Finite Group Databasebeta

← #14721634

Subgroups — #14721634

Order 2816.

1138 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13082382C211yesmaximal, characteristic
2#13077699C211yesmaximal, characteristic, complemented
3#13071579C211yesmaximal, characteristic, complemented
4#13067177C211yesmaximal, characteristic, complemented
5#13067176C211yesmaximal, characteristic, complemented
6#13067161C214yesmaximal, complemented
7#13067122C214yesmaximal, complemented
8#13067110C211yesmaximal, characteristic, complemented
9C11×#1502C211yesmaximal, characteristic, complemented
10C2×#67043—21no—
11C11×#532C2211yescharacteristic
12#10623364—21no—
13#10623258C412yescomplemented
14#10623249C412yescomplemented
15#10623208—22no—
16#10623128—22no—
17#10623127C2214yes—
18#10623084C412yescomplemented
19#10623083—22no—
20D4×Dic22C412yescomplemented
21#10623008C2211yescharacteristic
22C2×C4×Dic22C2211yescharacteristic
23C11×#431C2211yescharacteristic
24C4×C11×C4⋊C4C2211yescharacteristic
25#10622591—22no—
26#10622549C2211yescharacteristic
27C4×C4⋊Dic11—21no—
28#10622542—21no—
29#10622538C2211yescharacteristic
30#10622444—22no—
31#10622332C2214yes—
32#10622292—22no—
33#10622264C2212yes—
34#10622261C2212yes—
35C11×#327C2212yes—
36C11×#327C2212yes—
37#10622175C2214yes—
38#10622108C2214yes—
39#10622089C2212yes—
40#10622089C2212yes—
41C2×C11×#41—21no—
42#67043—44no—
43#67043—44no—
44C2×Q8×D11—22no—
45C2×#4341—24no—
46C2×#4341—22no—
47C2×#4339—21no—
48C2×#4339—22no—
49C22×Dic22—22no—
50C22×Dic22D411yescharacteristic
51C11×C4⋊Q8C2×C411yescharacteristic
52C11×#127C2×C411yescharacteristic
53C11×#124—21no—
54C11×#121—21no—
55C11×#121C2312yes—
56C4×D4×C11—21no—
57C4×D4×C11C2×C412yes—
58C2×C11×C4⋊C4—21no—
59C2×C42×C11—21no—
60C2×C42×C11C2311yescharacteristic
61#66999—22no—
62#66993C2×C414yes—
63#66992—24no—
64#66992—24no—
65#66991—22no—
66D4×Dic11C2×C414yes—
67#66983—21no—
68#66983—22no—
69#66983C2312yes—
70C2×C4⋊Dic11—21no—
71C2×C4⋊Dic11D411yescharacteristic
72#66977—22no—
73C2×Dic11⋊C4D412yes—
74C2×C4×Dic11—22no—
75C2×C4×Dic11C2312yes—
76#66949—24no—
77#66946C2×C414yes—
78#66941C2×C414yes—
79#66938—24no—
80#66937—24no—
81(C22⋊D11)⋊C4—24no—
82#66931—24no—
83#66928—21no—
84#66927C2×C411yescharacteristic
85C4×D44—21no—
86C4×D44C2×C412yes—
87#66922C2×C411yescharacteristic
88C4×Dic22C2×C412yes—
89C4×Dic22C2×C412yes—
90C4×Dic22D412yes—
91C4×Dic22D412yes—
92C11×C4⋊C8—21no—
93C11×C22⋊C8C2312yes—
94C4×C8×C11—21no—
95C11×#95C2311yescharacteristic
96C11×#95C2311yescharacteristic
97C11⋊#95—21no—
98C11⋊#95—22no—
99C11⋊#95—22no—
100C11⋊#95—22no—
101C11⋊#95C2312yes—
102C22⋊(C11⋊C8)C2312yes—
103#66879—24no—
104Dic11⋊C8—24no—
105C4⋊(C11⋊C8)—21no—
106C4×C11⋊C8—21no—
107#17631—111nomaximal, Sylow
108C11×#41—42no—
109C11×#41—42no—
110C2×Q8×C11—21no—
111C2×D4×C11—22no—
112C2×D4×C11—21no—
113C2×D4×C11—21no—
114C2×D4×C11C22×C412yes—
115C22×C4×C11—21no—
116C22×C4×C11—22no—
117C22×C4×C11—21no—
118C22×C4×C11C2×D411yescharacteristic
119C22×C4×C11C2×D411yescharacteristic
120C22×C4×C11C2411yescharacteristic
121C2×C22⋊D11—24no—
122C2×C22⋊D11—24no—
123C2×C22⋊D11—22no—
124C22×Dic11—24no—
125C22×Dic11—22no—
126C22×Dic11C2×D412yes—
127C22×Dic11C2×D412yes—
128Q8×D11—48no—
129#4341—48no—
130#4341—48no—
131#4341—48no—
132#4339—48no—
133#4339—42no—
134#4339—42no—
135C2×D44—21no—
136C2×D44—21no—
137C2×D44C22×C412yes—
138C2×C4×D11—24no—
139C2×C4×D11—22no—
140C2×Dic22—24no—
141C2×Dic22—21no—
142C2×Dic22—22no—
143C2×Dic22—22no—
144C2×Dic22—48no—
145C2×Dic22C22⋊C412yes—
146C2×Dic22C22⋊C412yes—
147C2×C8×C11—22no—
148C11×C4⋊C4—21no—
149C11×C4⋊C4—42no—
150C11×C4⋊C4C22×C412yes—
151C11×C22⋊C4—22no—
152C11×C22⋊C4—22no—
153C42×C11—42no—
154C42×C11C22×C411yescharacteristic
155C42×C11C22×C411yescharacteristic
156C42×C11C2×D411yescharacteristic
157C42×C11C2×D411yescharacteristic
158C22⋊Dic11—24no—
159(C2×D11)⋊C4—24no—
160(C2×D11)⋊C4—22no—
161(C2×D11)⋊C4—22no—
162C4⋊Dic11—21no—
163C4⋊Dic11—42no—
164C4⋊Dic11C22⋊C412yes—
165C4⋊Dic11C22⋊C412yes—
166C4⋊Dic11C22×C412yes—
167Dic11⋊C4—24no—
168Dic11⋊C4C22⋊C414yes—
169Dic11⋊C4C22⋊C414yes—
170C4×Dic11—44no—
171C4×Dic11C22×C414yes—
172C4×Dic11C2×D414yes—
173C2×C11⋊C8—22no—
174#3497—111no—
175#2983—111no—
176#2356—111no—
177#1517—111no—
178#1517—111no—
179#1514—114no—
180#1505—114no—
181#1502—111no—
182#1502D1111yescharacteristic, complemented
183C23×C11—22no—
184C23×C11C22×D411yescharacteristic
185C22×D11—22no—
186Q8×C11—42no—
187D4×C11—42no—
188D4×C11—42no—
189D4×C11—42no—
190D4×C11—42no—
191D4×C11—81no—
192C2×C4×C11—22no—
193C2×C4×C11—22no—
194C2×C4×C11—41no—
195C2×C4×C11—41no—
196C2×C4×C11—41no—
197C2×C4×C11—41no—
198C2×C4×C11—44no—
199C2×C4×C11—41no—
200C2×C4×C11—41no—
201C2×C4×C11—41no—
202C2×C4×C11C2×C22⋊C411yescharacteristic
203C2×C4×C11C2×C22⋊C411yescharacteristic
204C2×C4×C11C2×C22⋊C411yescharacteristic
205C2×C4×C11C2×C22⋊C411yescharacteristic
206C2×C4×C11AGL(1,ℤ/8)12yes—
207C2×C4×C11AGL(1,ℤ/8)12yes—
208C2×C4×C11C23×C411yescharacteristic
209C2×C4×C11C22×D411yescharacteristic
210C22⋊D11—48no—
211C22⋊D11—42no—
212C22⋊D11—42no—
213C22⋊D11—84no—
214C2×Dic11—22no—
215C2×Dic11—22no—
216C2×Dic11—22no—
217C2×Dic11—42no—
218C2×Dic11—48no—
219C2×Dic11—42no—
220C2×Dic11—42no—
221C2×Dic11—42no—
222C2×Dic11C2×C22⋊C414yes—
223C2×Dic11C2×C22⋊C414yes—
224D44—42no—
225D44—42no—
226D44—81no—
227C4×D11—48no—
228C4×D11—44no—
229Dic22—24no—
230Dic22—24no—
231Dic22—48no—
232Dic22—42no—
233C8×C11—42no—
234C11⋊C8—42no—
235C2×2−1+4—221no—
236#550—112no—
237#550—112no—
238D4×Q8—112no—
239D4×Q8—112no—
240#548—222no—
241#548—222no—
242#537—222no—
243#537—221no—
244#532—114no—
245#532—111no—
246#532C2×D1111yescharacteristic
247C2×C4×Q8—111no—
248#446—222no—
249#446—222no—
250#432—114no—
251#431—111no—
252#431C2×D1111yescharacteristic
253#391—221no—
254#387—222no—
255#383—112no—
256#382—111no—
257C4×C4⋊C4—221no—
258C4×C4⋊C4C2×D1111yescharacteristic
259#376—112no—
260#327—114no—
261#327—114no—
262#327—112no—
263#327—112no—
264#327C2×D1112yes—
265#327C2×D1112yes—
266C22×C11—21no—
267C22×C11—21no—
268C22×C11—42no—
269C22×C11—42no—
270C22×C11—42no—
271C22×C11C22×C22⋊C411yescharacteristic
272C22×C11C2×AGL(1,ℤ/8)11yescharacteristic
273C22×C11C2×AGL(1,ℤ/8)11yescharacteristic
274C2×D11—42no—
275C2×D11—42no—
276C2×D11—42no—
277C4×C11—21no—
278C4×C11—21no—
279C4×C11—21no—
280C4×C11—21no—
281C4×C11—42no—
282C4×C11—42no—
283C4×C11—81no—
284C4×C11#41812yes—
285C4×C11#41812yes—
286Dic11—24no—
287Dic11—24no—
288Dic11—42no—
289Dic11—82no—
2902−1+4—444no—
2912−1+4—444no—
292C2×#41—21no—
293C2×#41—222no—
294C2×#41—224no—
295C2×#41—222no—
296C2×#41—221no—
297C22×Q8—111no—
298C22×Q8—222no—
299C22×Q8—222no—
300C4⋊Q8—114no—
301C4⋊Q8—111no—
302C4⋊Q8C4×D1111yescharacteristic
303#127—114no—
304#127—111no—
305#127C4×D1111yescharacteristic
306#124—21no—
307#124—224no—
308#124—221no—
309C22⋊Q8—224no—
310C22⋊Q8—224no—
311C22⋊Q8—222no—
312C22⋊Q8—222no—
313#121—21no—
314#121—112no—
315#121—224no—
316#121—222no—
317#121—222no—
318#121—224no—
319#121—224no—
320#121—221no—
321#121C22×D1112yes—
322C4×Q8—112no—
323C4×Q8—112no—
324C4×Q8—112no—
325C4×Q8—112no—
326C4×D4—21no—
327C4×D4—114no—
328C4×D4—114no—
329C4×D4—112no—
330C4×D4—224no—
331C4×D4—221no—
332C4×D4C4×D1112yes—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—111no—
335C2×C4⋊C4—112no—
336C2×C4⋊C4—221no—
337C2×C42—21no—
338C2×C42—112no—
339C2×C42—222no—
340C2×C42C22×D1111yescharacteristic
341C4⋊C8—21no—
342C4⋊C8—224no—
343C4⋊C8—221no—
344C22⋊C8—112no—
345C22⋊C8C22×D1112yes—
346#97—224no—
347C4×C8—21no—
348C4×C8—221no—
349#95—112no—
350#95—222no—
351#95—222no—
352#95—222no—
353#95—221no—
354#95C22×D1111yescharacteristic
355#95C22×D1111yescharacteristic
356C2×C11—21no—
357C2×C11—21no—
358C2×C11—82no—
359C2×C11#288411yescharacteristic
360C2×C11C2×#41811yescharacteristic
361C2×C11#290111yescharacteristic
362D11—82no—
363#41—42no—
364#41—42no—
365#41—448no—
366#41—448no—
367#41—448no—
368#41—448no—
369#41—442no—
370#41—442no—
371C2×Q8—21no—
372C2×Q8—112no—
373C2×Q8—112no—
374C2×Q8—224no—
375C2×Q8—221no—
376C2×Q8—222no—
377C2×Q8—222no—
378C2×Q8—448no—
379C2×Q8—448no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—112no—
384C2×D4—224no—
385C2×D4—224no—
386C2×D4—222no—
387C2×D4—221no—
388C2×D4—221no—
389C2×D4C2×C4×D1112yes—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—112no—
394C22×C4—112no—
395C22×C4—224no—
396C22×C4—224no—
397C22×C4—222no—
398C22×C4—222no—
399C22×C4D4×D1111yescharacteristic
400C22×C4D4×D1111yescharacteristic
401C22×C4C23×D1111yescharacteristic
402C2×C8—22no—
403C2×C8—222no—
404C4⋊C4—21no—
405C4⋊C4—42no—
406C4⋊C4—114no—
407C4⋊C4—114no—
408C4⋊C4—112no—
409C4⋊C4—112no—
410C4⋊C4—112no—
411C4⋊C4—224no—
412C4⋊C4—221no—
413C4⋊C4—442no—
414C4⋊C4C2×C4×D1112yes—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—224no—
418C22⋊C4—224no—
419C22⋊C4—222no—
420C22⋊C4—222no—
421C42—42no—
422C42—114no—
423C42—114no—
424C42—444no—
425C42C2×C4×D1111yescharacteristic
426C42C2×C4×D1111yescharacteristic
427C42D4×D1111yescharacteristic
428C42D4×D1111yescharacteristic
429C11#1763111yescharacteristic, complemented, Sylow
430C23—22no—
431C23—222no—
432C23C2×D4×D1111yescharacteristic
433Q8—42no—
434Q8—224no—
435Q8—224no—
436Q8—448no—
437Q8—442no—
438D4—42no—
439D4—42no—
440D4—42no—
441D4—42no—
442D4—81no—
443D4—442no—
444D4—448no—
445D4—442no—
446D4—442no—
447D4—442no—
448D4—881no—
449D4—884no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—44no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—114no—
461C2×C4—114no—
462C2×C4—222no—
463C2×C4—222no—
464C2×C4—222no—
465C2×C4—448no—
466C2×C4—442no—
467C2×C4—448no—
468C2×C4—444no—
469C2×C4—442no—
470C2×C4—442no—
471C2×C4—442no—
472C2×C4C22⋊C4×D1111yescharacteristic
473C2×C4C22⋊C4×D1111yescharacteristic
474C2×C4C22⋊C4×D1111yescharacteristic
475C2×C4C22⋊C4×D1111yescharacteristic
476C2×C4#6696412yes—
477C2×C4#6696712yes—
478C2×C4C22×C4×D1111yescharacteristic
479C2×C4C2×D4×D1111yescharacteristic
480C8—42no—
481C8—442no—
482C22—21no—
483C22—21no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—442no—
488C22—442no—
489C22—442no—
490C22C2×C22⋊C4×D1111yescharacteristic
491C22C2×#6696411yescharacteristic
492C22C2×#6696711yescharacteristic
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—42no—
498C4—42no—
499C4—81no—
500C4—224no—
501C4—224no—
502C4—442no—
503C4—882no—
504C4#1062238312yes—
505C4#1062238312yes—
506C2—21no—
507C2—21no—
508C2—82no—
509C2—882no—
510C2#1307707811yescharacteristic
511C2C2×#1062238311yescharacteristic
512C2#1307716011yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1502C2
socleC22×C11C22×C22⋊C4
prime radical#13082382C2
cosocleC22×C4×C11C24
hypercenter#1502D11
hypocenterC11#17631
LCSG ▷ C2×C4×C11 ▷ C22×C11 ▷ C11
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
2. #13077699C2semidirect
3. #13071579C2semidirect
4. #13067177C2semidirect
5. #13067176C2semidirect
6. #13067161C2semidirect
7. #13067122C2semidirect
8. #13067110C2semidirect
9. C11×#1502C2semidirect
13. #10623258C4semidirect
14. #10623249C4semidirect
18. #10623084C4semidirect
20. D4×Dic22C4semidirect
182. #1502D11semidirect
429. C11#17631semidirect

Maximal subgroups

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

subgroupindex
#130823822
#130776992
#130715792
#130671772
#130671762
#130671612
#130671222
#130671102
C11×#15022
#1763111