Finite Group Databasebeta

← #14788838

Subgroups — #14788838

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
1C2×#10622703C212yesmaximal, complemented
2C2×#10622841C212yesmaximal
3C4×#66977C211yesmaximal, characteristic
4#13069859C212yesmaximal, complemented
5#13069716C212yesmaximal, complemented
6#13069540C212yesmaximal, complemented
7#13068596C211yesmaximal, characteristic
8#13068551C212yesmaximal
9C22×C11×C4⋊Q8C211yesmaximal, characteristic
10C23×Q8×C11C2212yes—
11C2×C11×C4⋊Q8—28no—
12C2×C11×C4⋊Q8—21no—
13C2×C11×C4⋊Q8—21no—
14C2×C11×C4⋊Q8C2212yes—
15C2×C11×C4⋊Q8C2212yes—
16C22×C11×C4⋊C4—21no—
17C22×C11×C4⋊C4—21no—
18C22×C42×C11C2211yescharacteristic
19C22×#4326C2214yes—
20C2×#66977C2211yescharacteristic
21C2×C4×Dic22C2211yescharacteristic
22#10622841—28no—
23C2×#66899C2214yes—
24C2×#66899C2214yes—
25C2×#66899C2214yes—
26#10622825C2211yescharacteristic
27#10622823—21no—
28C4×C22⋊Dic11—21no—
29C2×C22⋊(C11⋊C8)C2212yes—
30#10622703—28no—
31C4×C4⋊Dic11C2211yescharacteristic
32#10622546—21no—
33C4×Dic11⋊C4—21no—
34#10622542C2211yescharacteristic
35#10622541C2211yescharacteristic
36C2×C4⋊(C11⋊C8)C2212yes—
37#10622162C2212yes—
38C22×Q8×C11—28no—
39C22×Q8×C11—22no—
40C22×Q8×C11—24no—
41C22×Q8×C11—24no—
42C22×Q8×C11—44no—
43C22×Q8×C11C2314yes—
44C23×C4×C11—22no—
45C23×C4×C11D411yescharacteristic
46C23×C4×C11D411yescharacteristic
47C23×C4×C11C2311yescharacteristic
48C22×Dic22C2311yescharacteristic
49C11×C4⋊Q8—28no—
50C11×C4⋊Q8—28no—
51C11×C4⋊Q8—28no—
52C11×C4⋊Q8—24no—
53C11×C4⋊Q8D418yes—
54C2×C11×C4⋊C4—28no—
55C2×C11×C4⋊C4—28no—
56C2×C11×C4⋊C4—22no—
57C2×C11×C4⋊C4—22no—
58C2×C11×C4⋊C4—22no—
59C2×C11×C4⋊C4—22no—
60C2×C42×C11—24no—
61C2×C42×C11D411yescharacteristic
62C2×C42×C11D411yescharacteristic
63C2×C42×C11C2311yescharacteristic
64C2×C42×C11C2311yescharacteristic
65C2×C22⋊Dic11—21no—
66C2×C22⋊Dic11—21no—
67C2×#4326—28no—
68C2×#4326—28no—
69C2×#4326—44no—
70C2×C4⋊Dic11C2311yescharacteristic
71C2×C4⋊Dic11C2311yescharacteristic
72C2×C4⋊Dic11C2311yescharacteristic
73#66977—22no—
74#66977D414yes—
75C2×Dic11⋊C4—21no—
76C2×Dic11⋊C4—21no—
77C2×Dic11⋊C4—21no—
78C2×C4×Dic11—21no—
79C2×C4×Dic11—21no—
80C22×C11⋊C8C2314yes—
81C4×Dic22—24no—
82#66899—28no—
83#66899—28no—
84#66899—44no—
85C11⋊#95—21no—
86C11⋊#95—21no—
87C11⋊#95—21no—
88C11⋊#95—21no—
89C11⋊#95—21no—
90C22⋊(C11⋊C8)—24no—
91C4⋊(C11⋊C8)—24no—
92#24212—111nomaximal, Sylow
93C24×C11C2×D411yescharacteristic
94C2×Q8×C11—28no—
95C2×Q8×C11—24no—
96C2×Q8×C11—24no—
97C2×Q8×C11—24no—
98C2×Q8×C11—24no—
99C2×Q8×C11—24no—
100C2×Q8×C11—416no—
101C2×Q8×C11—416no—
102C2×Q8×C11—44no—
103C2×Q8×C11—44no—
104C2×Q8×C11C2×D418yes—
105C22×C4×C11—22no—
106C22×C4×C11—22no—
107C22×C4×C11—22no—
108C22×C4×C11—22no—
109C22×C4×C11—22no—
110C22×C4×C11—22no—
111C22×C4×C11—22no—
112C22×C4×C11—28no—
113C22×C4×C11—24no—
114C22×C4×C11—44no—
115C22×C4×C11—44no—
116C22×C4×C11Q1614yescomplemented
117C22×C4×C11C2×D411yescharacteristic
118C22×C4×C11C2×D411yescharacteristic
119C22×C4×C11C2×D412yes—
120C22×C4×C11C2×D411yescharacteristic
121C22×C4×C11C2×D411yescharacteristic
122C22×C4×C11C2×D411yescharacteristic
123C22×C4×C11#4112yes—
124C22×C4×C11C2411yescharacteristic
125C22×Dic11—21no—
126C22×Dic11—21no—
127C22×Dic11—21no—
128C22×Dic11—21no—
129C2×Dic22—22no—
130C2×Dic22—21no—
131C2×Dic22—22no—
132C2×Dic22C2×D412yes—
133C11×C4⋊C4—28no—
134C11×C4⋊C4—28no—
135C11×C4⋊C4—28no—
136C11×C4⋊C4—28no—
137C42×C11—22no—
138C42×C11—22no—
139C42×C11—22no—
140C42×C11C2×D414yes—
141C22⋊Dic11—24no—
142C22⋊Dic11—42no—
143#4326—416no—
144C4⋊Dic11—22no—
145C4⋊Dic11—21no—
146C4⋊Dic11—22no—
147C4⋊Dic11C2×D412yes—
148Dic11⋊C4—24no—
149Dic11⋊C4—42no—
150Dic11⋊C4—42no—
151C4×Dic11—42no—
152C4×Dic11—42no—
153C2×C11⋊C8—24no—
154C2×C11⋊C8—24no—
155C2×C11⋊C8—24no—
156C22×C4⋊Q8D1111yescharacteristic
157C2×#462—112no—
158C2×#451—112no—
159C4×C22⋊Q8—111no—
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—28no—
174Q8×C11—28no—
175Q8×C11—416no—
176Q8×C11—44no—
177Q8×C11—44no—
178C2×C4×C11—28no—
179C2×C4×C11—21no—
180C2×C4×C11—21no—
181C2×C4×C11—21no—
182C2×C4×C11—21no—
183C2×C4×C11—21no—
184C2×C4×C11—21no—
185C2×C4×C11—21no—
186C2×C4×C11—21no—
187C2×C4×C11—24no—
188C2×C4×C11—24no—
189C2×C4×C11—28no—
190C2×C4×C11—28no—
191C2×C4×C11—416no—
192C2×C4×C11—44no—
193C2×C4×C11—44no—
194C2×C4×C11#12012yes—
195C2×C4×C11#12012yes—
196C2×C4×C11#12112yes—
197C2×C4×C11C2×Q1612yes—
198C2×C4×C11C2×Q1612yes—
199C2×C4×C11#13712yes—
200C2×C4×C11#13712yes—
201C2×C4×C11C22×D411yescharacteristic
202C2×C4×C11C22×D412yes—
203C2×C4×C11C2×#4111yescharacteristic
204C2×C4×C11H(5,𝔽2)12yes—
205C2×Dic11—22no—
206C2×Dic11—22no—
207C2×Dic11—41no—
208C2×Dic11—41no—
209C2×Dic11—41no—
210C2×Dic11—41no—
211C2×Dic11—41no—
212C2×Dic11—41no—
213C2×Dic11—41no—
214C2×Dic11—41no—
215C2×Dic11—41no—
216C2×Dic11—41no—
217Dic22—24no—
218Dic22—42no—
219C11⋊C8—44no—
220C23×Q8C2×D1112yes—
221C22×Q16—114no—
222C2×C4⋊Q8—28no—
223C2×C4⋊Q8—21no—
224C2×C4⋊Q8—21no—
225C2×C4⋊Q8C2×D1112yes—
226C2×C4⋊Q8C2×D1112yes—
227C2×C22⋊Q8—111no—
228C2×C4×Q8—111no—
229C22×C4⋊C4—21no—
230C22×C4⋊C4—21no—
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#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#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—48no—
267C4×C11#45114yescomplemented
268C4×C11D4212yes—
269C4×C11#54612yes—
270Dic11—42no—
271Dic11—42no—
272Dic11—81no—
273Dic11—81no—
274C22×Q8—28no—
275C22×Q8—22no—
276C22×Q8—24no—
277C22×Q8—24no—
278C22×Q8—44no—
279C22×Q8—111no—
280C22×Q8C22×D1114yes—
281C23×C4—22no—
282C23×C4C22⋊D1111yescharacteristic
283C23×C4C22⋊D1111yescharacteristic
284C23×C4C22×D1111yescharacteristic
285C2×Q16—228no—
286C2×Q16—228no—
287C2×Q16—444no—
288C22×C8—114no—
289C4⋊Q8—28no—
290C4⋊Q8—28no—
291C4⋊Q8—28no—
292C4⋊Q8—24no—
293C4⋊Q8C22⋊D1118yes—
294C22⋊Q8—114no—
295C22⋊Q8—222no—
296C4×Q8—224no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—111no—
304C2×C4⋊C4—111no—
305C2×C4⋊C4—111no—
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×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—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×C22⋊D1111yescharacteristic
337C2×Q8—28no—
338C2×Q8—24no—
339C2×Q8—24no—
340C2×Q8—24no—
341C2×Q8—24no—
342C2×Q8—24no—
343C2×Q8—416no—
344C2×Q8—416no—
345C2×Q8—44no—
346C2×Q8—44no—
347C2×Q8—112no—
348C2×Q8—221no—
349C2×Q8—222no—
350C2×Q8—222no—
351C2×Q8C2×C22⋊D1118yes—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—28no—
360C22×C4—24no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—221no—
364C22×C4—221no—
365C22×C4—221no—
366C22×C4—221no—
367C22×C4#432614yescomplemented
368C22×C4D4×D1112yes—
369C22×C4#434112yes—
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—28no—
381C4⋊C4—28no—
382C4⋊C4—28no—
383C4⋊C4—28no—
384C4⋊C4—112no—
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—
398C42C2×C22⋊D1114yes—
399C11#2421211yescharacteristic, complemented, Sylow
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23#6699112yes—
405C23#6699512yes—
406C23C2×#432612yes—
407C23C22×C22⋊D1111yescharacteristic
408Q8—28no—
409Q8—28no—
410Q8—416no—
411Q8—44no—
412Q8—44no—
413Q8—224no—
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—24no—
425C2×C4—24no—
426C2×C4—28no—
427C2×C4—28no—
428C2×C4—416no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—222no—
432C2×C4—222no—
433C2×C4—441no—
434C2×C4—441no—
435C2×C4—441no—
436C2×C4—441no—
437C2×C4—441no—
438C2×C4—441no—
439C2×C4—441no—
440C2×C4—441no—
441C2×C4—441no—
442C2×C4—441no—
443C2×C4#6699112yes—
444C2×C4#6699512yes—
445C2×C4#6699512yes—
446C2×C4C2×#432612yes—
447C2×C4C2×#432612yes—
448C2×C4#6700612yes—
449C2×C4#6700612yes—
450C2×C4C2×D4×D1111yescharacteristic
451C2×C4C2×#434111yescharacteristic
452C2×C4#6703712yes—
453C2×C4C22×C22⋊D1112yes—
454C8—444no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22#1062270314yescomplemented
462C22C2×#6699111yescharacteristic
463C22#1062333611yescharacteristic
464C22C2×#6699511yescharacteristic
465C22C2×#6699511yescharacteristic
466C22C2×#6699511yescharacteristic
467C22C22×#432611yescharacteristic
468C22C2×#6700611yescharacteristic
469C4—22no—
470C4—22no—
471C4—22no—
472C4—22no—
473C4—48no—
474C4—442no—
475C4—442no—
476C4—881no—
477C4—881no—
478C4#1062284114yescomplemented
479C4D4×C22⋊D1112yes—
480C4#1062333712yes—
481C2—24no—
482C2#1307196611yescharacteristic
483C2C2×#1062284111yescharacteristic
484C2#1307785011yescharacteristic
485C2C2×#1062270311yescharacteristic
486C2#1307809911yescharacteristic
487C2#1307956411yescharacteristic
488C2#1307964311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC22×C11×C4⋊Q8C2
socleC23×C11C22×D4
prime radicalC24×C11C2×D4
hypercenterC22×C4⋊Q8D11
hypocenterC11#24212
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#10622703C2semidirect
4. #13069859C2semidirect
5. #13069716C2semidirect
6. #13069540C2semidirect
116. C22×C4×C11Q16semidirect
254. C22×C11#462semidirect
267. C4×C11#451semidirect
367. C22×C4#4326semidirect
399. C11#24212semidirect
461. C22#10622703semidirect
478. C4#10622841semidirect

Maximal subgroups

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

subgroupindex
C2×#106227032
C2×#106228412
C4×#669772
#130698592
#130697162
#130695402
#130685962
#130685512
C22×C11×C4⋊Q82
#2421211