Finite Group Databasebeta

← #14788844

Subgroups — #14788844

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×#10622644C212yesmaximal, complemented
3C2×#10622663C212yesmaximal
4#13069862C212yesmaximal, complemented
5#13069715C212yesmaximal, complemented
6#13069539C212yesmaximal, complemented
7#13068596C211yesmaximal, characteristic
8#13068550C212yesmaximal
9C4×C11×C22⋊Q8C211yesmaximal, characteristic
10C23×Dic22C2212yes—
11C2×C11×C22⋊Q8C2211yescharacteristic
12C2×C4×Q8×C11C2211yescharacteristic
13C22×C42×C11C2211yescharacteristic
14C22×#4326C2214yes—
15C22×C4⋊Dic11—21no—
16C22×C4⋊Dic11—21no—
17C2×#66922—21no—
18C2×#66922—21no—
19C2×#66922—28no—
20C2×#66922C2212yes—
21C2×#66922C2212yes—
22C11×#391C2211yescharacteristic
23C11×#389C2211yescharacteristic
24C11×#387—21no—
25C11×#385—21no—
26C11×#380C2211yescharacteristic
27C4×C11×C4⋊C4—21no—
28C4×C11×C4⋊C4C2211yescharacteristic
29C4×C11×C22⋊C4—21no—
30C2×C22⋊(C11⋊C8)C2212yes—
31#10622663—28no—
32#10622644—28no—
33C2×#66874C2214yes—
34C2×#66874C2214yes—
35C2×#66874C2214yes—
36C2×C4⋊(C11⋊C8)C2212yes—
37#10622162C2212yes—
38C22×Q8×C11C2311yescharacteristic
39C23×C4×C11D411yescharacteristic
40C23×C4×C11D411yescharacteristic
41C23×C4×C11C2311yescharacteristic
42C23×Dic11—22no—
43C22×Dic22—22no—
44C22×Dic22—28no—
45C22×Dic22—24no—
46C22×Dic22—24no—
47C22×Dic22—44no—
48C22×Dic22C2314yes—
49C11×C22⋊Q8—22no—
50C11×C22⋊Q8D414yes—
51C4×Q8×C11—24no—
52C2×C11×C4⋊C4—21no—
53C2×C11×C4⋊C4—21no—
54C2×C11×C4⋊C4—21no—
55C2×C11×C4⋊C4C2311yescharacteristic
56C2×C11×C4⋊C4C2311yescharacteristic
57C2×C11×C4⋊C4C2311yescharacteristic
58C2×C11×C22⋊C4—21no—
59C2×C11×C22⋊C4—21no—
60C2×C42×C11—24no—
61C2×C42×C11—21no—
62C2×C42×C11—21no—
63C2×C42×C11D411yescharacteristic
64C2×C42×C11D411yescharacteristic
65C2×C42×C11C2311yescharacteristic
66C2×C42×C11C2311yescharacteristic
67C2×#4326—28no—
68C2×#4326—28no—
69C2×#4326—44no—
70C2×C4⋊Dic11—28no—
71C2×C4⋊Dic11—28no—
72C2×C4⋊Dic11—22no—
73C2×C4⋊Dic11—22no—
74C2×C4⋊Dic11—22no—
75C2×C4⋊Dic11—22no—
76C22×C11⋊C8C2314yes—
77#66922—28no—
78#66922—28no—
79#66922—28no—
80#66922—24no—
81#66922D418yes—
82C11×#95—21no—
83C11×#95—21no—
84C11×#95—21no—
85C11×#95—21no—
86C11×#95—21no—
87C22⋊(C11⋊C8)—24no—
88#66874—28no—
89#66874—28no—
90#66874—44no—
91C4⋊(C11⋊C8)—24no—
92#24212—111nomaximal, Sylow
93C24×C11C2×D411yescharacteristic
94C2×Q8×C11—21no—
95C2×Q8×C11—22no—
96C2×Q8×C11—22no—
97C2×Q8×C11C2×D412yes—
98C22×C4×C11—22no—
99C22×C4×C11—22no—
100C22×C4×C11—22no—
101C22×C4×C11—22no—
102C22×C4×C11—22no—
103C22×C4×C11—22no—
104C22×C4×C11—22no—
105C22×C4×C11—21no—
106C22×C4×C11—21no—
107C22×C4×C11—21no—
108C22×C4×C11—21no—
109C22×C4×C11Q1614yescomplemented
110C22×C4×C11C2×D411yescharacteristic
111C22×C4×C11C2×D411yescharacteristic
112C22×C4×C11C2×D412yes—
113C22×C4×C11C2×D411yescharacteristic
114C22×C4×C11C2×D411yescharacteristic
115C22×C4×C11C2×D411yescharacteristic
116C22×C4×C11#4112yes—
117C22×C4×C11C2411yescharacteristic
118C22×Dic11—28no—
119C22×Dic11—24no—
120C22×Dic11—44no—
121C22×Dic11—44no—
122C2×Dic22—28no—
123C2×Dic22—24no—
124C2×Dic22—24no—
125C2×Dic22—24no—
126C2×Dic22—24no—
127C2×Dic22—24no—
128C2×Dic22—416no—
129C2×Dic22—416no—
130C2×Dic22—44no—
131C2×Dic22—44no—
132C2×Dic22C2×D418yes—
133C11×C4⋊C4—24no—
134C11×C4⋊C4—21no—
135C11×C4⋊C4—22no—
136C11×C4⋊C4—22no—
137C11×C4⋊C4—42no—
138C11×C4⋊C4—42no—
139C11×C4⋊C4C2×D412yes—
140C11×C22⋊C4—24no—
141C11×C22⋊C4—42no—
142C42×C11—22no—
143C42×C11—22no—
144C42×C11—22no—
145C42×C11—42no—
146C42×C11—42no—
147C42×C11C2×D414yes—
148#4326—416no—
149C4⋊Dic11—28no—
150C4⋊Dic11—28no—
151C4⋊Dic11—28no—
152C4⋊Dic11—28no—
153C2×C11⋊C8—24no—
154C2×C11⋊C8—24no—
155C2×C11⋊C8—24no—
156C22×C4⋊Q8—111no—
157C2×#462—112no—
158C2×#451—112no—
159C4×C22⋊Q8D1111yescharacteristic
160#2043—112no—
161#2016—112no—
162#1981—112no—
163#1797—111no—
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
173Q8×C11—24no—
174Q8×C11—42no—
175C2×C4×C11—28no—
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—21no—
183C2×C4×C11—21no—
184C2×C4×C11—24no—
185C2×C4×C11—24no—
186C2×C4×C11—28no—
187C2×C4×C11—22no—
188C2×C4×C11—22no—
189C2×C4×C11—41no—
190C2×C4×C11—41no—
191C2×C4×C11—41no—
192C2×C4×C11—41no—
193C2×C4×C11—41no—
194C2×C4×C11—41no—
195C2×C4×C11—41no—
196C2×C4×C11—41no—
197C2×C4×C11—41no—
198C2×C4×C11—41no—
199C2×C4×C11#12012yes—
200C2×C4×C11#12012yes—
201C2×C4×C11#12112yes—
202C2×C4×C11C2×Q1612yes—
203C2×C4×C11C2×Q1612yes—
204C2×C4×C11#13712yes—
205C2×C4×C11#13712yes—
206C2×C4×C11C22×D411yescharacteristic
207C2×C4×C11C22×D412yes—
208C2×C4×C11C2×#4111yescharacteristic
209C2×C4×C11H(5,𝔽2)12yes—
210C2×Dic11—28no—
211C2×Dic11—416no—
212C2×Dic11—44no—
213C2×Dic11—44no—
214Dic22—28no—
215Dic22—28no—
216Dic22—416no—
217Dic22—44no—
218Dic22—44no—
219C11⋊C8—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⋊Q8C2×D1111yescharacteristic
228C2×C4×Q8C2×D1111yescharacteristic
229C22×C4⋊C4—221no—
230C22×C4⋊C4—221no—
231C22×C42C2×D1111yescharacteristic
232#462—228no—
233#451—228no—
234C2×C4⋊C8—112no—
235C2×#103—114no—
236C2×#103—114no—
237C2×#103—114no—
238C2×C22⋊C8—112no—
239#391C2×D1111yescharacteristic
240#389C2×D1111yescharacteristic
241#387—21no—
242#385—21no—
243#380C2×D1111yescharacteristic
244C4×C4⋊C4—21no—
245C4×C4⋊C4C2×D1111yescharacteristic
246C4×C22⋊C4—21no—
247#336—112no—
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—42no—
267C4×C11—42no—
268C4×C11—81no—
269C4×C11—81no—
270C4×C11#45114yescomplemented
271C4×C11D4212yes—
272C4×C11#54612yes—
273Dic11—48no—
274C22×Q8—114no—
275C22×Q8—228no—
276C22×Q8—222no—
277C22×Q8—224no—
278C22×Q8—224no—
279C22×Q8—444no—
280C22×Q8C22×D1111yescharacteristic
281C23×C4—222no—
282C23×C4C22⋊D1111yescharacteristic
283C23×C4C22⋊D1111yescharacteristic
284C23×C4C22×D1111yescharacteristic
285C2×Q16—228no—
286C2×Q16—228no—
287C2×Q16—444no—
288C22×C8—114no—
289C4⋊Q8—118no—
290C4⋊Q8—228no—
291C4⋊Q8—228no—
292C4⋊Q8—228no—
293C4⋊Q8—224no—
294C22⋊Q8—22no—
295C22⋊Q8D4414yes—
296C4×Q8—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
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⋊C4C22×D1111yescharacteristic
307C2×C4⋊C4C22×D1111yescharacteristic
308C2×C4⋊C4C22×D1111yescharacteristic
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C42—24no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42C22⋊D1111yescharacteristic
315C2×C42C22⋊D1111yescharacteristic
316C2×C42C22×D1111yescharacteristic
317C2×C42C22×D1111yescharacteristic
318C4⋊C8—224no—
319#103—228no—
320#103—228no—
321#103—444no—
322C22⋊C8—224no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
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×C22⋊D1111yescharacteristic
337C2×Q8—21no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—118no—
341C2×Q8—228no—
342C2×Q8—224no—
343C2×Q8—224no—
344C2×Q8—224no—
345C2×Q8—224no—
346C2×Q8—224no—
347C2×Q8—4416no—
348C2×Q8—4416no—
349C2×Q8—444no—
350C2×Q8—444no—
351C2×Q8C2×D4412yes—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—228no—
364C22×C4—224no—
365C22×C4—444no—
366C22×C4—444no—
367C22×C4#432614yescomplemented
368C22×C4C2×D4412yes—
369C22×C4#433912yes—
370C22×C4C2×C22⋊D1111yescharacteristic
371C22×C4C2×C22⋊D1111yescharacteristic
372C22×C4C2×C22⋊D1111yescharacteristic
373C22×C4C2×C22⋊D1111yescharacteristic
374C22×C4C2×C22⋊D1111yescharacteristic
375C22×C4C23×D1111yescharacteristic
376Q16—4416no—
377C2×C8—224no—
378C2×C8—224no—
379C2×C8—224no—
380C4⋊C4—24no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—228no—
387C4⋊C4—228no—
388C4⋊C4—228no—
389C4⋊C4—228no—
390C4⋊C4C2×D4412yes—
391C22⋊C4—24no—
392C22⋊C4—42no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—42no—
397C42—42no—
398C42D4×D1114yes—
399C11#2421211yescharacteristic, complemented, Sylow
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23#6698312yes—
405C23C2×#432612yes—
406C23#6700412yes—
407C23C22×C22⋊D1111yescharacteristic
408Q8—24no—
409Q8—42no—
410Q8—228no—
411Q8—228no—
412Q8—4416no—
413Q8—444no—
414Q8—444no—
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—24no—
425C2×C4—24no—
426C2×C4—28no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—228no—
440C2×C4—4416no—
441C2×C4—444no—
442C2×C4—444no—
443C2×C4#6698312yes—
444C2×C4#6699012yes—
445C2×C4#6699012yes—
446C2×C4C2×#432612yes—
447C2×C4C2×#432612yes—
448C2×C4#6700412yes—
449C2×C4#6700412yes—
450C2×C4C22×D4411yescharacteristic
451C2×C4C2×#433911yescharacteristic
452C2×C4C2×D4×D1112yes—
453C2×C4#6704212yes—
454C8—444no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22#1062264414yescomplemented
462C22C2×#6698311yescharacteristic
463C22C2×#6699011yescharacteristic
464C22C22×#432611yescharacteristic
465C22C2×#6700411yescharacteristic
466C22C2×#6700411yescharacteristic
467C22C2×#6700411yescharacteristic
468C22#1062336111yescharacteristic
469C4—22no—
470C4—22no—
471C4—22no—
472C4—22no—
473C4—42no—
474C4—42no—
475C4—81no—
476C4—81no—
477C4—448no—
478C4#1062266314yescomplemented
479C4D4×D4412yes—
480C4#1062308112yes—
481C2—24no—
482C2#1307197211yescharacteristic
483C2C2×#1062266311yescharacteristic
484C2#1307784911yescharacteristic
485C2C2×#1062264411yescharacteristic
486C2#1307810311yescharacteristic
487C2#1307956111yescharacteristic
488C2#1307964011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC4×C11×C22⋊Q8C2
socleC23×C11C22×D4
prime radicalC24×C11C2×D4
hypercenterC4×C22⋊Q8D11
hypocenterC11#24212
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#10622644C2semidirect
4. #13069862C2semidirect
5. #13069715C2semidirect
6. #13069539C2semidirect
109. C22×C4×C11Q16semidirect
254. C22×C11#462semidirect
270. C4×C11#451semidirect
367. C22×C4#4326semidirect
399. C11#24212semidirect
461. C22#10622644semidirect
478. C4#10622663semidirect

Maximal subgroups

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

subgroupindex
C22×#669222
C2×#106226442
C2×#106226632
#130698622
#130697152
#130695392
#130685962
#130685502
C4×C11×C22⋊Q82
#2421211