—228no—
467C23—228no—
468C23—228no—
469C23—222no—
470C23—222no—
471C23—222no—
472C23—222no—
473C23—222no—
474C23—222no—
475C23—444no—
476C23—444no—
477C23—444no—
478C23—444no—
479C23C22×D4412yes—
480C23C2×#433912yes—
481C23C2×#433912yes—
482C23#6703712yes—
483C23C2×Q8×D1112yes—
484C23#41×D1112yes—
485C23#6704312yes—
486C23C24×D1111yescharacteristic
487Q8—4432no—
488Q8—4416no—
489C2×C4—28no—
490C2×C4—216no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—44no—
502C2×C4—48no—
503C2×C4—114no—
504C2×C4—114no—
505C2×C4—228no—
506C2×C4—228no—
507C2×C4—222no—
508C2×C4—222no—
509C2×C4—222no—
510C2×C4—222no—
511C2×C4—224no—
512C2×C4—222no—
513C2×C4—224no—
514C2×C4—222no—
515C2×C4—222no—
516C2×C4—224no—
517C2×C4—224no—
518C2×C4—224no—
519C2×C4—224no—
520C2×C4—444no—
521C2×C4—448no—
522C2×C4—448no—
523C2×C4—4416no—
524C2×C4#66928116yes—
525C2×C4C22×D4418yes—
526C2×C4C22×D4414yes—
527C2×C4C2×#433914yes—
528C2×C4C2×#433914yes—
529C2×C4C2×Q8×D1114yes—
530C22—224no—
531C22—224no—
532C22—224no—
533C22—442no—
534C22—442no—
535C22—4416no—
536C22—442no—
537C22—442no—
538C22C2×#6692814yes—
539C22#1062304214yes—
540C22#1062306414yes—
541C22#1062308514yes—
542C22#1062309214yes—
543C22Q8×D4414yes—
544C22#1062310914yes—
545C22C23×D4411yescharacteristic
546C22C22×#433911yescharacteristic
547C22C22×#433911yescharacteristic
548C22C2×#6703711yescharacteristic
549C22C22×Q8×D1111yescharacteristic
550C22C2×#41×D1111yescharacteristic
551C22C2×#6704311yescharacteristic
552C4—24no—
553C4—28no—
554C4—42no—
555C4—44no—
556C4—224no—
557C4—224no—
558C4—442no—
559C4—444no—
560C4—444no—
561C4C2×#6692818yes—
562C2—444no—
563C2#1307298418yesdirect product factor
564C2C22×#6692811yescharacteristic
565C2C2×#1062310911yescharacteristic
566C2C2×#1062309211yescharacteristic
567C2C2×#1062308511yescharacteristic
568C2C2×Q8×D4411yescharacteristic
569C2C2×#1062304211yescharacteristic
570C2C2×#1062306411yescharacteristic
subgroupG/H
centerC24C23×D11
commutator, cosocleC23×C11C25
FrattiniC23C24×D11
FittingC2×C4×C11×C4⋊C4C2
socleC24×C11C24
prime radicalC24×D11C23
hypercenterC2×C4×C4⋊C4D11
hypocenterC11C2×#2492
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#66941C2semidirect
3. #13072984C2direct (N × H)
4. C2×#10622272C2semidirect
6. C2×#10622542C2semidirect
7. C2×#10622542C2semidirect
8. C2×#10622587C2semidirect
12. C2×C4×C11×C4⋊C4C2semidirect
31. C2×#66941C22semidirect
32. C4×C11×C4⋊C4C22semidirect
42. #10622587C22semidirect
45. #10622542C22semidirect
46. #10622542C22semidirect
48. #10622272C22semidirect
83. C2×(C2×D11)⋊C4Q8semidirect
175. (C2×D11)⋊C4C2×Q8semidirect
190. C2×C4×C4⋊C4D11semidirect
283. C4×C4⋊C4C2×D11semidirect
382. C2×C11#2492semidirect
464. C11C2×#2492semidirect
563. C2#13072984direct (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×#669502
#130729842
C2×#106222722
C2×#106222862
C2×#106225422
C2×#106225872
C2×#106225532
C2×#106226002
C2×C4×(C2×D11)⋊C42
C2×C4×C11×C4⋊C42
C2×#249211