Finite Group Databasebeta

← C2×#13076669 #14960295

Subgroups — C2×#13076669 #14960295

Order 2816.

2566 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66942C211yesmaximal, characteristic, complemented
2C22×#66989C211yesmaximal, characteristic
3#13076669C2116yesmaximal, direct product factor
4C2×#10622562C212yesmaximal, complemented
5C2×#10622565C212yesmaximal, complemented
6C2×#10622561C212yesmaximal, complemented
7C2×#10622561C212yesmaximal, complemented
8C2×#10622269C211yesmaximal, characteristic, complemented
9C2×#10622592C211yesmaximal, characteristic, complemented
10C2×#10622835C211yesmaximal, characteristic
11C2×#10622263C211yesmaximal, characteristic, complemented
12C2×C11×#394C211yesmaximal, characteristic
13C23×D4×C11C2211yescharacteristic
14C23×Dic22C2211yescharacteristic
15C22×C11×C4⋊C4C2211yescharacteristic
16C22×C11×C22⋊C4C2212yes—
17C22×C11×C22⋊C4C2212yes—
18C22×C22⋊Dic11C2212yes—
19C22×C22⋊Dic11C2212yes—
20C22×C22⋊Dic11C2212yes—
21C22×C22⋊Dic11C2212yes—
22C2×#66989—28no—
23C2×#66989C2218yes—
24C22×C4⋊Dic11C2211yescharacteristic
25C22×C4⋊Dic11C2212yes—
26C22×Dic11⋊C4C2211yescharacteristic
27C22×Dic11⋊C4C2211yescharacteristic
28C22×Dic11⋊C4C2211yescharacteristic
29C22×Dic11⋊C4C2211yescharacteristic
30C22×Dic11⋊C4C2212yes—
31C22×Dic11⋊C4C2212yes—
32C22×C4×Dic11C2211yescharacteristic
33C2×#66942—28no—
34C2×#66942C2218yescomplemented
35C11×#394C2218yes—
36C2×C11×#95C2211yescharacteristic
37#10622835C2218yes—
38C2×C11⋊#95C2211yescharacteristic
39C2×C11⋊#95C2212yes—
40C2×C11⋊#95C2212yes—
41C2×C11⋊#95C2212yes—
42#10622592C2218yescomplemented
43#10622565C22116yescomplemented
44#10622562C22116yescomplemented
45#10622561C22116yescomplemented
46#10622561C22116yescomplemented
47#10622269C2218yescomplemented
48#10622263C2218yescomplemented
49C25×C11C2312yes—
50C22×D4×C11—24no—
51C22×D4×C11—44no—
52C22×D4×C11C2314yes—
53C23×C4×C11C2311yescharacteristic
54C23×C4×C11C2311yescharacteristic
55C23×C4×C11C2312yes—
56C23×C4×C11C2311yescharacteristic
57C23×Dic11C2311yescharacteristic
58C23×Dic11C2311yescharacteristic
59C23×Dic11C2312yes—
60C23×Dic11C2312yes—
61C23×Dic11C2311yescharacteristic
62C23×Dic11C2311yescharacteristic
63C22×Dic22—24no—
64C22×Dic22—44no—
65C22×Dic22C2314yes—
66C2×C11×C4⋊C4—24no—
67C2×C11×C4⋊C4C2314yes—
68C2×C11×C22⋊C4—44no—
69C2×C11×C22⋊C4—44no—
70C2×C11×C22⋊C4C2318yes—
71C2×C11×C22⋊C4C2318yes—
72C2×C22⋊Dic11—28no—
73C2×C22⋊Dic11—28no—
74C2×C22⋊Dic11—44no—
75C2×C22⋊Dic11—44no—
76C2×C22⋊Dic11C2318yes—
77C2×C22⋊Dic11C2318yes—
78C2×C22⋊Dic11C2318yes—
79C2×C22⋊Dic11C2318yes—
80#66989—232no—
81C2×C4⋊Dic11—24no—
82C2×C4⋊Dic11—44no—
83C2×C4⋊Dic11C2314yes—
84C2×C4⋊Dic11C2318yes—
85C2×Dic11⋊C4—24no—
86C2×Dic11⋊C4—24no—
87C2×Dic11⋊C4—24no—
88C2×Dic11⋊C4—24no—
89C2×Dic11⋊C4—44no—
90C2×Dic11⋊C4—44no—
91C2×Dic11⋊C4C2314yes—
92C2×Dic11⋊C4C2314yes—
93C2×Dic11⋊C4C2314yes—
94C2×Dic11⋊C4C2314yes—
95C2×Dic11⋊C4C2318yes—
96C2×Dic11⋊C4C2318yes—
97C2×C4×Dic11D418yes—
98C2×C4×Dic11C2314yes—
99#66942—232no—
100C11×#95C2314yes—
101C11⋊#95C2318yes—
102C11⋊#95C2314yes—
103C11⋊#95C2318yes—
104C11⋊#95C2318yes—
105C2×#2820—111nomaximal, Sylow
106C24×C11—24no—
107C24×C11—24no—
108C24×C11—24no—
109C24×C11—24no—
110C24×C11—24no—
111C24×C11—24no—
112C24×C11—24no—
113C24×C11C2414yes—
114C24×C11C2411yescharacteristic
115C2×D4×C11—28no—
116C2×D4×C11—48no—
117C2×D4×C11—416no—
118C22×C4×C11—24no—
119C22×C4×C11—22no—
120C22×C4×C11—22no—
121C22×C4×C11—24no—
122C22×C4×C11—24no—
123C22×C4×C11—24no—
124C22×C4×C11—22no—
125C22×C4×C11—22no—
126C22×C4×C11—24no—
127C22×C4×C11C2×D414yes—
128C22×C4×C11C2414yes—
129C22×C4×C11C2412yes—
130C22×C4×C11C2412yes—
131C22×C4×C11C2412yes—
132C22×Dic11—22no—
133C22×Dic11—22no—
134C22×Dic11—24no—
135C22×Dic11—22no—
136C22×Dic11—24no—
137C22×Dic11—24no—
138C22×Dic11—24no—
139C22×Dic11—24no—
140C22×Dic11—24no—
141C22×Dic11—22no—
142C22×Dic11—24no—
143C22×Dic11—22no—
144C22×Dic11—24no—
145C22×Dic11—22no—
146C22×Dic11C2×D414yes—
147C22×Dic11C2×D414yes—
148C22×Dic11C2414yes—
149C22×Dic11C2412yes—
150C22×Dic11C2414yes—
151C22×Dic11C2412yes—
152C22×Dic11C2412yes—
153C22×Dic11C2412yes—
154C2×Dic22—28no—
155C2×Dic22—48no—
156C2×Dic22—416no—
157C11×C4⋊C4—28no—
158C11×C22⋊C4—48no—
159C11×C22⋊C4—48no—
160C22⋊Dic11—216no—
161C22⋊Dic11—216no—
162C22⋊Dic11—48no—
163C22⋊Dic11—48no—
164C4⋊Dic11—28no—
165C4⋊Dic11—48no—
166Dic11⋊C4—28no—
167Dic11⋊C4—28no—
168Dic11⋊C4—28no—
169Dic11⋊C4—28no—
170Dic11⋊C4—48no—
171Dic11⋊C4—48no—
172C4×Dic11C2×D4116yes—
173C22×#125—111no—
174C22×#124—111no—
175#2820—1116no—
176C2×#399—112no—
177C2×#397—112no—
178C2×#396—112no—
179C2×#396—112no—
180C2×#395—111no—
181C2×#395—111no—
182C2×#394—111no—
183C2×#394D1111yescharacteristic
184C2×#382—111no—
185C23×C11—24no—
186C23×C11—24no—
187C23×C11—24no—
188C23×C11—24no—
189C23×C11—24no—
190C23×C11—24no—
191C23×C11—24no—
192C23×C11—48no—
193C23×C11—48no—
194C23×C11—48no—
195C23×C11—48no—
196C23×C11—48no—
197C23×C11—48no—
198C23×C11—48no—
199C23×C11C22×D412yes—
200C23×C11H(5,𝔽2)14yes—
201C23×C11H(5,𝔽2)12yes—
202C23×C112−1+414yes—
203C23×C112−1+412yes—
204C23×C11C2511yescharacteristic
205D4×C11—416no—
206C2×C4×C11—28no—
207C2×C4×C11—24no—
208C2×C4×C11—22no—
209C2×C4×C11—22no—
210C2×C4×C11—24no—
211C2×C4×C11—24no—
212C2×C4×C11—24no—
213C2×C4×C11—22no—
214C2×C4×C11—22no—
215C2×C4×C11—24no—
216C2×C4×C11—44no—
217C2×C4×C11—44no—
218C2×C4×C11—48no—
219C2×C4×C11C22×D414yes—
220C2×Dic11—28no—
221C2×Dic11—28no—
222C2×Dic11—22no—
223C2×Dic11—24no—
224C2×Dic11—22no—
225C2×Dic11—22no—
226C2×Dic11—22no—
227C2×Dic11—24no—
228C2×Dic11—24no—
229C2×Dic11—24no—
230C2×Dic11—24no—
231C2×Dic11—24no—
232C2×Dic11—24no—
233C2×Dic11—22no—
234C2×Dic11—22no—
235C2×Dic11—24no—
236C2×Dic11—48no—
237C2×Dic11—44no—
238C2×Dic11—44no—
239C2×Dic11—48no—
240C2×Dic11C22×D414yes—
241C2×Dic11C22×D414yes—
242Dic22—416no—
243C23×Q8—111no—
244C23×D4C2×D1111yescharacteristic
245C2×#125—118no—
246C2×#125—228no—
247C2×#124—118no—
248C2×#124—228no—
249C22×C4⋊C4—111no—
250C22×C4⋊C4—111no—
251C22×C4⋊C4—111no—
252C22×C4⋊C4—111no—
253C22×C4⋊C4—111no—
254C22×C4⋊C4—112no—
255C22×C4⋊C4—112no—
256C22×C4⋊C4—112no—
257C22×C4⋊C4C2×D1111yescharacteristic
258C22×C22⋊C4—112no—
259C22×C22⋊C4—112no—
260C22×C22⋊C4—112no—
261C22×C22⋊C4—112no—
262C22×C22⋊C4C2×D1112yes—
263C22×C22⋊C4C2×D1112yes—
264C22×C42—111no—
265#399—1116no—
266#397—1116no—
267#396—1116no—
268#396—1116no—
269#395—118no—
270#395—118no—
271#394—118no—
272#394C2×D1118yes—
273#382—118no—
274C2×#95—111no—
275C2×#95—112no—
276C2×#95—112no—
277C2×#95—112no—
278C2×#95C2×D1111yescharacteristic
279C22×C11—44no—
280C22×C11—44no—
281C22×C11—44no—
282C22×C11—44no—
283C22×C11—44no—
284C22×C11—44no—
285C22×C11—44no—
286C22×C11—816no—
287C22×C11#53514yes—
288C22×C11#53614yes—
289C22×C11#53714yes—
290C22×C11#56218yes—
291C22×C11#56318yes—
292C22×C11C23×D411yescharacteristic
293C22×C11C2×H(5,𝔽2)12yes—
294C22×C11C2×H(5,𝔽2)11yescharacteristic
295C22×C11C2×2−1+412yes—
296C22×C11C2×2−1+411yescharacteristic
297C4×C11—24no—
298C4×C11—42no—
299C4×C11—42no—
300C4×C11—44no—
301Dic11—24no—
302Dic11—24no—
303Dic11—44no—
304Dic11—42no—
305Dic11—42no—
306Dic11—44no—
307C25C22×D1112yes—
308C22×Q8—114no—
309C22×Q8—224no—
310C22×Q8—444no—
311C22×D4—24no—
312C22×D4—44no—
313C22×D4C22×D1114yes—
314C23×C4—111no—
315C23×C4—111no—
316C23×C4—111no—
317C23×C4—111no—
318C23×C4—112no—
319C23×C4—112no—
320C23×C4C22×D1111yescharacteristic
321C23×C4C22×D1111yescharacteristic
322C23×C4C22×D1112yes—
323C23×C4C22×D1111yescharacteristic
324#125—2232no—
325#124—2232no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—114no—
328C2×C4⋊C4—114no—
329C2×C4⋊C4—114no—
330C2×C4⋊C4—114no—
331C2×C4⋊C4—114no—
332C2×C4⋊C4—118no—
333C2×C4⋊C4—118no—
334C2×C4⋊C4—118no—
335C2×C4⋊C4—224no—
336C2×C4⋊C4—224no—
337C2×C4⋊C4—224no—
338C2×C4⋊C4—224no—
339C2×C4⋊C4—224no—
340C2×C4⋊C4—444no—
341C2×C4⋊C4—444no—
342C2×C4⋊C4—444no—
343C2×C4⋊C4C22×D1114yes—
344C2×C22⋊C4—44no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—118no—
347C2×C22⋊C4—118no—
348C2×C22⋊C4—118no—
349C2×C22⋊C4—118no—
350C2×C22⋊C4—228no—
351C2×C22⋊C4—228no—
352C2×C22⋊C4—444no—
353C2×C22⋊C4—444no—
354C2×C22⋊C4C22×D1118yes—
355C2×C22⋊C4C22×D1118yes—
356C2×C42—118no—
357C2×C42—114no—
358#95—114no—
359#95—118no—
360#95—118no—
361#95—118no—
362#95C22×D1114yes—
363C2×C11—84no—
364C2×C11#282018yescomplemented
365C2×C11C2×#53511yescharacteristic
366C2×C11C2×#53611yescharacteristic
367C2×C11C2×#53711yescharacteristic
368C2×C11C2×#56212yes—
369C2×C11C2×#56312yes—
370C24—24no—
371C24—24no—
372C24—24no—
373C24—24no—
374C24—24no—
375C24—24no—
376C24—24no—
377C24C23×D1114yes—
378C24C23×D1111yescharacteristic
379C2×Q8—228no—
380C2×Q8—448no—
381C2×Q8—4416no—
382C2×D4—28no—
383C2×D4—48no—
384C2×D4—416no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—114no—
395C22×C4—114no—
396C22×C4—112no—
397C22×C4—112no—
398C22×C4—114no—
399C22×C4—112no—
400C22×C4—114no—
401C22×C4—112no—
402C22×C4—222no—
403C22×C4—222no—
404C22×C4—224no—
405C22×C4—224no—
406C22×C4—222no—
407C22×C4—222no—
408C22×C4—222no—
409C22×C4—224no—
410C22×C4—224no—
411C22×C4—224no—
412C22×C4—224no—
413C22×C4—224no—
414C22×C4—222no—
415C22×C4—224no—
416C22×C4D4×D1114yes—
417C22×C4C23×D1114yes—
418C22×C4C23×D1112yes—
419C22×C4C23×D1112yes—
420C22×C4C23×D1112yes—
421C4⋊C4—28no—
422C4⋊C4—228no—
423C4⋊C4—228no—
424C4⋊C4—228no—
425C4⋊C4—228no—
426C4⋊C4—228no—
427C4⋊C4—448no—
428C4⋊C4—448no—
429C4⋊C4—448no—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C22⋊C4—2216no—
433C22⋊C4—2216no—
434C22⋊C4—448no—
435C22⋊C4—448no—
436C42—1116no—
437C11C2×#282011yescharacteristic, complemented, Sylow
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—48no—
446C23—48no—
447C23—48no—
448C23—48no—
449C23—48no—
450C23—48no—
451C23—48no—
452C23C2×D4×D1112yes—
453C23#6703714yes—
454C23#6703712yes—
455C23#6704312yes—
456C23#6704314yes—
457C23C24×D1111yescharacteristic
458Q8—4416no—
459D4—416no—
460C2×C4—28no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—48no—
473C2×C4—114no—
474C2×C4—114no—
475C2×C4—228no—
476C2×C4—222no—
477C2×C4—224no—
478C2×C4—222no—
479C2×C4—228no—
480C2×C4—224no—
481C2×C4—222no—
482C2×C4—222no—
483C2×C4—222no—
484C2×C4—224no—
485C2×C4—224no—
486C2×C4—224no—
487C2×C4—224no—
488C2×C4—224no—
489C2×C4—222no—
490C2×C4—224no—
491C2×C4—448no—
492C2×C4—444no—
493C2×C4—444no—
494C2×C4—448no—
495C2×C4C2×D4×D1114yes—
496C22—44no—
497C22—44no—
498C22—44no—
499C22—44no—
500C22—44no—
501C22—44no—
502C22—44no—
503C22—816no—
504C22#1062312314yes—
505C22#1062313914yes—
506C22#1062315218yes—
507C22#1062318218yes—
508C22#1062320614yes—
509C22C22×D4×D1111yescharacteristic
510C22C2×#6703712yes—
511C22C2×#6703711yescharacteristic
512C22C2×#6704312yes—
513C22C2×#6704311yescharacteristic
514C4—24no—
515C4—42no—
516C4—42no—
517C4—44no—
518C4—224no—
519C4—224no—
520C4—444no—
521C4—442no—
522C4—442no—
523C4—444no—
524C2—84no—
525C2#1307666918yesdirect product factor
526C2C2×#1062312311yescharacteristic
527C2C2×#1062320611yescharacteristic
528C2C2×#1062313911yescharacteristic
529C2C2×#1062315212yes—
530C2C2×#1062318212yes—
subgroupG/H
centerC24C23×D11
commutator, cosocleC23×C11C25
FrattiniC23C24×D11
FittingC2×C11×#394C2
socleC24×C11C24
prime radicalC23×D4×C11C22
hypercenterC2×#394D11
hypocenterC11C2×#2820
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
1. C22×#66942C2semidirect
3. #13076669C2direct (N × H)
4. C2×#10622562C2semidirect
5. C2×#10622565C2semidirect
6. C2×#10622561C2semidirect
7. C2×#10622561C2semidirect
8. C2×#10622269C2semidirect
9. C2×#10622592C2semidirect
11. C2×#10622263C2semidirect
34. C2×#66942C22semidirect
42. #10622592C22semidirect
43. #10622565C22semidirect
44. #10622562C22semidirect
45. #10622561C22semidirect
46. #10622561C22semidirect
47. #10622269C22semidirect
48. #10622263C22semidirect
364. C2×C11#2820semidirect
437. C11C2×#2820semidirect
525. C2#13076669direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#669422
C22×#669892
#130766692
C2×#106225622
C2×#106225652
C2×#106225612
C2×#106222692
C2×#106225922
C2×#106228352
C2×#106222632
C2×C11×#3942
C2×#282011