Finite Group Databasebeta

← C2×#13076650 #14959909

Subgroups — C2×#13076650 #14959909

Order 2816.

2478 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66949C211yesmaximal, characteristic
2C22×#66999C211yesmaximal, characteristic
3#13076650C2116yesmaximal, direct product factor
4C2×#10622271C212yesmaximal, complemented
5C2×#10622272C211yesmaximal, characteristic, complemented
6C2×#10622599C211yesmaximal, characteristic, complemented
7C2×#10622288C212yesmaximal
8C2×#10622606C212yesmaximal
9C2×#10622592C212yesmaximal, complemented
10C2×#10622842C211yesmaximal, characteristic, complemented
11C2×#10622277C211yesmaximal, characteristic
12C2×C11×#395C211yesmaximal, characteristic, complemented
13C23×Q8×C11C2211yescharacteristic
14C23×C4×D11C2211yescharacteristic
15C23×Dic22C2211yescharacteristic
16C22×C11×C4⋊C4C2211yescharacteristic
17C22×C11×C4⋊C4C2212yes—
18C2×#66999—28no—
19C2×#66999C2218yes—
20C22×(C2×D11)⋊C4C2211yescharacteristic
21C22×(C2×D11)⋊C4C2212yes—
22C22×(C2×D11)⋊C4C2211yescharacteristic
23C22×(C2×D11)⋊C4C2212yes—
24C22×C4⋊Dic11C2211yescharacteristic
25C22×Dic11⋊C4C2211yescharacteristic
26C22×Dic11⋊C4C2212yes—
27C22×Dic11⋊C4C2211yescharacteristic
28C22×Dic11⋊C4C2212yes—
29C2×#66949—28no—
30C2×#66949C2218yes—
31C11×#395C2218yescomplemented
32C2×C11×#95C2211yescharacteristic
33C2×C11×#95C2212yes—
34#10622842C2218yescomplemented
35C2×C11⋊#95C2211yescharacteristic
36C2×C11⋊#95C2212yes—
37C2×C11⋊#95C2212yes—
38C2×C11⋊#95C2212yes—
39C2×C11⋊#95C2212yes—
40C2×C11⋊#95C2212yes—
41C2×C11⋊#95C2212yes—
42#10622606C22116yes—
43#10622599C2218yescomplemented
44#10622592C22116yescomplemented
45#10622288C22116yes—
46#10622277C2218yes—
47#10622272C2218yescomplemented
48#10622271C22116yescomplemented
49C24×D11C2311yescharacteristic
50C22×Q8×C11—24no—
51C22×Q8×C11—44no—
52C22×Q8×C11C2314yes—
53C23×C4×C11C2312yes—
54C23×C4×C11C2311yescharacteristic
55C23×C4×C11C2312yes—
56C23×C4×C11C2311yescharacteristic
57C23×C4×C11C2311yescharacteristic
58C23×Dic11C2311yescharacteristic
59C23×Dic11C2311yescharacteristic
60C23×Dic11C2312yes—
61C23×Dic11C2312yes—
62C23×Dic11C2311yescharacteristic
63C22×C4×D11—44no—
64C22×C4×D11D418yes—
65C22×C4×D11C2314yes—
66C22×Dic22—24no—
67C22×Dic22—44no—
68C22×Dic22C2314yes—
69C2×C11×C4⋊C4—24no—
70C2×C11×C4⋊C4—44no—
71C2×C11×C4⋊C4C2314yes—
72C2×C11×C4⋊C4C2318yes—
73#66999—232no—
74C2×(C2×D11)⋊C4—24no—
75C2×(C2×D11)⋊C4—24no—
76C2×(C2×D11)⋊C4—28no—
77C2×(C2×D11)⋊C4—44no—
78C2×(C2×D11)⋊C4C2314yes—
79C2×(C2×D11)⋊C4C2314yes—
80C2×(C2×D11)⋊C4C2318yes—
81C2×(C2×D11)⋊C4C2318yes—
82C2×C4⋊Dic11—24no—
83C2×C4⋊Dic11C2314yes—
84C2×Dic11⋊C4—24no—
85C2×Dic11⋊C4—24no—
86C2×Dic11⋊C4—28no—
87C2×Dic11⋊C4—44no—
88C2×Dic11⋊C4C2314yes—
89C2×Dic11⋊C4C2318yes—
90C2×Dic11⋊C4C2314yes—
91C2×Dic11⋊C4C2318yes—
92#66949—232no—
93C11×#95C2318yes—
94C11×#95C2314yes—
95C11⋊#95C2318yes—
96C11⋊#95C2318yes—
97C11⋊#95C2318yes—
98C11⋊#95C2318yes—
99C11⋊#95C2318yes—
100C11⋊#95C2314yes—
101C11⋊#95C2318yes—
102C2×#2817—111nomaximal, Sylow
103C24×C11C2411yescharacteristic
104C23×D11—24no—
105C23×D11—24no—
106C23×D11—22no—
107C23×D11—22no—
108C23×D11C2×D414yes—
109C23×D11C2412yes—
110C2×Q8×C11—28no—
111C2×Q8×C11—48no—
112C2×Q8×C11—416no—
113C22×C4×C11—24no—
114C22×C4×C11—22no—
115C22×C4×C11—24no—
116C22×C4×C11—22no—
117C22×C4×C11—24no—
118C22×C4×C11—24no—
119C22×C4×C11—24no—
120C22×C4×C11—24no—
121C22×C4×C11—22no—
122C22×C4×C11—22no—
123C22×C4×C11—24no—
124C22×C4×C11—24no—
125C22×C4×C11C2×D414yes—
126C22×C4×C11C2414yes—
127C22×C4×C11C2412yes—
128C22×C4×C11C2412yes—
129C22×C4×C11C2414yes—
130C22×C4×C11C2412yes—
131C22×Dic11—22no—
132C22×Dic11—24no—
133C22×Dic11—22no—
134C22×Dic11—24no—
135C22×Dic11—24no—
136C22×Dic11—24no—
137C22×Dic11—22no—
138C22×Dic11—28no—
139C22×Dic11—24no—
140C22×Dic11—24no—
141C22×Dic11—22no—
142C22×Dic11C2×D414yes—
143C22×Dic11C2414yes—
144C22×Dic11C2412yes—
145C22×Dic11C2414yes—
146C22×Dic11C2412yes—
147C22×Dic11C2412yes—
148C2×C4×D11—416no—
149C2×C4×D11—48no—
150C2×C4×D11C2×D4116yes—
151C2×Dic22—28no—
152C2×Dic22—48no—
153C2×Dic22—416no—
154C11×C4⋊C4—28no—
155C11×C4⋊C4—48no—
156(C2×D11)⋊C4—28no—
157(C2×D11)⋊C4—28no—
158(C2×D11)⋊C4—216no—
159(C2×D11)⋊C4—48no—
160C4⋊Dic11—28no—
161Dic11⋊C4—28no—
162Dic11⋊C4—216no—
163Dic11⋊C4—28no—
164Dic11⋊C4—48no—
165C22×C22⋊Q8—111no—
166C22×C22⋊Q8—111no—
167#2817—1116no—
168C2×#400—111no—
169C2×#400—112no—
170C2×#400—111no—
171C2×#397—112no—
172C2×#397—112no—
173C2×#395—111no—
174C2×#395—112no—
175C2×#395D1111yescharacteristic, complemented
176C2×#381—111no—
177C23×C11C22×D412yes—
178C23×C11H(5,𝔽2)14yes—
179C23×C112−1+414yes—
180C23×C112−1+412yes—
181C23×C112−1+412yes—
182C23×C11C2511yescharacteristic
183C22×D11—28no—
184C22×D11—28no—
185C22×D11—28no—
186C22×D11—24no—
187C22×D11—22no—
188C22×D11—24no—
189C22×D11—22no—
190C22×D11—48no—
191C22×D11—48no—
192C22×D11C22×D414yes—
193Q8×C11—416no—
194C2×C4×C11—28no—
195C2×C4×C11—24no—
196C2×C4×C11—22no—
197C2×C4×C11—24no—
198C2×C4×C11—22no—
199C2×C4×C11—24no—
200C2×C4×C11—24no—
201C2×C4×C11—24no—
202C2×C4×C11—24no—
203C2×C4×C11—22no—
204C2×C4×C11—24no—
205C2×C4×C11—22no—
206C2×C4×C11—24no—
207C2×C4×C11—44no—
208C2×C4×C11—48no—
209C2×C4×C11—44no—
210C2×C4×C11—48no—
211C2×C4×C11C22×D414yes—
212C2×Dic11—28no—
213C2×Dic11—24no—
214C2×Dic11—22no—
215C2×Dic11—24no—
216C2×Dic11—22no—
217C2×Dic11—24no—
218C2×Dic11—24no—
219C2×Dic11—22no—
220C2×Dic11—28no—
221C2×Dic11—24no—
222C2×Dic11—24no—
223C2×Dic11—22no—
224C2×Dic11—44no—
225C2×Dic11—48no—
226C2×Dic11—44no—
227C2×Dic11—48no—
228C2×Dic11C22×D414yes—
229C4×D11—416no—
230Dic22—416no—
231C23×Q8—111no—
232C23×Q8C2×D1111yescharacteristic
233C24×C4—111no—
234C2×C22⋊Q8—118no—
235C2×C22⋊Q8—118no—
236C2×C22⋊Q8—228no—
237C2×C22⋊Q8—228no—
238C22×C4⋊C4—112no—
239C22×C4⋊C4—111no—
240C22×C4⋊C4—111no—
241C22×C4⋊C4—111no—
242C22×C4⋊C4—112no—
243C22×C4⋊C4C2×D1111yescharacteristic
244C22×C4⋊C4C2×D1112yes—
245C22×C22⋊C4—111no—
246C22×C22⋊C4—112no—
247C22×C22⋊C4—111no—
248C22×C22⋊C4—112no—
249#400—118no—
250#400—1116no—
251#400—118no—
252#397—1116no—
253#397—1116no—
254#395—118no—
255#395—1116no—
256#395C2×D1118yescomplemented
257#381—118no—
258C2×#95—112no—
259C2×#95—112no—
260C2×#95—112no—
261C2×#95—112no—
262C2×#95—111no—
263C2×#95—112no—
264C2×#95—112no—
265C2×#95C2×D1112yes—
266C2×#95C2×D1111yescharacteristic
267C22×C11#53414yes—
268C22×C11#53614yes—
269C22×C11#53614yes—
270C22×C11#56318yes—
271C22×C11#56318yes—
272C22×C11C23×D411yescharacteristic
273C22×C11C2×H(5,𝔽2)12yes—
274C22×C11C2×2−1+411yescharacteristic
275C22×C11C2×2−1+412yes—
276C22×C11C2×2−1+411yescharacteristic
277C2×D11—24no—
278C2×D11—24no—
279C2×D11—24no—
280C2×D11—416no—
281C2×D11—44no—
282C2×D11—44no—
283C4×C11—24no—
284C4×C11—44no—
285C4×C11—42no—
286C4×C11—42no—
287C4×C11—44no—
288Dic11—24no—
289Dic11—42no—
290Dic11—42no—
291Dic11—44no—
292Dic11—44no—
293C25—111no—
294C22×Q8—24no—
295C22×Q8—44no—
296C22×Q8—114no—
297C22×Q8—224no—
298C22×Q8—444no—
299C22×Q8C22×D1114yes—
300C23×C4—111no—
301C23×C4—118no—
302C23×C4—114no—
303C23×C4—111no—
304C23×C4—112no—
305C23×C4—111no—
306C23×C4—112no—
307C23×C4—444no—
308C23×C4C22×D1112yes—
309C23×C4C22×D1111yescharacteristic
310C23×C4C22×D1111yescharacteristic
311C23×C4C22×D1112yes—
312C23×C4C22×D1111yescharacteristic
313C22⋊Q8—2232no—
314C22⋊Q8—2232no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—118no—
318C2×C4⋊C4—114no—
319C2×C4⋊C4—114no—
320C2×C4⋊C4—114no—
321C2×C4⋊C4—118no—
322C2×C4⋊C4—224no—
323C2×C4⋊C4—228no—
324C2×C4⋊C4—224no—
325C2×C4⋊C4—224no—
326C2×C4⋊C4—444no—
327C2×C4⋊C4C22×D1114yes—
328C2×C4⋊C4C22×D1118yes—
329C2×C22⋊C4—114no—
330C2×C22⋊C4—118no—
331C2×C22⋊C4—114no—
332C2×C22⋊C4—118no—
333C2×C22⋊C4—224no—
334C2×C22⋊C4—224no—
335C2×C22⋊C4—228no—
336C2×C22⋊C4—444no—
337#95—118no—
338#95—114no—
339#95—118no—
340#95—118no—
341#95—118no—
342#95—118no—
343#95—118no—
344#95C22×D1114yes—
345#95C22×D1118yes—
346C2×C11#281718yescomplemented
347C2×C11C2×#53411yescharacteristic
348C2×C11C2×#53611yescharacteristic
349C2×C11C2×#53611yescharacteristic
350C2×C11C2×#56312yes—
351C2×C11C2×#56312yes—
352D11—44no—
353C24—114no—
354C24—112no—
355C24—224no—
356C24—224no—
357C24—222no—
358C24—222no—
359C24C23×D1111yescharacteristic
360C2×Q8—28no—
361C2×Q8—48no—
362C2×Q8—416no—
363C2×Q8—228no—
364C2×Q8—448no—
365C2×Q8—4416no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—114no—
379C22×C4—112no—
380C22×C4—1116no—
381C22×C4—114no—
382C22×C4—114no—
383C22×C4—112no—
384C22×C4—112no—
385C22×C4—224no—
386C22×C4—224no—
387C22×C4—222no—
388C22×C4—222no—
389C22×C4—224no—
390C22×C4—224no—
391C22×C4—228no—
392C22×C4—224no—
393C22×C4—224no—
394C22×C4—222no—
395C22×C4—222no—
396C22×C4—4416no—
397C22×C4—448no—
398C22×C4D4×D1114yes—
399C22×C4C23×D1114yes—
400C22×C4C23×D1112yes—
401C22×C4C23×D1112yes—
402C22×C4C23×D1114yes—
403C22×C4C23×D1112yes—
404C4⋊C4—28no—
405C4⋊C4—48no—
406C4⋊C4—228no—
407C4⋊C4—2216no—
408C4⋊C4—228no—
409C4⋊C4—228no—
410C4⋊C4—448no—
411C22⋊C4—228no—
412C22⋊C4—228no—
413C22⋊C4—2216no—
414C22⋊C4—448no—
415C11C2×#281711yescharacteristic, complemented, Sylow
416C23—114no—
417C23—228no—
418C23—228no—
419C23—228no—
420C23—224no—
421C23—222no—
422C23—224no—
423C23—222no—
424C23—448no—
425C23—448no—
426C23C2×D4×D1112yes—
427C23#6703714yes—
428C23#6704014yes—
429C23#6704012yes—
430C23#6704312yes—
431C23C24×D1111yescharacteristic
432Q8—416no—
433Q8—4416no—
434C2×C4—28no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—48no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—48no—
451C2×C4—114no—
452C2×C4—228no—
453C2×C4—224no—
454C2×C4—222no—
455C2×C4—224no—
456C2×C4—222no—
457C2×C4—224no—
458C2×C4—224no—
459C2×C4—222no—
460C2×C4—228no—
461C2×C4—224no—
462C2×C4—224no—
463C2×C4—222no—
464C2×C4—4416no—
465C2×C4—448no—
466C2×C4—448no—
467C2×C4—444no—
468C2×C4—444no—
469C2×C4C2×D4×D1114yes—
470C22—224no—
471C22—224no—
472C22—224no—
473C22—444no—
474C22—444no—
475C22—4416no—
476C22#1062312014yes—
477C22#1062315914yes—
478C22#1062317718yes—
479C22#1062325514yes—
480C22#1062326618yes—
481C22C22×D4×D1111yescharacteristic
482C22C2×#6703712yes—
483C22C2×#6704011yescharacteristic
484C22C2×#6704012yes—
485C22C2×#6704311yescharacteristic
486C4—24no—
487C4—44no—
488C4—42no—
489C4—42no—
490C4—44no—
491C4—224no—
492C4—442no—
493C4—442no—
494C4—444no—
495C4—444no—
496C2—444no—
497C2#1307665018yesdirect product factor
498C2C2×#1062312011yescharacteristic
499C2C2×#1062315911yescharacteristic
500C2C2×#1062325511yescharacteristic
501C2C2×#1062326612yes—
502C2C2×#1062317712yes—
subgroupG/H
centerC24C23×D11
commutator, cosocleC23×C11C25
FrattiniC23C24×D11
FittingC2×C11×#395C2
socleC24×C11C24
prime radicalC24×D11C23
hypercenterC2×#395D11
hypocenterC11C2×#2817
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
3. #13076650C2direct (N × H)
4. C2×#10622271C2semidirect
5. C2×#10622272C2semidirect
6. C2×#10622599C2semidirect
9. C2×#10622592C2semidirect
10. C2×#10622842C2semidirect
12. C2×C11×#395C2semidirect
31. C11×#395C22semidirect
34. #10622842C22semidirect
43. #10622599C22semidirect
44. #10622592C22semidirect
47. #10622272C22semidirect
48. #10622271C22semidirect
175. C2×#395D11semidirect
256. #395C2×D11semidirect
346. C2×C11#2817semidirect
415. C11C2×#2817semidirect
497. C2#13076650direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#669492
C22×#669992
#130766502
C2×#106222712
C2×#106222722
C2×#106225992
C2×#106222882
C2×#106226062
C2×#106225922
C2×#106228422
C2×#106222772
C2×C11×#3952
C2×#281711