Finite Group Databasebeta

← #14756264

Subgroups — #14756264

Order 2816.

1126 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66872C211yesmaximal, characteristic, complemented
2C2×#10622579C211yesmaximal, characteristic
3C2×#10622615C211yesmaximal, characteristic
4#13071204C211yesmaximal, characteristic
5#13069250C214yesmaximal, complemented
6#13069243C214yesmaximal, complemented
7#13068337C211yesmaximal, characteristic, complemented
8#13068313C211yesmaximal, characteristic, complemented
9C11×#2274C211yesmaximal, characteristic, complemented
10C22×#4339C2211yescharacteristic
11C2×C11×#117C2211yescharacteristic
12C2×C11×#117C2211yescharacteristic
13C22×C11×C4⋊C4C2211yescharacteristic
14C22×C4⋊Dic11C2211yescharacteristic
15C22×#4318C2211yescharacteristic
16C23×C11⋊C8C2211yescharacteristic
17C4×C11×C4⋊C4—22no—
18C4×C11×C4⋊C4C2214yes—
19C4×C11×C22⋊C4—22no—
20#10622615C418yescomplemented
21#10622600—22no—
22C2×#66874C2212yes—
23C2×#66874C2212yes—
24#10622579—24no—
25C2×#66873C2212yes—
26C2×#66873C2212yes—
27C2×#66872—24no—
28C2×#66872C2214yes—
29#10622571—22no—
30#10622552C2212yes—
31#10622542C2212yes—
32#10622167C2214yes—
33#10622162C2214yes—
34C23×C4×C11—21no—
35C23×C4×C11C2311yescharacteristic
36C22×C22⋊D11—21no—
37C23×Dic11—21no—
38C2×#4339—22no—
39C2×#4339—22no—
40C2×#4339—22no—
41C2×#4339—22no—
42C2×#4339—22no—
43C2×#4339C2×C414yes—
44C22×D44C2311yescharacteristic
45C22×C4×D11—21no—
46C22×Dic22C2311yescharacteristic
47C11×#117—22no—
48C11×#117—22no—
49C11×#117—22no—
50C11×#117C2×C414yes—
51C2×C11×C4⋊C4—21no—
52C2×C11×C4⋊C4—22no—
53C2×C11×C4⋊C4D414yes—
54C2×C11×C4⋊C4D412yes—
55C2×C11×C4⋊C4C2312yes—
56C2×C11×C4⋊C4C2312yes—
57C2×C11×C22⋊C4—21no—
58C2×C11×C22⋊C4D412yes—
59C2×C42×C11—21no—
60C2×C42×C11—24no—
61C2×C42×C11C2312yes—
62C2×(C2×D11)⋊C4—24no—
63C2×C4⋊Dic11—22no—
64C2×C4⋊Dic11—22no—
65C2×C4⋊Dic11—21no—
66C2×C4⋊Dic11C2312yes—
67C2×#4318—22no—
68C2×#4318—22no—
69C2×#4318—22no—
70C22×C11⋊C8—22no—
71C22×C11⋊C8—22no—
72C22×C11⋊C8C2×C414yes—
73C22×C11⋊C8C2312yes—
74C22×C11⋊C8C2312yes—
75C11×#95—24no—
76C11⋊#95—24no—
77#66874—24no—
78#66874C2×C418yescomplemented
79#66873—24no—
80#66873C2×C418yescomplemented
81#66872—28no—
82#21068—111nomaximal, Sylow
83C24×C11C2×D411yescharacteristic
84C23×D11—21no—
85C22×C4×C11—22no—
86C22×C4×C11—22no—
87C22×C4×C11—24no—
88C22×C4×C11—82no—
89C22×C4×C11C22×C412yes—
90C22×C4×C11C2×D414yes—
91C22×C4×C11C2×D411yescharacteristic
92C22×C4×C11C2×D412yes—
93C22×C4×C11#4112yes—
94C22×C4×C11#4112yes—
95C22×C4×C11#4112yes—
96C22×C4×C11#4112yes—
97C22×C4×C11C2411yescharacteristic
98C2×C22⋊D11—24no—
99C2×C22⋊D11—42no—
100C2×C22⋊D11—42no—
101C22×Dic11—21no—
102C22×Dic11—22no—
103C22×Dic11—42no—
104C22×Dic11—82no—
105#4339—28no—
106#4339—28no—
107#4339—44no—
108#4339—44no—
109C2×D44—21no—
110C2×D44—21no—
111C2×D44—21no—
112C2×D44—21no—
113C2×D44—21no—
114C2×D44C22×C412yes—
115C2×C4×D11—24no—
116C2×C4×D11—42no—
117C2×C4×D11—42no—
118C2×Dic22—21no—
119C2×Dic22—21no—
120C2×Dic22—21no—
121C2×Dic22—21no—
122C2×Dic22—21no—
123C2×Dic22C22×C412yes—
124C11×C4⋊C4—24no—
125C11×C4⋊C4—24no—
126C11×C4⋊C4—22no—
127C11×C4⋊C4—22no—
128C11×C4⋊C4C22×C414yes—
129C11×C4⋊C4C2×D414yes—
130C11×C22⋊C4—24no—
131C11×C22⋊C4—24no—
132C42×C11—24no—
133C42×C11—22no—
134C42×C11—22no—
135C42×C11—84no—
136(C2×D11)⋊C4—84no—
137C4⋊Dic11—22no—
138C4⋊Dic11—22no—
139C4⋊Dic11—42no—
140#4318—44no—
141C2×C11⋊C8—28no—
142C2×C11⋊C8—22no—
143C2×C11⋊C8—22no—
144C2×C11⋊C8—22no—
145C2×C11⋊C8—22no—
146C2×C11⋊C8—22no—
147C2×C11⋊C8C22×C414yes—
148C22×#106—111no—
149C2×#417—111no—
150C2×#416—111no—
151#2311—111no—
152#2274D1111yescharacteristic, complemented
153#1927—114no—
154#1926—114no—
155#1736—111no—
156#1729—111no—
157C23×C11—22no—
158C23×C11—22no—
159C23×C11C4×D412yes—
160C23×C11#12112yes—
161C23×C11#12312yes—
162C23×C11C22×D411yescharacteristic
163C22×D11—22no—
164C22×D11—41no—
165C22×D11—41no—
166C22×D11—42no—
167C22×D11—42no—
168C2×C4×C11—24no—
169C2×C4×C11—24no—
170C2×C4×C11—24no—
171C2×C4×C11—24no—
172C2×C4×C11—24no—
173C2×C4×C11—42no—
174C2×C4×C11—44no—
175C2×C4×C11—42no—
176C2×C4×C11—84no—
177C2×C4×C11#11714yes—
178C2×C4×C11C4×D414yes—
179C2×C4×C11C4×D412yes—
180C2×C4×C11#12112yes—
181C2×C4×C11#12312yes—
182C2×C4×C11#12414yes—
183C2×C4×C11#12614yes—
184C2×C4×C11C23×C411yescharacteristic
185C2×C4×C11C22×D411yescharacteristic
186C2×C4×C11C2×#4111yescharacteristic
187C2×C4×C11C2×#4111yescharacteristic
188C2×C4×C11C2×#4111yescharacteristic
189C2×C4×C11C2×#4111yescharacteristic
190C22⋊D11—48no—
191C2×Dic11—22no—
192C2×Dic11—41no—
193C2×Dic11—41no—
194C2×Dic11—42no—
195C2×Dic11—44no—
196C2×Dic11—84no—
197D44—22no—
198D44—22no—
199D44—41no—
200D44—41no—
201C4×D11—48no—
202Dic22—22no—
203Dic22—22no—
204Dic22—41no—
205Dic22—41no—
206C11⋊C8—24no—
207C11⋊C8—42no—
208C22×#41—111no—
209C22×M16—111no—
210C23×C8—111no—
211C2×#117C2×D1111yescharacteristic
212C2×#117C2×D1111yescharacteristic
213C22×C4⋊C4—111no—
214C22×C4⋊C4C2×D1111yescharacteristic
215C2×#106—114no—
216C2×#106—224no—
217#417—224no—
218#416—118no—
219C2×#103—112no—
220C2×#103—112no—
221C2×#102—112no—
222C2×#102—112no—
223#391—112no—
224#390—112no—
225#386—222no—
226#380—222no—
227C4×C4⋊C4—22no—
228C4×C4⋊C4C2×D1114yes—
229C4×C22⋊C4—22no—
230#340—114no—
231#336—114no—
232C22×C11—24no—
233C22×C11—24no—
234C22×C11—22no—
235C22×C11—22no—
236C22×C11#38814yes—
237C22×C11C2×C4×D411yescharacteristic
238C22×C11C2×#12111yescharacteristic
239C22×C11C2×#12311yescharacteristic
240C22×C11#57712yes—
241C22×C11#57712yes—
242C2×D11—44no—
243C2×D11—42no—
244C2×D11—82no—
245C2×D11—81no—
246C2×D11—81no—
247C4×C11—44no—
248C4×C11—44no—
249C4×C11—44no—
250C4×C11#38814yes—
251C4×C11C2×#11711yescharacteristic
252C4×C11C2×C4×D411yescharacteristic
253C4×C11C2×#12411yescharacteristic
254C4×C11C2×#12611yescharacteristic
255Dic11—42no—
256Dic11—82no—
257C2×#41—114no—
258C2×#41—222no—
259C2×#41—222no—
260C2×#41—222no—
261C2×#41—222no—
262C2×#41—222no—
263C22×Q8—111no—
264C22×D4—111no—
265C22×D4—221no—
266C23×C4—21no—
267C23×C4—221no—
268C23×C4—221no—
269C23×C4C22×D1111yescharacteristic
270C2×M16—222no—
271C2×M16—222no—
272C2×M16—222no—
273C22×C8—114no—
274C22×C8—112no—
275C22×C8—112no—
276C22×C8—222no—
277C22×C8—222no—
278#117—22no—
279#117—22no—
280#117—22no—
281#117C4×D1114yes—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—112no—
285C2×C4⋊C4—222no—
286C2×C4⋊C4—222no—
287C2×C4⋊C4—221no—
288C2×C4⋊C4D4414yes—
289C2×C4⋊C4C22⋊D1112yes—
290C2×C4⋊C4C22×D1112yes—
291C2×C4⋊C4C22×D1112yes—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—224no—
294C2×C22⋊C4C22⋊D1112yes—
295C2×C42—21no—
296C2×C42—24no—
297C2×C42C22×D1112yes—
298#106—228no—
299#103—118no—
300#103—224no—
301#102—118no—
302#102—224no—
303#95—24no—
304#95—224no—
305C2×C11—24no—
306C2×C11C2×#38811yescharacteristic
307C2×C11#295212yes—
308C2×C11#304211yescharacteristic
309C2×C11#304311yescharacteristic
310C2×C11#309912yes—
311D11—82no—
312C24—221no—
313C24C2×C22⋊D1111yescharacteristic
314#41—228no—
315#41—228no—
316#41—444no—
317#41—444no—
318C2×Q8—112no—
319C2×Q8—221no—
320C2×Q8—221no—
321C2×Q8—221no—
322C2×Q8—221no—
323C2×Q8—221no—
324C2×D4—112no—
325C2×D4—224no—
326C2×D4—221no—
327C2×D4—221no—
328C2×D4—221no—
329C2×D4—221no—
330C2×D4—221no—
331C2×D4—442no—
332C2×D4—442no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—82no—
337C22×C4—221no—
338C22×C4—224no—
339C22×C4—222no—
340C22×C4—442no—
341C22×C4—442no—
342C22×C4—442no—
343C22×C4—882no—
344C22×C4C2×C4×D1112yes—
345C22×C4C2×D4412yes—
346C22×C4#433912yes—
347C22×C4#433912yes—
348C22×C4#433912yes—
349C22×C4#433912yes—
350C22×C4C2×C22⋊D1114yes—
351C22×C4C2×C22⋊D1111yescharacteristic
352C22×C4C23×D1111yescharacteristic
353M16—444no—
354C2×C8—114no—
355C2×C8—228no—
356C2×C8—222no—
357C2×C8—222no—
358C2×C8—222no—
359C2×C8—222no—
360C2×C8—222no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—222no—
366C4⋊C4—222no—
367C4⋊C4—442no—
368C4⋊C4C2×C4×D1114yes—
369C4⋊C4C2×D4414yes—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—884no—
373C42—24no—
374C42—22no—
375C42—22no—
376C42—84no—
377C11#2106811yescharacteristic, complemented, Sylow
378C23—22no—
379C23—22no—
380C23—222no—
381C23—441no—
382C23—441no—
383C23—442no—
384C23—442no—
385C23C4×C22⋊D1112yes—
386C23#6698212yes—
387C23#6698312yes—
388C23C22×C22⋊D1111yescharacteristic
389Q8—222no—
390Q8—222no—
391Q8—441no—
392Q8—441no—
393D4—222no—
394D4—222no—
395D4—448no—
396D4—441no—
397D4—441no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—84no—
407C2×C4—222no—
408C2×C4—441no—
409C2×C4—441no—
410C2×C4—448no—
411C2×C4—442no—
412C2×C4—444no—
413C2×C4—884no—
414C2×C4#6692514yes—
415C2×C4C4×D4414yes—
416C2×C4#6692814yes—
417C2×C4#6692914yes—
418C2×C4C4×C22⋊D1112yes—
419C2×C4#6698212yes—
420C2×C4#6698312yes—
421C2×C4C22×C4×D1111yescharacteristic
422C2×C4C22×D4411yescharacteristic
423C2×C4C2×#433911yescharacteristic
424C2×C4C2×#433911yescharacteristic
425C2×C4C2×#433911yescharacteristic
426C2×C4C2×#433911yescharacteristic
427C8—224no—
428C8—442no—
429C22—24no—
430C22—22no—
431C22—22no—
432C22—24no—
433C22—442no—
434C22—444no—
435C22—882no—
436C22—881no—
437C22—881no—
438C22#1062255314yes—
439C22C2×C4×C22⋊D1111yescharacteristic
440C22C2×#6698211yescharacteristic
441C22C2×#6698311yescharacteristic
442C22#1062336712yes—
443C22#1062336812yes—
444C4—44no—
445C4—44no—
446C4—44no—
447C4—442no—
448C4—882no—
449C4#1062255314yes—
450C4C2×#6692511yescharacteristic
451C4C2×C4×D4411yescharacteristic
452C4C2×#6692811yescharacteristic
453C4C2×#6692911yescharacteristic
454C2—24no—
455C2—882no—
456C2C2×#1062255311yescharacteristic
457C2#1307749712yes—
458C2#1307812811yescharacteristic
459C2#1307813711yescharacteristic
460C2#1307860512yes—
subgroupG/H
centerC23C22×C22⋊D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#2274C2
socleC23×C11C22×D4
prime radicalC22×#4339C22
cosocleC22×C4×C11C24
hypercenter#2274D11
hypocenterC11#21068
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2274

Complements

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

NH ≅ G/Ntype
1. C22×#66872C2semidirect
5. #13069250C2semidirect
6. #13069243C2semidirect
7. #13068337C2semidirect
8. #13068313C2semidirect
9. C11×#2274C2semidirect
20. #10622615C4semidirect
78. #66874C2×C4semidirect
80. #66873C2×C4semidirect
152. #2274D11semidirect
377. C11#21068semidirect

Maximal subgroups

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

subgroupindex
C22×#668722
C2×#106225792
C2×#106226152
#130712042
#130692502
#130692432
#130683372
#130683132
C11×#22742
#2106811