Finite Group Databasebeta

← C2×#13074780 #14920577

Subgroups — C2×#13074780 #14920577

Order 2816.

2354 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66932C211yesmaximal, characteristic
2C22×#66943C211yesmaximal, characteristic, complemented
3#13074780C2116yesmaximal, direct product factor
4C2×#10622595C211yesmaximal, characteristic, complemented
5C2×#10622274C211yesmaximal, characteristic, complemented
6C2×#10622593C212yesmaximal, complemented
7C2×#10622566C211yesmaximal, characteristic
8C2×#10622561C212yesmaximal
9C2×#10622598C212yesmaximal, complemented
10C2×#10622564C212yesmaximal
11C2×#10622262C211yesmaximal, characteristic, complemented
12C2×C11×#399C211yesmaximal, characteristic
13C22×C11×C4⋊C4C2211yescharacteristic
14C22×C11×C4⋊C4C2212yes—
15C22×C11×C22⋊C4C2211yescharacteristic
16C22×C11×C22⋊C4C2212yes—
17C22×C22⋊Dic11C2211yescharacteristic
18C22×C22⋊Dic11C2211yescharacteristic
19C22×C22⋊Dic11C2212yes—
20C22×C4⋊Dic11C2212yes—
21C22×C4⋊Dic11C2211yescharacteristic
22C22×C4⋊Dic11C2211yescharacteristic
23C22×C4⋊Dic11C2212yes—
24C22×Dic11⋊C4C2212yes—
25C22×Dic11⋊C4C2211yescharacteristic
26C22×Dic11⋊C4C2211yescharacteristic
27C22×Dic11⋊C4C2212yes—
28C22×C4×Dic11C2212yes—
29C22×C4×Dic11C2211yescharacteristic
30C2×#66943—28no—
31C2×#66943C2218yescomplemented
32C2×#66932—28no—
33C2×#66932C2218yes—
34C11×#399C2218yes—
35C2×C11×#95C2211yescharacteristic
36C2×C11⋊#95C2211yescharacteristic
37C2×C11⋊#95C2212yes—
38C2×C11⋊#95C2212yes—
39C2×C11⋊#95C2212yes—
40C2×C11⋊#95C2212yes—
41#10622598C22116yescomplemented
42#10622595C2218yescomplemented
43#10622593C22116yescomplemented
44#10622566C2218yes—
45#10622564C22116yes—
46#10622561C22116yes—
47#10622274C2218yescomplemented
48#10622262C2218yescomplemented
49C25×C11C2311yescharacteristic
50C23×C4×C11C2311yescharacteristic
51C23×C4×C11C2312yes—
52C23×C4×C11C2311yescharacteristic
53C23×C4×C11C2312yes—
54C23×Dic11C2312yes—
55C23×Dic11C2311yescharacteristic
56C23×Dic11C2312yes—
57C23×Dic11C2311yescharacteristic
58C23×Dic11C2311yescharacteristic
59C23×Dic11C2311yescharacteristic
60C2×C11×C4⋊C4—24no—
61C2×C11×C4⋊C4—28no—
62C2×C11×C4⋊C4C2314yes—
63C2×C11×C4⋊C4C2318yes—
64C2×C11×C22⋊C4—24no—
65C2×C11×C22⋊C4—28no—
66C2×C11×C22⋊C4C2314yes—
67C2×C11×C22⋊C4C2318yes—
68C2×C22⋊Dic11—24no—
69C2×C22⋊Dic11—24no—
70C2×C22⋊Dic11—44no—
71C2×C22⋊Dic11C2314yes—
72C2×C22⋊Dic11C2314yes—
73C2×C22⋊Dic11C2318yes—
74C2×C4⋊Dic11—24no—
75C2×C4⋊Dic11—24no—
76C2×C4⋊Dic11—28no—
77C2×C4⋊Dic11—44no—
78C2×C4⋊Dic11C2314yes—
79C2×C4⋊Dic11C2314yes—
80C2×C4⋊Dic11C2318yes—
81C2×C4⋊Dic11C2318yes—
82C2×Dic11⋊C4—24no—
83C2×Dic11⋊C4—28no—
84C2×Dic11⋊C4—24no—
85C2×Dic11⋊C4—28no—
86C2×Dic11⋊C4C2314yes—
87C2×Dic11⋊C4C2314yes—
88C2×Dic11⋊C4C2318yes—
89C2×Dic11⋊C4C2318yes—
90C2×C4×Dic11—44no—
91C2×C4×Dic11D418yes—
92C2×C4×Dic11C2318yes—
93C2×C4×Dic11C2314yes—
94#66943—232no—
95#66932—232no—
96C11×#95C2314yes—
97C11⋊#95C2314yes—
98C11⋊#95C2318yes—
99C11⋊#95C2318yes—
100C11⋊#95C2318yes—
101C11⋊#95C2318yes—
102C2×#2664—111nomaximal, Sylow
103C24×C11—22no—
104C24×C11—24no—
105C24×C11—22no—
106C24×C11—24no—
107C24×C11—22no—
108C24×C11C2412yes—
109C24×C11C2411yescharacteristic
110C22×C4×C11—24no—
111C22×C4×C11—24no—
112C22×C4×C11—24no—
113C22×C4×C11—22no—
114C22×C4×C11—24no—
115C22×C4×C11—24no—
116C22×C4×C11—24no—
117C22×C4×C11—24no—
118C22×C4×C11C2×D414yes—
119C22×C4×C11#4118yes—
120C22×C4×C11C2412yes—
121C22×C4×C11C2414yes—
122C22×C4×C11C2412yes—
123C22×C4×C11C2414yes—
124C22×Dic11—22no—
125C22×Dic11—24no—
126C22×Dic11—22no—
127C22×Dic11—24no—
128C22×Dic11—24no—
129C22×Dic11—22no—
130C22×Dic11—24no—
131C22×Dic11—22no—
132C22×Dic11—24no—
133C22×Dic11—22no—
134C22×Dic11—24no—
135C22×Dic11—22no—
136C22×Dic11—24no—
137C22×Dic11—24no—
138C22×Dic11C2×D414yes—
139C22×Dic11C2×D414yes—
140C22×Dic11C2414yes—
141C22×Dic11C2412yes—
142C22×Dic11C2414yes—
143C22×Dic11C2412yes—
144C22×Dic11C2412yes—
145C22×Dic11C2412yes—
146C11×C4⋊C4—28no—
147C11×C4⋊C4—216no—
148C11×C22⋊C4—28no—
149C11×C22⋊C4—216no—
150C22⋊Dic11—28no—
151C22⋊Dic11—28no—
152C22⋊Dic11—48no—
153C4⋊Dic11—28no—
154C4⋊Dic11—216no—
155C4⋊Dic11—28no—
156C4⋊Dic11—48no—
157Dic11⋊C4—28no—
158Dic11⋊C4—216no—
159Dic11⋊C4—28no—
160Dic11⋊C4—216no—
161C4×Dic11—48no—
162C4×Dic11C2×D4116yes—
163C22×#126—111no—
164C22×#125—111no—
165#2664—1116no—
166C2×#400—111no—
167C2×#399D1111yescharacteristic
168C2×#398—111no—
169C2×#398—112no—
170C2×#397—111no—
171C2×#396—112no—
172C2×#389—112no—
173C2×#388—112no—
174C2×C4×C4⋊C4—111no—
175C23×C11—22no—
176C23×C11—24no—
177C23×C11—22no—
178C23×C11—22no—
179C23×C11—24no—
180C23×C11—44no—
181C23×C11—44no—
182C23×C11—44no—
183C23×C11—48no—
184C23×C11—48no—
185C23×C11C22×D412yes—
186C23×C11C2×#4114yes—
187C23×C11H(5,𝔽2)12yes—
188C23×C112−1+412yes—
189C23×C112−1+414yes—
190C23×C11C2511yescharacteristic
191C2×C4×C11—28no—
192C2×C4×C11—216no—
193C2×C4×C11—24no—
194C2×C4×C11—24no—
195C2×C4×C11—24no—
196C2×C4×C11—22no—
197C2×C4×C11—24no—
198C2×C4×C11—24no—
199C2×C4×C11—24no—
200C2×C4×C11—24no—
201C2×C4×C11—44no—
202C2×C4×C11—48no—
203C2×C4×C11C22×D414yes—
204C2×C4×C11C2×#4118yes—
205C2×Dic11—28no—
206C2×Dic11—28no—
207C2×Dic11—24no—
208C2×Dic11—22no—
209C2×Dic11—22no—
210C2×Dic11—22no—
211C2×Dic11—24no—
212C2×Dic11—22no—
213C2×Dic11—24no—
214C2×Dic11—24no—
215C2×Dic11—22no—
216C2×Dic11—24no—
217C2×Dic11—24no—
218C2×Dic11—22no—
219C2×Dic11—24no—
220C2×Dic11—24no—
221C2×Dic11—44no—
222C2×Dic11—48no—
223C2×Dic11—44no—
224C2×Dic11—48no—
225C2×Dic11C22×D414yes—
226C2×Dic11C22×D414yes—
227C2×#126—118no—
228C2×#126—228no—
229C2×#125—118no—
230C2×#125—228no—
231C22×C4⋊C4—112no—
232C22×C4⋊C4—112no—
233C22×C4⋊C4—111no—
234C22×C4⋊C4—111no—
235C22×C4⋊C4—111no—
236C22×C4⋊C4—112no—
237C22×C4⋊C4—111no—
238C22×C4⋊C4—112no—
239C22×C4⋊C4C2×D1112yes—
240C22×C4⋊C4C2×D1111yescharacteristic
241C22×C22⋊C4—111no—
242C22×C22⋊C4—111no—
243C22×C22⋊C4—112no—
244C22×C22⋊C4C2×D1112yes—
245C22×C22⋊C4C2×D1111yescharacteristic
246C22×C42—111no—
247C22×C42—112no—
248#400—118no—
249#399C2×D1118yes—
250#398—1116no—
251#398—118no—
252#397—118no—
253#396—1116no—
254#389—1116no—
255#388—1116no—
256C4×C4⋊C4—118no—
257C2×#95—112no—
258C2×#95—112no—
259C2×#95—111no—
260C2×#95—112no—
261C2×#95—112no—
262C2×#95C2×D1111yescharacteristic
263C22×C11—42no—
264C22×C11—42no—
265C22×C11—42no—
266C22×C11—44no—
267C22×C11—44no—
268C22×C11—88no—
269C22×C11#53214yes—
270C22×C11#53614yes—
271C22×C11#53714yes—
272C22×C11#53918yes—
273C22×C11#54118yes—
274C22×C11C23×D411yescharacteristic
275C22×C11C22×#4112yes—
276C22×C11C2×H(5,𝔽2)11yescharacteristic
277C22×C11C2×2−1+411yescharacteristic
278C22×C11C2×2−1+412yes—
279C4×C11—24no—
280C4×C11—28no—
281C4×C11—42no—
282C4×C11—44no—
283Dic11—24no—
284Dic11—24no—
285Dic11—42no—
286Dic11—44no—
287Dic11—42no—
288Dic11—44no—
289C25C22×D1111yescharacteristic
290C23×C4—111no—
291C23×C4—111no—
292C23×C4—111no—
293C23×C4—112no—
294C23×C4—111no—
295C23×C4—112no—
296C23×C4C22×D1111yescharacteristic
297C23×C4C22×D1112yes—
298C23×C4C22×D1111yescharacteristic
299C23×C4C22×D1112yes—
300#126—2232no—
301#125—2232no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—118no—
305C2×C4⋊C4—118no—
306C2×C4⋊C4—114no—
307C2×C4⋊C4—114no—
308C2×C4⋊C4—114no—
309C2×C4⋊C4—114no—
310C2×C4⋊C4—118no—
311C2×C4⋊C4—118no—
312C2×C4⋊C4—228no—
313C2×C4⋊C4—228no—
314C2×C4⋊C4—224no—
315C2×C4⋊C4—224no—
316C2×C4⋊C4—224no—
317C2×C4⋊C4—224no—
318C2×C4⋊C4—228no—
319C2×C4⋊C4—444no—
320C2×C4⋊C4C22×D1114yes—
321C2×C4⋊C4C22×D1118yes—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—28no—
324C2×C22⋊C4—114no—
325C2×C22⋊C4—114no—
326C2×C22⋊C4—118no—
327C2×C22⋊C4—224no—
328C2×C22⋊C4—224no—
329C2×C22⋊C4—444no—
330C2×C22⋊C4C22×D1114yes—
331C2×C22⋊C4C22×D1118yes—
332C2×C42—118no—
333C2×C42—114no—
334C2×C42—118no—
335C2×C42—444no—
336#95—114no—
337#95—118no—
338#95—118no—
339#95—118no—
340#95—118no—
341#95C22×D1114yes—
342C2×C11—82no—
343C2×C11#266418yescomplemented
344C2×C11C2×#53211yescharacteristic
345C2×C11C2×#53611yescharacteristic
346C2×C11C2×#53711yescharacteristic
347C2×C11C2×#53912yes—
348C2×C11C2×#54112yes—
349C24—22no—
350C24—24no—
351C24—22no—
352C24—22no—
353C24—24no—
354C24C23×D1112yes—
355C24C23×D1111yescharacteristic
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—114no—
365C22×C4—114no—
366C22×C4—112no—
367C22×C4—112no—
368C22×C4—114no—
369C22×C4—112no—
370C22×C4—112no—
371C22×C4—114no—
372C22×C4—224no—
373C22×C4—222no—
374C22×C4—222no—
375C22×C4—222no—
376C22×C4—222no—
377C22×C4—224no—
378C22×C4—224no—
379C22×C4—222no—
380C22×C4—224no—
381C22×C4—224no—
382C22×C4—224no—
383C22×C4—222no—
384C22×C4—224no—
385C22×C4—224no—
386C22×C4D4×D1114yes—
387C22×C4#434118yes—
388C22×C4C23×D1112yes—
389C22×C4C23×D1114yes—
390C22×C4C23×D1112yes—
391C22×C4C23×D1114yes—
392C4⋊C4—28no—
393C4⋊C4—216no—
394C4⋊C4—2216no—
395C4⋊C4—2216no—
396C4⋊C4—228no—
397C4⋊C4—228no—
398C4⋊C4—228no—
399C4⋊C4—228no—
400C4⋊C4—2216no—
401C4⋊C4—448no—
402C22⋊C4—28no—
403C22⋊C4—216no—
404C22⋊C4—228no—
405C22⋊C4—228no—
406C22⋊C4—448no—
407C42—1116no—
408C42—448no—
409C11C2×#266411yescharacteristic, complemented, Sylow
410C23—22no—
411C23—24no—
412C23—22no—
413C23—22no—
414C23—24no—
415C23—44no—
416C23—48no—
417C23—44no—
418C23—44no—
419C23—48no—
420C23C2×D4×D1112yes—
421C23C2×#434114yes—
422C23#6704012yes—
423C23#6704212yes—
424C23#6704314yes—
425C23C24×D1111yescharacteristic
426C2×C4—28no—
427C2×C4—24no—
428C2×C4—216no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—44no—
437C2×C4—48no—
438C2×C4—114no—
439C2×C4—114no—
440C2×C4—228no—
441C2×C4—228no—
442C2×C4—224no—
443C2×C4—222no—
444C2×C4—222no—
445C2×C4—222no—
446C2×C4—222no—
447C2×C4—224no—
448C2×C4—224no—
449C2×C4—222no—
450C2×C4—224no—
451C2×C4—224no—
452C2×C4—224no—
453C2×C4—222no—
454C2×C4—224no—
455C2×C4—224no—
456C2×C4—444no—
457C2×C4—448no—
458C2×C4—444no—
459C2×C4—448no—
460C2×C4C2×D4×D1114yes—
461C2×C4C2×#434118yes—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—44no—
466C22—44no—
467C22—88no—
468C22#1062315618yes—
469C22#1062316014yes—
470C22#1062317914yes—
471C22#1062319618yes—
472C22#1062324714yes—
473C22C22×D4×D1111yescharacteristic
474C22C22×#434112yes—
475C22C2×#6704011yescharacteristic
476C22C2×#6704211yescharacteristic
477C22C2×#6704312yes—
478C4—24no—
479C4—28no—
480C4—42no—
481C4—44no—
482C4—224no—
483C4—224no—
484C4—442no—
485C4—442no—
486C4—444no—
487C4—444no—
488C2—82no—
489C2#1307478018yesdirect product factor
490C2C2×#1062324711yescharacteristic
491C2C2×#1062317911yescharacteristic
492C2C2×#1062316011yescharacteristic
493C2C2×#1062319612yes—
494C2C2×#1062315612yes—
subgroupG/H
centerC24C23×D11
commutator, cosocleC23×C11C25
FrattiniC23C24×D11
FittingC2×C11×#399C2
socleC24×C11C24
prime radicalC25×C11C23
hypercenterC2×#399D11
hypocenterC11C2×#2664
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
2. C22×#66943C2semidirect
3. #13074780C2direct (N × H)
4. C2×#10622595C2semidirect
5. C2×#10622274C2semidirect
6. C2×#10622593C2semidirect
9. C2×#10622598C2semidirect
11. C2×#10622262C2semidirect
31. C2×#66943C22semidirect
41. #10622598C22semidirect
42. #10622595C22semidirect
43. #10622593C22semidirect
47. #10622274C22semidirect
48. #10622262C22semidirect
343. C2×C11#2664semidirect
409. C11C2×#2664semidirect
489. C2#13074780direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#669322
C22×#669432
#130747802
C2×#106225952
C2×#106222742
C2×#106225932
C2×#106225662
C2×#106225612
C2×#106225982
C2×#106225642
C2×#106222622
C2×C11×#3992
C2×#266411