Finite Group Databasebeta

← #14756956

Subgroups — #14756956

Order 2816.

890 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#4325C211yesmaximal, characteristic
2#13071267C211yesmaximal, characteristic, complemented
3#13070976C211yesmaximal, characteristic
4#13069253C212yesmaximal, complemented
5#13069197C212yesmaximal, complemented
6#13068947C212yesmaximal
7#13068878C212yesmaximal
8C4×#66872C211yesmaximal, characteristic, complemented
9#13068442C211yesmaximal, characteristic
10#13068369C211yesmaximal, characteristic, complemented
11C11×#1763C211yesmaximal, characteristic, complemented
12C2×C4×Q8×C11C2211yescharacteristic
13C22×#4325C2211yescharacteristic
14C2×#66946C2212yes—
15C2×C4×D44C2211yescharacteristic
16C2×C11×#103C2212yes—
17C2×C11×#97C2211yescharacteristic
18C4×C11×C4⋊C4C2211yescharacteristic
19#10622843C2212yes—
20C2×#66899C2211yescharacteristic
21C2×#66885C2212yes—
22C4×#4325C418yescomplemented
23#10622598C2212yes—
24#10622594—22no—
25C4⋊C4×Dic11C2212yes—
26C2×#66873C2211yescharacteristic
27C2×#66872C2212yes—
28C2×#66872C2211yescharacteristic
29C4×(C2×D11)⋊C4—22no—
30#10622550—21no—
31#10622549—21no—
32C4×C4⋊Dic11C2211yescharacteristic
33#10622545C2211yescharacteristic
34C2×C4×C11⋊C8C2211yescharacteristic
35#10622279—21no—
36D11×#95—21no—
37C11×#340C2212yes—
38#10622189C2212yes—
39#10622188C2212yes—
40#10622188C2212yes—
41#10622167C2212yes—
42C22×Q8×C11C2311yescharacteristic
43C22×D44C2311yescharacteristic
44C22×C4×D11—21no—
45C22×C4×D11—22no—
46C22×C8×C11C2312yes—
47C4×Q8×C11—22no—
48C4×Q8×C11C2×C414yescomplemented
49C2×C11×C4⋊C4—21no—
50C2×C11×C4⋊C4C2311yescharacteristic
51C2×C11×C4⋊C4C2312yes—
52C2×C42×C11—21no—
53C2×C42×C11—21no—
54C2×C42×C11C2311yescharacteristic
55C2×#4325—24no—
56C2×#4325C2×C414yes—
57C2×(C2×D11)⋊C4—21no—
58C2×(C2×D11)⋊C4—22no—
59C2×C4⋊Dic11—22no—
60C2×C4⋊Dic11C2312yes—
61C2×C4⋊Dic11C2311yescharacteristic
62C2×C4×Dic11—21no—
63C2×C4×Dic11—22no—
64C2×C4×Dic11C2312yes—
65C22×C11⋊C8C2311yescharacteristic
66C22×C11⋊C8C2311yescharacteristic
67#66946—24no—
68#66946—24no—
69C4×D44—22no—
70C4×D44C2×C414yes—
71C11×#103—24no—
72C11×#97—22no—
73C11×#95—21no—
74#66899C2×C414yescomplemented
75C11⋊#95—21no—
76C11⋊#95—22no—
77C11⋊#95—22no—
78C11⋊#95—22no—
79C11⋊#95—22no—
80C11⋊#95D411yescharacteristic
81C11⋊#95D411yescharacteristic
82C11⋊#95D412yes—
83#66885—24no—
84#66873C2×C414yes—
85#66872—24no—
86#66872C2×C414yescomplemented
87C4×C11⋊C8C2×C414yescomplemented
88#21133—111nomaximal, Sylow
89C23×D11—21no—
90C2×Q8×C11—21no—
91C2×Q8×C11—22no—
92C2×Q8×C11—22no—
93C2×Q8×C11C22×C412yes—
94C22×C4×C11—21no—
95C22×C4×C11—21no—
96C22×C4×C11—22no—
97C22×C4×C11C2×D411yescharacteristic
98C22×C4×C11C2×D411yescharacteristic
99C22×C4×C11C2411yescharacteristic
100C22×Dic11—21no—
101C22×Dic11—22no—
102C22×Dic11C2×D412yes—
103C22×Dic11C2×D412yes—
104C2×D44—21no—
105C2×D44—22no—
106C2×D44C22×C412yes—
107C2×D44#4114yes—
108C2×C4×D11—28no—
109C2×C4×D11—24no—
110C2×C4×D11—82no—
111C2×C4×D11—82no—
112C2×C4×D11—82no—
113C2×C8×C11—22no—
114C2×C8×C11—22no—
115C2×C8×C11—22no—
116C11×C4⋊C4—24no—
117C11×C4⋊C4—22no—
118C11×C4⋊C4—21no—
119C11×C4⋊C4C22×C412yes—
120C11×C4⋊C4#4114yes—
121C42×C11—24no—
122C42×C11—81no—
123C42×C11C22×C412yes—
124C42×C11#4112yes—
125#4325—28no—
126(C2×D11)⋊C4—28no—
127(C2×D11)⋊C4—24no—
128C4⋊Dic11—21no—
129C4⋊Dic11—22no—
130C4⋊Dic11—22no—
131C4⋊Dic11—82no—
132C4⋊Dic11C22×C412yes—
133C4×Dic11—22no—
134C4×Dic11—22no—
135C4×Dic11—81no—
136C4×Dic11—82no—
137C2×C11⋊C8—22no—
138C2×C11⋊C8—22no—
139C2×C11⋊C8C22×C412yes—
140C2×C11⋊C8C22×C412yes—
141C2×C4×SD16—111no—
142#2320—111no—
143#2275—111no—
144#1927—112no—
145#1919—112no—
146#1877—112no—
147#1864—112no—
148C4×#106—111no—
149#1763D1111yescharacteristic, complemented
150#1762—111no—
151#1746—111no—
152C23×C11C22×D411yescharacteristic
153C22×D11—22no—
154C22×D11—24no—
155C22×D11—41no—
156C22×D11—42no—
157C22×D11—41no—
158Q8×C11—22no—
159Q8×C11—22no—
160Q8×C11—42no—
161C2×C4×C11—21no—
162C2×C4×C11—21no—
163C2×C4×C11—22no—
164C2×C4×C11—22no—
165C2×C4×C11—22no—
166C2×C4×C11—24no—
167C2×C4×C11—42no—
168C2×C4×C11—42no—
169C2×C4×C11—41no—
170C2×C4×C11—42no—
171C2×C4×C11—41no—
172C2×C4×C11C4×D412yes—
173C2×C4×C11C4×D412yes—
174C2×C4×C11C23×C411yescharacteristic
175C2×C4×C11C2×#4112yes—
176C2×C4×C11C2×#4111yescharacteristic
177C2×C4×C11C2×#4112yes—
178C2×Dic11—22no—
179C2×Dic11—22no—
180C2×Dic11—22no—
181C2×Dic11—22no—
182C2×Dic11—22no—
183C2×Dic11—22no—
184C2×Dic11—22no—
185C2×Dic11—42no—
186C2×Dic11—42no—
187C2×Dic11—42no—
188C2×Dic11—42no—
189D44—22no—
190D44—24no—
191D44#11714yes—
192C4×D11—84no—
193C4×D11—88no—
194C8×C11—42no—
195C11⋊C8—22no—
196C11⋊C8—22no—
197C22×SD16—111no—
198C2×C4×Q8C2×D1111yescharacteristic
199C2×C4×D4—112no—
200C2×C4×D4—111no—
201C4×SD16—118no—
202C2×#106—111no—
203C2×#106—112no—
204C2×#103—111no—
205C2×#103C2×D1112yes—
206C2×#102—111no—
207C2×#102—112no—
208C2×#97C2×D1111yescharacteristic
209C2×C4×C8—111no—
210#391—112no—
211#389—112no—
212#387—222no—
213#383—221no—
214#382—221no—
215C4×C4⋊C4—112no—
216C4×C4⋊C4—111no—
217C4×C4⋊C4C2×D1111yescharacteristic
218C4×C22⋊C4—221no—
219C4×C22⋊C4—222no—
220#376—111no—
221C2×#95—221no—
222#340—112no—
223#340—112no—
224#340C2×D1112yes—
225#336—112no—
226#336—112no—
227C22×C11C2×C4×D411yescharacteristic
228C22×C11#52512yes—
229C22×C11#57711yescharacteristic
230C22×C11#57712yes—
231C22×C11#57711yescharacteristic
232C2×D11—24no—
233C2×D11—44no—
234C2×D11—44no—
235C2×D11—42no—
236C2×D11—42no—
237C4×C11—22no—
238C4×C11—22no—
239C4×C11—44no—
240C4×C11—42no—
241C4×C11—42no—
242C4×C11C2×#11711yescharacteristic
243C4×C11C4×#4111yescharacteristic
244C4×C11#52912yes—
245Dic11—42no—
246Dic11—42no—
247Dic11—42no—
248Dic11—82no—
249C22×Q8C22×D1111yescharacteristic
250C22×D4—111no—
251C23×C4—222no—
252C23×C4—221no—
253C2×SD16—114no—
254C2×SD16—224no—
255C22×C8—111no—
256C22×C8—111no—
257C22×C8C22×D1112yes—
258C4×Q8—22no—
259C4×Q8C4×D1114yescomplemented
260C4×D4—114no—
261C4×D4—224no—
262C4×D4—224no—
263C4×D4—222no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—111no—
266C2×C4⋊C4—112no—
267C2×C4⋊C4—222no—
268C2×C4⋊C4C22×D1111yescharacteristic
269C2×C4⋊C4C22×D1112yes—
270C2×C22⋊C4—222no—
271C2×C22⋊C4—221no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—112no—
275C2×C42—222no—
276C2×C42—221no—
277C2×C42C22×D1111yescharacteristic
278#106—114no—
279#106—224no—
280#103—24no—
281#103—114no—
282#102—114no—
283#102—224no—
284#97—22no—
285C4×C8—114no—
286#95—21no—
287#95—111no—
288#95—111no—
289#95—112no—
290#95—222no—
291#95—222no—
292#95—222no—
293#95—222no—
294#95—221no—
295C2×C11#229011yescharacteristic
296C2×C11#294911yescharacteristic
297C2×C11#295211yescharacteristic
298C2×C11#312312yes—
299C2×C11#312512yes—
300D11—44no—
301C24—221no—
302C2×Q8—21no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8C2×C4×D1112yes—
306C2×D4—112no—
307C2×D4—114no—
308C2×D4—222no—
309C2×D4—221no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—112no—
314C22×C4—112no—
315C22×C4—228no—
316C22×C4—221no—
317C22×C4—224no—
318C22×C4—222no—
319C22×C4—882no—
320C22×C4—882no—
321C22×C4—882no—
322C22×C4D4×D1111yescharacteristic
323C22×C4D4×D1111yescharacteristic
324C22×C4C23×D1111yescharacteristic
325SD16—228no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—112no—
330C2×C8—112no—
331C2×C8—222no—
332C2×C8—222no—
333C4⋊C4—24no—
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—112no—
337C4⋊C4—221no—
338C4⋊C4—222no—
339C4⋊C4—222no—
340C4⋊C4—882no—
341C4⋊C4C2×C4×D1112yes—
342C4⋊C4#433914yes—
343C22⋊C4—228no—
344C22⋊C4—224no—
345C42—24no—
346C42—81no—
347C42—222no—
348C42—222no—
349C42—881no—
350C42—882no—
351C42C2×C4×D1112yes—
352C42#434112yes—
353C11#2113311yescharacteristic, complemented, Sylow
354C23—222no—
355C23—224no—
356C23—441no—
357C23—442no—
358C23—441no—
359C23C2×D4×D1111yescharacteristic
360Q8—22no—
361Q8—22no—
362Q8—42no—
363D4—114no—
364D4—222no—
365D4—224no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—222no—
378C2×C4—222no—
379C2×C4—222no—
380C2×C4—222no—
381C2×C4—222no—
382C2×C4—222no—
383C2×C4—222no—
384C2×C4—442no—
385C2×C4—442no—
386C2×C4—442no—
387C2×C4—442no—
388C2×C4—884no—
389C2×C4—888no—
390C2×C4(C22⋊D11)⋊C412yes—
391C2×C4(C22⋊D11)⋊C412yes—
392C2×C4C22×C4×D1111yescharacteristic
393C2×C4C2×#433912yes—
394C2×C4C2×#434111yescharacteristic
395C2×C4#41×D1112yes—
396C8—42no—
397C8—222no—
398C8—222no—
399C22—224no—
400C22—444no—
401C22—444no—
402C22—442no—
403C22—442no—
404C22C2×(C22⋊D11)⋊C411yescharacteristic
405C22#1062307812yes—
406C22#1062330611yescharacteristic
407C22#1062330912yes—
408C22#1062330911yescharacteristic
409C4—22no—
410C4—22no—
411C4—44no—
412C4—42no—
413C4—42no—
414C4—442no—
415C4—442no—
416C4—442no—
417C4—882no—
418C4C4×#433911yescharacteristic
419C4D11×#11711yescharacteristic
420C4#1062308812yes—
421C2—444no—
422C2#1307106211yescharacteristic
423C2#1307747111yescharacteristic
424C2#1307749511yescharacteristic
425C2#1307886712yes—
426C2#1307889112yes—
subgroupG/H
centerC23C2×D4×D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1763C2
socleC23×C11C22×D4
prime radicalC22×D44C23
cosocleC22×C4×C11C24
hypercenter#1763D11
hypocenterC11#21133
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
2. #13071267C2semidirect
4. #13069253C2semidirect
5. #13069197C2semidirect
8. C4×#66872C2semidirect
10. #13068369C2semidirect
11. C11×#1763C2semidirect
22. C4×#4325C4semidirect
48. C4×Q8×C11C2×C4semidirect
74. #66899C2×C4semidirect
86. #66872C2×C4semidirect
87. C4×C11⋊C8C2×C4semidirect
149. #1763D11semidirect
259. C4×Q8C4×D11semidirect
353. C11#21133semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#43252
#130712672
#130709762
#130692532
#130691972
#130689472
#130688782
C4×#668722
#130684422
#130683692
C11×#17632
#2113311