Finite Group Databasebeta

← #14788845

Subgroups — #14788845

Order 2816.

1430 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66922C211yesmaximal, characteristic
2C2×#10622462C212yesmaximal, complemented
3C2×#10622352C212yesmaximal
4C4×#66977C211yesmaximal, characteristic
5#13069863C212yesmaximal, complemented
6#13069718C212yesmaximal, complemented
7#13069538C212yesmaximal, complemented
8#13068552C212yesmaximal
9C11×#1797C211yesmaximal, characteristic
10C23×Dic22C2212yes—
11C22×C42×C11C2211yescharacteristic
12C22×C4⋊Dic11—21no—
13C22×C4⋊Dic11—21no—
14C2×#66977C2211yescharacteristic
15C22×Dic44C2214yes—
16C2×#66922—21no—
17C2×#66922—21no—
18C2×#66922—28no—
19C2×#66922C2212yes—
20C2×#66922C2212yes—
21C2×C4×Dic22C2211yescharacteristic
22C2×C11×C4⋊C8C2212yes—
23C2×C11×C22⋊C8C2212yes—
24#10622825C2211yescharacteristic
25#10622823—21no—
26C4×C22⋊Dic11—21no—
27C2×#66880C2214yes—
28C2×#66880C2214yes—
29C2×#66880C2214yes—
30C4×C4⋊Dic11C2211yescharacteristic
31#10622546—21no—
32C4×Dic11⋊C4—21no—
33#10622542C2211yescharacteristic
34#10622541C2211yescharacteristic
35#10622462—28no—
36#10622352—28no—
37C11×#336C2212yes—
38C23×C4×C11D411yescharacteristic
39C23×C4×C11D411yescharacteristic
40C23×C4×C11C2311yescharacteristic
41C23×Dic11—22no—
42C22×Dic22—22no—
43C22×Dic22—28no—
44C22×Dic22—24no—
45C22×Dic22—24no—
46C22×Dic22—44no—
47C22×Dic22C2311yescharacteristic
48C22×Dic22C2314yes—
49C22×C8×C11C2314yes—
50C2×C42×C11—24no—
51C2×C42×C11D411yescharacteristic
52C2×C42×C11D411yescharacteristic
53C2×C42×C11C2311yescharacteristic
54C2×C42×C11C2311yescharacteristic
55C2×C22⋊Dic11—21no—
56C2×C22⋊Dic11—21no—
57C2×C4⋊Dic11—28no—
58C2×C4⋊Dic11—28no—
59C2×C4⋊Dic11—22no—
60C2×C4⋊Dic11—22no—
61C2×C4⋊Dic11—22no—
62C2×C4⋊Dic11—22no—
63C2×C4⋊Dic11C2311yescharacteristic
64C2×C4⋊Dic11C2311yescharacteristic
65C2×C4⋊Dic11C2311yescharacteristic
66#66977—22no—
67#66977D414yes—
68C2×Dic11⋊C4—21no—
69C2×Dic11⋊C4—21no—
70C2×Dic11⋊C4—21no—
71C2×C4×Dic11—21no—
72C2×C4×Dic11—21no—
73C2×Dic44—28no—
74C2×Dic44—28no—
75C2×Dic44—44no—
76#66922—28no—
77#66922—28no—
78#66922—28no—
79#66922—24no—
80#66922D418yes—
81C4×Dic22—24no—
82C11×C4⋊C8—24no—
83C11×C22⋊C8—24no—
84C11⋊#95—21no—
85C11⋊#95—21no—
86C11⋊#95—21no—
87C11⋊#95—21no—
88C11⋊#95—21no—
89#66880—28no—
90#66880—28no—
91#66880—44no—
92#24212—111nomaximal, Sylow
93C24×C11C2×D411yescharacteristic
94C22×C4×C11—22no—
95C22×C4×C11—22no—
96C22×C4×C11—22no—
97C22×C4×C11—22no—
98C22×C4×C11—22no—
99C22×C4×C11—22no—
100C22×C4×C11—22no—
101C22×C4×C11Q1614yescomplemented
102C22×C4×C11C2×D411yescharacteristic
103C22×C4×C11C2×D411yescharacteristic
104C22×C4×C11C2×D412yes—
105C22×C4×C11C2×D411yescharacteristic
106C22×C4×C11C2×D411yescharacteristic
107C22×C4×C11C2×D411yescharacteristic
108C22×C4×C11#4112yes—
109C22×C4×C11C2411yescharacteristic
110C22×Dic11—28no—
111C22×Dic11—21no—
112C22×Dic11—21no—
113C22×Dic11—24no—
114C22×Dic11—21no—
115C22×Dic11—21no—
116C22×Dic11—44no—
117C22×Dic11—44no—
118C2×Dic22—28no—
119C2×Dic22—24no—
120C2×Dic22—22no—
121C2×Dic22—24no—
122C2×Dic22—24no—
123C2×Dic22—21no—
124C2×Dic22—24no—
125C2×Dic22—22no—
126C2×Dic22—24no—
127C2×Dic22—416no—
128C2×Dic22—416no—
129C2×Dic22—44no—
130C2×Dic22—44no—
131C2×Dic22C2×D418yes—
132C2×Dic22C2×D412yes—
133C2×C8×C11—24no—
134C2×C8×C11—24no—
135C2×C8×C11—24no—
136C42×C11—22no—
137C42×C11—22no—
138C42×C11—22no—
139C42×C11C2×D414yes—
140C22⋊Dic11—24no—
141C22⋊Dic11—42no—
142C4⋊Dic11—28no—
143C4⋊Dic11—28no—
144C4⋊Dic11—28no—
145C4⋊Dic11—28no—
146C4⋊Dic11—22no—
147C4⋊Dic11—21no—
148C4⋊Dic11—22no—
149C4⋊Dic11C2×D412yes—
150Dic11⋊C4—24no—
151Dic11⋊C4—42no—
152Dic11⋊C4—42no—
153C4×Dic11—42no—
154C4×Dic11—42no—
155Dic44—416no—
156C22×C4⋊Q8—111no—
157C2×#462—112no—
158C2×#451—112no—
159C4×C22⋊Q8—111no—
160#2043—112no—
161#2016—112no—
162#1981—112no—
163#1797D1111yescharacteristic
164#1787—112no—
165C23×C11—21no—
166C23×C11—21no—
167C23×C11—21no—
168C23×C11—21no—
169C23×C11#12112yes—
170C23×C11C2×Q1612yes—
171C23×C11#13712yes—
172C23×C11C22×D411yescharacteristic
173C2×C4×C11—28no—
174C2×C4×C11—21no—
175C2×C4×C11—21no—
176C2×C4×C11—21no—
177C2×C4×C11—21no—
178C2×C4×C11—21no—
179C2×C4×C11—21no—
180C2×C4×C11—21no—
181C2×C4×C11—21no—
182C2×C4×C11—24no—
183C2×C4×C11—24no—
184C2×C4×C11—28no—
185C2×C4×C11#12012yes—
186C2×C4×C11#12012yes—
187C2×C4×C11#12112yes—
188C2×C4×C11C2×Q1612yes—
189C2×C4×C11C2×Q1612yes—
190C2×C4×C11#13712yes—
191C2×C4×C11#13712yes—
192C2×C4×C11C22×D411yescharacteristic
193C2×C4×C11C22×D412yes—
194C2×C4×C11C2×#4111yescharacteristic
195C2×C4×C11H(5,𝔽2)12yes—
196C2×Dic11—28no—
197C2×Dic11—22no—
198C2×Dic11—22no—
199C2×Dic11—416no—
200C2×Dic11—41no—
201C2×Dic11—44no—
202C2×Dic11—44no—
203C2×Dic11—41no—
204C2×Dic11—41no—
205C2×Dic11—41no—
206C2×Dic11—41no—
207C2×Dic11—41no—
208C2×Dic11—41no—
209C2×Dic11—41no—
210C2×Dic11—41no—
211C2×Dic11—41no—
212Dic22—24no—
213Dic22—28no—
214Dic22—28no—
215Dic22—416no—
216Dic22—44no—
217Dic22—42no—
218Dic22—44no—
219C8×C11—44no—
220C23×Q8—112no—
221C22×Q16—114no—
222C2×C4⋊Q8—112no—
223C2×C4⋊Q8—112no—
224C2×C4⋊Q8—228no—
225C2×C4⋊Q8—221no—
226C2×C4⋊Q8—221no—
227C2×C22⋊Q8—111no—
228C2×C4×Q8—111no—
229C22×C4⋊C4—221no—
230C22×C4⋊C4—221no—
231C22×C42C2×D1111yescharacteristic
232#462—228no—
233#451—228no—
234C2×C4⋊C8C2×D1112yes—
235C2×#103—114no—
236C2×#103—114no—
237C2×#103—114no—
238C2×C22⋊C8C2×D1112yes—
239#391—111no—
240#389—111no—
241#387—221no—
242#385—221no—
243#380—111no—
244C4×C4⋊C4—111no—
245C4×C4⋊C4—221no—
246C4×C22⋊C4—221no—
247#336C2×D1112yes—
248C22×C11—22no—
249C22×C11—22no—
250C22×C11—22no—
251C22×C11—22no—
252C22×C11—22no—
253C22×C11—22no—
254C22×C11#46214yescomplemented
255C22×C11C2×#12011yescharacteristic
256C22×C11C2×#12111yescharacteristic
257C22×C11#53511yescharacteristic
258C22×C11C22×Q1611yescharacteristic
259C22×C11C2×#13711yescharacteristic
260C22×C11C2×#13711yescharacteristic
261C22×C11C2×#13711yescharacteristic
262C4×C11—22no—
263C4×C11—22no—
264C4×C11—22no—
265C4×C11—22no—
266C4×C11#45114yescomplemented
267C4×C11D4212yes—
268C4×C11#54612yes—
269Dic11—48no—
270Dic11—42no—
271Dic11—42no—
272Dic11—81no—
273Dic11—81no—
274C22×Q8—111no—
275C22×Q8—114no—
276C22×Q8—228no—
277C22×Q8—222no—
278C22×Q8—224no—
279C22×Q8—224no—
280C22×Q8—444no—
281C23×C4—222no—
282C23×C4D4411yescharacteristic
283C23×C4D4411yescharacteristic
284C23×C4C22×D1111yescharacteristic
285C2×Q16—228no—
286C2×Q16—228no—
287C2×Q16—444no—
288C22×C8C22×D1114yes—
289C4⋊Q8—118no—
290C4⋊Q8—228no—
291C4⋊Q8—228no—
292C4⋊Q8—228no—
293C4⋊Q8—224no—
294C22⋊Q8—114no—
295C22⋊Q8—222no—
296C4×Q8—224no—
297C2×C4⋊C4—111no—
298C2×C4⋊C4—111no—
299C2×C4⋊C4—111no—
300C2×C4⋊C4—228no—
301C2×C4⋊C4—228no—
302C2×C4⋊C4—222no—
303C2×C4⋊C4—222no—
304C2×C4⋊C4—222no—
305C2×C4⋊C4—222no—
306C2×C4⋊C4—221no—
307C2×C4⋊C4—221no—
308C2×C4⋊C4—221no—
309C2×C22⋊C4—221no—
310C2×C22⋊C4—221no—
311C2×C42—24no—
312C2×C42—221no—
313C2×C42—221no—
314C2×C42D4411yescharacteristic
315C2×C42D4411yescharacteristic
316C2×C42C22×D1111yescharacteristic
317C2×C42C22×D1111yescharacteristic
318C4⋊C8—24no—
319#103—228no—
320#103—228no—
321#103—444no—
322C22⋊C8—24no—
323#95—221no—
324#95—221no—
325#95—221no—
326#95—221no—
327#95—221no—
328C2×C11—24no—
329C2×C11#240811yescharacteristic
330C2×C11C2×#45111yescharacteristic
331C2×C11#300411yescharacteristic
332C2×C11C2×#46211yescharacteristic
333C2×C11#303911yescharacteristic
334C2×C11#320911yescharacteristic
335C2×C11#321811yescharacteristic
336C24C2×D4411yescharacteristic
337C2×Q8—118no—
338C2×Q8—112no—
339C2×Q8—228no—
340C2×Q8—224no—
341C2×Q8—221no—
342C2×Q8—224no—
343C2×Q8—222no—
344C2×Q8—224no—
345C2×Q8—224no—
346C2×Q8—222no—
347C2×Q8—224no—
348C2×Q8—4416no—
349C2×Q8—4416no—
350C2×Q8—444no—
351C2×Q8—444no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—228no—
360C22×C4—221no—
361C22×C4—221no—
362C22×C4—224no—
363C22×C4—221no—
364C22×C4—221no—
365C22×C4—444no—
366C22×C4—444no—
367C22×C4Dic4414yescomplemented
368C22×C4C2×D4411yescharacteristic
369C22×C4C2×D4411yescharacteristic
370C22×C4C2×D4411yescharacteristic
371C22×C4C2×D4411yescharacteristic
372C22×C4C2×D4411yescharacteristic
373C22×C4D4×D1112yes—
374C22×C4#434312yes—
375C22×C4C23×D1111yescharacteristic
376Q16—4416no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C4⋊C4—112no—
381C4⋊C4—228no—
382C4⋊C4—228no—
383C4⋊C4—228no—
384C4⋊C4—228no—
385C4⋊C4—224no—
386C4⋊C4—221no—
387C4⋊C4—222no—
388C4⋊C4—222no—
389C4⋊C4—442no—
390C4⋊C4—442no—
391C22⋊C4—224no—
392C22⋊C4—442no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—442no—
397C42—442no—
398C42D4×D1114yes—
399C11#2421211yescharacteristic, complemented, Sylow
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23#6694812yes—
405C23C2×Dic4412yes—
406C23#6696212yes—
407C23C22×D4411yescharacteristic
408Q8—224no—
409Q8—228no—
410Q8—228no—
411Q8—4416no—
412Q8—444no—
413Q8—444no—
414Q8—442no—
415C2×C4—28no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—28no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—228no—
428C2×C4—222no—
429C2×C4—222no—
430C2×C4—4416no—
431C2×C4—441no—
432C2×C4—441no—
433C2×C4—441no—
434C2×C4—444no—
435C2×C4—444no—
436C2×C4—441no—
437C2×C4—441no—
438C2×C4—441no—
439C2×C4—441no—
440C2×C4—441no—
441C2×C4—441no—
442C2×C4—441no—
443C2×C4#6693512yes—
444C2×C4#6693512yes—
445C2×C4#6694812yes—
446C2×C4C2×Dic4412yes—
447C2×C4C2×Dic4412yes—
448C2×C4#6696212yes—
449C2×C4#6696212yes—
450C2×C4C2×D4×D1111yescharacteristic
451C2×C4C2×D4×D1112yes—
452C2×C4C2×#434311yescharacteristic
453C2×C4#6704212yes—
454C8—44no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22#1062246214yescomplemented
462C22C2×#6693511yescharacteristic
463C22C2×#6694811yescharacteristic
464C22#1062305611yescharacteristic
465C22C22×Dic4411yescharacteristic
466C22C2×#6696211yescharacteristic
467C22C2×#6696211yescharacteristic
468C22C2×#6696211yescharacteristic
469C4—22no—
470C4—22no—
471C4—22no—
472C4—22no—
473C4—442no—
474C4—448no—
475C4—442no—
476C4—881no—
477C4—881no—
478C4#1062235214yescomplemented
479C4#1062314012yes—
480C4#1062316112yes—
481C2—24no—
482C2#1307197311yescharacteristic
483C2C2×#1062235211yescharacteristic
484C2#1307785311yescharacteristic
485C2C2×#1062246211yescharacteristic
486C2#1307810411yescharacteristic
487C2#1307956611yescharacteristic
488C2#1307964511yescharacteristic
subgroupG/H
centerC23C22×D44
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC11×#1797C2
socleC23×C11C22×D4
prime radicalC24×C11C2×D4
hypercenter#1797D11
hypocenterC11#24212
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
2. C2×#10622462C2semidirect
5. #13069863C2semidirect
6. #13069718C2semidirect
7. #13069538C2semidirect
101. C22×C4×C11Q16semidirect
254. C22×C11#462semidirect
266. C4×C11#451semidirect
367. C22×C4Dic44semidirect
399. C11#24212semidirect
461. C22#10622462semidirect
478. C4#10622352semidirect

Maximal subgroups

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

subgroupindex
C22×#669222
C2×#106224622
C2×#106223522
C4×#669772
#130698632
#130697182
#130695382
#130685522
C11×#17972
#2421211